Hyperbolic Functions 4th Ed
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OSMANIA UNIVERSITY LIBRARY Call
Na
Accession No.
Author Title
This book should be returned on or before the date last marked below.
MATHEMATICAL MONOGRAPHS EDITED BY
The Late Mansfield Merriman and Robert
S.
Woodward
Octavo, Cloth No.
1.
No.
2.
Modern Mathematics. By DAVID EUGENE SMITH. History of
Synthetic Projective Geometry.
By the Lato GEOROK BRUCE HALSTED. No.
3.
Determinants. the Late LAENAS GIFFORD WELD. Hyperbolic Functions. By the Late JAMES Me AH ON.
By
No.
4.
No.
5.
M
Harmonic Functions. By WILLIAM E. BYERLY. Grassmann's Space Analysis. By EDWARD W. HYDE. Probability and Theory of Errors.
No.
6.
No.
7.
No.
8.
No.
9. Differential
By the Late KOBEKT S. WOODWARD. Vector Analysis and Quaternions. By the Late ALEXANDER MACFARLANE. Equations.
By the Late WILLIAM WOOLSEY JOHNSON. No. 10. The Solution of Equations. By the Late MANSFIELD MERRIMAN. No. 11. Functions of a Complex Variable. By THOMAS
S.
FISKE.
No. 12. The Theory of Relativity. By ROBERT D. CARMICUAEL. No. 13. The Theory of Numbers. By ROBERT D. CARMICUAEL. No. 14. Algebraic Invariant". By LEONARD E. DICKBON. No. 16. Diophantine Analysis. By ROBERT D. CARMICUAEL. No. 17. Ten British Mathematicians. By the Late ALEXANDER MACFARLANE. No. 18. Elliptic Integrals.
By HARRIS HANCOCK. No. 19. Empirical Formulas. By THEODORE R. RUNNING. No. 20. Ten British Physicists. By the Lato ALEXANDER MACFARLANE. No. 21. The Dynamics of the Airplane. By KENNETH P. WILLIAMS.
PUBLISHED BY
JOHN WILEY &
SONS,
CHAPMAN & HALL,
Inc.,
Limited,
NEW YORK LONDON
MATHEMATICAL MONOGRAPHS. EDITED BY
MANSFIELD MERRIMAN AND ROBERT
No.
WOODWARD.
S.
4.
HYPERBOLIC FUNCTIONS.
JAMES McMAHON. LAIE
PkUi-LXs
,K
OF AlAlHl'MAIIO IN
FOURTH
COKNKU
1 N T
IVKKM
I
Y.
EDITION. ENLARGED.
NEW YORK: JOHN WILEY & SONS. LONDON:
CHAPMAN & HALL,
LIMITED.
COPYRIGHT,
1896,
Y
MANSFIELD MKRRIMAN
AND
RORERT
UNDPK THE T
I
S.
WOODWARD
LS
HIGHER MATHEMATICS. First Edition, September, 1896.
becond Edition, January, 1898, Third Edition, August, 1900. Fourth Edition, January,
Printed in U. S.
1906*
j
PRESS Of
RAUNWORTH CO INC BOOK MANUFACTURERS BROOKLYN, NEWYOHK ft.
,
EDITORS' PREFACE.
THE of which
volume
called
Higher Mathematics, the
was published
first
edition
in 1896, contained eleven chapters
by
eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent
The
colleges.
that given in
to
and engineering
classical
volume
publication of that
npw
is
discontinued
*
and the chapters are issued in separate form. In these reissues it will generally be found that the monographs" arc enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but that
it
may
prove advantageous
mathematical
to the series
from time
warrant
Among
to
is
also thought
literature.
It is the intention of the publishers
monographs same seems
it
to readers in special lines of
it.
and
editors to
add other
to time, if the call for the
the topics which are under
consideration are those of elliptic functions, the theory of
num-
and
non-
bers, the
group theory, the
Euclidean geometry;
calculus of
possibly also
variations,
monographs on branches of
astronomy, mechanics, and mathematical physics It is the
hope of the editors that
this
form of
may be included. publication may
tend to promote mathematical study and research qyer a wider field than that which the former volume has occupied. December, 1905. iii
AUTHOR'S PREFACE.
This compendium of hyperbolic trigonometry was first published as a chapter in Mernman and Woodward's Higher Mathematics. There is reason to believe that it supplies a need, being adapted to two or three different types of readers. College students who have
had elementary courses in trigonometry, analytic geometry, and differential and integral calculus, and who wi.sh to know something of the on account of its important and historic relahyperbolic* trigonometry tions to each of those branches, will,
it is hoped, find these relations in the first half of the and a way comprehensive simple presented work. Readers who have some interest in imagmaries are then intro-
in
to the more general trigonometry of the the circular and hyperbolic functions merge into ents, the singly periodic functions, having cither For those who also wish to view inary period.
duced
complex plane, where one class of transcenda real or a pure imag-
the subject in some of practical relations, numerous applications have be n selected so as to illustrate the various parts of the theory, and to show its use to the physicist and engineer, appropriate numerical tables being supplied for f
its
these purposes. With all these things in mind, much thought has been given to the mode ot approaching the subject, and to the presentation of fundamental notions, and it is hoped that some improvements are discerni-
For instance, it has been customary to define the hyperbolic ble. functions in relation to a sector of the rectangular hyperbola, and to take the initial radius of the sector coincident with the principal radius of the curve; in the present work, these and similar restrictions are discarded in the interest of analogy and generality, with a gain in symmetry and simplicity, and the functions are defined as certain charac-
belonging to any sector of any hyperbola. Such definiconnection with the fruitful notion of correspondence of points on comes, lead to simple and general proofs of the addition-theorems, from which easily follow the con version- formulas, the derivatives, the teristic ratios
tions, in
Maclaurin expansions, and the exponential expresMons. The proofs are so arranged as to apply equally to the circular functions, regarded as the characteristic ratios belonging to any elliptic sector. For those, however, who may wish to start with the exponential expressions as the definitions of the hyperbolic functions, the appropriate order of procedure is indicated cm page 25, and a direct mode of bringing such
exponential definitions into geometrical relation with the hyperbolic sector is ^.hown in the Appendix. December, igo^
CONTENTS.
ART.
r.
2.
3.
4. 5.
6. 7.
8.
Q.
10.
n. 12.
13 14. 15.
16.
17 18 ig.
20.
21.
22 23 24.
25.
26.
27 28.
29.
30.
31
32 33. 34.
35.
CORRESPONDENCE OF POINTS ON CONICS AREAS OF CORRESPONDING TRIANGLES AKEVS OF CORRESPONDING SECTORS CHARACTERISTIC KAIIOS OF SI-CTORIAL MEASURES RATIOS EXPRESS* D \S TRIANGLE-MEASURhS FUNCTIONAL KM AI IONS IOK KLIIPSE FUNCTIONAL RFLATIONS FOR HMFKHOLA KELAIIONS HFI \\M-N HYPERBOLIC UNCTIONS VAKIAIIONS OK THE HNPIRBOLIC FUNCTIONS. .
.
Page
9 10
.
IO
.
11
u
.
12
I-
...
ANTI HYPKRHOI ic FUNCTIONS FUNCTIONS OF SUMS AND DIH-ERENCFS CONVERSION FORMULAS LIMITING RATIOS DERIVATIVES OF HYPERBOIIC FUNCTIONS DERIVATINES OF ANII-HYPI-RBOLIC FUNCTIONS FXPANSION OF HYPERHOLIC FUNCTIONS KXPQNKNIIAL EXPRESSIONS EXPANSION OF ANTI UNCTIONS LOGARITHMIC EXPRESSION OF ANTI-JUNCTIONS THE (iUDKRMANIAN FUNCTION CIRCULAR FUNCTIONS OF (IUDERMANIAN (iUDERMANFAN A.\c;LE DERIVAFIM-S OF GUDERMANIAN AND INVERSE SERIES FOR (JUDERMANIAN AND ITS INVERSE GRAPHS OF HYPERBOLIC FUNCTIONS ELEMENTARY INTEGRALS FUNCTIONS OF COMPLEX NUMBERS ADDITION THEOREMS M>R COMPLEXES FUNCTIONS OF PURE IMAGINARIES FUNCTIONS OF x + iy IN THE FORM A' * lY
14
....
.
16
.
.
16
18
...
.
22
23 24
-i-
.
28 2()
....
.30 31
32 35
38
.
....
THE THE THE THE THE
CATENARY CATENARY OF UNIFORM STRENGTH ELASTIC CATENARY TRACTORY LOXODROME
25
27 28
...
.
19
20
.
...
/
o
.
.
.... ....
...
.
,
.
.
.
40 41
43
47 49
50 .
51
52
6
CONTENTS.
ART. 36 37.
38. 39.
TABLE
COMBINED FLEXURE AND TENSION ALTERNATING CURRENTS MISCELLANEOUS APPLICATIONS EXPLANATION OF TABLES
53 55
60 62
HYPERBOLIC FUNCTIONS VALUES OF COSH (x+ iy) AND SINH (x+ iy) III. VALUES OF gdw AND 0* LOG COSH u IV. VALUES OF gdw, LOG SINH
64 66
I.
II.
70
70
,
APPENDIX. HISTORICAL AND BIBLIOGRAPHICAL EXPONENTIAL EXPRESSIONS AS DEFINITIONS
INDEX
....
71
72
73
HYPERBOLIC FUNCTIONS. ART.
To
CORRKSPONDKNCE OK POINTS ON CONICS.
1.
prepare the
way
for a general
bolic functions a preliminary discussion
between hyperbolic to apply at the
and
the circle;
sectors.
treatment of the hyperis given on the relations
The method adopted
is
such as
same time
to sectors of the ellipse, including the analogy of the hyperbolic and circular
functions will be obvious at every step, since the same set of equations can be read in connection with either the hypeibola
or the ellipse.*
It
is
convenient to begin with the theory of
correspondence of points on two central conies of like species, i.Cc either both ellipses or both hypeibolas.
To
OB l
obtain a definition of corresponding points,
be conjugate
l
of
radii
a.
central
let
0,/f,,
conic, and O^A^ OJi^
radii of any other central conic of the same species; f\ be two points on the curves; and let their coordinates referred to the respective pairs of conjugate directions
conjugate let
P
be
(-*,, J,), (*,.;',);
l
,
then,
by
analytic geometry,
*The hyperbolic functions are not so named on account of any analogy " The with what are termed Elliptic Functions. elliptic integrals, and thence the elliptic functions, derive their
name from
cians at the rectification of the ellipse.
disadvantage;
.
.
.
.
.
the early attempts of mathemati.
To
a certain extent this
note cosh u sinh u, etc., by analogy with which the
merely
the
Functions,
circular functions cos #, sin 0, p. 175.)
is
a
because we employ the name hyperbolic function to de-
etc.
elliptic .
."
functions would be (Greenhill,
Elliptic
HYPhKJIOLIC FUNCTIONS.
8
Now
if
the points
P
l
,
1\ be so situated that
* (2)
the equalities referring to sign as well as magnitude, then P9 are called corresponding points in the two systems. If Qi be another pair of correspondents, then the sector and
angle P^O^Q,
are
when to
/\O
l
,
tri-
respectively with the Tiiese definitions will apply also
y
Q
9
.
the conies coincide, the points
any two
Q
to correspond
said
sector and tsiangle
P19
/'
,
I\ being then referred
pairs of conjugate diameters of the
same
conic.
between corresponding areas it " is convenient to adopt the following use of the word measure": The measure of any area connected with a given central conic In discussing the relations
the ratio which
bears to the constant area of the triangle formed by two conjugate diameters of the same conic.
is
I r or
it
example, the measure of the sector A^O^P^ sector A,O,PI triangle""^, (9JJ5J
is
the ratio
AREAS OF CORRESPONDING SECTORS.
and
to be
is
A^O^
A
and
common
The let
regarded as positive or negative according as O ^ are at the same or opposite sides of their l
l
initial line.
ART.
For,
l
9
AREAS OF CORRESPONDING TRIANGLES.
2.
areas of corresponding triangles have equal measures. the coordinates of />, <2, be (^ l ,y l ) (-*/, J'/) anc* let t
those of their correspondents /',, <2a be (.r,, j' 9 ), (,r/, j'/); let the triangles />,>, <2, 1\O^Q^ be T19 7",, and let the measuring tri,
A OH
K
and their angles a?,, r^ of both magnitude account then, by analytic geometry, taking angles
1
1
and direction 7*
:=
,
1
A^OJ\
be
A",
and
of angles, areas,
JU.lV- -OPS'"
A'
Jrt7
Therefore, by
ART.
.sin
ft>,
;
lines,
_
,
J//
^
u>
-
-
(2),
3.
tJ
,
_^^ a
/;
L'
6
\
(3)
AREAS OF CORRESPONDING SECTORS.
The For
areas of corresponding sectors h.ive equal measures. conceive the sectors S 19 Sv divided up into infinitesimal
corresponding sectors then the respective infinitesimal corresponding triangles have equal measures (Art. 2); but the given sectors are the limits of the sums of these infinitesimal ;
triangles,
hence
5
-
K
= S
*
In particular, the sectors ures; for the It
may
initial
points
A^O.P^
A A t ,
, x (4)
AY
t
t
A OP t
y
9
have equal meas-
are corresponding points.
be proved conversely by an obvious reductio ad
points of two equal-measured sectors correspond, then their terminal points correspond.
absurdum that
if
the
initial
O,A 19 Ot A 9 be the initial lines of two equal-measured sec^rs whose terminal radii are O P^ O9 Ptt Thus
if
any
radii
V
HYPERBOLIC FUNCTIONS.
10
P P
are corresponding points referred respectively to then 19 t the pairs of conjugate directions O^A^ O V B^ and that is,
Prove that the sector /\0,(?, is bisected by the line P,Q (Refer the points 19 Q 19 recommon axis of x and to the two Q 19 opposite conjugate directions as axis of y and show that are then corresponding points.) Prob. 2. Prove that the measure of a circular sector is equal to the radian measure of its angle. Prob. 3. Find the measure of an elliptic quadrant, and of the sector included by conjugate radii. Prob.
I.
joining <9, to the mid-point of spectively, to the median as
X
P
.
9
P
9
ART.
CHARACTERISTIC RATIOS OF SECTORIAL MEASURES.
4.
A OP =S
Let
1
1
1
be any sector of a central conic; draw O^l l9 i.e. parallel to the tangent at A MJ*i = }'M O A = rf, and the conjugate radius l
/^J/, ordinatc to
OM
let
l
OJ>,
.r,,
l
= ^\
t
t
l
then the ratios
;
,
l
xja v9 yjb
v
are called the charac-
the given sectorial measure S //Cr These ratios are constant both in magnitude and sign for all sectors
teristic ratios of
t
same measure and species wherever these may be situHence there exists a functional relation beated (Art. 3). tween the sectorial measure and each of its characteristic
of the
ratios.
ART.
5.
RATIOS EXPRESSED AS TRIANGLE-MEASURES.
The triangle of a sector and its complementary triangle are measured by the two characteristic ratios. For, let the triangle A^O.P^ and its complementary triangle P O B be denoted by 1
r
if
TV; then
1
1
"
K^ T'
^&\b\ sin lrb
t
x
ctfj
b^
o>,
x.
(5) t
sin
FUNCTIONAL RELATIONS FOR
ART.
The
11
FUNCTIONAL RELATIONS FOR ELLIPSE.
6.
functional relations that exist between the sectorial
measure and each of for
ELLIPSE.
all
its
characteristic
ratios
are
the same
in-
elliptic,
r~~i.^
eluding circular, sectors (Art. 4).
sT-
B,
s^
Let/*,,
1\ be corresponding
points on an ellipse and a circle, referred to the conjugate di-
0,A^ O B l9 and
rections
t
right angles
;
let
the angle
A^O^R^
0,
A OJ\ = 9
~L*
1
A,
the latter pair being at in radian measure; then (6)
, tf,
5, 5, ^ = cos.;., j -V^sm^; .
a
9
.-.
At
***
[a.
=
hence, in the ellipse, by Art. 3, (7)
Prob.
Also find
Prob.
measure Prob.
Given id
4.
AiOiPi.
the measure of the elliptic sector
5.
=
4,
/fr,
ratios of
3, G?
an
=
60.
elliptic sector
whose
is ITT.
Write down the relation between an
6.
ART.
The
\a\; find
area when- 0,
Find the characteristic
its triangle.
its
=
its
7.
(See Art.
elliptic sector
and
5.)
FUNCTIONAL RELATIONS FOR HYPERBOLA.
functional relations between a sectorial measure and
characteristic ratios in the case of the hyperbola
may be
written in the form
S.
'
v.
S.
-*M
and these express that the ratio of the two lines on the left is a certain definite function of the ratio of the two areas on the right.
These functions are
called
by analogy the hyperbolic
HYPERBOLIC FUNCTIONS. cosine and the hyperbolic sine.
Thus, writing u for
SJKV
the
two equations
= a
cosh
If b
//,
t
=
sinh u
($\ v >
l
serve to define the hyperbolic cosine and sine of a given sectorial measure u and the hyperbolic tangent, cotangent, secant, ;
and cosecant are then defined as follows .
^
tanh
;/
=
sinh ---
?/
.
-
;
u
cotli
,
cosh
:
cosh//
=
--
sinh
;/
//'
(9) i
sech
//
=
T
T t
.
The names
csch u
,
cosh
.
sinh
//
of these functions
or "hyper-cosine," etc.
!
(See
"
//
J
be read " h-cosine,"
may
"
angloid
or
"hyperbolic
angle," p. 73-)
ART.
RELATIONS AMONG HYPERBOLIC FUNCTIONS.
8.
the six functions there are five independent relawhen the numerical value of one of the functions
Among
tions, so that is
given, the values of the other five can be found.
these relations consist of the four defining equations fifth is derived from the equation of the hyperbola
giving sinh*//
cosh*//
By
a combination of
ary relations
by
may
cosh* w, sinh
1
//,
some
I.
(10)
of these equations other subsidi-
and applying tanh' u
coth* (9),
(10),
(9),
give
= sech* = csch
;/, )
(u)
I
(u)
i
2
//
//.
)
will readily serve to express the
value of any function in terms of any other. when tanh u is given,
coth
//
=
(9).
of
The
be obtained; thus, dividing (10) successively
I
Equations
=
Four
-
tanh u
,
sech u
=
A/I
For example,
tanh*//,
RELATIONS BETWEEN HYPERBOLIC FUNCTIONS.
=
,
cosh?/
=
,
csch u
I
sinn//
,
V
I
\/
I
.
.
-----
=-__
tailll
V
tanh*//
13 //
-,
tanh'//
I
tanh*//
---
-
--
tanh
//
The ambiguity in the sign of the square root may usually The functions be removed by the following considerations cosh//, scch // are always positive, because the primary char:
acteristic ratio
O,M
the abscissa
is
.*,/
OA
positive, since the initial line
O
are similarly directed from
l
}
and
l
on which-
,
ever branch of the hyperbola I\ may be situated; but the functions si nh //, tanh it, coth u, csch //, involve the other charac-
y^ //
and is
b^
which
yjb^
teristic ratio
is
positive or negative
have the same or opposite
signs,
i.e.,
according as as the measure
positive or negative; hence these four functions are either
Thus all positive or all negative. tions sinh //, tanh //, csch //, coth ;/,
when any one
of the func-
given magnitude and sign, there is no ambiguity in the value of any of the six but when either cosh // or sech // is hyperbolic functions is there ambiguity as to whether the other four functions given, in
is
;
shall
be
all
positive or
all
negative.
_
The hyperbolic tangent may be expressed two
For draw the tangent
lines.
AC ~ t\ then y x a y tanh u = 4 :- = .--
line ^
i
b
a
x
b
= a..- = t
t
f
-.
(
The hyperbolic tangent
is
^.
.
I3
as the ratio of
)0
-
i
*
.
the measure of the triangle
OAC.
For
Thus the
OAC OAS sector AOP
are proportional to
y
;/,
at
**
t
3)
*
and the triangles AOP, FOB,
sinh
sinhw
,
cosh
> >
.
,
//,
tanh u (eqs.
tanh//.
5,
13)
;
AOC, hence (14)
HYPERBOLIC FUNCTIONS.
14 Prob.
Prob.
8.
<
i,
Prob.
9.
tanh u
Express
7.
Given cosh
//
=
2,
all
the hyperbolic functions in terms ot sinh
Prove from eqs. sech u
u.
find the values of the other functions.
<
n,
10,
that
cosh//> sinh
,
cosh*/>i,
i.
In the figure of Art.
i, let
OA-2, OB-\,
AOB =
60,
and area of sector AOP => $\ find the sectorial measure, and the two characteristic ratios, in the elliptic sector, and also in the hyperbolic sector; and find the area of the triangle AOP. (Use tables of cos, bin, cosh, sinh.)
Prob. 10.
Show
coth u
that
9
sech
//,
csch u
may each be
ex-
P
pressed as the ratio of two lines, as follows: Let the tangent at m make on the conjugate axes OA, OB, intercepts OS n\ in let the tangent at B^ to the conjugate hyperbola, meet y
=
OT =
}
OP
making BR =
/;
~
coth u Prob. IT.
The
this for
Modify
R
then sech u
//a,
measure of
the ellipse.
=
csch
m/a,
segment AMP
Modify
=
//
njb.
sinh u cosh u
is
u.
10-14, and probs.
also eqs.
8, 10.
ART.
9.
VARIATIONS OF THE HYPERBOLIC FUNCTIONS.
Since the values of the hyperbolic functions depend only on the sectorial measure, it is convenient, in tracing their variations,
half
one whose rectangular hyperbola,
to consider only sectors
of a
of
conjugate radii are equal, and to take the principal axis OA as the common initial line
of
all
the
sectors.
The
sectorial
measure u assumes every value from through
o, to -f- oo
,
oo,
as the terminal point
P
comes in from infinity on the lower branch, and passes to infinity on the upper
branch; that is, as the terminal line OP swings from the lower asymptotic position
but as
y is
=
x* to the
proved
P passes to
in
upper one,
y
= x.
Art. 17, that the sector
It
is
here assumed,
AOP becomes
infinite
infinity.
Since the functions cosh
//,
sinh
//,
tanh
u, for
any position
VARIATIONS OF THE HYPERBOLIC FUNCTIONS. of OP) are equal to the ratios of x, y, a, it is evident from the figure that
cosh o
=
= 0,
sinh
I,
/,
15
to the principal radius
= 0,
tanh
(15)
and that as u increases towards positive infinity, cosh it, sinh u are positive and become infinite, but tanh// approaches unity as a limit
thus
;
cosh
=
co
sinh
oo,
00=00,
tanh
=
oo
i.
(16)
Again, as // changes from zero towards the negative side, cosh u is positive and increases from unity to infinity, but sinh u
negative and
is
increases numerically from
and tanh
zero to a
negative and negative infinite, numerically from zero to negative unity hence //
is
inci eases
also
;
cosh
oo )
(
=
oo
sinh
,
(
oo
For intermediate values
)
of
= //
oo
tanh
,
(
oo)
A
are tabulated at the end of this chapter.
manner
I.
(17)
the numerical values of these
functions can be found from the formulas of Arts.
their
=
16, 17,
and
general idea of
of variation can be obtained from the curves in
which the sectorial measure u
represented by the abscissa, and the values of the functions cosh //, sinh //, etc., Art. 25,
in
is
are represented by the ordinate.
The
relations
between the functions of
;/
and of u are
evident from the definitions, as indicated above, and
Art.
in
8.
Thus cosh sech
tanh
() = (
u)
(
u)
= =
cosh
u,
sinh
-f sech
//,
csch
tanh
//,
coth
-{-
(
+
=
(//)
Prob. 12. Trace the changes in sech oo to oo. Show that sinh
from
//)
(
//)
?/,
,
=
coth
cosh
sinh u,
\
csch
;/,
>
coth
//.
(18)
;
csch u, as u passes are infinites of the
#, //
same order when u ,
is infinite. (It will appear in Art. 17 that sinh cosh u are infinites of an order infinitely higher than the order
of u.)
Prob. 13. Applying eq. (12) to figure, page 14, prove tanh u, tan
A OP.
=
HYPERBOLIC FUNCTIONS.
16
ART.
The
x -
equations
=
cosh
y
1
y -r
=
t
sinh
7
//,
=
tanh
u, etc.,
^ 1
-,
t
=
sinli~ T,
u,
by the inverse notation u^cosh"
also be expressed
may
=
ANTI-HYPERBOLIC FUNCTIONS.
10.
tanh"
1
-r,
etc.,
which
be read: "#
may
is
the sectorial measure whose hyperbolic cosine is the ratio x to a" etc. or " u is the anti-h-cosine of x/a" etc. ;
Since there are two values of
u,
correspond to a given value of cosh
with opposite signs, that it follows that if u be
//,
determined from the equation cosh u = ;, where m is a given number greater than unity, u is a two-valued function of m.
The symbol
m
will be used to denote the positive value ;;/. the equation cosh u Similarly the will stand for the positive value of // that
cosh"
1
=
of u that satisfies
symbol sech"
1
;;/
=
The
m. the equation sech. u functions sinh' ;//, tanh' ;//, coth"
satisfies
1
1
1
as the sign of m.
Hence
all
signs of the other csch" ;;/, are the same 1
;//,
of the anti-hyperbolic functions
numbers are one-valued.
of real
Prob. 14. Prove the following relations:
cosh'
1
m=
sinh' 1
a
V;;/
i,
sinh"
1
;;/
=
cosh" 1
upper or lower sign being used according as 1 1 Modify these relations for sin" , cos" negative. iie
;//
a
V;;/ is
+
i,
positive or
.
=
OB =
AOB =
Prob. 15. In figure, Art. i, let OA 2, 60; find i, the area of the hyperbolic sector A OP, and of the segment 1 if the abscissa of is 3. (Find cosh" from the tables for cosh.)
AMP,
P
ART. (a)
11.
To sinh
prove the difference-formulas = sinh u cosh v cosh (// ?>)
cosh
Let
FUNCTIONS OF SUMS AND DIFFERENCES.
(//
v)
=
cosh u cosh v
sinh
;/
sinh #,
//
sinh
v.
OA
AOQ
be any radius of a hyperbola, and let the sectors AOP, v is the measure of the have the measures u, v\ then u
Let OB, OQ' be the radii conjugate to OA, OQ\ be (*,,.?,), (x,y\ (*',/) the coordinates of P, Q, with reference to the axes OA, OB\ then
sector
and
QOP.
let
Q
FUNCTIONS OF SUMS AND DIFFERENCES. sinh
=
-*)
toLQr = triangle
sinh
A
^*
= sinh
17
QO/
A
s>
^ sn sin a)
u cosh v
cosh u sinh v\
w
., cosh (
N
v)
=
,
cosh
QOP = triangle ^ - * POQ' [Art. A/A
sector
r .
5.
sn ^
,
but
x
(20)
since Q, Q' are extremities of conjugate radii
cosh
v)
(
= cosh
;/
cosh
sinh
?>
;
hence
//
sinh
z/.
In the figures u is positive and v is positive or negative. Other figures may be drawn with n negative, and the language in
the text will apply to
all.
In the case of elliptic sectors,
drawn, and the same language the second that except equation of (20) will be x' /a similar figures
may be
l
will
apply,
=
j
therefore
(b)
sin
(//
v)
cos
(//
z/)
To prove
= sin u cos v = cos u cos v
;/
sin
u sin
sin v> v.
the sum-formulas
sinh (u
+ v) = sinh w cosh v
cosh
-j-
(
-f-
cos
v)
=
cosh
cosh
These equations follow from
z/
'
-f-
w sinh osh u cosh
-f-
inh u sinh sinh
(19)
z/,
)
f z/. )
by changing v
into
v,
HYPERBOLIC FUNCTIONS.
18
and then
for sinh(
v\ cosh(
sinh v, coshfl
writing
#),
(Art. 9, eqs. (18)). /
T To
\
(c)' v
^ ^
^i
v);
sin
=
v)
T-
-
v
,
(22) ; v
.
tanh v
tanh
i
Writing tanh (u
tanh v
tanh u
=
\
/
i
prove that tanh (u v v *
expanding and dividing
!2,
^-
numerator and denominator by cosh u cosh
ob-
v, eq. (22) is
tained.
Prob.
1.
cosh
3.
i
4.
tanh \u
2//
=
+ cosh .
cosh*#
=
//
.
6.
= =
.
Sinll
2U
sinh yi
8.
=
cosh (u -f
3, find
cosh"
^//,
+ 2
=
// --
-
cosh
=
3 //
.
=
2
sinh
-,
COSh
i//.
i\*
1
J.
r-r-.
tanh u
i
cosh 3^
^/,
I.
8
\cosh u -f- i/ 4- tanh' w
2//
8
cosh' #
2
/cosh u
= T
//
+ 4 sinh
3 sinh
u
i i
.
tanh
2 sinh*
//
sinh u
//
tanh
i
i -\-
cosh cosh u
.
i
.
=
4 cosh'w
3 cosh u.
'
.
,
.
i
Generalize (8); and show also what it becomes 9 2 a cosh'jc sm*y sinh'jt: -f- sin /. 10. sinh jc cos y 9.
u. cosh" sinh"
1
cosh'
///
1
sinh"
///
when
u=v=
?'). .
. ,
=
+
12.
v).
+ tanh \u cosh w + sinh u = tanh \u (cosh + sinh ;/)(cosh v + sinh e^)=cosh (u -\-v)-\- sinh (u + .
7. '
2
sinh
-f-
sinh
=
.
i
5.
cosh v
2,
Prob. 17. Prove the following identities: 2 sinh # cosh u. sinh 211
2.
99
=
Given cosh u
1 6.
1
//
1
//
=.
=
Prob. 18.
What
Prob.
Modify the
cosh'
sinh'
1
1
^/// y
a
y
^/////
I
(/// 7
i -j-
y
)( wa ~"i)ji
+w
a
j.
modifications of signs are required in (21), (22), in order to pass to circular functions ? 19.
ART.
To
12.
identities of Prob. 17 for the
same purpose.
CONVERSION FORMULAS.
prove that cosh //,+ cosh u9
cosh sinh
w, j
sinh w.
= 2 cosh J(f/,+ *') cosh ^(u =
l
u9),
w + sinh + sinh u^ = 2 K. + cosh 4X iX ^) = 2 cosh i(i + *) sinh J(. cosh
#,
2 sinh sinli
//,
(//,
(//,
1
?',)
(*/,
a ),
*i).
,) )
sin:i
*
,)-
(23) JI
J
LIMITING RATIOS.
From
the addition formulas
cosh (u
+ v + cosh
cosh (w
-f~
si
nh
sinh (w
cosu ( u
^)
=
(//,
(
w
v
-f-
2 cosh
w
2 sinh
u
>
i
;/
2 si"h
)
(//
"" s i nn
-|" t; )
and then by writing u z;
= v = v) = = v) v)
(
+ ^) + sinh
(
follows that
it
)
19
2
cosh
v
=.
a
cosh
v,
sinh ^, /*
7*
,
cosh
z/,
sinh 7% //
= J(//
j
-j-
;/
s) t
these equations take the form required.
f ),
Prob. 20. In passing to circular functions, show that the only modification to be made in the conversion formulas is in the algebraic sign of the right-hand member of the second formula. __
.
---+ cosh
cosh -
...
.
Prob. 21. Simplify Prob. 22. Prove
211
.
-\-
<\v
=
sinh* j>
sinh*.*:
,
,
sinh
coshV cosh*^ Simplify coshV cosV +
sinh
Prob. 23. Simplify Prob. 24.
ART.
To
471
'
cosh 4^
sinh (x
y).
sinh'.r sinh*^. sinh*jc sin
?
_y.
u approaches zero, of
find the limit, as
sinh u
tanh u
~^~~'
~T~'
which are then indeterminate
By
(.v -\-y)
+ cosh_J-.
LIMITING RATIOS.
13.
eq. (14), sinh u
cosh ----- 211 cosh 2U
47^
-
.
sinh 2U
>
//
>
form.
in
tanh u
and
;
if
sinh u and tanh u
be successively divided by each term of these inequalities,
it
follows that I
<
sinh u
--u
.
sech u
but when
w^o,
cosh u
lim.
sinh u
=
u
u
o
~
<
<
,
cosh
-- <
tanh
_
m =
\\ '
//
=
sech
i,
u
u,
.
I,
i:
hence
tanh u
-
.
HYPERBOLIC FUNCTIONS.
ART.
To
DERIVATIVES OF HYPERBOLIC FUNCTIONS.
14.
prove that
=
^(sinh u)
(^
du //(cosh u)
cosh u
=
T
du
smh
'
, '
du (25)
,,*
,
//(cscb
s\
= sinh 4j/ = sinb = 2 cosh
Let
y
=
= cosh
ft.
limit of
sinh u
Aii)
4(2 w
+ ^) sinh %Au, .sinh 4Jft
.
(ft \
Jft
Take the
csch u coth
ft,
(ft -f-
Ay
ft,
csch* u,
//)
-j
(/) (a)
sech u tanh
^/(coth w)
to /
=
//(sech u)
a) \
+ -kAii) j
i
both
Ay
^j^
sides, as
_
dy
lim. cosh
-
.
/
//(sinh
u)
(ft
Au ~
=
o,
and put
;/)
cosh
ft,
sinh -
lim
//(sinh
then
(b)
^
Similar to
ft)
3
(see Art. 13)
;
=
cosh
.
(a).
//(tanh u} __ ~~
3W
[
sinh w
// "
cosh w
cosh* u
sinh*
cosh* u
w
=
nr#
COSIl
=
sech* ^.
DERIVATIVES OF HYPERBOLIC FUNCTIONS. (d)
Similar to
(c).
--
'
(/)
-
d
\
^
21
du
Similar to
~
i
T du
^
i
sinh i
cosh u
cohh
*
=
seen
//
tanh
u.
//
(c).
thus appears that the functions sinh ?/, cosh u reproduce themselves in two differentiations and, similarly, that the circular functions sin?/, cos// produce their opposites in two It
;
In this connection
differentiations.
it
may
be noted that the
frequent appearance of the hyperbolic (and circular) functions the solution of physical problems is chiefly due to the fact
in
that they answer the question What function has its second derivative equal to a positive (or negative) constant multiple :
of the function itself? (See Probs. 28-30.)
y = cosh mx
is
An
answer such as
not, however, to be understood as asserting that
mx
is an actual sectorial measure and^v its characteristic ratio but only that the relation between the numbers mx and y is the same as the known relation between the measure of a hyper-
;
bolic sector
and
its
characteristic ratio;
and that the numerical
value of y could be found from a table of hyperbolic cosines. Prob. 25 Show that for circular functions the only modifications required are in the algebraic signs of (#), (
Prob. 27. Find the derivative of tanh u independently of the derivatives of sinh
//,
cosh
Prob. 28. Eliminate
equation y = A cosh mx
u.
the constants by differentiation from the -B sinh mx, and prove that d*y/dx* m*y.
=
-f-
Prob. 29. Eliminate the constants from the equation
=A that d *y/dx* = y
and prove
Prob. 30. Write tial
equations
down
cos
mx
+ B sin mx,
my.
the most general solutions of the differen-
22
HYPERBOLIC FUNCTIONS.
ART.
DERIVATIVES OF ANTI-HYPERBOLIC FUNCTIONS.
15.
rf(sinh"'
x)
~
Tx
i
r ~fe +\'
1
^(cosh"* x) ~~
~&
i '
V*"" 11 "!
4tanh~'
"I
i
__
)
(26)
v
,
'
dx
I
~
i
*?
1
I
^/(csch-
x \x Let
=
=
#
Vi
sinh w
Similar to
(c)
Let
(^/)
u
dfo
tanh"
tanh
Similar to
=
Vi
x
+
then
^r,
if)du
= 1
-*"
sinh
du
du
>
/*
//,
-
=
(i
^r
=
tanh
;/,
^w
^X//,
=
dx
=
d
i
t
(cosh-
,
i\ 1 -!=-/ i
1
sech"
dtr/i
\*
1
-
I
I
(e).
Prob. 31. Prove
_
x
i
^r
= cosh w
//
;r'.
(c).
^)_^~
(/) Similar to
1
2
1
.
then
^r,
r
(a).
=
(i
_^ ^ ^/(sech" ,
1
8
-f-
(6)
=
sinh"
+
i
i
+
__
i/(cot"' x) ~" JP*'
^
i
J
=
i
du
EXPANSION OF HYPERBOLIC FUNCTIONS.
23
Prob. 32. Prove a
rfsinh
.
.
tanh-
1
dx
a -=
# -
=
a
..
.
a
-^
a'
' ,
"1
x -
.
,
1
rf/coth-
,
-,
a
*'_]*<
--= - -,adx
*
#
"~|
J*>
!
1
Prob. 33. Find ^(sech" x) independently of cosh~ x. Prob. 34.
When
tanh"
1
x
1 prove that coth"
is real,
nary, and conversely; except when x
_
_
,
For
16.
this
/(*)
=
.
-j-, ~^
x
x
,
when ^
purpose take Maclaurin's Theorem,
= AO) +
/(o)
+
J,
y
;/
(o)
+ ~ ^/"'(o) +
.
I,
sinh u
hence similarly, or
by
cosh u
By means cosh
00.
= sinh u, f(u) = cosh K, /"() = sinh = 0, /'(o) = cosh o = /(o) = sinh
then
u,
ima 5i
EXPANSION OF HYPERBOLIC FUNCTIONS.
and put /()
and
is
i.
.
Prob. 35. Evaluate
ART.
=
= u +~ u + 3
u'
+
.
.
.
;
.
-,
ir f
.
.
.,
.;
(27)
differentiation,
= +4 i
f
+ ^ + ---4
(28)
of these series the numerical values of sinh
//,
can be computed and tabulated for successive values of
the independent variable u. They are convergent for all values of u, because the ratio of the //th term to the preceding is in
the
first
u*/(2n
i)(2// 2), and in the second case both of which ratios can be made less than
case u*/(2n 2)(2
3),
unity by taking n large enough, no matter what value u has. Lagrange's remainder shows equivalence of function and series.
24
HYPERBOLIC FUNCTIONS.
From
these series the following can be obtained by division
tanh u
=u-
:
'
-
4
.
(2
These four developments are seldom used, as there is no observable law in the coefficients, and as the functions tanh //, sech
;/,
coth
computed
//,
csch
can be found directly from the previously
//,
values of cosh u, sinh
Show
Prob. 36.
u.
that these six developments can be adapted to
the circular functions by changing the alternate signs.
ART.
EXPONENTIAL EXPRESSIONS.
17.
Adding and subtracting cosh u
+ sinh u =
I
-f-
//
-f-
~u*
~u
-|
-i
,
sinh
//
=
I
u
I
.
-u
-{
I
a
^
.
tanh
=
Me
tt
-u
where the symbol
x
in
~'
= V' 2
">
**'
.
t u sech
=
The analogous exponential cos u
I
+ T" u
= i(^M
u
,
3
sinh u
+ e~ \
--= e*-e-*
~u"
-}-
sn #
-f-
.
.
.
=
.
.
.
= e~ u
u
e
9
*r *
'
hence cosh u
3
j
*
cosh u
give the identities
(27), (28)
4
e'
--
u
\
2
h
-
'
,
9
etc.
(30)
I
expressions for sin u, cos u are
=
.
22
stands for the result of substituting 0* for
the exponential development
This
will
be more
numbers, Arts. 28, 29.
fully
explained
in treating of
complex
EXPANSION OF ANTI-FUNCTIONS.
#5
Prob. 37. Show that the properties of the hyperbolic functions could be placed on a purely algebraic basis by starting with equafor example, verify the identities tions (30) as their definitions :
;
sinh a
cosh u
sinh
-
u)
( a
u
=
sinh (u +?>)
i,
- =m mu)-
r
cosh
sinh u,
(//)
= sinh
=
cosh u,
u cosh v + cosh u sinh
v,
--,--.- = n? sinh a
a
cosh
;;///.
di?
Prob. 38. Prove (cosh u
+ sinh
(sinh
n
u)
=
/////)
.
,
cosh nu
.
+ sinh
nu.
Prob. 39. Assuming from Art. 14 that cosh sinh u satisfy the differential equation dty/dfo* j, whose general solution may be ,
=
=
+ Be~ where A B are arbitrary constants show how to determine A, B in order to derive the expressions for cosh
written
y
u
Ae u
>
y
;
,
sinh u, respectively.
[Use
Show how
Prob. 40.
eq. (15).]
to construct a table of exponential funcvice versa.
from a table of hyperbolic sines and cosines, and sinh u). Prob. 41. Prove u = log, (cosh u
tions
+
Prob. 42. Show that the area of any hyperbolic sector when its terminal line is one of the asymptotes.
2*-
Prob. 43.
From
w
=cosh
1
( cos h
^)
u the relation 2 cosh u -= e
;/tf
prove
+ #cosh (n2)u+\n(ni) cosh
and examine the last term when n is odd or even. Find also the corresponding expression for 2*~
ART. .
-^
(n
n
l
(sinh u)
.
EXPANSION OF ANTI-FUNCTIONS.
18.
--x) = dx
dfsinrr
c Since
infinite
u
e"
-f-
is
1
-
-j
-i
= *
= (i /
I
\
i
i
x*}-*
-f-
34
i
+
3 5
i
246
24
y< "T"
-**
'
hence, by integration, i
sinh-
i 1
*
-+ 2467'
^ x% = * ---h-~3 --i
!
i
23245
!
3 5
*T
i
/
\
(30 w/
the integration-constant being zero, since sinh" x vanishes with x. This series is convergent, and can be used in compu1
HYPERBOLIC FUNCTIONS.
$6
:ation, r
.
>
i,
Another series, convergent when only when x < i. is obtained by writing the above derivative in the form
-L _I x = C+log b *+2 2X* 2
sinh-'
"
vhere
is
A
3 + I2 466x' | I
f
v ( VJ '
'
the integration-constant, which will be shown in
be equal to log,
\rt. 19 to
3 J 44X
'
1
2.
development of similar form
dx
_ ~^
is
iV
_i/ ~ vt '
,
1 2
obtained for cosh~'^r; for
l-LllJ-J-ilS I ^~ ^ ^
"1
,
J:
a
2
t 4 X
2 4 6
X
f
'
'
'J'
icnce :osh-'
n
x=C+\ogx-^ ^.-..., 22^',--^ 2 4 4** ---I 2 4 6 6*' .
to
which
C
is
again equal to log, 2 [Art.
>rder that the function cosh"'.r
vergent, hence
it is
may
when x exceeds
ess than unity; but
be
(33) VJO/
Prob. 46]. In not be real, 19,
x must
unity, this series
is
con-
always available for computation.
Again, ind hence
From jech"
1
x
tanh"
1
x = x + - x* -\- -x* -\- -x
(32), (33), (34) are derived
= cosh"
1
+
.
.
. ,
(34)
:
1
-fl-l^-L!^,.... 2.2 2.4.4 2.4.6.6
(3S) Vt"'
LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS.
,,
cscrr'
jr
u = sinh"
-* x
i 1
=
II
l
x
2
I
+-2 i
ix
.
3 ^ I 4 5*
*
3 -
27
*
5
I
,+
2467*'
^-tog^r+^-i^ + Iil^-...; 2.2 2.4.42.4.6.6 coth-'*
= tanh-
I *
1
Show
Prob. 44.
=1+ *
+
,
'
6
'
3*'
+
-L
+ '
.
..
.
(36)'
v (37) '
TX*
5^r
that the series for tanh"
1
jc,
coth' 1
sech" 1
jc,
A:,
are always available for computation.
Show
Prob. 45.
that one or other of the
inverse hyperbolic cosecant
ART.
19.
is
two developments of the
available.
LOGARITHMIC EXPRESSION OF ANTI FUNCTIONS.
x
Let
cosh
= sinh u\ u -f- sinh u = e", "'
then Vx*
//,
i
Similarly,
= cosh u, = cosh' ^, = log (x + sinh" ^ = log (x -f i/^+'T).
Also
sech" ! ;r
therefore
x-4- Vx*
I
1
and
V*'
I
(38)
).
1
=
=
cosh" 1
'II* -*=s,nh-'-
.
,
x
let
I
r
therefore
= tanh
*
I
^**
=
#
L
log
1^
"
*
**
I
/
\
(4I )
~" ^~w _M
,
_ + = =^ ^" /* u
JT
-
i
^r
2u
= log^i
coth" 1 ^
and
x
,
tanh- 1 ^-^^ log i
sinh"
1
^
log
x
(38), (39), that,
#:Mog
2,
~~~
cosh""
1
^
-
.
(43)
I
when .r^
and hence show that the integration-constants 2.
(42)
;
1
Show from
Prob. 46.
x
= tanh' - = i log ?-l. JT
equal to log
(40)
,
= log-X_^_.
1
Again,
(39)
log
oo,
x- log
2,
in (32), (33) are
each
28
HYPERBOLIC FUNCTIONS. l
Prob. 47. Derive from (42) the series for tanh
x
given in (34).
Prob. 48. Prove the identities: 2
= x=
log sec
.r
log tan
i
!#
loc x=^2 tann
X+I
=tann -i*~^ i_-sinn X +1 7
1
tanh" tan ^x\ log esc x
2
1
tanh" cos 2#
ART.
20.
in Arts.
=
2
J
tanh"
1
sinh" cot 2x
== cosn
"y(x~\~x~ V
" )i
!
tan'^nr
=
-[-
i#);
cosh" 1 esc 2X.
THE GUDERMANIAN FUNCTION.
The coirespondence cussed
=
x
~v\x
of sectors of the It is
1-4.
correspondence that
may
same
now convenient
exist
species was disto treat of the
between sectors of
different
species.
Pv PI on any hyperbola and ellipse, are said to with reference to two pairs of conjugates O A correspond Two
points
,
l
0,#,
,
l
,
and O^AI, O^B^j respectively, when *i/<*i
=
<**/*
l
Afl^Pi are then also said to
(44)
The sectors A^O^P.^ correspond. Thus corresponding
and when y ,y^ have the same
sign.
sectors of central conies of different species are of the
same
sign and have their primary characteristic ratios reciprocal. Hence there is a fixed functional relation between their re-
The elliptic sectorial measure is called spective measures. the gudermanian of the corresponding hyperbolic sectorial measure, and the latter the anti-gudermanian of the former. This relation is expressed by
SJK, or
ART.
The
v
= gd
=
gd s,/*;
and
;/,
u
= gd~
!
z/.
(45)
CIRCULAR FUNCTIONS OF GUDERMANIAN.
21.
six hyperbolic functions of u are expressible in
of the six circular functions of
x
= cosh
x //,
its
gudermanian
= cos v,
;
terms
for since
(see Arts. 6, 7)
which u, v are the measures of corresponding hyperbolic and elliptic sectors,
in
GUDERMANIAN ANGLE.
29
= sec v, [eq. (44)] = tan v, sinh u = t/secV tanh # = tan #/sec v = sin #, coth w = esc = cos?;, sech csch u = cot cosh u
hence
I
(46)
z>,
//
z>.
The gudermanian
sometimes useful in computation for be given, v can be found from a table of natural tangents, and the other circular functions of v will give instance,
if
sinh
is
;
//
Other uses
the remaining hyperbolic functions of u. function are given in Arts. 2226, 32-36. Prob. 49. Prove that gd u
=
!
sec~ (cosh u)
=
of this
1
tan~ (sinh u)
= cos'^sech u) =sin~ (tanh w), gd" ^ = cosh" '(sec v) = sinh" (tan v) = sech"" (cos v) = tanh" ^sin v). = = o, gd oo ^TT gd o gd( oo) = -oo. gd- 0=0, gd-^flr) =00, gd- (-i7r) 1
1
Prob. 50. Prove
1
1
Prob. 51. Prove
TT,
?
1
Prob
Show
52.
Prob. 53. tion tanh
\u
that gd
From
= tan
the
1
//
first
and gd" 1 v are odd functions of identity in 4, Prob.
u, v.
17, derive the rela-
\v.
Prob. 54. Prove J
tanh~ (tan u)
=
ART. If
! % gd 2U, and tan~ (tanh x)
22.
% gd
-1
2JC.
GUDERMANIAN ANGLE
a circle be used instead of the ellipse of Art. 20, the
gudermanian of the hyperbolic sectorial measure will be equal to the radian measure of the angle of the corresponding circulai sector (see eq. (6), and Art. called the gudermanian angle
3, ;
Prob. 2). This angle will be but the gudermanian function v,
merely a number, or ratio and this number is equal to the radian measure of the gudermanian angle 0, which is itself usually tabulated in degree measure thus as
above defined,
is
;
;
/7r
(47)
HYPERBOLIC FUNCTIONS.
'10
Show that
Prob. 55.
the gudermanian angle of u
may be
construct*
ed as follows:
Take the
OA
principal radius
/
initial
of an equilateral hyperbola, as the as the terminal line, and
OP
line, of the
from
M,
sector
whose measure
is
u\
the foot of the ordinate of
MT
P, draw tangent to the circle whose diameter is the transverse axis; then
AOTis
the angle required.* Show that the angle
Prob. 56.
never exceeds 90.
Art.
and Prob.
9,
The
Prob. 57.
M
A
bisector of angle
bisects the sector
and the
53, Art. 21),
line
AOT
A OP
(see Prob. 13, (See Prob. i, Art. 3.)
AP.
Prob. 58. This bisector is parallel to TP, and the points 2\P are in line with the point diametrically opposite to A. Prob. 59. The tangent at P passes through the foot of the
oidinate of
7 and 1
,
Prob. 60.
ART.
23.
The
TM on the tangent at A. APM half the gudermanian angle.
intersects
angle
is
DERIVATIVES OF GUDERMANIAN AND INVERSE.
= sec v = v tan vdv = sec vdv = d(gd~ v} = dv = =
Let then sec
gd
cosh
Again, therefore
=
1
gd-
v,
,
sinh udu, du,
l
therefore
u
u,
sec vdv.
cos
vdu
sech
;/
(48)
=
sech u du,
du.
(49)
Prob. 61. Differentiate:
y
=
sinh u
y
=
tanh u sech u
gd
u,
+
gd
u,
y y
= =
sin v
+ gd"
tan v sec v
1
v,
-f-
gd"
1
#.
and denoted by /. *This angle was called by Gudermann the longitude of His inverse symbol was U.; thus u = H.(/tt). (Crelle's Journal, vol. 6, 1830.) ,
Lambert, who introduced the angle 6, named it the transcendent angle. (Hist, I'acad. roy de Berlin, 1761). Hottel (Nouvelles Annales, vol. 3, 1864)
de
called it the hyperbolic amplitude of u, and wrote it amh u, in analogy with the Cayley (Elliptic amplitude of an elliptic function, as shown in Prob. 62. Functions, 1876) made the usage uniform by attaching to the angle the name
of the mathematician
theory of
elliptic
who had used
it
extensively in
functions of modulus unity.
tabulation and in the
GUbERMANIAN AND
SERIES FOR
ITS INVERSE.
Prob. 62. Writing the "elliptic integral of the
1
kind"
first
in
the form
K
*#_
f+
U
Vi-
J
being called the modulus, and
= show
am
u,
that, in the special case
u cos
am
u
= =
gd" 0, sech
and that thus the
tan
w,
ART.
=
when K u
=
is,
i,
gd
u
sin
//,
am
//
=
tanh
if,
sinh #;
functions sin
elliptic
the hyperbolic functions,
the amplitude; that
(mod. K),
am am
1
K* sin* 0'
am
degenerate into
etc.,
//,
when the modulus
is
unity.*
SERIES FOR GUDERMANIAN AND ITS INVERSE.
24.
Substitute for sech
//,
sec v in (49), (48) their expansions,
Art. 16, and integrate, then u gd u
X + - rfh" + W + frS +1^ +
= gd-V = v +
1
T&U'
(50) .
.
(51)
.
No
constants of integration appear, since gd u vanishes with and gd" ^ with v. These series are seldom used in compu1
,
tation, as
gd u
is
and hyperbolic gd u
and
means
best found and tabulated by
of natural tangents
=
tan-^sinh
of tables
from the equation
sines, //),
a table of the direct function can be used to furnish the
numerical values of the inverse function
;
or the latter can be
obtained from the equation, gd~'^
To
=
sinh "'(tan v)
=
f
cosh~ (sec
obtain a logarithmic expression for gd"" ' l
gd' v =11, * The relation gd u
=am
u,
(mod.
i),
z>).
1
?',
let
v = gd u, led Hottel to
name
the function
gd
u,
In this the hyperbolic amplitude of u, and to write itamh u (see note, Art. 22). connection Cayley expressed the functions tanh u, sech u. sinh u in the form
gd u, cos gd u, tan gd u, and wrote them sg u, eg //, tg u, to correspond with the abbreviations sn u, en u, dn u for sin am u, cos am u, tan am u.
sin
=
=
sn u, (mod. i); etc. sg u note that neither the elliptic nor the hyperbol'c functions received their names on account of the relation existing between them in a
Thus tanh u It is
well to
special case.
(See foot-note,
p.
7
)
HYPERBOLIC FUNCTIONS.
32
= cosh u, = tan v cosh u
sec v
}-
f
-\-
sin
v
,
= gd
l
v,
=
^3
i
.
U
sinh u 9
=
.
,
tan
r
-f '
,
,
x
* 'f Jf/)
z;)
u ~]
Prob. 64. Prove that gd u
when w
v)'
+ +
log. tan (iff
8d u
i * 17 * Prob. 63. Evaluate
T3
sinh
v
/t
sin (Jff
z;
=
=. eu ,
cos
I .
cos
order,
tan v
sec v
therefore
-
%&~ lv -
.
J=o sin
w
,
is
(52)
*/).
~
~i
V
v ~\ .
.
Jz/=o
an infinitesimal of the
fifth
== o.
Prob. 65. Prove the relations iff
+ %v=
ART.
25.
u tan~ e l
9
iff
^v
=
tan"V~*.
GRAPHS OF HYPERBOLIC FUNCTIONS.
Drawing two rectangular axes, and laying down a series of points whose abscissas represent, on any convenient scale, successive values of the sectorial measure, and whose ordinates represent, preferably on the same scale, the corre-
sponding
values
of
the
function to be plotted, the
out by this series of points will be a locus traced
graphical representation of the variation of the function as the sectorial meas-
A
GRAPHS OF THE HYPERBOLIC FUNCTIONS.
The equations
ure varies.
sian notation are
y= y= y= y=
B C
D
= sech x y = csch x y = coth x
cosh x,
y
sinh x,
tanh x,
gd
numerical value of cosh
//,
;
;
;
.r.
written for the scctorial measure
.r is
etc.
and y for the thus to be noted that the
It is
//,
y are numbers, or ratios, and that the equation merely expresses that the relation between the
variables x,
x numbers x and y == cosh
tween a
Dotted Lines.
Full Lines.
A
y
of the curves in the ordinary carte-
:
Fig.
Here
33
taken to be the same as the relation be-
is
measure and
sectorial
The
characteristic ratio.
its
tanh u are given in the //, //, tables at the end of this chapter for values of u between o and For greater values they may be computed from the devel4. numerical values of cosh
opments
The
sinh
of Art. 16.
curves exhibit graphically the relations
sech u
=
:
=-
u
csch"
,
cosh u
<
cosh u
= tanh u) = cosh o = = oo oo cosh sinh
(
)
(
I,
)
(
,
,
=
coth u
>
tanh
I,
cosh
sinh u,
tanh
sinh
gd
u,
= 0,
sinh
oo)
(
(
;/
)
//)
tanh o
=
>
(
00,
-
tanh u
sinh u
sech u
I,
-
:
I,
gd u
= cosh = gd o,
tanh
;
//,
//,
csch (o) (
;
etc.;
=00
= oo)
,
etc.;
i.etc.
=
sinh x is given by the equation slope of the curve y cosh x, showing that it is always positive, and that dy/dx the curve becomes more nearly vertical as x becomes infinite.
The
=
Its direction
of curvature
proving that the curve
is
= sinh x,
obtained from d*y/dx* concave downward when x is
is
nega-
The point of inflexion is tive, and upward when x is positive. inflexional and the at the origin, tangent bisects the angle between the axes.
34
HYPERBOLIC FUNCTIONS.
The
direction of curvature of the locus =. sech x(2 tanh
by d?y/dx*
7
x
y
= sech x
is
and thus the curve
i),
cave downwards or
given con-
is
upwards
a according as 2 tanh
;r
is
i
negative or positive. The inflexions occur at the points
= tanh- 707, = .881, = y .707 and the slopes of x
1
.
;
,
the
inflexional
The curve y \
rapidly than
\ -
x
cross at the points
csch
x
is
approaches the
x
.T.
=
The
x
=
3,7
is
=
10 only I, but it is not till/ csch x,y=. sinh x curves y .
so small as
is
it
axis of y, for \\hen
c that
=
asymptotic to both axes, but approaches the axis of x more
-i--
\
are
tangents
.881,^
=
=
I.
Prob. 66. Find the direction of curvature, the inflexional tantanh x. gd^,.r gent, and the asymptotes of the curves y Prob. 67. Show that there is no inflexion-point on the curves
=
y
=
cosh Prob.
jc,
y
68.
=
coth
Show
y = tanh #
=
jr.
that
any
line
^
= mx + n
meets the curve
Hence prove that in either three real points or one. mx -f n has either three real roots or one. the equation tanh x From the figure give an approximate solution of the equation tanh x
=x
=
i.
ELEMENTARY INTEGRALS. Prob. 69. Solve the equations:
gd x
=x
=x+
x
cosh
35
x
2\ sinh
=
Jo:;
\n.
Prob. 70. Show which of the graphs represent even functions, and which of them represent odd ones.
ART.
The 14,
ELEMENTARY INTEGRALS.
26.
following useful indefinite integrals follow from Arts.
15,23: Hyperbolic.
Circular.
u du
=
cosh
u,
fsin u du
=
2.
/"cosh u du
=
sinh u,
fcos u du
=
3.
/tanh u du
=
log cosh
/ tan u du
=
4.
/coth w
=
log sinh
5.
/csch udu
1.
Ainh
f///
= 6.
j
sech
dx
ax .
9
/dx -y-
,
=
gd
=
S ' nh
=
///*l/^
,
= log tanh sinh-^csch
rfw
//,
.
^///
sec
//
log cos u,
=
log tan -,
==
sty li*
V/ Va*-=i x
= -tanh- .x- >tyf
,
1
dx
-
a
//,
log sin
^
t/
,
du
,
=
dx
^
x**
sin
=
f
" i*
\
*/ *///
/esc K
y
cosh~ a
cot
;/),
,
,
,
y
cos
//,
cosh -'(esc
),
!
gd~
//,
=
SU1
=
cos"
"
x
x
*.
a
,
i
^r-tan- ,^ 1
a
* Forms 7-12 are preferable to the respective logarithmic expressions on account of the close analogy with the circular forms, and also
(Art. 19).
because they involve functions that are directly tabulated. appears more
clearly in 13-20.
This advantage
36
HYPERBOLIC FUNCTIONS.
-
r -d*
J ^"1 ^
"i
f
r
x
i
= -COth-
, 1
/ a V
-,
#
^J^> a
dx 7-j ?
+x
=
a 1
dx
/dx+
=====
x Va*
From derived
13
-
*'
;r
I
= #- csch-
1
--,
-=
a V^r - a
a
may be
_.
l
Vac-b*
=
I
= "7=I
Va
dx
cos
<J
Vtfac ~
rt^r
.
4" ^
I
;
^ "egative.
T7^="' vb ac
^^r 4- b
"
ZT
=
+ _, ^ positive, ac
rt:^-
.
,
cosh" Vrt
"
f
tanh
~
V
Thus,
4
ac
<
"+
/A
~ ac
b
2~coth=-coth~ (^~2)=cothv 1
l
.
,
^-4^+3
;
1
3
^J 4
= tanh~ (.5)tanh- (.3333)=. 5494 1
1
^r / - ------r-
^-^ t
.3466^.2028.^
2 6
25
/
I
-
these fundamental integrals the following
r J ~,
t/8
x
.
:
17
/
i
dkr
/*
^/ ^
x
i
-COt-'-, a
a
I ^tanh-'o -
=
tanh-'(jr v
2)
'J
=-
^
tanh'Y.s) v -5494-
(By interpreting these two integrals as areas, show graphically that the first is positive, and the second negative.)
--/dx
^__
(a-x)Vx-b
*For tanh- 1 (.5)
2 _. _ ____
Va-b
,
-
Ixb
,
tanh~ A /
\
interpolate between tanh (.54)
T, a-6'
=
.4930, tanh (.56)
(see tables, pp. 64, 65); and similarly for tanh- 1 (.3333).
=
.5080
ELEMENTARY INTEGRALS.
= 2
or
Vb
tan
~
Xb
.
dx
or
,
V
a
the real form to be taken.
x
(Put
2
.
Vab
^/ba
bx -
1
A
=
b
,
and apply
z*,
9, 10.)
Ibx
I
.
coth-
,
com
..
.
2
------
or
2
\ / -7 b ~~ a
/
-7
V
2
b-a' ,
__ tan
or \
f
~
a -b
-
* 1] a-b*
/*
,
\ / \J
the real form to be taken.
-^ 2
By means
-
a
^
^)'
i
- -"'
1
coslr -.
a
2
of a reduction-formula this integral
is
made
easily
to depend on
It may also be obtained by transforming 8. the expression into hyperbolic functions by the assumption x a cosh u, when the integral takes the form
=
/c? sinh
1
ndu=.
I
[*
j (cosh 2u
\)du
= ~/i (sinh a
2u cosh
which gives 17 on replacing a cosh
;/
by
^r,
2//) //
and a sinh
//, //
by
The geometrical
3
interpretation of the result is evident, as it expresses that the area of a rectangular-hyperbolic segment AMP\s the difference between a triangle
(x*
rt
)*.
OMP
and a sector OAP. 18.
19.
(j^
20.
ysec
- x^i* = -*(a* - *)* + -V
+
f
fl
)*rfr
3
= -x(x* + 2
)*
+
sin-
sinh-
= J\\ + tan 0)*rf tan = tan 0(i + tan* 0)* + = 4 sec tan + i gd" a
0^/0
1
21.
/ sech
1
f
flf= i sech
Prob. 71.
What
Prob. 72.
Show
is
tanh w
-f-
^ gd
1
sinh- (tan 0) 0//.
the geometrical interpretation of 18, 19?
that / (0,*
1
+ 2 ^ + ^)^ reduces
to 17, 18, 19,
H\PRBOLIC FUNCTIONS.
38
when a
respectively:
and when a
is
positive, with ac
Prob. 73. Prove
< &*
positive, with ac
is
>
;
when a
is
negative;
If.
/ sinh u tanh u du
=
sinh u
cosh u coth u du
=
cosh u
gd
u,
+ log tanh
.
2
Prob. 74. Integrate
/ = $px,
if s be the length of arc the angle which the tangent line makes with the vertical tangent, prove that the intrinsic equation of 3 ! tan the curve is ds/d<(> +/gd~ 0. a p sec 2p sec 0, s a Prob. 76. The polar equation of a parabola being r sec 0,
Prob. 75. In the parabola
measured from the vertex, and
=
=
=
focus as pole, express s in terms of 0. Prob. 77. Find the intrinsic equation of the curve y/a and of the curve y/a log sec #/0.
referred to
its
= cosh x/a
=
t
Prob. 78. Investigate a formula of reduction for / cosh* #
also integrate
1
tanh" 1 a: ^r, (sinh" jr)Vjr; and
show that the ordinary methods
of reduction for
m n / cp$ xs\n xdx
can be applied to / cosh* x sinh n x dx.
ART.
FUNCTIONS OF COMPLEX NUMBERS.
27.
As vector quantities are of frequent occurence in Mathematical Physics; and as the numerical measure of a vector in terms of a standard vector is a complex number of the iOrm x-\-iy>
in
which
becomes necessary
x,
j
are real, and
in treating of
ations to consider the
meaning
any
i
stands for
of these operations
formed on such generalized numbers.*
The
*The
it
is
r,
when
geometrical
tions of cosh//, sinh u, given in Art. 7, being then applicable,
V
it
class of functional oper-
per-
defini-
no longer
necessary to assign to each of the symbols
u e of vectors in electrical theory
VP
shown
in
Bedell and Crehore's
The advantage published in 1892). of introducing the complex measures of such vectors into the differential equaAlternating Currents, Chaps, xiv-xx
(first
tions is shown by Stein metz, Proc. Elec. Congress, 1893; while the additional convenience of expressing the solution in hyperbolic functions of these complex
numbers
is
exemplified
Engineers, April 1895*
by Kennelly, Proc. (See below, Art. 37.)
American
Institute
Electrical
FUNCTIONS OF COMPLEX NUMBERS.
39
cosh (x -f- iy\ sinh (x -\- iy), a suitable algebraic meaning, which should be consistent with the known algebraic values of cosher, sinh x> and include these values as a particular case
when y
= o.
The meanings
assigned should also, if possible, be such as to permit the addition-formulas of Art. 1 1 to be made general, with all the consequences that flow from them.
Such ments in
definitions are furnished
by the algebraic developwhich are convergent for all values of //, real
Art. 16,
Thus the
or complex. are to be
COSh (*
+ /=
sinh (*
+
I
=
i
definitions of cosh (x
-f- iy),
sinh (x
-f- iy)
+jj(
(x
+
+ -,(* + z' +
iy)
.
.
.
*
From
these series the numerical
values of cosh (x
-\-
iy\
sinh (x-\-iy) could be computed to any degree of approximaIn general the results will come tion, when x and y are given. out in the complex form*
The
cosh (x
+ iy) = a + i&
sinh (x
-f- iy)
=^
other functions are defined as
9
-f- *V/.
Art.
in
7,
eq. (9).
Prob. 79. Prove from these definitions that, whatever u
cosh
d
is
=
cosh u
T-jCOsh *It
)
(
=
cosh
to be borne in
y,
d sinh u,
=w
/
sinh
(
sinh u
j* 1
mind
braic operators which convert
cosh mu,
j-
9
i/)
.
sinh
=
may
be,
sinh u,
= cosh
#,
w = nf sinh
w//.f
symbols cosh, sinh, here stand for algeone number into another; or which, in the Ian*
that the
guage of vector-analysis, change one vector into another, by stretching and turning. f
The generalized hyperbolic functions
matical Physics as
where 0, Art. 37.)
///,
usually present themselves in Mathe-
the solution of the differential equation dP0/
u are complex numbers, the measures of vector
=
quantities.
m*fa (See
40
HYPERBOLIC FUNCTIONS.
ART.
ADDITION-THEOREMS FOR COMPLEXES.
28.
The addition-theorems
+
for cosh (u
etc.,
f')
be derived as follows.
complex numbers, may as real numbers, then, by Art. n, cosh (u
hence
i
-f-
+
.=
v)
w
cosh u cosh
)+...=(i
z/-|~
where
#,
v are u v
First take
9
sinh u sinh v\
+ f + ...)(i + --,*'+.
.
.)
This equation is true when //, v are any real numbers. It must, then, be an algebraic identity. For, compare the terms of the rih degree in the letters //, v on each side. Those on the
left
r
arc f
(w-f-
?>)
;
and those on the
when
right,
collected,
form an rth-degree function which is numerically equal to the former for more than r values of u when v is constant, and for
more than
r values of v
of the rth degree
tions of u
and
z>.*
when
Thus the equation above is
true for
values of
all
;/
//,
written v,
=
cosh
and by changing v into
?;,
cosh
(;/
-f- ?')
?')
--
In a similar manner sinh (u
Hence the terms
constant.
v)
=
cosh is
an algebraic identity, and
is
whether
writing for each side its symbol,
cosh (u
is
on each side are algebraically identical funcSimilarly for the terms of any other degree.
it
real or
complex.
Then
follows that
;/
cosh v
;/
cosh v
-j-
sinh u sinh v\
(53)
sinh u sinh
(54)
v.
found
sinh
?/
cosh u sinh
cosh v
v.
(55)
In particular, for a complex argument, '
cosh (x sinh (x
*"
If
= cosh x cosh iy = sinh x cosh iy iy)
iy)
sinh
x
sinh
y/>
)
cosh
x
sinh
iy.
)
two rth-degree functions of a single variable be equal
values of the variable, then they are equal for algebraically identical."
all
f
for
(5 6)
more than r
values of the variable, and are
FUNCTIONS OF PURE iMAGlNARlES.
41
Prob. 79. Show, by a similar process of generalization,* that if exp u f be defined by their developments in powers of then, whatever u may be,
sin #, cos Uj u,
sin
cos
+
(//
exp
+ + v) z;
(
(//
= sin u cos v + cos u sin = cos u cos ~~ sm # sin = exp exp
v)
?;
)
v, z/,
z f.
i/
Prob. 80. Prove that the following are identities:
cosh
2
8
sinh u
14
cosh
//
+
sinh w
cosh u cosh sinh
ART.
29.
//
//
sinh u
= =
|
= = exp = exp i,
//,
u),
(
+ exp
[exp
J[exp u
(
//)],
exp(
)].
FUNCTIONS OF PURE IMAGINARIES.
In the defining identities
cosh u
sinh
//
=
f
i
H
f//
2.
= + //
r//
cosh i> =: sinh
i>=i>
by
division,
-
-f
//
-}-
j
=
i
.
.
.,
then
+ -I(i' -f J
tanh iy
.,
j
ty,
'
and,
.
6
-;/"
/ -f ~y -
i
.
4.
3*
put for u the pure imaginary
+
4
H
6
(/
.
.
+
.
.
= .
cos 7,
(57)
=*' sin 7,
(58)
.
'
tan y.
(59)
* This method of generalization is sometimes called the principle of the " permanence of equivalence of forms." It is not, however, strictly speaking, a
"
principle," but a method; for, the validity of the generalization
has to be
demonstrated, for any particular form, by means of the principle of the algebraic identity of polynomials enunciated in the preceding foot-note. (See
Annals of Mathematics, Vol. 6, p. 81.) f The *ymbol exp u stands for "exponential function of u," which cal with f when u is real.
is
identi-
HYPERBOLIC FUNCTIONS.
These formulas serve to interchange hyperbolic and
The
functions.
circular
hyperbolic cosine of a pure imaginary
is real,
and the hyperbolic sine and tangent are pure imaginaries.
The tanh
//,
following table exhibits the variation of sinh u, cosh u, exp ;/, as u takes a succession of pure imaginary values.
* In this table .7
is
Prob. 81. Prove the following identities cos y
=
cosh
sin.}'
=
- sinh
cos ^ cos
+
i
/
j>
cos
iy
iy
=
written for | 4/2,
=
i[exp
=
-.[exp
iy
+ exp
/>
.707 ....
:
(
exp
/))],
/],
(
= cosh iy + sinh ry = exp sin y = cosh /y sinh ry = exp sin i> = sinh ^. /y = cosh y, sin
j^
iy, (
/y),
i
Equating the respective real and imaginary paits on n each' side of the equation cos ny * sin ny i sin y) , (cos y express cos ny in powers of cos 7, sin y ; and hence derive the corProb. 82
=
+
responding expression for cosh Prob. 83.
Show
+
ny.
that, in the identities (57)
and
replaced by a general complex, and hence that sinh (x
iy)
=
/ sin
(y
^
ix) 9
(58),
y may be
FUNCTIONS OF X cosh (x
x
(or
/
cos
(.#
/>)
X
THE FORM
= cos (y ^ />), = sinh ^ = cosh (7 ^ ix). /
(>'
the product-series
for sin
-\- t
43
Y.
wr),
x
derive that
for
:
sn * sinh
ART.
By
/
sin
From
Prob. 84. sinh
-f iy IN
*
= *i -
= *i + ,i + --.i +
FUNCTIONS OF x
30.
+ iy
IN
.
.
..
THE FORM Jf+i'K
the addition-formulas,
= cosh x cosh iy sinh x sinh = sinh cosh y/ cosh x sinh y% sinh (x iy) sinh /j = sin y, cosh iy = cos y, but = cosh cos y -\-i sinh sin y, hence cosh y/) = sinh x cos sin ^. i cosh sinh y/) = # + #, sinh (x -\-iy) c-\- id, Thus cosh (x iy) sin y, # = cosh cos y, # = sinh [ sin j. cos y, d = cosh c = sinh cosh (x
-f-
iy)
-f-
or
-\-
ijr,
-f-
*
(;tr
.r
.ar
-{-
if
-f-
^r
j;
or
From
.#
j/ -|-
(.r -f-
then
)
^r
(60
)
these expressions the complex tables at the end of
this chapter
have been computed.
= X-\- iY\ let the iy, Z Writing cosh z^=-Z, where s = x Z be on represented Argand diagrams, in complex numbers z, the usual way, by the points whose coordinates are (x, y\ (Jf, F); and let the point z move parallel to the ^-axis, on a
+
x
=
Z
will describe an ellipse ;, then the point whose equation, obtained by eliminating y between the equasinh m sin y, is tions cosh m cos^,
given line
X=
Y=
(cosh
mf
"^
and which, as the parameter con focal
ellipses,
__ / Mu (sinh ;
*M \
=I
'
*# varies, represents a series of the distance between whose foci is unity.
HYPERBOLIC FUNCTIONS.
44
the point z move parallel to the .r-axis, on a given will describe an hyperbola whose equa;/, the point tion, obtained by eliminating the variable x from the equations X=- cosh x cos ;/, Y sinh x sin ;/, is Similarly,
if
line/ =
Z
=
X*
_
F
a
_ a
(cos
;/)'
and which, as the parameter
(sin w) ;/.
varies, represents a series uf
hyperbolas con focal with the former series of
ellipses.
These two systems of curves, when accurately drawn at close intervals on the Z plane, constitute a chart of the hypciand the numerical value of cosh (in -f- in) can br bolic cosine ;
m
read off at the intersection of the ellipse whose parameter is with the hyperbola whose parameter is w.* A similar chart can
be drawn for sinh (x+iy), as indicated in Prob. 85. Periodicity of Hyperbolic Functions.
The
and cosh u have the pure imaginary period sinh (u + 2in) =sinh u cos m+i cosh u cosh (u + 2ix) =cosh u cos 2n + i sinh u
The
functions sinh
argument u
is
The
sin sin
u
For
= sinh 27r = cosh 27:
u, u.
u and cosh u each change sign when the
increased by the half period in.
sinh (w-HVr) =sinh
cosh (w
functions sinh
2ix.
u cos n + i cosh u cos n + i sinh u
= cosh u -H'TT)
For
sin TT=
sinh w,
sin TT=
cosh u.
u has the period in. For, it follows from by dividing member by member, that tanh (u+in) =tanh u.
function tanh
the last two identities,
By
a similar use of the addition formulas sinh (u + ^in)
=i
cosh u,
cosh (u + J/TT)
it
=i
is
shown
that
sinh u.
By means
of these periodic, half-periodic, and quarter-periodic the relations, hyperbolic functions of x+iy are easily expressible in /terms of functions of x + iy', in which y' is less than JTT. * Such a chart is given by Kennelly, Proc. A. I. E. E., April 1895, and is used by him to obtain the numerical values of cosh (x iy) sinh (x ty) which present themselves as the measures of certain vector quantities in the theory of
+
alternating currents.
and of y between o and
+
t
(See Art. 37.) The chart is constructed for values of x 1.2; but it is available for all values of y on account of
the periodicity of the functions.
t
FUNCTIONS OF X + iy IN THE FORM
The
X+iY.
45
hyperbolic functions are classed in the modern fanction-
theory of a complex variable as functions that are singly periodic with a pure imaginary period, just as the circular functions are singly periodic with a real period,
and the
elliptic
functions are
doubly periodic with both a real and a pure imaginary period. Multiple Values of Inverse Hyperbolic Functions. It follows from the periodicity of the direct functions that the inverse
m
functions sinh" 1
and cosh" 1
m
have each an
ular value of sinh"" 1
than \n nor
less
m
than
which has the
number
indefinite
That
of values arranged in a series at intervals of zin.
partic-
coefficient of i not greater
called the principal value of sinh" 1
m\ and that particular value of cosh"" 1 m which has the coefficient of i not greater than n nor less than zero is called the principal value
When
of cosh" 1 ^.
\K
is
necessary to distinguish between the general value and the principal value the symbol of the former thus will be capitalized it
is
;
Sinh""
1
m = sinh" m + zirn, Cosh" m = cosh"" m -f 2irx, Tanh" m = tanh" m +irx 1
1
1
1
1
y
in
which
r is
integer, positive or negative.
any
Complex Roots
of Cubic Equations.
It is
well
known
that
when
the roots of a cubic equation are all real they are expressible in terms of circular functions. Analogous hyperbolic expressions
are easily found
when two
of the roots are complex.
Let the
cubic, with second term removed, be written
Consider
x=r
first
the case in which b has the positive sign.
sinh u, substitute, and divide
sinh
3
by
r
3 ,
u + ^ sinh u = - ,.
3 Comparison with the formula sinh u + l sinh
3* gives
whence therefore
7* r r=2&*,
Let
then
=3 7' 4
P r
=
sinh3=-Tj,
x= 20* sinh
i I
r~' 4 M=-sinh~ 1 T}j O
I 1
c\
sinn" TI
)
>
=J
sinh yi
HYPERBOLIC FUNCTIONS.
46 in
which the sign of
Now
b*
be taken the same as the sign of
to
is
the principal value of sinhr 1
let
^,
found from the
n2in
then two of the imaginary values are
be n\
x
of
three values
- and
are 26* sinh
2 ft* sinIj(-
\3
3
two reduce
last
Mfsinh
to
i\/3 cosh
tables,
hence the
t
).
The
3/
j.
O I
U
\
c*
let the coefficient of x be negative and equal to 36. then be shown similarly that the substitution x=r sin
Next,
may
It
leads to the three solutions
sin,
26*
6* (sin
3
These roots are
#=rcoshw
3
real
when c%b*.
all
where
\/3 cos-),
\
w= sin"
1
TT. 0*
37 If
c>b*,
the substitution
leads to the solution
#= 26* cosh
(-cosh"~ l
JT),
which gives the three roots 26* cosh -, 3
in
6*
(
cosh
which the sign of Prob. 85.
Show
-
wherein n = cosh" 1 TT , ) & 3/ to be taken the same as the sign of
/V T sin h "~ *
3
\
b* is
i
that the chart of cosh (x
-f- iy)
c.
can be adapted
+ /?), by turning through a right angle; also to sin (x +iy). sinh 2r 4- /sin 2V n ^ -. Prob. 86. Prove the identity tanh (x + ty) = cosh 2x cos 2y = + *A be written in the " modulus Prob. 87. If cosh (x + and amplitude" form as r(cos + /sin 0), = exp /^, then
to sinh (x
,
.
.
.
.
.
/
.
.
v
-f-
tv),
/
r*
=a
tan ^
= cosh = ^/a = tanh 7
jc
x
= cos ^ 9
2
-f b*
sin'j^
tan
sinh* x,
j>.
Prob. 88. Find the modulus and amplitude of sinh (x Prob. 89.
Prob. 90. .
*
sin"
01
=
*
Show
that the period of exp
When .
/
.
cosh
real
is
;;/
_,J
and
>
i,
01 is real
and
cos" 1
m.
<
i,
cosh"
1
m=
i
iy).
is ia.
2
When
+
cos"
1
01.
m =
i cosh""
1
m
%
tHE CAfENAfcY.
ART.
47
THE CATENARY.
31.
A flexible
inextensible string is suspended from two fixed and takes up a position of equilibrium under the points, action of gravity.
curve
in
w be
;
of the portion
T
required to find the equation of the
It is
it
hangs. the weight of unit length, and s the length of arc measured from the lowest point A then ws is the weight
Let
AP
which
AP.
This
is
balanced by the terminal tensions,
H
acting in the tangent line at P, and
Resolving horizontally and vertically gives
tangent.
=
T cos which
in
is
//,
c
is
=
T'sin
ws,
the inclination of the tangent at P\ hence
tan0 wheie
the horizontal
in
written for
=
H/w,
constant horizontal tension
;
_= ws
s _,
the length whose weight therefore
is
the
ds
x c
which axis of
is
.,,*.,* = = smh-, smh c c
s
1
= dy y = ax -7
c
,
,*
cosh -, c
*
the required equation of the catenary, referred to an at a distance c below A.
x drawn
The following trigonometric method illustrates the use of the gudermanian The " intrinsic equation," s c tan 0, c ds c ds cos hence sec sec* d dx, 0, gives
=
:
=
;
dy ~dss\\\ 0,=rsec whence y/c = sec %
=
tan 0*/0; thus
=
x=c
gd~* 0,
= sec gd x/c = cosh */* = tan gd ;r/r = sinh ^/^. sfc A
;
^=
and
chain whose length is 30 feet is suspended from two points 20 feet apart in the same horizontal; find the parameter c, and the depth of the lowest
Numerical Exercise.
nninK
HYPERBOLIC FUNCTIONS.
48
The equation by putting lo/c
s/c
= z,
=
ining the intersection of it appears that the root
To
find a closer
in the
= sinh lo/V, which, = sinh z. By examthe graphs of y = sinh#, y = of this equation z = 1.6, nearly.
sinh x/c gives i$/c may be written 1.5^
1.5,3:,
is
approximation to the sinh z
lorm/(s) /(l.6o)
/(i.62) ^(1.64)
= = =
1.5$
=
root, write the
by the
o, then,
2.3756
2.4000
2.4276
2.4300
2.4806
2.4600
= = =
equation
tables,
.0244, .0024,
-f .0206;
whence, by interpolation, it is found that /( 1.6221) = o, and The ordinate of either of # = 1.6221, c 10/2 = 6.1649.
=
the fixed points
=
is
given
cosh x/c
~
from tables; hence
y
y/c
vertex
=y
c
=
by the equation
cosh lO/c
=
=. 16.2174,
cosh 1.6221
=
2.6306,
and required depth of the
10.0525 feet.*
Prob. 91. In the above numerical problem, find the inclination of the terminal tangent to the horizon.
MN
be drawn from the foot of the Prob. 92. If a perpendicular 1 is equal to the conordinate to the tangent at 7 prove that Hence show that is equal to the arc AJ\ stant c, and that
MN
,
NP
-W is the involute of the catenary, and has the propthat the erty length of the tangent, from the point of contact to the axis of jc, is constant. (This is the characteristic property of the the locus of
tractory).
Prob. 93.
The
tension
T at
is equal to the weight of a equal to the ordinate y of that
any point
portion of the string whose length
is
point.
An arch in the form of an inverted catenary f is 30 wide and 10 feet high; show that the length of the arch can be
Prob. 94 feet
2
obtained from the equations cosh z
z
3
=
i,
25
=
3O sinh
.
z
* See a similar problem in Chap. I, Art. 7. " For the iheory of this form of arch, see " Arch in the Encyclopaedia
f
Britannica.
.
CATENARY OF UNIFORM STRENGTH.
ART.
49
CATENARY OF UNIFORM STRENGTH.
32.
the area of the normal section at any point be made proportional to the tension at that point, there will then be a If
constant tension per unit of area, and the tendency to break be the same at all points. To find the equation of the
will
curve of equilibrium under gravity, consider the equilibrium of an element PP whose length isrfi, and whose weight is gpoads, 1
where weight
the section at P, and p the uniform density. This balanced by the difference of the vertical components
GO is is
of the tensions at
Pand P' d( T sin
,
d(
therefore
=
T cos
T= //sec 0.
0)
T cos 0)
ff,
Again,
hence
if
then by hypothesis cy/c0
= gpoods, =o ;
the tension at the lowest point, and G? O be the section at the lowest point,
=
T/ff=
sec 0, and the
first
equation
becomes
or
where
//
sin 0)
c d tan
=
= gp&)
sec
whose weight
ca )
lowest point
;
ds,
c stands for the constant
(of section
sec
is
ff/gpoo^ the length of string equal to the tension at the
hence,
ds
=
c sec 0^/0,
s/c
=
gd-'0>
the intrinsic equation of the catenary of uniform strength.
Also
hence
dx
x
= ds =
=
cos
cfa
y=
c */0,
in
=
= ds sin
= c tan
d
;
c log sec 0,
and thus the Cartesian equation y/c
dy
is
log sec x/c,
which the axis of x
is
the tangent at the lowest point.
Prob. 95. Using the same data as in Art. 3i find the parameter and the depth of the lowest point. (The equation x/c = gd s/c which, by putting i$/e = z, becomes IQ/C = gd i$/t gives c
9
HYPERBOLIC FUNCTIONS.
50
From
If the graph it is seen that z is nearly 1.8. f z, then, from the tables of the gudermanian at the
%dz~$z.
= gd z
f(z)
end of
this chapter,
= + .0432, 2 667 = + .0072, /(i.po) 1.2739 1.2881 1.3000 = .0119, 7(1.95) = whence, by interpolation, z 1.9189 and c= 7.8170. Again, y/c = log* sec x/c but x/c = lo/V = 1.2793; and 1.2793 radians = 73 '7' 55"; hence j= 7.8170 X .54153X2.3026 = 9.7472, the /(i.8o)
= = =
1.2000
1.2432
~
i
;
required depth.) Prob. 96. Find the inclination of the terminal tangent.
Show
Prob. 97.
oo,
that the curve has
two
Prob. 98. Prove that the law of the tension T, and of the section at a distance s, measured from the lowest point along the
curve,
is
T=& = cobh ,
H and show that is
vertical asymptotes.
in the
3.48 times the
Prob.
99.
s c
;
above numerical example the terminal section
minimum
section.
Prove that the radius of curvature is given by Also that the weight of the arc s is given by in which s is measured from the vertex.
= c cosh s/c. W = H sinh s/c,
ART.
An
33.
THE ELASTIC CATENARY.
uniform section and density in its natususpended from two points. Find its equation of
elastic string of
ral state is
equilibrium. Let the element d
=
+
hence the weight of the stretched element gpaods/(i
+ IT).
d(Ts\r\ 0)
T cos
and hence in
which
#/(tan /*
ds,
=^
Accordingly, as before,
=gpvds/(
= H = gpooc, sec 0), 0) = ds/(i + I*
stands for A/f, the extension at the lowest point
;
THE TRACTORY.
= <:(sec'
ds
therefore
= tan +
j/V
ju(sec
51 *
1 -j" A
sec 0)d?0
+ gd"
tan
1
[prob. 20, p. 37
0),
the intrinsic equation of the curve, and reduces to that The coordinates x, y of the common catenary when yu o.
which
is
=
may ting
be expressed
in
terms of the single parameter
= ds cos = r(sec + = ^(sec* + dy = ds sin
dx
=
1
/*
8
}*
sec 0) sin
Whence
*/0.
-4-
/*
tan 1 0.
These equations are more convenient than the eliminating 0, which is somewhat complicated.
AKT.
To
34.
by put-
0X0,
= sec
tan 0,
gd"
sec*
result of
THE TRACTORY.*
the equation of the curve which possesses the property that the length of the tangent from the point of contact to the axis of x is confind
stant.
Let
PT
9
secutive
P'T' be two
tangents
PT=P'T' = = /; draw TS
and
c,
to/"r'; then is
let
conthat
OT
perpendicular if
that
evident
such
PP' = ds, it ST' differs
M
from ds by an infinitesimal of a higher order. Let /Tmake with OA, the axis of y\ then (to the first order of an angle
= TS = TT cos 0; that = cos t = c gd~'0, sin 0), y c sin 0, = ^(gd~
infinitesimals) PTd
ctt
x
=/
This
is
;
and
dt, !
c cos 0.
a convenient single-parameter form, which gives
*This curve p. 242)
is,
in
is used in Schiele's anti-friction pivot (Minchln's Statics, Vol. I, the theory of the skew circular arch, the horizontal projection
of the joints being a tractory. (See "Arch/' Encyclopaedia Britannica.) gd t/c furnishes a convenient method of plotting the curve. equation
=
all
The
HYPERBOLIC FUNCTIONS.
5
values of x>
pressed in the same form, ds
= ST' = dt sin
is
=
=
At the point ^4, o, Cartesian equation, obtained
= gd*
=
.r
^
0^/0,
= o,
j
o,
= log, sec 0. * = o, j=.
ex-
s,
<:
by eliminating
*
/
1
-\ ^/
=
0,
cosh"
1
The
is
-
A
y
/i
?-.
\
c
//
.2658^,
x
=
/
=
1.0360^.
Show
Prob. 101.
=
=
=
,
Prob. 100. Given
7
c tan
sin [cos'
-)
v
value of
be put for //, and be taken as independent variable, // tanh //, j//^: sech w, 5/^ gd log cosh //. ;r/
If
==
[cos"
The
i/T.
found from the relation
1
1
o to
increases from
as
y
show that At what point
= 74 = c?
2C,
35', s
=
1.3249^,
is /
that the evolute of the tractory
is
the catenary.
(See Prob. 92.) o'
Prob. 102. Find the radius of curvature of the tractory in terms derive the intrinsic equation of the involute.
0; and
ART.
!n
THE LOXODROME.
35.
On the surface of a sphere a curve starts from the equator a given direction and cuts all the meridians at the same angle.
To
find its
in latitude-and
ordinates
equation longitude co-
:
Let the loxodrome cross
two consecutive meridians
AM, AN'm
^
MN= dx, RQ = dy,
all in
let
PR be
radian measure
;
the points/3 Q\ ,
a parallel of lati-
and
let
the angle
MOP=RPQ = a\ then tan a = RQfPR, but PR = MN cos MP* hence dx tan a = dy sec/, and x tan a = gd~ y, there !
no integration-constant since y vanishes with x quired equation
is
y *
;
=
gd (x tan
a).
Jones, Trigonometry (Ithaca, 1890), p. 185.
being thus the re-
COMBINED FLEXURE AND TENSION.
To
OP:
find the length of the arc
= dy esc
ds
To
whence
of,
Integrate the equation
= y esc a.
s
suppose a ship
illustrate numerically,
53
sails
northeast,
from a point on the equator, until her difference of longitude 45, find her latitude and distance
is
:
Here tan a
=
andj/
I,
= y 1/2 =
radians: s
= gd
1.0114
= gd \n =
x
radii.
The
gd(.;854)
= .7152
latitude in degrees
is
40.980.
the ship set out from latitude y^ the formula must be modified as follows: Integrating the above differential equaIf
and (x^y^ gives
tion between the limits (x^ j,) (*,
- *,) tan
a
=
gd" >,
- gd-'j/,;
hence gd" /a = gd~'j, -f- (X, x^ tan <*, from which the final latitude can be found when the initial latitude and the differ!
The
ence of longitude are given.
~7i)
(/a
a
csc
radii,
Mercator's Chart. parallel straight lines, line y'
= x tan
distance sailed
a radius being 60
X
is
equal to
i8o/7r nautical miles.
In this projection the meridians are and the loxodrome becomes the straight
hence the relations between the coordinates of
ar,
corresponding points on the plane and sphere are x'
y
= gd~ y.
is
tabulated under the
y"
;
Thus the
the values of
latitude
name
y and
y
of
"
= x,
l
magnified into gd y, which " meridional part for latitude is
of y' being given in minutes.
A chart
constructed accurately from the tables can be used to furnish graphical solutions of problems like the one proposed above. Prob. 103. Find the distance on a
(30
E) and (30
N, 20
ART.
A beam other,
and
36.
that is
plied at the
assumed by
S,
40
rhumb
line
between the points
E).
COMBINED FLEXURE AND TENSION. is
b'ult-in at
one end
carries a load
P at
the
Q
ap-
also subjected to a horizontal tensile force
same point; its
to find the equation of the curve
neutral surface: Let x>
y
be any point of the
HYPERBOLIC FUNCTIONS.
54
end as
elastic curve, referred to the free
origin, then the
Px. ing moment for this point is Qy notation of the theory of flexure,*
A cosh nx + B sinh nx y = mx + A cosh nx + /? sinh
whence that
u =.
is,
The
9
B
arbitrary constants A,
At
terminal conditions.
must be
Hence, with the usual
mx = u, anddy/t/x* = d*u/dx*>
which, on putting/
zero,
becomes
[probs. 28, 30 #.
are to be determined by the
= o, y = o
x
the free end
;
hence
A
and
= #/;r /? sinh nx -J- = /# 4- nB cosh # y
-f-
y
ax but at the fixed end,
x=
and dy/dx
/,
5= y^ =
hence
--
/,
f^ sinh
/^r
;
=. o,
cosh
w/
and accordingly
To
bend-
nx
-
cosh nl?
.
obtain the deflection of the loaded end, find the ordinate
of the fixed
end by putting x deflection
Prob. 104.
Compute
=
giving
/,
= m(l -- tanh nl).
the deflection of a cast-iron beam,
2X2
inches section, and 6 feet span, built-in at one end and carrying a load of 100 pounds at the other end, the beam being subjected to a horizontal tension of 8000 pounds.
E=
15
X
deflection
*
io
6 ,
Q = 8000, P =
= sV(7 2 "" 5
tan ^
I
100
-44)
;
/ = 4/3, = 1/50, m = 1/80, 44-^9) = -34 inches.]
[In this case
hence n
= A(7 2
Menriman, Mechanics of Materials (New York,
1
1895), pp. 70-77, 267-269,
ALTERNATING CURRENTS.
55
Prob. 105. If the load be uniformly distributed over the beam, say per linear unit, prove that the differential equation is
w
=
or
and that the solution is>'= A cosh nx
Show
how
also
-f
v
-///*,
B sinh nx + MX* +
n
determine the arbitrary constants.
to
ART.
37.
ALTERNATING CURRENTS.*
In the general problem treated the cable or wire
is
regarded
as having resistance, distributed capacity, self-induction,
leakage
and
although some of these may be zero in special line will also be considered to feed into a receiver
;
cases.
The
circuit
of
in
and the general solution will inwhich the receiving end is either
The
electromotive force may, without
any description
clude the particular cases
grounded or insulated.
;
be taken as a simple harmonic function of the time, because any periodic function can be expressed in a Fourier series of simple harmonics.f The E.M.F. and the
loss of generality,
which
may differ in phase by any angle, will be have given values at the terminals of the receiver supposed to circuit; and the problem then is to determine the E.M.F.
current,
and current that must be kept up at the generator terminals and also to express the values of these quantities at any intermediate point, distant x from the receiving end the four
;
;
line-constants being supposed
known,
viz.:
= resistance, in ohms per mile, = coefficient of self-induction, in henrys per mile, c = capacity, in farads per mile, g = coefficient of leakage, in mhos per mile. J r
/
It is
shown
in
standard works
* See references in footnote, Art. 27. t
This
that
if
any simple harmonic
t Byerly,
Harmonic Functions. on the Application
article follows the notation of Kennelly's Treatise
of Hyperbolic Functions to Electrical Engineering Problems, p. 70.
Thomson and Sound, Vol.
I.
Tait, Natural Philosophy, Vol.
p. 20; Bedell
I.
p. 40; Raleigh,
and Crehore, Alternating Currents,
Theory
p. 214.
of
HYPERBOLIC FUNCTIONS. function a sin
a and angle 27T/G0,
be represented by a vector of length then two simple harmonics of the same period -f- 0)
(<*>
6,
but having different values of the phase-angle
combined by E.M.F. and the current
adding their representative vectors.
any point of the from the receiving end, are of the form e
in
=e
which the
are
all
at
sin (&?/ -f #)
t
maximum
functions of x.
sented by the vectors are the
=
*
*i
snl (<&*
can be
0,
Now
the
circuit, distant
+
x
(64)
#')>
and the phase-angles 0, ff' These simple harmonics will be reprevalues /
*\,
1
e /&,
iJO
t
9
whose numerical measures
;
complexes e (cos tt -{-/sin #)*, i, (cos ff -\- j sin #'), be denoted by e, L The relations between e and i may be obtained from the ordinary equations f
which
l
will
c
for, since de/dt
will
=
de
de
di
w^
cos
(o>/
-'
+ 0) =
=n + w^
di l
sin (ut
>
+ + ^
TT),
then
be represented by the vector otfi/0+ \K\ and
by the sum
two vectors ge /6, cwe^/Q -f- ^x whose numerical measures are the complexes ge, juce\ and similarly for de/dx in the second equation thus the relations between of the
l
;
;
the complexes
jx
e, i
=
are
(g
g = + ;0i.
+ ;0&
(f
(66)J
i. symbol/ is used, instead of i, for V and Crehore, Alternating Currents, p. 181. The sign of dx is changed, because x is measured from the receiving end. The coefficient of leakage, g, is usually taken zero, but is here retained for generality and sym-
* In electrical theory the f Bedell
metry. t
These relations have the advantage of not involving the time. Steinmetz them from first principles without using the variable /. For instance,
derives
he regards to
i,
with
r
+jwl
as a generalized resistance-coefficient,
i.
dielectric
which, when applied
phase with /, and part in quadrature Kennelly calls r -f jul the conductor impedance; and g -f- juc the admittance; the reciprocal of which is the dielectric impedance.
gives an E.M.F., part of which
is
in
ALTERNATING CURRENTS
57
Differentiating and substituting give
are similar functions of x to be distinguished terminal values. only by their It is now convenient to define two constants z by the * equations
and thus
I
e,
9
,
cf
and the
=
+
(r
j o)l) (g
+ juc}
differential equations
ZQ
,
e
=A
(68)
;
-"
<w
j
+ JB sinh a^,
cosh ax
jV)
then be written
may
-* the solutions of which are
= a/ (g +
I
=
;
^4
cosh
+ B' sinh ojc,
ajc
wherein only two of the four constants are arbitrary; for substituting in either of the equations (66), and equating coefficients, give
(g
whence
Next
+ MM = B B' = A/z <*
=
(S
A = B/zQ f
Q,
.
the assigned terminal values of e, i, at the reo gives E A, by E, / then putting x f the thus soB and whence B z 7, A', general /2 let
ceiver be denoted
I
-
'>
lution
=
;
=
=
=
;
is
e
f
= E cosh a# + z I sinh a#, Q
= / cosh ojc H
isinh a^, ^o
* Professor Kennelly calls
impedance of the
a
and Art.
27, foot-note.
(7)
r J
the attenuation-constant, and
line.
t See Art. 14, Probs. 28-30;
]
z
the
surge-
HYPERBOLIC FUNCTIONS'.
68
these expressions could be thrown into the ordir nary complex form -}-jY', by putting for the let-\-jY, ters their complex values, and applying the addition-theorems If desired,
X
X
X
The
% Y, X' 9 quantities Y' would then be expressed as functions of x and the repre where e? X*-\-Y*, sentative vectors of e, i, would be ^/0,
for the hyperbolic sine
and cosine.
;
i? = X" +
Y'\ tan 8
=
2,^0',
=
7 Y/X, tanT
= ~Y'/X'.
For purposes of numerical computation, however, the formulas (70) are the most convenient, when either a chart,* or a table,! of cosh
,
sinh
,
is
available, for
complex values of
u.
=
Prob. io6.J Given the four line-constants: r 2ohms per i == 20 millihenrys per mile, c microfarad 1/2 per mile, o; and given w, the angular velocity of E.M.F. to be 2000
=
mile,
g
=
radians per second; then
= 40 ohms, conductor reactance per mile; = + /w/ 2 + 407 ohms, conductor impedance per mile; we = .001 mho, dielectric susceptance per mile; g + /we = .001; mho, dielectric admittance per mile; = iooo/ ohms, dielectric impedance per mile; (g + ;w)"~ a = (r+ /w/) (g + /we) = .04 + .oo2/, which is the measure w/
r
l
2
of .04005/177
a = measure
8';
therefore X
=
+
.0050 .2000;, an abstract coefficient per mile, of dimensions [length]" l ,
So
of .2001/88
= <x/(g + /we) =
200
-
34
s/
ohms.
Next let the assigned terminal conditions at the receiver be: I = o (line insulated); and E = iooo volts, whose phase may be taken as the standard (or zero) phase; then at any distance x,
by
(70),
e
= E cosh ax,
I
=
E*
sinh
ax
t
2o
in
which ax
is ai>
Suppose
it is
must be kept up
abstract complex.
required to find the E.M.F. and current that at a generator 100 miles away; then
* Art. 30, foot-note. J
The data
p. 38).
for
this
t See Table II.
example are
taken from
Kennelly's artidf
(1.
c.
ALTERNATING CURRENTS.
59
+ 207), I = 200(40 /)"' sinh + 207), = cosh (.5 + 2o/ 6nj) (.5 + 207) = cosh + 1.157) = .4600 + -475 /
e .= 1000 cosh (.5
(.5
but, by page 44, cosh .
(.5
obtained from Table II, by interpolation between cosh and cosh (.5 -f 1.27); hence
(.5
+
!!/)
~ 460 + 4757 =
l
and hence '
=
(*
+ /)( 2126 +
I.028/)
=
=
= "^('495 = (cos 0' + /sin 0'), = 1495 sec #'/t6oi = 45', i,
f
where log tan
10.7427,
e
let it
=
79
/,
and magnitude of required current.
5.25 amperes, the phase
Next
ff'
=
be required to find e at x
1000 cosh (.04
by subtracting tween sinh (o
^JT/,
+
Similarly
07")
+
1.67")
10007* sinh (.04
and applying page 44. and sinh (o -|- .if) gives
sinh (o
+
037)
sinh
+
.037)
(.1
Interpolation between the sinh (.04
Hence/ =7(40.02
=
8; then
last
+
-f-
.037),
Interpolation be-
= ooooo + 02 99S/ = .10004 + .030047'. '
two gives
3/)
+ 29.997)=
= .04002 + .029997. 29.99+40.027" =^,(cos
0+j sin
#)>
where log tan
= .12530, 0=
126
51',*,
=
29.99 sec I2 6
51'
= 5-i
volts.
Again,
let it
be required to find
e at
x
=
16; here
= 1000 cosh (.08 + .067"), 3-27") = 1.0020 -f .0067; = but cosh (o + .067") .9970 + o/, cosh (.1 + .o6/) cosh (.08 + -o6/) = i.ooio +.00487", hence e= iooi+4.8/= ^,(cos 6^+/sin ^), and = 180 17', t = 1001 volts. Thus at a distance of about where e
1000 cosh (.08 H-
t
16 miles the E.M.F.
is
the
same
as at the receiver, but in opposite
HYPERBOLIC FUNCTIONS.
60 Since e
phase.
proportional to cosh (.005
is
+ .2j)x
9
the value of
which the phase is exactly 180 is n/.2 = 15.7. Similarly the phase of the E.M.F. at x = 7.85 is 90. There is agreement in phase at any two points whose distance apart is 31.4 miles.
x
for
In conclusion take the more general terminal conditions in line feeds into a receiver circuit, and suppose the current in advance of the elecis to be kept at 50 amperes, in a phase 40 sin 40) tromotive force; then / 5o(cos 40 38.30 32.14/2
which the
=
+/
and
+
substituting the constants in (70) gives
e=
+ ,2j)x + (7821 + 62367)
icoo cosh (.005
=
460-] 4757
+ .2j)x
sinh (.005
4288+98417 ^(cos #+7
-4748+93667=
sin #),
where 0= 113 33',
If
i
\ 2 ii
/
where n 2
=
= o, g = o, / = o; = wrc/2, w = r/2wc;
then a
=
and the
2
t
(i
+ j)n,
solution
2
is
2nx
+ cos 2nx,
tan 6
= tan nx tanh nx,
E I/cosh 2nx
cos 2nx,
tan 0'
=
E ^cosh
2Wi
tan nx coth nx.
1 08. If self-induction and capacity be zero, and the rebe insulated, show that the graph of the electromotive end ceiving force is a catenary if g j* o, a line if g = o. Prob. 109. Neglecting leakage and capacity, prove that the
Prob.
solution of equations (66)
= /,
is I
E+ r
e
(r
+ jul)Ix.
Prob. no. If x be measured from the sending end, show equations (65), (66) are to be modified; and prove that
= E cosh ax zjo sinh ax, I = I where E I refer to the sending end.
cosh ax
e
-- E
how
sinh ax,
z
.
c
ART. I.
The
38.
MISCELLANEOUS APPLICATIONS.
length of the arc of the logarithmic curve
y= M(cos\\ //+logtanh The
2.
s
=
ture
+ 2), where sinh u =
In the hyperbola x*/tt is
p=
a
(#
sinh
8
measure of the sector 4. In
M= I/log
;/
+
b*
A OP,
f/b*
is
0.
=
i
cosh* u)*/a6
i.e.
= a* = r = ati is
y u
a, sinh
length of arc of the spiral of Archimedes
#(sinh 2u 3.
which
#), in
cosh u
the radius of curva;
in
= x/a,
which sinh u
//
is
the
=y/b.
an oblate spheroid, the superficial area of the zone
MISCELLANEOUS APPLICATIONS.
61
between the equator and a parallel plane at a distance y is 5 = irP(sinh 2u 2u)/2e wherein b is the axial radius, e eccen-
+
tricity, sinh u
J
= ey/p
and
/ parameter of generating ellipse. length of the arc of the parabola 2px, measured from the vertex of the curve, is /= ^/(siiih 2u in which 2//), t
y=
The
5.
sinh u
~y/p
+
= tan
where
0,
is
the inclination of the termi-
nal tangent to the initial one. 6.
The
centre of gravity of this arc
given by
is
1
3/^r
nr/^cosh u
i),
64/;7
and the surface of a paraboloid 7.
The moment
minal ordinate
of revolution
of inertia of the
/=
is
p* (sinh 4;*
2x\
^[xKx
same -f-
is
5= 2n yL
arc about
wP*N\
where
its ter,
p
is
the mass of unit length, and
yV= 8.
The
J-
sinh 2u
in its
which
= jr/f
is
2u -f
^(sinli ;/
s ' nn
6.
centre of gravity of the arc of a catenary measured
from the lowest point 4/j/
4+ iV
sinh
;
/ = /^ rV (
2#),
/JF= c\u sinh
and the moment
terminal abscissa b
given by
cosh u
-f- i),
of inertia of this arc
about
is
sin ^ 3 W
~H t sin ' 1
;/
~~
u cos ' 1
7
')'
Applications to the vibrations of bars are given in RayVol. I, art. 170; to the torsion of leigh, Theory of Sound, in Love, Elasticity, pp. 166-74; to the flow of heat prisms 9.
Byerly, Fourier Series, pp. 75-81; to wave motion in fluids in Rayleigh, Vol. I, Appendix, p. 477, and in
and
electricity in
Bassett,
Hydrodynamics,
arts.
120,
384; to the
theory of
in Maxwell, Electricity, arts. potential in Byerly p. 135, and Non-Euclidian to geometry and many other subjects '172-4;
Gunther, Hyperbelfunktionen, Chaps. V and VI. Several numerical examples are worked out in Laxsant, Essai sur lea
in
fonctions hyperboliques.
HYPERBOLIC FUNCTIONS.
(&
ART. In Table sinh
,
cosh
I
#,
EXPLANATION OF TABLES.
39.
the numerical values of the hyperbolic functions tanh u are tabulated for values of u increasing
from o to 4 at intervals of may be used.
in
When
.02.
u exceeds
4,
Table IV
Table II gives hyperbolic functions of complex arguments, which cosh (x
=a
iy)
and the values of
sinh (x
t& 9
#, b, c,
d
if)
are tabulated
ranging separately from o to
and of
y
When
interpolation
is
necessary
1.5
=c
wf,
values
for
may be performed
it
of
at intervals of in
x .1.
three
For example, to find cosh (.82-}- 1.341): First find stages. cosh (.82+ l-30 by keeping/ at 1.3 and interpolating between the entries under x
=
.8
=
and*
.9
;
next find cosh
(.82 -f l-4*)
by keeping y at 1.4 and interpolating between the entries under x = .8 and jr = .9, as before then by interpolation between cosh (.82 1.41) find cosh( .82 i-34*) 1.31) and cosh (.82 ;
+
in
which x
of j/,
however
sinh (x It
great,
.82.
+ 2** = sinh^r,
occurs
in practice.
x
is
= cosh
cosh (x-\- 2tn)
)
does not apply when
dom
+
+
The table is available for all values means of the formulas on page 44: by
kept at
is
greater than
1.5,
x, etc.
but this case
This table can also be used as a
sel-
com
plex table of circular functions, for cos (y
ix)
= a ^p ib,
sin
(y
and, moreover, the exponential function
exp
(
x
i
=^
^r
ix) is
=
given
i(*
by
^/),
which the signs of c and */are to be taken the same as the sign of jr, and the sign of i on the right is to be the product of in
the signs of
x and
of i on the
left.
(See Appendix, C.)
Table III gives the values of v= gd and of the guder0= 180 v/n> as u changes from o to I at inter,
manian angle
EXPLANATION OF TABLES. from
vals of .02,
intervals of
%
6iJ
to 2 at intervals of .05, and from 2 to 4 at
I
i.
IV are given the values of gd log sinh u, log u as increases from to 6 at of intervals u, .1, from 6 to 4 of intervals and from at to of at intervals .2, 7 .5. 7 9 In Table
,
cosh
In the rare cases in which sary, reference
Glaisher,
may
made
be
more extensive
tables are neces-
to the tables* of
Gudermann, Gudermanthe independent variable, and
and Geipel and Kilgour.
In the
first
the
ian angle (written k) is taken as increases from o to roo grades at intervals of .01, the corresponding value of u (written Lk) being tabulated. In the usual
which the table
case, in
entered with the value of
is
//,
it
gives
by interpolation the value of the gudermanian angle, whose circular functions would then give the hyperbolic functions of
When
//.
u
is
large, this angle
is
so nearly right that inter-
To remedy
this inconvenience Gupolation dermann's second table gives directly log sinh u, log cosh //, log tanh u, to nine figures, for values of u varying by .001 from 2 to 5, and by .01 from 5 to 12. is
not reliable.
and c~* to nine sigfrom o to i, by .01 from O from o to $00. From these
Glaisher has tabulated the values of
x
nificant figures, as
to
2,
by
.1
from o to
varies 10,
by .001 and by I
the values of cosh x, sinh
x
e*
t
.
are easily obtained.
Geipel imd Kilgour's handbook gives the values of cosh;t, sinh r, to seven figures, as x varies by .01 from o to 4. There are also extensive tables by Forti, Gronau, Vassal, and there are four-place tables in Byerly's Callet, and Houel Fourier Series, and in Wheeler's Trigonometry, (See Ap;
pendix, C.) In the following tables a dash over a that the number has been increased. Gudermann
final digit indicates
in Crelle's Journal, vols. 6-9, 1831-2 (published separately Theorie der hyperbolischen Functionen, Berlin, 1833). Glaisher in Cambridge Phil. Trans., vol. 13, 1881. Geipel and Kilgour's Electrical Handbook.
under the
title
HYPERBOLIC FUNCTIONS. TABLE
I.
HYPERBOLIC FUNCTIONS.
TABLES. TABLE
I.
HYPERBOLIC FUNCTIONS.
65
HYPERBOLIC FUNCTIONS.
66 TABLE
II.
VALUES OF COSH
(JT
+ iy)
AND SINH
(x -f
iy).
TABLES.
TABLE
II.
VALUES OF COSH
(x
+ fy)
6?
AND SINH
(x
+
iy).
HYPERBOLIC FUNCTIONS.
68 TABLE
II.
VALUES OF COSH
(x
+ iy)
AND SINK (x
+
iy).
69
TABLES. TABLE
II.
VALUES OF COSH(*
+
iy)
AND SINH(*
+
iy.)
70
HYPERBOLIC FUNCTIONS. TABLE
III.
TABLE
IV.
APPENDIX.
A.
HISTORICAL AND BIBLIOGRAPHICAL.
What is probably the earliest suggestion of the analogy between the sector of the circle and that of the hyperbola is found in Newton's Principia (Bk. 2, prop. 8 et seq.) in connection with the solution of a
On
dynamical problem.
the analytical side, the
first
hint of the modi-
and cosine is seen in Roger Cotes' Harmonica Mensurarum where he suggests the possibility of modifying the expression
fied sine
(1722), for the area of the prolate spheroid so as to give that of the oblate one,
by a
certain use of the operator
\/i. The
actual inventor of the
hyperbolic trigonometry was Vincenzo Riccati, SJ. (Opuscula ad res Phys. et Math, pertinens, Bononiae, 1757). He adopted the notation Sh.<, Ch.< for the hyperbolic functions, and Sc.<, Cc.0 for the circular ones.
He
proved the addition theorem geometrically and derived Soon after, Daviet
a construction for the solution of a cubic equation.
de Foncenex showed tions
the
the use of
by work
resting
how
to interchange circular
and hyperbolic func-
V
i, and gave the analogue of De Moivre's theorem, more on analogy, however, than on clear definition
(Reflex, sur les quant, imag., Miscel.
Turin
Soc.,
Tom.
Heinrich Lambert systematized the subject, and gave the
i).
Johann
serial devel-
opments and the exponential expressions. He adopted the notation sinh ft, etc., and introduced the transcendent angle, now called the gudermanian, using it in computation and in the construction of tables (1.
c.
page
30).
The important
history of the subject
is
place occupied by indicated on page 30.
Gudermann
in the
of the circular and hyperbolic trigonometry naturally considerable a part in the controversy regarding the doctrine played
The analogy
of imaginaries, which occupied so tury,
and which gave
much
birtU to the
attention in the eighteenth cen-
modern theory
of functions of the
HYPERBOLIC FUNCTIONS.
72
In the growth of the general complex theory, the
complex variable. importance of the
"
singly periodic functions"
was gradually developed by such log. et trig., Florence, 1782);
became
still
clearer,
and
writers as Ferroni (Magnit. expon.
Dirksen (Organon der tran. Anal, Ber-
lin, 1845); Schellbach (Die einfach. period, funkt., Crelle, 1854); Ohm (Versuch eines volk. conseq. Syst. der Math., Nurnberg, 1855); Hoiiel
(Theor. des quant, complex, Paris, 1870). in systematizing
helped
them
and tabulating these
Many
other writers have
functions,
and
in
adapting
The
following works may be espeGronau mentioned: (Tafeln, 1862, Theor. und Anwend., 1865); cially Forti (Tavoli e teoria, 1870); Laisant (Essai, 1874); Gunther (Die to a variety of applications.
Lehre
.
.
.
,
1881).
The last-named work
and bibliography with numerous
contains a very full history Professor A. G. Greenapplications.
various places in his writings, has shown the importance of both and inverse hyperbolic functions, and has done much to popularize their use (see Did. and Int. Calc., 1891). The following articles
hill, in
the direct
on fundamental conceptions should be noticed: Macfarlane, On the definitions of the trigonometric functions (Papers on Space Analysis,
N.
Y., 1894); Ilaskell,
functions (Bull.
On
N. Y. M.
the introduction of the notion of hyperbolic Attention has been called in Soc., 1895).
30 and 37 to the work of Arthur E. Kennelly in applying the hyperbolic complex theory to the plane vectors which present them-
Arts.
selves in the theory of alternating currents; and his chart has been described on page 44 as a useful substitute for a numerical complex It may be worth mentioning in this table (Proc. A. I. E. E., 1895).
connection that the present writer's complex table in Art. 39 is believed to be the earliest of its kind for any function of the general argument
x
+ iy. B.
(See Appendix, C.)
EXPONENTIAL EXPRESSIONS AS DEFINITIONS.
For those who wish to start with the exponential expressions as the u and cosh u, as indicated on page 25, it is here pro-
definitions of sinh
posed to show
how
these definitions can be easily brought into direct
geometrical relation with the hyperbolic sector in the form #/a=cosh 2 2 S/Kj y/b sinh S/K, by making use of the identity cosh u sinh w= i,"
and the differential relations d cosh tt=sinh u du d sinh w==cosh u du, which are themselves immediate consequences of those exponential Let 0-4, the initial radius of the hyperbolic sector, be definitions. y
EXPONENTIAL EXPRESSIONS AS DEFINITIONS. taken as axis of #, and
OB=b,
its
AOB=w,
angle
73
conjugate radius OB as axis of y\ let OA = a, and area of triangle AOB=K, then JRT
Ja&sinw. Let the coordinates of a point P on the hyperbola be x and y, then x2/a2 yi/b2 =i. Comparison of this equation with the 2 sinh 2 u=i permits the two assumptions #/a=cosh u identity cosh
and y/b=smh
u,
wherein u
is
a single auxiliary variable; and
remains to give a geometrical interpretation to u=S/K, wherein S is the area of the sector OAP. of a second point
POQ POQ
Q
be
x+dx
,
and
it
now
to prove that
Let the coordinates
and
y-\-Ay^ then the area of the triangle is, by analytic geometry, %(xJy ydx)sin aj. Now the sector a ratio whose limit is unity, hence the bears to the triangle
POQ
differential of the
Ja&sin w(cosh
u=S/K,
2
S may be written dS=$(xdy y= u)du=Kdu. By integration S=Ku, hence
sector
sinh2
the sectorial measure (p. 10)
geometrical relations
C.
;
this establishes the
fundamental
#/a=cosh S/K, ;y/6=sinh S/K.
RECENT TABLES AND APPLICATIONS.
The most
extensive
tables
of
hyperbolic
functions of real
arguments are those published by the Smithsonian Institution, prepared by G. F. Becker and C, E. Van Orstrand (IQOQ). For complex arguments the most elaborate tables are those of Professor A. E. Kennelly: "Tables of Complex Hyperbolic " Circular Functions (Harvard University Press, 1914).
Three-digit
tables of sinh
and cosh
of
x+iy, up to
and
x= i and
by W.
E. Miller in a paper, .01, given by steps y=i " Formulae, Constants, and Hyperbolic Functions for Transmission" in the General Electric Review Supplement, Schenline Problems are
of
ectady, N. Y.,
May,
1910.
There are interesting applications and an extensive bibliography " The Application of Hyperbolic in Professor Kennelly 's treatise on " Functions to Electrical Engineering Problems (University of
London
Press, 1912).
It should
be noted that this author uses the term
"
hyperbolic hyperbolic sectorial measure," the analogy being due " " for the circle and ellipse sectorial measure to the fact that the "
Bangle
is
"
for
an actual angle
(p.
n).
been suggested by Professor
The convenient term "angloid" has S.
Epsteen.
INDEX.
Addition-theorems, pages 16, 40.
Admittance of
dielectric, 56.
Algebraic identity, 41. Alternating currents, 38, 46, 55. Ambiguity of value, 13, 16, 45.
Amplitude, hyperbolic, 31. of
complex number, 46.
Anti-gudermanian,
Complementary triangles, Complex numbers, 38-46.
Applications of, 55-60. Tables, 62, 66.
Conductor resistance and impedance, 58.
Construction for gudermanian, 30. of charts, 43. of graphs, 32.
28, 30, 47, 51, 52.
Anti-hyperbolic functions, 16, 22, 25, 29,
Convergence, 23, 25. Conversion-formulas,
35. 45-
Applications, 46
et seq.
18.
Corresponding points on conies,
Arch, 48, 51.
sectors
Areas, 8, 9, 14, 36, 37, 60.
Argand diagram,
10.
and
7,
28.
triangles, 9, 28.
Currents, alternating, 55.
43, 58.
Curvature, 50, 52, 60. Bassett's
Hydrodynamics, 61.
Cotes, reference to, 71.
Beams, flexure of, 54. Becker and Van Orstrand, 73. Bedell and Crehore, 38, 56.
Deflection of beams, 54.
Byerly's Fourier Series,
Difference formula, 16.
etc., 61, 63.
Derived functions,
20, 22, 30.
Differential equation, 21, 25, 47, 49, 51, Callet's Tables, 63.
52, 57-
Capacity of conductor, 55. Catenary, 47. of uniform strength, 49. Cayley's Elliptic Functions, 30, 31.
Electromotive force, 55, 58. Elimination of constants, 21.
Center of gravity, 61.
Ellipses, chart of confocal, 43.
Characteristic ratios, 10.
Elliptic functions, 7, 30, 31.
Elastic, 48.
*
Dirksen's Organon, 71. Distributed load, 55.
Chart of hyperbolic functions, 44, 58.
integrals, 7, 31.
Mercator's, 53. Circular functions, 7,
sectors, 7, 31.
u,
14, 18, 21, 24,
29, 35> 4i, 43-
of complex numbers, 39, 41, 42. of gudermanian, 28
Equations, Differential (see).
Numerical, 35, 48, 50. Evolute of tractory, 52.
Expansion
in series, 23, 25, 31.
76
INDEX.
Exponential expressions, 24, 25, 72.
Hyperbolic functions of complex
num-
bers, 38 et seq.
Ferroni, reference
relations
to, 71.
among, 12. gudermanian,
relations to
Flexure of beams, 53.
Foncenex, reference to, 71. Forti's Tavoli e teoria, 63, 71.
29. relations to circular functions, 29, 42. tables of, 64 et seq.
Fourier
variation of, 20.
series, 55, 61.
Function, anti-gudermanian (see). anti-hyperbolic (see). circular (see).
Imaginary, see complex.
Impedance,
34.
Integrals, 35. elliptic (see).
gudermanian
Interchange of hyperbolic and circular (see).
hyperbolic, defined, n. of complex numbers, 38. of pure imaginaries, 41.
of
sum and
difference, 16.
functions, 42.
Interpolation, 30, 48, 50, 59, 62. Intrinsic equation, 38, 47, 49, 51.
Involute of catenary, 48. of tractory, 50.
periodic, 44.
Jones' Trigonometiy, 52.
Geipel and Kilgour's Electrical Hand-
book, 63.
Kcnnelly on alternating currents, 38, 58. Kennelly's chart, 46, 58; treatise, 73.
Generalization, 41.
Geometrical interpretation, 37. treatment of hyperbolic
Liisant's Essai,
etc., 61, 71.
Lambert's notation, 30.
functions, jet seq., 16.
place in the history, 70. of conductor, 55.
Glaisher's exponential tables, 63.
Leakage
Graphs, 32.
Limiting ratios, 19, 23, 32. Logarithmic curve, 60.
Greenhill's Calculus, 72.
expressions, 27, 32.
Elliptic Functions, 7.
Love's
Gronau's Tafeln, 63, 72. Theor. und An wend.,
Gudermann's
72.
notation, 30.
Gudermanian, angle,
Loxodrome,
function, 28, 31, 34,47, 53, 63, 70. etc., 61, 71.
et seq.,
30, 37, 44, 60.
Hyperbolic functions, defined, addition-theorems for, 16. applications of, 46
W.
Modulus,
Moment
E., Tables, etc., 73
31, 46.
of inertia, 61.
Multiple values, 13, 16, 45.
n.
et seq.
derivatives of, 20.
expansions
definitions, 72.
Maxwell's Electricity, 61. Measure, defined, 8; of sector, 9 Mercator's chart, 53. Miller,
Haskell on fundamental notions, 72. HoiiePs notation, etc., 30, 31, 71.
Hyperbola, 7
52.
Macfarlane on
29.
Gunther's Die Lehre,
elasticity, 61.
of, 23.
Newton, reference
to, 71.
Numbers, complex, 38
Ohm,
of, 32.
integrals involving, 35.
seq.
reference to, 71.
Operators, generalized, 39, 56.
exponential expressions for, 24.
graphs
et
Parabola, 38, 61. Periodicity, 44, 62.
et seq.
INDEX. Permanence of equivalence, Phase angle, 56, 59.
41.
Physical problems, 21, 38, 47 Potential theory, 61.
Self-induction of conductor, 55. Series, 23, 31.
et seq.
Product -series, 43. Pure imaginary, 41.
Spheroid, area of oblate, 58 Spiral of Archimedes, 60.
Steinmetz on alternating currents, 38. Susceptance of dielectric, 58.
Ratios, characteristic, 10.
Tables, 62, 73. limiting, 19.
of Sound, 61.
Rayleigh's Theory Reactance of conductor, 58.
Reduction formula, 37, 38. Relations
among
functions, 12, 29, 42.
Resistance of conductor. 56.
Rhumb
line, 53. Riccati's place in the history, 71.
Schellbach, reference
Terminal conditions, 54,
58, 60.
Tractory, 48, 5 1.
Van Orstrand, C. E., Tables, 73. Variation of hyperbolic functions, 14* Vassal's Tables, 63. Vectors, 38, 56. Vibrations of bars, 61,
to, 71.
Sectors of conies, 9, 28.
Wheeler's Trigonometry, 6l.
OSMANIA UNIVERSITY LIBRARY Call
Na
Accession No.
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By HARRIS HANCOCK. No. 19. Empirical Formulas. By THEODORE R. RUNNING. No. 20. Ten British Physicists. By the Lato ALEXANDER MACFARLANE. No. 21. The Dynamics of the Airplane. By KENNETH P. WILLIAMS.
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NEW YORK LONDON
MATHEMATICAL MONOGRAPHS. EDITED BY
MANSFIELD MERRIMAN AND ROBERT
No.
WOODWARD.
S.
4.
HYPERBOLIC FUNCTIONS.
JAMES McMAHON. LAIE
PkUi-LXs
,K
OF AlAlHl'MAIIO IN
FOURTH
COKNKU
1 N T
IVKKM
I
Y.
EDITION. ENLARGED.
NEW YORK: JOHN WILEY & SONS. LONDON:
CHAPMAN & HALL,
LIMITED.
COPYRIGHT,
1896,
Y
MANSFIELD MKRRIMAN
AND
RORERT
UNDPK THE T
I
S.
WOODWARD
LS
HIGHER MATHEMATICS. First Edition, September, 1896.
becond Edition, January, 1898, Third Edition, August, 1900. Fourth Edition, January,
Printed in U. S.
1906*
j
PRESS Of
RAUNWORTH CO INC BOOK MANUFACTURERS BROOKLYN, NEWYOHK ft.
,
EDITORS' PREFACE.
THE of which
volume
called
Higher Mathematics, the
was published
first
edition
in 1896, contained eleven chapters
by
eleven authors, each chapter being independent of the others, but all supposing the reader to have at least a mathematical training equivalent
The
colleges.
that given in
to
and engineering
classical
volume
publication of that
npw
is
discontinued
*
and the chapters are issued in separate form. In these reissues it will generally be found that the monographs" arc enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but that
it
may
prove advantageous
mathematical
to the series
from time
warrant
Among
to
is
also thought
literature.
It is the intention of the publishers
monographs same seems
it
to readers in special lines of
it.
and
editors to
add other
to time, if the call for the
the topics which are under
consideration are those of elliptic functions, the theory of
num-
and
non-
bers, the
group theory, the
Euclidean geometry;
calculus of
possibly also
variations,
monographs on branches of
astronomy, mechanics, and mathematical physics It is the
hope of the editors that
this
form of
may be included. publication may
tend to promote mathematical study and research qyer a wider field than that which the former volume has occupied. December, 1905. iii
AUTHOR'S PREFACE.
This compendium of hyperbolic trigonometry was first published as a chapter in Mernman and Woodward's Higher Mathematics. There is reason to believe that it supplies a need, being adapted to two or three different types of readers. College students who have
had elementary courses in trigonometry, analytic geometry, and differential and integral calculus, and who wi.sh to know something of the on account of its important and historic relahyperbolic* trigonometry tions to each of those branches, will,
it is hoped, find these relations in the first half of the and a way comprehensive simple presented work. Readers who have some interest in imagmaries are then intro-
in
to the more general trigonometry of the the circular and hyperbolic functions merge into ents, the singly periodic functions, having cither For those who also wish to view inary period.
duced
complex plane, where one class of transcenda real or a pure imag-
the subject in some of practical relations, numerous applications have be n selected so as to illustrate the various parts of the theory, and to show its use to the physicist and engineer, appropriate numerical tables being supplied for f
its
these purposes. With all these things in mind, much thought has been given to the mode ot approaching the subject, and to the presentation of fundamental notions, and it is hoped that some improvements are discerni-
For instance, it has been customary to define the hyperbolic ble. functions in relation to a sector of the rectangular hyperbola, and to take the initial radius of the sector coincident with the principal radius of the curve; in the present work, these and similar restrictions are discarded in the interest of analogy and generality, with a gain in symmetry and simplicity, and the functions are defined as certain charac-
belonging to any sector of any hyperbola. Such definiconnection with the fruitful notion of correspondence of points on comes, lead to simple and general proofs of the addition-theorems, from which easily follow the con version- formulas, the derivatives, the teristic ratios
tions, in
Maclaurin expansions, and the exponential expresMons. The proofs are so arranged as to apply equally to the circular functions, regarded as the characteristic ratios belonging to any elliptic sector. For those, however, who may wish to start with the exponential expressions as the definitions of the hyperbolic functions, the appropriate order of procedure is indicated cm page 25, and a direct mode of bringing such
exponential definitions into geometrical relation with the hyperbolic sector is ^.hown in the Appendix. December, igo^
CONTENTS.
ART.
r.
2.
3.
4. 5.
6. 7.
8.
Q.
10.
n. 12.
13 14. 15.
16.
17 18 ig.
20.
21.
22 23 24.
25.
26.
27 28.
29.
30.
31
32 33. 34.
35.
CORRESPONDENCE OF POINTS ON CONICS AREAS OF CORRESPONDING TRIANGLES AKEVS OF CORRESPONDING SECTORS CHARACTERISTIC KAIIOS OF SI-CTORIAL MEASURES RATIOS EXPRESS* D \S TRIANGLE-MEASURhS FUNCTIONAL KM AI IONS IOK KLIIPSE FUNCTIONAL RFLATIONS FOR HMFKHOLA KELAIIONS HFI \\M-N HYPERBOLIC UNCTIONS VAKIAIIONS OK THE HNPIRBOLIC FUNCTIONS. .
.
Page
9 10
.
IO
.
11
u
.
12
I-
...
ANTI HYPKRHOI ic FUNCTIONS FUNCTIONS OF SUMS AND DIH-ERENCFS CONVERSION FORMULAS LIMITING RATIOS DERIVATIVES OF HYPERBOIIC FUNCTIONS DERIVATINES OF ANII-HYPI-RBOLIC FUNCTIONS FXPANSION OF HYPERHOLIC FUNCTIONS KXPQNKNIIAL EXPRESSIONS EXPANSION OF ANTI UNCTIONS LOGARITHMIC EXPRESSION OF ANTI-JUNCTIONS THE (iUDKRMANIAN FUNCTION CIRCULAR FUNCTIONS OF (IUDERMANIAN (iUDERMANFAN A.\c;LE DERIVAFIM-S OF GUDERMANIAN AND INVERSE SERIES FOR (JUDERMANIAN AND ITS INVERSE GRAPHS OF HYPERBOLIC FUNCTIONS ELEMENTARY INTEGRALS FUNCTIONS OF COMPLEX NUMBERS ADDITION THEOREMS M>R COMPLEXES FUNCTIONS OF PURE IMAGINARIES FUNCTIONS OF x + iy IN THE FORM A' * lY
14
....
.
16
.
.
16
18
...
.
22
23 24
-i-
.
28 2()
....
.30 31
32 35
38
.
....
THE THE THE THE THE
CATENARY CATENARY OF UNIFORM STRENGTH ELASTIC CATENARY TRACTORY LOXODROME
25
27 28
...
.
19
20
.
...
/
o
.
.
.... ....
...
.
,
.
.
.
40 41
43
47 49
50 .
51
52
6
CONTENTS.
ART. 36 37.
38. 39.
TABLE
COMBINED FLEXURE AND TENSION ALTERNATING CURRENTS MISCELLANEOUS APPLICATIONS EXPLANATION OF TABLES
53 55
60 62
HYPERBOLIC FUNCTIONS VALUES OF COSH (x+ iy) AND SINH (x+ iy) III. VALUES OF gdw AND 0* LOG COSH u IV. VALUES OF gdw, LOG SINH
64 66
I.
II.
70
70
,
APPENDIX. HISTORICAL AND BIBLIOGRAPHICAL EXPONENTIAL EXPRESSIONS AS DEFINITIONS
INDEX
....
71
72
73
HYPERBOLIC FUNCTIONS. ART.
To
CORRKSPONDKNCE OK POINTS ON CONICS.
1.
prepare the
way
for a general
bolic functions a preliminary discussion
between hyperbolic to apply at the
and
the circle;
sectors.
treatment of the hyperis given on the relations
The method adopted
is
such as
same time
to sectors of the ellipse, including the analogy of the hyperbolic and circular
functions will be obvious at every step, since the same set of equations can be read in connection with either the hypeibola
or the ellipse.*
It
is
convenient to begin with the theory of
correspondence of points on two central conies of like species, i.Cc either both ellipses or both hypeibolas.
To
OB l
obtain a definition of corresponding points,
be conjugate
l
of
radii
a.
central
let
0,/f,,
conic, and O^A^ OJi^
radii of any other central conic of the same species; f\ be two points on the curves; and let their coordinates referred to the respective pairs of conjugate directions
conjugate let
P
be
(-*,, J,), (*,.;',);
l
,
then,
by
analytic geometry,
*The hyperbolic functions are not so named on account of any analogy " The with what are termed Elliptic Functions. elliptic integrals, and thence the elliptic functions, derive their
name from
cians at the rectification of the ellipse.
disadvantage;
.
.
.
.
.
the early attempts of mathemati.
To
a certain extent this
note cosh u sinh u, etc., by analogy with which the
merely
the
Functions,
circular functions cos #, sin 0, p. 175.)
is
a
because we employ the name hyperbolic function to de-
etc.
elliptic .
."
functions would be (Greenhill,
Elliptic
HYPhKJIOLIC FUNCTIONS.
8
Now
if
the points
P
l
,
1\ be so situated that
* (2)
the equalities referring to sign as well as magnitude, then P9 are called corresponding points in the two systems. If Qi be another pair of correspondents, then the sector and
angle P^O^Q,
are
when to
/\O
l
,
tri-
respectively with the Tiiese definitions will apply also
y
Q
9
.
the conies coincide, the points
any two
Q
to correspond
said
sector and tsiangle
P19
/'
,
I\ being then referred
pairs of conjugate diameters of the
same
conic.
between corresponding areas it " is convenient to adopt the following use of the word measure": The measure of any area connected with a given central conic In discussing the relations
the ratio which
bears to the constant area of the triangle formed by two conjugate diameters of the same conic.
is
I r or
it
example, the measure of the sector A^O^P^ sector A,O,PI triangle""^, (9JJ5J
is
the ratio
AREAS OF CORRESPONDING SECTORS.
and
to be
is
A^O^
A
and
common
The let
regarded as positive or negative according as O ^ are at the same or opposite sides of their l
l
initial line.
ART.
For,
l
9
AREAS OF CORRESPONDING TRIANGLES.
2.
areas of corresponding triangles have equal measures. the coordinates of />, <2, be (^ l ,y l ) (-*/, J'/) anc* let t
those of their correspondents /',, <2a be (.r,, j' 9 ), (,r/, j'/); let the triangles />,>, <2, 1\O^Q^ be T19 7",, and let the measuring tri,
A OH
K
and their angles a?,, r^ of both magnitude account then, by analytic geometry, taking angles
1
1
and direction 7*
:=
,
1
A^OJ\
be
A",
and
of angles, areas,
JU.lV- -OPS'"
A'
Jrt7
Therefore, by
ART.
.sin
ft>,
;
lines,
_
,
J//
^
u>
-
-
(2),
3.
tJ
,
_^^ a
/;
L'
6
\
(3)
AREAS OF CORRESPONDING SECTORS.
The For
areas of corresponding sectors h.ive equal measures. conceive the sectors S 19 Sv divided up into infinitesimal
corresponding sectors then the respective infinitesimal corresponding triangles have equal measures (Art. 2); but the given sectors are the limits of the sums of these infinitesimal ;
triangles,
hence
5
-
K
= S
*
In particular, the sectors ures; for the It
may
initial
points
A^O.P^
A A t ,
, x (4)
AY
t
t
A OP t
y
9
have equal meas-
are corresponding points.
be proved conversely by an obvious reductio ad
points of two equal-measured sectors correspond, then their terminal points correspond.
absurdum that
if
the
initial
O,A 19 Ot A 9 be the initial lines of two equal-measured sec^rs whose terminal radii are O P^ O9 Ptt Thus
if
any
radii
V
HYPERBOLIC FUNCTIONS.
10
P P
are corresponding points referred respectively to then 19 t the pairs of conjugate directions O^A^ O V B^ and that is,
Prove that the sector /\0,(?, is bisected by the line P,Q (Refer the points 19 Q 19 recommon axis of x and to the two Q 19 opposite conjugate directions as axis of y and show that are then corresponding points.) Prob. 2. Prove that the measure of a circular sector is equal to the radian measure of its angle. Prob. 3. Find the measure of an elliptic quadrant, and of the sector included by conjugate radii. Prob.
I.
joining <9, to the mid-point of spectively, to the median as
X
P
.
9
P
9
ART.
CHARACTERISTIC RATIOS OF SECTORIAL MEASURES.
4.
A OP =S
Let
1
1
1
be any sector of a central conic; draw O^l l9 i.e. parallel to the tangent at A MJ*i = }'M O A = rf, and the conjugate radius l
/^J/, ordinatc to
OM
let
l
OJ>,
.r,,
l
= ^\
t
t
l
then the ratios
;
,
l
xja v9 yjb
v
are called the charac-
the given sectorial measure S //Cr These ratios are constant both in magnitude and sign for all sectors
teristic ratios of
t
same measure and species wherever these may be situHence there exists a functional relation beated (Art. 3). tween the sectorial measure and each of its characteristic
of the
ratios.
ART.
5.
RATIOS EXPRESSED AS TRIANGLE-MEASURES.
The triangle of a sector and its complementary triangle are measured by the two characteristic ratios. For, let the triangle A^O.P^ and its complementary triangle P O B be denoted by 1
r
if
TV; then
1
1
"
K^ T'
^&\b\ sin lrb
t
x
ctfj
b^
o>,
x.
(5) t
sin
FUNCTIONAL RELATIONS FOR
ART.
The
11
FUNCTIONAL RELATIONS FOR ELLIPSE.
6.
functional relations that exist between the sectorial
measure and each of for
ELLIPSE.
all
its
characteristic
ratios
are
the same
in-
elliptic,
r~~i.^
eluding circular, sectors (Art. 4).
sT-
B,
s^
Let/*,,
1\ be corresponding
points on an ellipse and a circle, referred to the conjugate di-
0,A^ O B l9 and
rections
t
right angles
;
let
the angle
A^O^R^
0,
A OJ\ = 9
~L*
1
A,
the latter pair being at in radian measure; then (6)
, tf,
5, 5, ^ = cos.;., j -V^sm^; .
a
9
.-.
At
***
[a.
=
hence, in the ellipse, by Art. 3, (7)
Prob.
Also find
Prob.
measure Prob.
Given id
4.
AiOiPi.
the measure of the elliptic sector
5.
=
4,
/fr,
ratios of
3, G?
an
=
60.
elliptic sector
whose
is ITT.
Write down the relation between an
6.
ART.
The
\a\; find
area when- 0,
Find the characteristic
its triangle.
its
=
its
7.
(See Art.
elliptic sector
and
5.)
FUNCTIONAL RELATIONS FOR HYPERBOLA.
functional relations between a sectorial measure and
characteristic ratios in the case of the hyperbola
may be
written in the form
S.
'
v.
S.
-*M
and these express that the ratio of the two lines on the left is a certain definite function of the ratio of the two areas on the right.
These functions are
called
by analogy the hyperbolic
HYPERBOLIC FUNCTIONS. cosine and the hyperbolic sine.
Thus, writing u for
SJKV
the
two equations
= a
cosh
If b
//,
t
=
sinh u
($\ v >
l
serve to define the hyperbolic cosine and sine of a given sectorial measure u and the hyperbolic tangent, cotangent, secant, ;
and cosecant are then defined as follows .
^
tanh
;/
=
sinh ---
?/
.
-
;
u
cotli
,
cosh
:
cosh//
=
--
sinh
;/
//'
(9) i
sech
//
=
T
T t
.
The names
csch u
,
cosh
.
sinh
//
of these functions
or "hyper-cosine," etc.
!
(See
"
//
J
be read " h-cosine,"
may
"
angloid
or
"hyperbolic
angle," p. 73-)
ART.
RELATIONS AMONG HYPERBOLIC FUNCTIONS.
8.
the six functions there are five independent relawhen the numerical value of one of the functions
Among
tions, so that is
given, the values of the other five can be found.
these relations consist of the four defining equations fifth is derived from the equation of the hyperbola
giving sinh*//
cosh*//
By
a combination of
ary relations
by
may
cosh* w, sinh
1
//,
some
I.
(10)
of these equations other subsidi-
and applying tanh' u
coth* (9),
(10),
(9),
give
= sech* = csch
;/, )
(u)
I
(u)
i
2
//
//.
)
will readily serve to express the
value of any function in terms of any other. when tanh u is given,
coth
//
=
(9).
of
The
be obtained; thus, dividing (10) successively
I
Equations
=
Four
-
tanh u
,
sech u
=
A/I
For example,
tanh*//,
RELATIONS BETWEEN HYPERBOLIC FUNCTIONS.
=
,
cosh?/
=
,
csch u
I
sinn//
,
V
I
\/
I
.
.
-----
=-__
tailll
V
tanh*//
13 //
-,
tanh'//
I
tanh*//
---
-
--
tanh
//
The ambiguity in the sign of the square root may usually The functions be removed by the following considerations cosh//, scch // are always positive, because the primary char:
acteristic ratio
O,M
the abscissa
is
.*,/
OA
positive, since the initial line
O
are similarly directed from
l
}
and
l
on which-
,
ever branch of the hyperbola I\ may be situated; but the functions si nh //, tanh it, coth u, csch //, involve the other charac-
y^ //
and is
b^
which
yjb^
teristic ratio
is
positive or negative
have the same or opposite
signs,
i.e.,
according as as the measure
positive or negative; hence these four functions are either
Thus all positive or all negative. tions sinh //, tanh //, csch //, coth ;/,
when any one
of the func-
given magnitude and sign, there is no ambiguity in the value of any of the six but when either cosh // or sech // is hyperbolic functions is there ambiguity as to whether the other four functions given, in
is
;
shall
be
all
positive or
all
negative.
_
The hyperbolic tangent may be expressed two
For draw the tangent
lines.
AC ~ t\ then y x a y tanh u = 4 :- = .--
line ^
i
b
a
x
b
= a..- = t
t
f
-.
(
The hyperbolic tangent
is
^.
.
I3
as the ratio of
)0
-
i
*
.
the measure of the triangle
OAC.
For
Thus the
OAC OAS sector AOP
are proportional to
y
;/,
at
**
t
3)
*
and the triangles AOP, FOB,
sinh
sinhw
,
cosh
> >
.
,
//,
tanh u (eqs.
tanh//.
5,
13)
;
AOC, hence (14)
HYPERBOLIC FUNCTIONS.
14 Prob.
Prob.
8.
<
i,
Prob.
9.
tanh u
Express
7.
Given cosh
//
=
2,
all
the hyperbolic functions in terms ot sinh
Prove from eqs. sech u
u.
find the values of the other functions.
<
n,
10,
that
cosh//> sinh
,
cosh*/>i,
i.
In the figure of Art.
i, let
OA-2, OB-\,
AOB =
60,
and area of sector AOP => $\ find the sectorial measure, and the two characteristic ratios, in the elliptic sector, and also in the hyperbolic sector; and find the area of the triangle AOP. (Use tables of cos, bin, cosh, sinh.)
Prob. 10.
Show
coth u
that
9
sech
//,
csch u
may each be
ex-
P
pressed as the ratio of two lines, as follows: Let the tangent at m make on the conjugate axes OA, OB, intercepts OS n\ in let the tangent at B^ to the conjugate hyperbola, meet y
=
OT =
}
OP
making BR =
/;
~
coth u Prob. IT.
The
this for
Modify
R
then sech u
//a,
measure of
the ellipse.
=
csch
m/a,
segment AMP
Modify
=
//
njb.
sinh u cosh u
is
u.
10-14, and probs.
also eqs.
8, 10.
ART.
9.
VARIATIONS OF THE HYPERBOLIC FUNCTIONS.
Since the values of the hyperbolic functions depend only on the sectorial measure, it is convenient, in tracing their variations,
half
one whose rectangular hyperbola,
to consider only sectors
of a
of
conjugate radii are equal, and to take the principal axis OA as the common initial line
of
all
the
sectors.
The
sectorial
measure u assumes every value from through
o, to -f- oo
,
oo,
as the terminal point
P
comes in from infinity on the lower branch, and passes to infinity on the upper
branch; that is, as the terminal line OP swings from the lower asymptotic position
but as
y is
=
x* to the
proved
P passes to
in
upper one,
y
= x.
Art. 17, that the sector
It
is
here assumed,
AOP becomes
infinite
infinity.
Since the functions cosh
//,
sinh
//,
tanh
u, for
any position
VARIATIONS OF THE HYPERBOLIC FUNCTIONS. of OP) are equal to the ratios of x, y, a, it is evident from the figure that
cosh o
=
= 0,
sinh
I,
/,
15
to the principal radius
= 0,
tanh
(15)
and that as u increases towards positive infinity, cosh it, sinh u are positive and become infinite, but tanh// approaches unity as a limit
thus
;
cosh
=
co
sinh
oo,
00=00,
tanh
=
oo
i.
(16)
Again, as // changes from zero towards the negative side, cosh u is positive and increases from unity to infinity, but sinh u
negative and
is
increases numerically from
and tanh
zero to a
negative and negative infinite, numerically from zero to negative unity hence //
is
inci eases
also
;
cosh
oo )
(
=
oo
sinh
,
(
oo
For intermediate values
)
of
= //
oo
tanh
,
(
oo)
A
are tabulated at the end of this chapter.
manner
I.
(17)
the numerical values of these
functions can be found from the formulas of Arts.
their
=
16, 17,
and
general idea of
of variation can be obtained from the curves in
which the sectorial measure u
represented by the abscissa, and the values of the functions cosh //, sinh //, etc., Art. 25,
in
is
are represented by the ordinate.
The
relations
between the functions of
;/
and of u are
evident from the definitions, as indicated above, and
Art.
in
8.
Thus cosh sech
tanh
() = (
u)
(
u)
= =
cosh
u,
sinh
-f sech
//,
csch
tanh
//,
coth
-{-
(
+
=
(//)
Prob. 12. Trace the changes in sech oo to oo. Show that sinh
from
//)
(
//)
?/,
,
=
coth
cosh
sinh u,
\
csch
;/,
>
coth
//.
(18)
;
csch u, as u passes are infinites of the
#, //
same order when u ,
is infinite. (It will appear in Art. 17 that sinh cosh u are infinites of an order infinitely higher than the order
of u.)
Prob. 13. Applying eq. (12) to figure, page 14, prove tanh u, tan
A OP.
=
HYPERBOLIC FUNCTIONS.
16
ART.
The
x -
equations
=
cosh
y
1
y -r
=
t
sinh
7
//,
=
tanh
u, etc.,
^ 1
-,
t
=
sinli~ T,
u,
by the inverse notation u^cosh"
also be expressed
may
=
ANTI-HYPERBOLIC FUNCTIONS.
10.
tanh"
1
-r,
etc.,
which
be read: "#
may
is
the sectorial measure whose hyperbolic cosine is the ratio x to a" etc. or " u is the anti-h-cosine of x/a" etc. ;
Since there are two values of
u,
correspond to a given value of cosh
with opposite signs, that it follows that if u be
//,
determined from the equation cosh u = ;, where m is a given number greater than unity, u is a two-valued function of m.
The symbol
m
will be used to denote the positive value ;;/. the equation cosh u Similarly the will stand for the positive value of // that
cosh"
1
=
of u that satisfies
symbol sech"
1
;;/
=
The
m. the equation sech. u functions sinh' ;//, tanh' ;//, coth"
satisfies
1
1
1
as the sign of m.
Hence
all
signs of the other csch" ;;/, are the same 1
;//,
of the anti-hyperbolic functions
numbers are one-valued.
of real
Prob. 14. Prove the following relations:
cosh'
1
m=
sinh' 1
a
V;;/
i,
sinh"
1
;;/
=
cosh" 1
upper or lower sign being used according as 1 1 Modify these relations for sin" , cos" negative. iie
;//
a
V;;/ is
+
i,
positive or
.
=
OB =
AOB =
Prob. 15. In figure, Art. i, let OA 2, 60; find i, the area of the hyperbolic sector A OP, and of the segment 1 if the abscissa of is 3. (Find cosh" from the tables for cosh.)
AMP,
P
ART. (a)
11.
To sinh
prove the difference-formulas = sinh u cosh v cosh (// ?>)
cosh
Let
FUNCTIONS OF SUMS AND DIFFERENCES.
(//
v)
=
cosh u cosh v
sinh
;/
sinh #,
//
sinh
v.
OA
AOQ
be any radius of a hyperbola, and let the sectors AOP, v is the measure of the have the measures u, v\ then u
Let OB, OQ' be the radii conjugate to OA, OQ\ be (*,,.?,), (x,y\ (*',/) the coordinates of P, Q, with reference to the axes OA, OB\ then
sector
and
QOP.
let
Q
FUNCTIONS OF SUMS AND DIFFERENCES. sinh
=
-*)
toLQr = triangle
sinh
A
^*
= sinh
17
QO/
A
s>
^ sn sin a)
u cosh v
cosh u sinh v\
w
., cosh (
N
v)
=
,
cosh
QOP = triangle ^ - * POQ' [Art. A/A
sector
r .
5.
sn ^
,
but
x
(20)
since Q, Q' are extremities of conjugate radii
cosh
v)
(
= cosh
;/
cosh
sinh
?>
;
hence
//
sinh
z/.
In the figures u is positive and v is positive or negative. Other figures may be drawn with n negative, and the language in
the text will apply to
all.
In the case of elliptic sectors,
drawn, and the same language the second that except equation of (20) will be x' /a similar figures
may be
l
will
apply,
=
j
therefore
(b)
sin
(//
v)
cos
(//
z/)
To prove
= sin u cos v = cos u cos v
;/
sin
u sin
sin v> v.
the sum-formulas
sinh (u
+ v) = sinh w cosh v
cosh
-j-
(
-f-
cos
v)
=
cosh
cosh
These equations follow from
z/
'
-f-
w sinh osh u cosh
-f-
inh u sinh sinh
(19)
z/,
)
f z/. )
by changing v
into
v,
HYPERBOLIC FUNCTIONS.
18
and then
for sinh(
v\ cosh(
sinh v, coshfl
writing
#),
(Art. 9, eqs. (18)). /
T To
\
(c)' v
^ ^
^i
v);
sin
=
v)
T-
-
v
,
(22) ; v
.
tanh v
tanh
i
Writing tanh (u
tanh v
tanh u
=
\
/
i
prove that tanh (u v v *
expanding and dividing
!2,
^-
numerator and denominator by cosh u cosh
ob-
v, eq. (22) is
tained.
Prob.
1.
cosh
3.
i
4.
tanh \u
2//
=
+ cosh .
cosh*#
=
//
.
6.
= =
.
Sinll
2U
sinh yi
8.
=
cosh (u -f
3, find
cosh"
^//,
+ 2
=
// --
-
cosh
=
3 //
.
=
2
sinh
-,
COSh
i//.
i\*
1
J.
r-r-.
tanh u
i
cosh 3^
^/,
I.
8
\cosh u -f- i/ 4- tanh' w
2//
8
cosh' #
2
/cosh u
= T
//
+ 4 sinh
3 sinh
u
i i
.
tanh
2 sinh*
//
sinh u
//
tanh
i
i -\-
cosh cosh u
.
i
.
=
4 cosh'w
3 cosh u.
'
.
,
.
i
Generalize (8); and show also what it becomes 9 2 a cosh'jc sm*y sinh'jt: -f- sin /. 10. sinh jc cos y 9.
u. cosh" sinh"
1
cosh'
///
1
sinh"
///
when
u=v=
?'). .
. ,
=
+
12.
v).
+ tanh \u cosh w + sinh u = tanh \u (cosh + sinh ;/)(cosh v + sinh e^)=cosh (u -\-v)-\- sinh (u + .
7. '
2
sinh
-f-
sinh
=
.
i
5.
cosh v
2,
Prob. 17. Prove the following identities: 2 sinh # cosh u. sinh 211
2.
99
=
Given cosh u
1 6.
1
//
1
//
=.
=
Prob. 18.
What
Prob.
Modify the
cosh'
sinh'
1
1
^/// y
a
y
^/////
I
(/// 7
i -j-
y
)( wa ~"i)ji
+w
a
j.
modifications of signs are required in (21), (22), in order to pass to circular functions ? 19.
ART.
To
12.
identities of Prob. 17 for the
same purpose.
CONVERSION FORMULAS.
prove that cosh //,+ cosh u9
cosh sinh
w, j
sinh w.
= 2 cosh J(f/,+ *') cosh ^(u =
l
u9),
w + sinh + sinh u^ = 2 K. + cosh 4X iX ^) = 2 cosh i(i + *) sinh J(. cosh
#,
2 sinh sinli
//,
(//,
(//,
1
?',)
(*/,
a ),
*i).
,) )
sin:i
*
,)-
(23) JI
J
LIMITING RATIOS.
From
the addition formulas
cosh (u
+ v + cosh
cosh (w
-f~
si
nh
sinh (w
cosu ( u
^)
=
(//,
(
w
v
-f-
2 cosh
w
2 sinh
u
>
i
;/
2 si"h
)
(//
"" s i nn
-|" t; )
and then by writing u z;
= v = v) = = v) v)
(
+ ^) + sinh
(
follows that
it
)
19
2
cosh
v
=.
a
cosh
v,
sinh ^, /*
7*
,
cosh
z/,
sinh 7% //
= J(//
j
-j-
;/
s) t
these equations take the form required.
f ),
Prob. 20. In passing to circular functions, show that the only modification to be made in the conversion formulas is in the algebraic sign of the right-hand member of the second formula. __
.
---+ cosh
cosh -
...
.
Prob. 21. Simplify Prob. 22. Prove
211
.
-\-
<\v
=
sinh* j>
sinh*.*:
,
,
sinh
coshV cosh*^ Simplify coshV cosV +
sinh
Prob. 23. Simplify Prob. 24.
ART.
To
471
'
cosh 4^
sinh (x
y).
sinh'.r sinh*^. sinh*jc sin
?
_y.
u approaches zero, of
find the limit, as
sinh u
tanh u
~^~~'
~T~'
which are then indeterminate
By
(.v -\-y)
+ cosh_J-.
LIMITING RATIOS.
13.
eq. (14), sinh u
cosh ----- 211 cosh 2U
47^
-
.
sinh 2U
>
//
>
form.
in
tanh u
and
;
if
sinh u and tanh u
be successively divided by each term of these inequalities,
it
follows that I
<
sinh u
--u
.
sech u
but when
w^o,
cosh u
lim.
sinh u
=
u
u
o
~
<
<
,
cosh
-- <
tanh
_
m =
\\ '
//
=
sech
i,
u
u,
.
I,
i:
hence
tanh u
-
.
HYPERBOLIC FUNCTIONS.
ART.
To
DERIVATIVES OF HYPERBOLIC FUNCTIONS.
14.
prove that
=
^(sinh u)
(^
du //(cosh u)
cosh u
=
T
du
smh
'
, '
du (25)
,,*
,
//(cscb
s\
= sinh 4j/ = sinb = 2 cosh
Let
y
=
= cosh
ft.
limit of
sinh u
Aii)
4(2 w
+ ^) sinh %Au, .sinh 4Jft
.
(ft \
Jft
Take the
csch u coth
ft,
(ft -f-
Ay
ft,
csch* u,
//)
-j
(/) (a)
sech u tanh
^/(coth w)
to /
=
//(sech u)
a) \
+ -kAii) j
i
both
Ay
^j^
sides, as
_
dy
lim. cosh
-
.
/
//(sinh
u)
(ft
Au ~
=
o,
and put
;/)
cosh
ft,
sinh -
lim
//(sinh
then
(b)
^
Similar to
ft)
3
(see Art. 13)
;
=
cosh
.
(a).
//(tanh u} __ ~~
3W
[
sinh w
// "
cosh w
cosh* u
sinh*
cosh* u
w
=
nr#
COSIl
=
sech* ^.
DERIVATIVES OF HYPERBOLIC FUNCTIONS. (d)
Similar to
(c).
--
'
(/)
-
d
\
^
21
du
Similar to
~
i
T du
^
i
sinh i
cosh u
cohh
*
=
seen
//
tanh
u.
//
(c).
thus appears that the functions sinh ?/, cosh u reproduce themselves in two differentiations and, similarly, that the circular functions sin?/, cos// produce their opposites in two It
;
In this connection
differentiations.
it
may
be noted that the
frequent appearance of the hyperbolic (and circular) functions the solution of physical problems is chiefly due to the fact
in
that they answer the question What function has its second derivative equal to a positive (or negative) constant multiple :
of the function itself? (See Probs. 28-30.)
y = cosh mx
is
An
answer such as
not, however, to be understood as asserting that
mx
is an actual sectorial measure and^v its characteristic ratio but only that the relation between the numbers mx and y is the same as the known relation between the measure of a hyper-
;
bolic sector
and
its
characteristic ratio;
and that the numerical
value of y could be found from a table of hyperbolic cosines. Prob. 25 Show that for circular functions the only modifications required are in the algebraic signs of (#), (
Prob. 27. Find the derivative of tanh u independently of the derivatives of sinh
//,
cosh
Prob. 28. Eliminate
equation y = A cosh mx
u.
the constants by differentiation from the -B sinh mx, and prove that d*y/dx* m*y.
=
-f-
Prob. 29. Eliminate the constants from the equation
=A that d *y/dx* = y
and prove
Prob. 30. Write tial
equations
down
cos
mx
+ B sin mx,
my.
the most general solutions of the differen-
22
HYPERBOLIC FUNCTIONS.
ART.
DERIVATIVES OF ANTI-HYPERBOLIC FUNCTIONS.
15.
rf(sinh"'
x)
~
Tx
i
r ~fe +\'
1
^(cosh"* x) ~~
~&
i '
V*"" 11 "!
4tanh~'
"I
i
__
)
(26)
v
,
'
dx
I
~
i
*?
1
I
^/(csch-
x \x Let
=
=
#
Vi
sinh w
Similar to
(c)
Let
(^/)
u
dfo
tanh"
tanh
Similar to
=
Vi
x
+
then
^r,
if)du
= 1
-*"
sinh
du
du
>
/*
//,
-
=
(i
^r
=
tanh
;/,
^w
^X//,
=
dx
=
d
i
t
(cosh-
,
i\ 1 -!=-/ i
1
sech"
dtr/i
\*
1
-
I
I
(e).
Prob. 31. Prove
_
x
i
^r
= cosh w
//
;r'.
(c).
^)_^~
(/) Similar to
1
2
1
.
then
^r,
r
(a).
=
(i
_^ ^ ^/(sech" ,
1
8
-f-
(6)
=
sinh"
+
i
i
+
__
i/(cot"' x) ~" JP*'
^
i
J
=
i
du
EXPANSION OF HYPERBOLIC FUNCTIONS.
23
Prob. 32. Prove a
rfsinh
.
.
tanh-
1
dx
a -=
# -
=
a
..
.
a
-^
a'
' ,
"1
x -
.
,
1
rf/coth-
,
-,
a
*'_]*<
--= - -,adx
*
#
"~|
J*>
!
1
Prob. 33. Find ^(sech" x) independently of cosh~ x. Prob. 34.
When
tanh"
1
x
1 prove that coth"
is real,
nary, and conversely; except when x
_
_
,
For
16.
this
/(*)
=
.
-j-, ~^
x
x
,
when ^
purpose take Maclaurin's Theorem,
= AO) +
/(o)
+
J,
y
;/
(o)
+ ~ ^/"'(o) +
.
I,
sinh u
hence similarly, or
by
cosh u
By means cosh
00.
= sinh u, f(u) = cosh K, /"() = sinh = 0, /'(o) = cosh o = /(o) = sinh
then
u,
ima 5i
EXPANSION OF HYPERBOLIC FUNCTIONS.
and put /()
and
is
i.
.
Prob. 35. Evaluate
ART.
=
= u +~ u + 3
u'
+
.
.
.
;
.
-,
ir f
.
.
.,
.;
(27)
differentiation,
= +4 i
f
+ ^ + ---4
(28)
of these series the numerical values of sinh
//,
can be computed and tabulated for successive values of
the independent variable u. They are convergent for all values of u, because the ratio of the //th term to the preceding is in
the
first
u*/(2n
i)(2// 2), and in the second case both of which ratios can be made less than
case u*/(2n 2)(2
3),
unity by taking n large enough, no matter what value u has. Lagrange's remainder shows equivalence of function and series.
24
HYPERBOLIC FUNCTIONS.
From
these series the following can be obtained by division
tanh u
=u-
:
'
-
4
.
(2
These four developments are seldom used, as there is no observable law in the coefficients, and as the functions tanh //, sech
;/,
coth
computed
//,
csch
can be found directly from the previously
//,
values of cosh u, sinh
Show
Prob. 36.
u.
that these six developments can be adapted to
the circular functions by changing the alternate signs.
ART.
EXPONENTIAL EXPRESSIONS.
17.
Adding and subtracting cosh u
+ sinh u =
I
-f-
//
-f-
~u*
~u
-|
-i
,
sinh
//
=
I
u
I
.
-u
-{
I
a
^
.
tanh
=
Me
tt
-u
where the symbol
x
in
~'
= V' 2
">
**'
.
t u sech
=
The analogous exponential cos u
I
+ T" u
= i(^M
u
,
3
sinh u
+ e~ \
--= e*-e-*
~u"
-}-
sn #
-f-
.
.
.
=
.
.
.
= e~ u
u
e
9
*r *
'
hence cosh u
3
j
*
cosh u
give the identities
(27), (28)
4
e'
--
u
\
2
h
-
'
,
9
etc.
(30)
I
expressions for sin u, cos u are
=
.
22
stands for the result of substituting 0* for
the exponential development
This
will
be more
numbers, Arts. 28, 29.
fully
explained
in treating of
complex
EXPANSION OF ANTI-FUNCTIONS.
#5
Prob. 37. Show that the properties of the hyperbolic functions could be placed on a purely algebraic basis by starting with equafor example, verify the identities tions (30) as their definitions :
;
sinh a
cosh u
sinh
-
u)
( a
u
=
sinh (u +?>)
i,
- =m mu)-
r
cosh
sinh u,
(//)
= sinh
=
cosh u,
u cosh v + cosh u sinh
v,
--,--.- = n? sinh a
a
cosh
;;///.
di?
Prob. 38. Prove (cosh u
+ sinh
(sinh
n
u)
=
/////)
.
,
cosh nu
.
+ sinh
nu.
Prob. 39. Assuming from Art. 14 that cosh sinh u satisfy the differential equation dty/dfo* j, whose general solution may be ,
=
=
+ Be~ where A B are arbitrary constants show how to determine A, B in order to derive the expressions for cosh
written
y
u
Ae u
>
y
;
,
sinh u, respectively.
[Use
Show how
Prob. 40.
eq. (15).]
to construct a table of exponential funcvice versa.
from a table of hyperbolic sines and cosines, and sinh u). Prob. 41. Prove u = log, (cosh u
tions
+
Prob. 42. Show that the area of any hyperbolic sector when its terminal line is one of the asymptotes.
2*-
Prob. 43.
From
w
=cosh
1
( cos h
^)
u the relation 2 cosh u -= e
;/tf
prove
+ #cosh (n2)u+\n(ni) cosh
and examine the last term when n is odd or even. Find also the corresponding expression for 2*~
ART. .
-^
(n
n
l
(sinh u)
.
EXPANSION OF ANTI-FUNCTIONS.
18.
--x) = dx
dfsinrr
c Since
infinite
u
e"
-f-
is
1
-
-j
-i
= *
= (i /
I
\
i
i
x*}-*
-f-
34
i
+
3 5
i
246
24
y< "T"
-**
'
hence, by integration, i
sinh-
i 1
*
-+ 2467'
^ x% = * ---h-~3 --i
!
i
23245
!
3 5
*T
i
/
\
(30 w/
the integration-constant being zero, since sinh" x vanishes with x. This series is convergent, and can be used in compu1
HYPERBOLIC FUNCTIONS.
$6
:ation, r
.
>
i,
Another series, convergent when only when x < i. is obtained by writing the above derivative in the form
-L _I x = C+log b *+2 2X* 2
sinh-'
"
vhere
is
A
3 + I2 466x' | I
f
v ( VJ '
'
the integration-constant, which will be shown in
be equal to log,
\rt. 19 to
3 J 44X
'
1
2.
development of similar form
dx
_ ~^
is
iV
_i/ ~ vt '
,
1 2
obtained for cosh~'^r; for
l-LllJ-J-ilS I ^~ ^ ^
"1
,
J:
a
2
t 4 X
2 4 6
X
f
'
'
'J'
icnce :osh-'
n
x=C+\ogx-^ ^.-..., 22^',--^ 2 4 4** ---I 2 4 6 6*' .
to
which
C
is
again equal to log, 2 [Art.
>rder that the function cosh"'.r
vergent, hence
it is
may
when x exceeds
ess than unity; but
be
(33) VJO/
Prob. 46]. In not be real, 19,
x must
unity, this series
is
con-
always available for computation.
Again, ind hence
From jech"
1
x
tanh"
1
x = x + - x* -\- -x* -\- -x
(32), (33), (34) are derived
= cosh"
1
+
.
.
. ,
(34)
:
1
-fl-l^-L!^,.... 2.2 2.4.4 2.4.6.6
(3S) Vt"'
LOGARITHMIC EXPRESSION OF ANTI-FUNCTIONS.
,,
cscrr'
jr
u = sinh"
-* x
i 1
=
II
l
x
2
I
+-2 i
ix
.
3 ^ I 4 5*
*
3 -
27
*
5
I
,+
2467*'
^-tog^r+^-i^ + Iil^-...; 2.2 2.4.42.4.6.6 coth-'*
= tanh-
I *
1
Show
Prob. 44.
=1+ *
+
,
'
6
'
3*'
+
-L
+ '
.
..
.
(36)'
v (37) '
TX*
5^r
that the series for tanh"
1
jc,
coth' 1
sech" 1
jc,
A:,
are always available for computation.
Show
Prob. 45.
that one or other of the
inverse hyperbolic cosecant
ART.
19.
is
two developments of the
available.
LOGARITHMIC EXPRESSION OF ANTI FUNCTIONS.
x
Let
cosh
= sinh u\ u -f- sinh u = e", "'
then Vx*
//,
i
Similarly,
= cosh u, = cosh' ^, = log (x + sinh" ^ = log (x -f i/^+'T).
Also
sech" ! ;r
therefore
x-4- Vx*
I
1
and
V*'
I
(38)
).
1
=
=
cosh" 1
'II* -*=s,nh-'-
.
,
x
let
I
r
therefore
= tanh
*
I
^**
=
#
L
log
1^
"
*
**
I
/
\
(4I )
~" ^~w _M
,
_ + = =^ ^" /* u
JT
-
i
^r
2u
= log^i
coth" 1 ^
and
x
,
tanh- 1 ^-^^ log i
sinh"
1
^
log
x
(38), (39), that,
#:Mog
2,
~~~
cosh""
1
^
-
.
(43)
I
when .r^
and hence show that the integration-constants 2.
(42)
;
1
Show from
Prob. 46.
x
= tanh' - = i log ?-l. JT
equal to log
(40)
,
= log-X_^_.
1
Again,
(39)
log
oo,
x- log
2,
in (32), (33) are
each
28
HYPERBOLIC FUNCTIONS. l
Prob. 47. Derive from (42) the series for tanh
x
given in (34).
Prob. 48. Prove the identities: 2
= x=
log sec
.r
log tan
i
!#
loc x=^2 tann
X+I
=tann -i*~^ i_-sinn X +1 7
1
tanh" tan ^x\ log esc x
2
1
tanh" cos 2#
ART.
20.
in Arts.
=
2
J
tanh"
1
sinh" cot 2x
== cosn
"y(x~\~x~ V
" )i
!
tan'^nr
=
-[-
i#);
cosh" 1 esc 2X.
THE GUDERMANIAN FUNCTION.
The coirespondence cussed
=
x
~v\x
of sectors of the It is
1-4.
correspondence that
may
same
now convenient
exist
species was disto treat of the
between sectors of
different
species.
Pv PI on any hyperbola and ellipse, are said to with reference to two pairs of conjugates O A correspond Two
points
,
l
0,#,
,
l
,
and O^AI, O^B^j respectively, when *i/<*i
=
<**/*
l
Afl^Pi are then also said to
(44)
The sectors A^O^P.^ correspond. Thus corresponding
and when y ,y^ have the same
sign.
sectors of central conies of different species are of the
same
sign and have their primary characteristic ratios reciprocal. Hence there is a fixed functional relation between their re-
The elliptic sectorial measure is called spective measures. the gudermanian of the corresponding hyperbolic sectorial measure, and the latter the anti-gudermanian of the former. This relation is expressed by
SJK, or
ART.
The
v
= gd
=
gd s,/*;
and
;/,
u
= gd~
!
z/.
(45)
CIRCULAR FUNCTIONS OF GUDERMANIAN.
21.
six hyperbolic functions of u are expressible in
of the six circular functions of
x
= cosh
x //,
its
gudermanian
= cos v,
;
terms
for since
(see Arts. 6, 7)
which u, v are the measures of corresponding hyperbolic and elliptic sectors,
in
GUDERMANIAN ANGLE.
29
= sec v, [eq. (44)] = tan v, sinh u = t/secV tanh # = tan #/sec v = sin #, coth w = esc = cos?;, sech csch u = cot cosh u
hence
I
(46)
z>,
//
z>.
The gudermanian
sometimes useful in computation for be given, v can be found from a table of natural tangents, and the other circular functions of v will give instance,
if
sinh
is
;
//
Other uses
the remaining hyperbolic functions of u. function are given in Arts. 2226, 32-36. Prob. 49. Prove that gd u
=
!
sec~ (cosh u)
=
of this
1
tan~ (sinh u)
= cos'^sech u) =sin~ (tanh w), gd" ^ = cosh" '(sec v) = sinh" (tan v) = sech"" (cos v) = tanh" ^sin v). = = o, gd oo ^TT gd o gd( oo) = -oo. gd- 0=0, gd-^flr) =00, gd- (-i7r) 1
1
Prob. 50. Prove
1
1
Prob. 51. Prove
TT,
?
1
Prob
Show
52.
Prob. 53. tion tanh
\u
that gd
From
= tan
the
1
//
first
and gd" 1 v are odd functions of identity in 4, Prob.
u, v.
17, derive the rela-
\v.
Prob. 54. Prove J
tanh~ (tan u)
=
ART. If
! % gd 2U, and tan~ (tanh x)
22.
% gd
-1
2JC.
GUDERMANIAN ANGLE
a circle be used instead of the ellipse of Art. 20, the
gudermanian of the hyperbolic sectorial measure will be equal to the radian measure of the angle of the corresponding circulai sector (see eq. (6), and Art. called the gudermanian angle
3, ;
Prob. 2). This angle will be but the gudermanian function v,
merely a number, or ratio and this number is equal to the radian measure of the gudermanian angle 0, which is itself usually tabulated in degree measure thus as
above defined,
is
;
;
/7r
(47)
HYPERBOLIC FUNCTIONS.
'10
Show that
Prob. 55.
the gudermanian angle of u
may be
construct*
ed as follows:
Take the
OA
principal radius
/
initial
of an equilateral hyperbola, as the as the terminal line, and
OP
line, of the
from
M,
sector
whose measure
is
u\
the foot of the ordinate of
MT
P, draw tangent to the circle whose diameter is the transverse axis; then
AOTis
the angle required.* Show that the angle
Prob. 56.
never exceeds 90.
Art.
and Prob.
9,
The
Prob. 57.
M
A
bisector of angle
bisects the sector
and the
53, Art. 21),
line
AOT
A OP
(see Prob. 13, (See Prob. i, Art. 3.)
AP.
Prob. 58. This bisector is parallel to TP, and the points 2\P are in line with the point diametrically opposite to A. Prob. 59. The tangent at P passes through the foot of the
oidinate of
7 and 1
,
Prob. 60.
ART.
23.
The
TM on the tangent at A. APM half the gudermanian angle.
intersects
angle
is
DERIVATIVES OF GUDERMANIAN AND INVERSE.
= sec v = v tan vdv = sec vdv = d(gd~ v} = dv = =
Let then sec
gd
cosh
Again, therefore
=
1
gd-
v,
,
sinh udu, du,
l
therefore
u
u,
sec vdv.
cos
vdu
sech
;/
(48)
=
sech u du,
du.
(49)
Prob. 61. Differentiate:
y
=
sinh u
y
=
tanh u sech u
gd
u,
+
gd
u,
y y
= =
sin v
+ gd"
tan v sec v
1
v,
-f-
gd"
1
#.
and denoted by /. *This angle was called by Gudermann the longitude of His inverse symbol was U.; thus u = H.(/tt). (Crelle's Journal, vol. 6, 1830.) ,
Lambert, who introduced the angle 6, named it the transcendent angle. (Hist, I'acad. roy de Berlin, 1761). Hottel (Nouvelles Annales, vol. 3, 1864)
de
called it the hyperbolic amplitude of u, and wrote it amh u, in analogy with the Cayley (Elliptic amplitude of an elliptic function, as shown in Prob. 62. Functions, 1876) made the usage uniform by attaching to the angle the name
of the mathematician
theory of
elliptic
who had used
it
extensively in
functions of modulus unity.
tabulation and in the
GUbERMANIAN AND
SERIES FOR
ITS INVERSE.
Prob. 62. Writing the "elliptic integral of the
1
kind"
first
in
the form
K
*#_
f+
U
Vi-
J
being called the modulus, and
= show
am
u,
that, in the special case
u cos
am
u
= =
gd" 0, sech
and that thus the
tan
w,
ART.
=
when K u
=
is,
i,
gd
u
sin
//,
am
//
=
tanh
if,
sinh #;
functions sin
elliptic
the hyperbolic functions,
the amplitude; that
(mod. K),
am am
1
K* sin* 0'
am
degenerate into
etc.,
//,
when the modulus
is
unity.*
SERIES FOR GUDERMANIAN AND ITS INVERSE.
24.
Substitute for sech
//,
sec v in (49), (48) their expansions,
Art. 16, and integrate, then u gd u
X + - rfh" + W + frS +1^ +
= gd-V = v +
1
T&U'
(50) .
.
(51)
.
No
constants of integration appear, since gd u vanishes with and gd" ^ with v. These series are seldom used in compu1
,
tation, as
gd u
is
and hyperbolic gd u
and
means
best found and tabulated by
of natural tangents
=
tan-^sinh
of tables
from the equation
sines, //),
a table of the direct function can be used to furnish the
numerical values of the inverse function
;
or the latter can be
obtained from the equation, gd~'^
To
=
sinh "'(tan v)
=
f
cosh~ (sec
obtain a logarithmic expression for gd"" ' l
gd' v =11, * The relation gd u
=am
u,
(mod.
i),
z>).
1
?',
let
v = gd u, led Hottel to
name
the function
gd
u,
In this the hyperbolic amplitude of u, and to write itamh u (see note, Art. 22). connection Cayley expressed the functions tanh u, sech u. sinh u in the form
gd u, cos gd u, tan gd u, and wrote them sg u, eg //, tg u, to correspond with the abbreviations sn u, en u, dn u for sin am u, cos am u, tan am u.
sin
=
=
sn u, (mod. i); etc. sg u note that neither the elliptic nor the hyperbol'c functions received their names on account of the relation existing between them in a
Thus tanh u It is
well to
special case.
(See foot-note,
p.
7
)
HYPERBOLIC FUNCTIONS.
32
= cosh u, = tan v cosh u
sec v
}-
f
-\-
sin
v
,
= gd
l
v,
=
^3
i
.
U
sinh u 9
=
.
,
tan
r
-f '
,
,
x
* 'f Jf/)
z;)
u ~]
Prob. 64. Prove that gd u
when w
v)'
+ +
log. tan (iff
8d u
i * 17 * Prob. 63. Evaluate
T3
sinh
v
/t
sin (Jff
z;
=
=. eu ,
cos
I .
cos
order,
tan v
sec v
therefore
-
%&~ lv -
.
J=o sin
w
,
is
(52)
*/).
~
~i
V
v ~\ .
.
Jz/=o
an infinitesimal of the
fifth
== o.
Prob. 65. Prove the relations iff
+ %v=
ART.
25.
u tan~ e l
9
iff
^v
=
tan"V~*.
GRAPHS OF HYPERBOLIC FUNCTIONS.
Drawing two rectangular axes, and laying down a series of points whose abscissas represent, on any convenient scale, successive values of the sectorial measure, and whose ordinates represent, preferably on the same scale, the corre-
sponding
values
of
the
function to be plotted, the
out by this series of points will be a locus traced
graphical representation of the variation of the function as the sectorial meas-
A
GRAPHS OF THE HYPERBOLIC FUNCTIONS.
The equations
ure varies.
sian notation are
y= y= y= y=
B C
D
= sech x y = csch x y = coth x
cosh x,
y
sinh x,
tanh x,
gd
numerical value of cosh
//,
;
;
;
.r.
written for the scctorial measure
.r is
etc.
and y for the thus to be noted that the
It is
//,
y are numbers, or ratios, and that the equation merely expresses that the relation between the
variables x,
x numbers x and y == cosh
tween a
Dotted Lines.
Full Lines.
A
y
of the curves in the ordinary carte-
:
Fig.
Here
33
taken to be the same as the relation be-
is
measure and
sectorial
The
characteristic ratio.
its
tanh u are given in the //, //, tables at the end of this chapter for values of u between o and For greater values they may be computed from the devel4. numerical values of cosh
opments
The
sinh
of Art. 16.
curves exhibit graphically the relations
sech u
=
:
=-
u
csch"
,
cosh u
<
cosh u
= tanh u) = cosh o = = oo oo cosh sinh
(
)
(
I,
)
(
,
,
=
coth u
>
tanh
I,
cosh
sinh u,
tanh
sinh
gd
u,
= 0,
sinh
oo)
(
(
;/
)
//)
tanh o
=
>
(
00,
-
tanh u
sinh u
sech u
I,
-
:
I,
gd u
= cosh = gd o,
tanh
;
//,
//,
csch (o) (
;
etc.;
=00
= oo)
,
etc.;
i.etc.
=
sinh x is given by the equation slope of the curve y cosh x, showing that it is always positive, and that dy/dx the curve becomes more nearly vertical as x becomes infinite.
The
=
Its direction
of curvature
proving that the curve
is
= sinh x,
obtained from d*y/dx* concave downward when x is
is
nega-
The point of inflexion is tive, and upward when x is positive. inflexional and the at the origin, tangent bisects the angle between the axes.
34
HYPERBOLIC FUNCTIONS.
The
direction of curvature of the locus =. sech x(2 tanh
by d?y/dx*
7
x
y
= sech x
is
and thus the curve
i),
cave downwards or
given con-
is
upwards
a according as 2 tanh
;r
is
i
negative or positive. The inflexions occur at the points
= tanh- 707, = .881, = y .707 and the slopes of x
1
.
;
,
the
inflexional
The curve y \
rapidly than
\ -
x
cross at the points
csch
x
is
approaches the
x
.T.
=
The
x
=
3,7
is
=
10 only I, but it is not till/ csch x,y=. sinh x curves y .
so small as
is
it
axis of y, for \\hen
c that
=
asymptotic to both axes, but approaches the axis of x more
-i--
\
are
tangents
.881,^
=
=
I.
Prob. 66. Find the direction of curvature, the inflexional tantanh x. gd^,.r gent, and the asymptotes of the curves y Prob. 67. Show that there is no inflexion-point on the curves
=
y
=
cosh Prob.
jc,
y
68.
=
coth
Show
y = tanh #
=
jr.
that
any
line
^
= mx + n
meets the curve
Hence prove that in either three real points or one. mx -f n has either three real roots or one. the equation tanh x From the figure give an approximate solution of the equation tanh x
=x
=
i.
ELEMENTARY INTEGRALS. Prob. 69. Solve the equations:
gd x
=x
=x+
x
cosh
35
x
2\ sinh
=
Jo:;
\n.
Prob. 70. Show which of the graphs represent even functions, and which of them represent odd ones.
ART.
The 14,
ELEMENTARY INTEGRALS.
26.
following useful indefinite integrals follow from Arts.
15,23: Hyperbolic.
Circular.
u du
=
cosh
u,
fsin u du
=
2.
/"cosh u du
=
sinh u,
fcos u du
=
3.
/tanh u du
=
log cosh
/ tan u du
=
4.
/coth w
=
log sinh
5.
/csch udu
1.
Ainh
f///
= 6.
j
sech
dx
ax .
9
/dx -y-
,
=
gd
=
S ' nh
=
///*l/^
,
= log tanh sinh-^csch
rfw
//,
.
^///
sec
//
log cos u,
=
log tan -,
==
sty li*
V/ Va*-=i x
= -tanh- .x- >tyf
,
1
dx
-
a
//,
log sin
^
t/
,
du
,
=
dx
^
x**
sin
=
f
" i*
\
*/ *///
/esc K
y
cosh~ a
cot
;/),
,
,
,
y
cos
//,
cosh -'(esc
),
!
gd~
//,
=
SU1
=
cos"
"
x
x
*.
a
,
i
^r-tan- ,^ 1
a
* Forms 7-12 are preferable to the respective logarithmic expressions on account of the close analogy with the circular forms, and also
(Art. 19).
because they involve functions that are directly tabulated. appears more
clearly in 13-20.
This advantage
36
HYPERBOLIC FUNCTIONS.
-
r -d*
J ^"1 ^
"i
f
r
x
i
= -COth-
, 1
/ a V
-,
#
^J^> a
dx 7-j ?
+x
=
a 1
dx
/dx+
=====
x Va*
From derived
13
-
*'
;r
I
= #- csch-
1
--,
-=
a V^r - a
a
may be
_.
l
Vac-b*
=
I
= "7=I
Va
dx
cos
<J
Vtfac ~
rt^r
.
4" ^
I
;
^ "egative.
T7^="' vb ac
^^r 4- b
"
ZT
=
+ _, ^ positive, ac
rt:^-
.
,
cosh" Vrt
"
f
tanh
~
V
Thus,
4
ac
<
"+
/A
~ ac
b
2~coth=-coth~ (^~2)=cothv 1
l
.
,
^-4^+3
;
1
3
^J 4
= tanh~ (.5)tanh- (.3333)=. 5494 1
1
^r / - ------r-
^-^ t
.3466^.2028.^
2 6
25
/
I
-
these fundamental integrals the following
r J ~,
t/8
x
.
:
17
/
i
dkr
/*
^/ ^
x
i
-COt-'-, a
a
I ^tanh-'o -
=
tanh-'(jr v
2)
'J
=-
^
tanh'Y.s) v -5494-
(By interpreting these two integrals as areas, show graphically that the first is positive, and the second negative.)
--/dx
^__
(a-x)Vx-b
*For tanh- 1 (.5)
2 _. _ ____
Va-b
,
-
Ixb
,
tanh~ A /
\
interpolate between tanh (.54)
T, a-6'
=
.4930, tanh (.56)
(see tables, pp. 64, 65); and similarly for tanh- 1 (.3333).
=
.5080
ELEMENTARY INTEGRALS.
= 2
or
Vb
tan
~
Xb
.
dx
or
,
V
a
the real form to be taken.
x
(Put
2
.
Vab
^/ba
bx -
1
A
=
b
,
and apply
z*,
9, 10.)
Ibx
I
.
coth-
,
com
..
.
2
------
or
2
\ / -7 b ~~ a
/
-7
V
2
b-a' ,
__ tan
or \
f
~
a -b
-
* 1] a-b*
/*
,
\ / \J
the real form to be taken.
-^ 2
By means
-
a
^
^)'
i
- -"'
1
coslr -.
a
2
of a reduction-formula this integral
is
made
easily
to depend on
It may also be obtained by transforming 8. the expression into hyperbolic functions by the assumption x a cosh u, when the integral takes the form
=
/c? sinh
1
ndu=.
I
[*
j (cosh 2u
\)du
= ~/i (sinh a
2u cosh
which gives 17 on replacing a cosh
;/
by
^r,
2//) //
and a sinh
//, //
by
The geometrical
3
interpretation of the result is evident, as it expresses that the area of a rectangular-hyperbolic segment AMP\s the difference between a triangle
(x*
rt
)*.
OMP
and a sector OAP. 18.
19.
(j^
20.
ysec
- x^i* = -*(a* - *)* + -V
+
f
fl
)*rfr
3
= -x(x* + 2
)*
+
sin-
sinh-
= J\\ + tan 0)*rf tan = tan 0(i + tan* 0)* + = 4 sec tan + i gd" a
0^/0
1
21.
/ sech
1
f
flf= i sech
Prob. 71.
What
Prob. 72.
Show
is
tanh w
-f-
^ gd
1
sinh- (tan 0) 0//.
the geometrical interpretation of 18, 19?
that / (0,*
1
+ 2 ^ + ^)^ reduces
to 17, 18, 19,
H\PRBOLIC FUNCTIONS.
38
when a
respectively:
and when a
is
positive, with ac
Prob. 73. Prove
< &*
positive, with ac
is
>
;
when a
is
negative;
If.
/ sinh u tanh u du
=
sinh u
cosh u coth u du
=
cosh u
gd
u,
+ log tanh
.
2
Prob. 74. Integrate
/ = $px,
if s be the length of arc the angle which the tangent line makes with the vertical tangent, prove that the intrinsic equation of 3 ! tan the curve is ds/d<(> +/gd~ 0. a p sec 2p sec 0, s a Prob. 76. The polar equation of a parabola being r sec 0,
Prob. 75. In the parabola
measured from the vertex, and
=
=
=
focus as pole, express s in terms of 0. Prob. 77. Find the intrinsic equation of the curve y/a and of the curve y/a log sec #/0.
referred to
its
= cosh x/a
=
t
Prob. 78. Investigate a formula of reduction for / cosh* #
also integrate
1
tanh" 1 a: ^r, (sinh" jr)Vjr; and
show that the ordinary methods
of reduction for
m n / cp$ xs\n xdx
can be applied to / cosh* x sinh n x dx.
ART.
FUNCTIONS OF COMPLEX NUMBERS.
27.
As vector quantities are of frequent occurence in Mathematical Physics; and as the numerical measure of a vector in terms of a standard vector is a complex number of the iOrm x-\-iy>
in
which
becomes necessary
x,
j
are real, and
in treating of
ations to consider the
meaning
any
i
stands for
of these operations
formed on such generalized numbers.*
The
*The
it
is
r,
when
geometrical
tions of cosh//, sinh u, given in Art. 7, being then applicable,
V
it
class of functional oper-
per-
defini-
no longer
necessary to assign to each of the symbols
u e of vectors in electrical theory
VP
shown
in
Bedell and Crehore's
The advantage published in 1892). of introducing the complex measures of such vectors into the differential equaAlternating Currents, Chaps, xiv-xx
(first
tions is shown by Stein metz, Proc. Elec. Congress, 1893; while the additional convenience of expressing the solution in hyperbolic functions of these complex
numbers
is
exemplified
Engineers, April 1895*
by Kennelly, Proc. (See below, Art. 37.)
American
Institute
Electrical
FUNCTIONS OF COMPLEX NUMBERS.
39
cosh (x -f- iy\ sinh (x -\- iy), a suitable algebraic meaning, which should be consistent with the known algebraic values of cosher, sinh x> and include these values as a particular case
when y
= o.
The meanings
assigned should also, if possible, be such as to permit the addition-formulas of Art. 1 1 to be made general, with all the consequences that flow from them.
Such ments in
definitions are furnished
by the algebraic developwhich are convergent for all values of //, real
Art. 16,
Thus the
or complex. are to be
COSh (*
+ /=
sinh (*
+
I
=
i
definitions of cosh (x
-f- iy),
sinh (x
-f- iy)
+jj(
(x
+
+ -,(* + z' +
iy)
.
.
.
*
From
these series the numerical
values of cosh (x
-\-
iy\
sinh (x-\-iy) could be computed to any degree of approximaIn general the results will come tion, when x and y are given. out in the complex form*
The
cosh (x
+ iy) = a + i&
sinh (x
-f- iy)
=^
other functions are defined as
9
-f- *V/.
Art.
in
7,
eq. (9).
Prob. 79. Prove from these definitions that, whatever u
cosh
d
is
=
cosh u
T-jCOsh *It
)
(
=
cosh
to be borne in
y,
d sinh u,
=w
/
sinh
(
sinh u
j* 1
mind
braic operators which convert
cosh mu,
j-
9
i/)
.
sinh
=
may
be,
sinh u,
= cosh
#,
w = nf sinh
w//.f
symbols cosh, sinh, here stand for algeone number into another; or which, in the Ian*
that the
guage of vector-analysis, change one vector into another, by stretching and turning. f
The generalized hyperbolic functions
matical Physics as
where 0, Art. 37.)
///,
usually present themselves in Mathe-
the solution of the differential equation dP0/
u are complex numbers, the measures of vector
=
quantities.
m*fa (See
40
HYPERBOLIC FUNCTIONS.
ART.
ADDITION-THEOREMS FOR COMPLEXES.
28.
The addition-theorems
+
for cosh (u
etc.,
f')
be derived as follows.
complex numbers, may as real numbers, then, by Art. n, cosh (u
hence
i
-f-
+
.=
v)
w
cosh u cosh
)+...=(i
z/-|~
where
#,
v are u v
First take
9
sinh u sinh v\
+ f + ...)(i + --,*'+.
.
.)
This equation is true when //, v are any real numbers. It must, then, be an algebraic identity. For, compare the terms of the rih degree in the letters //, v on each side. Those on the
left
r
arc f
(w-f-
?>)
;
and those on the
when
right,
collected,
form an rth-degree function which is numerically equal to the former for more than r values of u when v is constant, and for
more than
r values of v
of the rth degree
tions of u
and
z>.*
when
Thus the equation above is
true for
values of
all
;/
//,
written v,
=
cosh
and by changing v into
?;,
cosh
(;/
-f- ?')
?')
--
In a similar manner sinh (u
Hence the terms
constant.
v)
=
cosh is
an algebraic identity, and
is
whether
writing for each side its symbol,
cosh (u
is
on each side are algebraically identical funcSimilarly for the terms of any other degree.
it
real or
complex.
Then
follows that
;/
cosh v
;/
cosh v
-j-
sinh u sinh v\
(53)
sinh u sinh
(54)
v.
found
sinh
?/
cosh u sinh
cosh v
v.
(55)
In particular, for a complex argument, '
cosh (x sinh (x
*"
If
= cosh x cosh iy = sinh x cosh iy iy)
iy)
sinh
x
sinh
y/>
)
cosh
x
sinh
iy.
)
two rth-degree functions of a single variable be equal
values of the variable, then they are equal for algebraically identical."
all
f
for
(5 6)
more than r
values of the variable, and are
FUNCTIONS OF PURE iMAGlNARlES.
41
Prob. 79. Show, by a similar process of generalization,* that if exp u f be defined by their developments in powers of then, whatever u may be,
sin #, cos Uj u,
sin
cos
+
(//
exp
+ + v) z;
(
(//
= sin u cos v + cos u sin = cos u cos ~~ sm # sin = exp exp
v)
?;
)
v, z/,
z f.
i/
Prob. 80. Prove that the following are identities:
cosh
2
8
sinh u
14
cosh
//
+
sinh w
cosh u cosh sinh
ART.
29.
//
//
sinh u
= =
|
= = exp = exp i,
//,
u),
(
+ exp
[exp
J[exp u
(
//)],
exp(
)].
FUNCTIONS OF PURE IMAGINARIES.
In the defining identities
cosh u
sinh
//
=
f
i
H
f//
2.
= + //
r//
cosh i> =: sinh
i>=i>
by
division,
-
-f
//
-}-
j
=
i
.
.
.,
then
+ -I(i' -f J
tanh iy
.,
j
ty,
'
and,
.
6
-;/"
/ -f ~y -
i
.
4.
3*
put for u the pure imaginary
+
4
H
6
(/
.
.
+
.
.
= .
cos 7,
(57)
=*' sin 7,
(58)
.
'
tan y.
(59)
* This method of generalization is sometimes called the principle of the " permanence of equivalence of forms." It is not, however, strictly speaking, a
"
principle," but a method; for, the validity of the generalization
has to be
demonstrated, for any particular form, by means of the principle of the algebraic identity of polynomials enunciated in the preceding foot-note. (See
Annals of Mathematics, Vol. 6, p. 81.) f The *ymbol exp u stands for "exponential function of u," which cal with f when u is real.
is
identi-
HYPERBOLIC FUNCTIONS.
These formulas serve to interchange hyperbolic and
The
functions.
circular
hyperbolic cosine of a pure imaginary
is real,
and the hyperbolic sine and tangent are pure imaginaries.
The tanh
//,
following table exhibits the variation of sinh u, cosh u, exp ;/, as u takes a succession of pure imaginary values.
* In this table .7
is
Prob. 81. Prove the following identities cos y
=
cosh
sin.}'
=
- sinh
cos ^ cos
+
i
/
j>
cos
iy
iy
=
written for | 4/2,
=
i[exp
=
-.[exp
iy
+ exp
/>
.707 ....
:
(
exp
/))],
/],
(
= cosh iy + sinh ry = exp sin y = cosh /y sinh ry = exp sin i> = sinh ^. /y = cosh y, sin
j^
iy, (
/y),
i
Equating the respective real and imaginary paits on n each' side of the equation cos ny * sin ny i sin y) , (cos y express cos ny in powers of cos 7, sin y ; and hence derive the corProb. 82
=
+
responding expression for cosh Prob. 83.
Show
+
ny.
that, in the identities (57)
and
replaced by a general complex, and hence that sinh (x
iy)
=
/ sin
(y
^
ix) 9
(58),
y may be
FUNCTIONS OF X cosh (x
x
(or
/
cos
(.#
/>)
X
THE FORM
= cos (y ^ />), = sinh ^ = cosh (7 ^ ix). /
(>'
the product-series
for sin
-\- t
43
Y.
wr),
x
derive that
for
:
sn * sinh
ART.
By
/
sin
From
Prob. 84. sinh
-f iy IN
*
= *i -
= *i + ,i + --.i +
FUNCTIONS OF x
30.
+ iy
IN
.
.
..
THE FORM Jf+i'K
the addition-formulas,
= cosh x cosh iy sinh x sinh = sinh cosh y/ cosh x sinh y% sinh (x iy) sinh /j = sin y, cosh iy = cos y, but = cosh cos y -\-i sinh sin y, hence cosh y/) = sinh x cos sin ^. i cosh sinh y/) = # + #, sinh (x -\-iy) c-\- id, Thus cosh (x iy) sin y, # = cosh cos y, # = sinh [ sin j. cos y, d = cosh c = sinh cosh (x
-f-
iy)
-f-
or
-\-
ijr,
-f-
*
(;tr
.r
.ar
-{-
if
-f-
^r
j;
or
From
.#
j/ -|-
(.r -f-
then
)
^r
(60
)
these expressions the complex tables at the end of
this chapter
have been computed.
= X-\- iY\ let the iy, Z Writing cosh z^=-Z, where s = x Z be on represented Argand diagrams, in complex numbers z, the usual way, by the points whose coordinates are (x, y\ (Jf, F); and let the point z move parallel to the ^-axis, on a
+
x
=
Z
will describe an ellipse ;, then the point whose equation, obtained by eliminating y between the equasinh m sin y, is tions cosh m cos^,
given line
X=
Y=
(cosh
mf
"^
and which, as the parameter con focal
ellipses,
__ / Mu (sinh ;
*M \
=I
'
*# varies, represents a series of the distance between whose foci is unity.
HYPERBOLIC FUNCTIONS.
44
the point z move parallel to the .r-axis, on a given will describe an hyperbola whose equa;/, the point tion, obtained by eliminating the variable x from the equations X=- cosh x cos ;/, Y sinh x sin ;/, is Similarly,
if
line/ =
Z
=
X*
_
F
a
_ a
(cos
;/)'
and which, as the parameter
(sin w) ;/.
varies, represents a series uf
hyperbolas con focal with the former series of
ellipses.
These two systems of curves, when accurately drawn at close intervals on the Z plane, constitute a chart of the hypciand the numerical value of cosh (in -f- in) can br bolic cosine ;
m
read off at the intersection of the ellipse whose parameter is with the hyperbola whose parameter is w.* A similar chart can
be drawn for sinh (x+iy), as indicated in Prob. 85. Periodicity of Hyperbolic Functions.
The
and cosh u have the pure imaginary period sinh (u + 2in) =sinh u cos m+i cosh u cosh (u + 2ix) =cosh u cos 2n + i sinh u
The
functions sinh
argument u
is
The
sin sin
u
For
= sinh 27r = cosh 27:
u, u.
u and cosh u each change sign when the
increased by the half period in.
sinh (w-HVr) =sinh
cosh (w
functions sinh
2ix.
u cos n + i cosh u cos n + i sinh u
= cosh u -H'TT)
For
sin TT=
sinh w,
sin TT=
cosh u.
u has the period in. For, it follows from by dividing member by member, that tanh (u+in) =tanh u.
function tanh
the last two identities,
By
a similar use of the addition formulas sinh (u + ^in)
=i
cosh u,
cosh (u + J/TT)
it
=i
is
shown
that
sinh u.
By means
of these periodic, half-periodic, and quarter-periodic the relations, hyperbolic functions of x+iy are easily expressible in /terms of functions of x + iy', in which y' is less than JTT. * Such a chart is given by Kennelly, Proc. A. I. E. E., April 1895, and is used by him to obtain the numerical values of cosh (x iy) sinh (x ty) which present themselves as the measures of certain vector quantities in the theory of
+
alternating currents.
and of y between o and
+
t
(See Art. 37.) The chart is constructed for values of x 1.2; but it is available for all values of y on account of
the periodicity of the functions.
t
FUNCTIONS OF X + iy IN THE FORM
The
X+iY.
45
hyperbolic functions are classed in the modern fanction-
theory of a complex variable as functions that are singly periodic with a pure imaginary period, just as the circular functions are singly periodic with a real period,
and the
elliptic
functions are
doubly periodic with both a real and a pure imaginary period. Multiple Values of Inverse Hyperbolic Functions. It follows from the periodicity of the direct functions that the inverse
m
functions sinh" 1
and cosh" 1
m
have each an
ular value of sinh"" 1
than \n nor
less
m
than
which has the
number
indefinite
That
of values arranged in a series at intervals of zin.
partic-
coefficient of i not greater
called the principal value of sinh" 1
m\ and that particular value of cosh"" 1 m which has the coefficient of i not greater than n nor less than zero is called the principal value
When
of cosh" 1 ^.
\K
is
necessary to distinguish between the general value and the principal value the symbol of the former thus will be capitalized it
is
;
Sinh""
1
m = sinh" m + zirn, Cosh" m = cosh"" m -f 2irx, Tanh" m = tanh" m +irx 1
1
1
1
1
y
in
which
r is
integer, positive or negative.
any
Complex Roots
of Cubic Equations.
It is
well
known
that
when
the roots of a cubic equation are all real they are expressible in terms of circular functions. Analogous hyperbolic expressions
are easily found
when two
of the roots are complex.
Let the
cubic, with second term removed, be written
Consider
x=r
first
the case in which b has the positive sign.
sinh u, substitute, and divide
sinh
3
by
r
3 ,
u + ^ sinh u = - ,.
3 Comparison with the formula sinh u + l sinh
3* gives
whence therefore
7* r r=2&*,
Let
then
=3 7' 4
P r
=
sinh3=-Tj,
x= 20* sinh
i I
r~' 4 M=-sinh~ 1 T}j O
I 1
c\
sinn" TI
)
>
=J
sinh yi
HYPERBOLIC FUNCTIONS.
46 in
which the sign of
Now
b*
be taken the same as the sign of
to
is
the principal value of sinhr 1
let
^,
found from the
n2in
then two of the imaginary values are
be n\
x
of
three values
- and
are 26* sinh
2 ft* sinIj(-
\3
3
two reduce
last
Mfsinh
to
i\/3 cosh
tables,
hence the
t
).
The
3/
j.
O I
U
\
c*
let the coefficient of x be negative and equal to 36. then be shown similarly that the substitution x=r sin
Next,
may
It
leads to the three solutions
sin,
26*
6* (sin
3
These roots are
#=rcoshw
3
real
when c%b*.
all
where
\/3 cos-),
\
w= sin"
1
TT. 0*
37 If
c>b*,
the substitution
leads to the solution
#= 26* cosh
(-cosh"~ l
JT),
which gives the three roots 26* cosh -, 3
in
6*
(
cosh
which the sign of Prob. 85.
Show
-
wherein n = cosh" 1 TT , ) & 3/ to be taken the same as the sign of
/V T sin h "~ *
3
\
b* is
i
that the chart of cosh (x
-f- iy)
c.
can be adapted
+ /?), by turning through a right angle; also to sin (x +iy). sinh 2r 4- /sin 2V n ^ -. Prob. 86. Prove the identity tanh (x + ty) = cosh 2x cos 2y = + *A be written in the " modulus Prob. 87. If cosh (x + and amplitude" form as r(cos + /sin 0), = exp /^, then
to sinh (x
,
.
.
.
.
.
/
.
.
v
-f-
tv),
/
r*
=a
tan ^
= cosh = ^/a = tanh 7
jc
x
= cos ^ 9
2
-f b*
sin'j^
tan
sinh* x,
j>.
Prob. 88. Find the modulus and amplitude of sinh (x Prob. 89.
Prob. 90. .
*
sin"
01
=
*
Show
that the period of exp
When .
/
.
cosh
real
is
;;/
_,J
and
>
i,
01 is real
and
cos" 1
m.
<
i,
cosh"
1
m=
i
iy).
is ia.
2
When
+
cos"
1
01.
m =
i cosh""
1
m
%
tHE CAfENAfcY.
ART.
47
THE CATENARY.
31.
A flexible
inextensible string is suspended from two fixed and takes up a position of equilibrium under the points, action of gravity.
curve
in
w be
;
of the portion
T
required to find the equation of the
It is
it
hangs. the weight of unit length, and s the length of arc measured from the lowest point A then ws is the weight
Let
AP
which
AP.
This
is
balanced by the terminal tensions,
H
acting in the tangent line at P, and
Resolving horizontally and vertically gives
tangent.
=
T cos which
in
is
//,
c
is
=
T'sin
ws,
the inclination of the tangent at P\ hence
tan0 wheie
the horizontal
in
written for
=
H/w,
constant horizontal tension
;
_= ws
s _,
the length whose weight therefore
is
the
ds
x c
which axis of
is
.,,*.,* = = smh-, smh c c
s
1
= dy y = ax -7
c
,
,*
cosh -, c
*
the required equation of the catenary, referred to an at a distance c below A.
x drawn
The following trigonometric method illustrates the use of the gudermanian The " intrinsic equation," s c tan 0, c ds c ds cos hence sec sec* d
=
:
=
;
dy ~dss\\\ 0,=rsec whence y/c = sec %
=
tan 0*/0; thus
=
x=c
gd~* 0,
= sec gd x/c = cosh */* = tan gd ;r/r = sinh ^/^. sfc A
;
^=
and
chain whose length is 30 feet is suspended from two points 20 feet apart in the same horizontal; find the parameter c, and the depth of the lowest
Numerical Exercise.
nninK
HYPERBOLIC FUNCTIONS.
48
The equation by putting lo/c
s/c
= z,
=
ining the intersection of it appears that the root
To
find a closer
in the
= sinh lo/V, which, = sinh z. By examthe graphs of y = sinh#, y = of this equation z = 1.6, nearly.
sinh x/c gives i$/c may be written 1.5^
1.5,3:,
is
approximation to the sinh z
lorm/(s) /(l.6o)
/(i.62) ^(1.64)
= = =
1.5$
=
root, write the
by the
o, then,
2.3756
2.4000
2.4276
2.4300
2.4806
2.4600
= = =
equation
tables,
.0244, .0024,
-f .0206;
whence, by interpolation, it is found that /( 1.6221) = o, and The ordinate of either of # = 1.6221, c 10/2 = 6.1649.
=
the fixed points
=
is
given
cosh x/c
~
from tables; hence
y
y/c
vertex
=y
c
=
by the equation
cosh lO/c
=
=. 16.2174,
cosh 1.6221
=
2.6306,
and required depth of the
10.0525 feet.*
Prob. 91. In the above numerical problem, find the inclination of the terminal tangent to the horizon.
MN
be drawn from the foot of the Prob. 92. If a perpendicular 1 is equal to the conordinate to the tangent at 7 prove that Hence show that is equal to the arc AJ\ stant c, and that
MN
,
NP
-W is the involute of the catenary, and has the propthat the erty length of the tangent, from the point of contact to the axis of jc, is constant. (This is the characteristic property of the the locus of
tractory).
Prob. 93.
The
tension
T at
is equal to the weight of a equal to the ordinate y of that
any point
portion of the string whose length
is
point.
An arch in the form of an inverted catenary f is 30 wide and 10 feet high; show that the length of the arch can be
Prob. 94 feet
2
obtained from the equations cosh z
z
3
=
i,
25
=
3O sinh
.
z
* See a similar problem in Chap. I, Art. 7. " For the iheory of this form of arch, see " Arch in the Encyclopaedia
f
Britannica.
.
CATENARY OF UNIFORM STRENGTH.
ART.
49
CATENARY OF UNIFORM STRENGTH.
32.
the area of the normal section at any point be made proportional to the tension at that point, there will then be a If
constant tension per unit of area, and the tendency to break be the same at all points. To find the equation of the
will
curve of equilibrium under gravity, consider the equilibrium of an element PP whose length isrfi, and whose weight is gpoads, 1
where weight
the section at P, and p the uniform density. This balanced by the difference of the vertical components
GO is is
of the tensions at
Pand P' d( T sin
,
d(
therefore
=
T cos
T= //sec 0.
0)
T cos 0)
ff,
Again,
hence
if
then by hypothesis cy/c0
= gpoods, =o ;
the tension at the lowest point, and G? O be the section at the lowest point,
=
T/ff=
sec 0, and the
first
equation
becomes
or
where
//
sin 0)
c d tan
=
= gp&)
sec
whose weight
ca )
lowest point
;
ds,
c stands for the constant
(of section
sec
is
ff/gpoo^ the length of string equal to the tension at the
hence,
ds
=
c sec 0^/0,
s/c
=
gd-'0>
the intrinsic equation of the catenary of uniform strength.
Also
hence
dx
x
= ds =
=
cos
cfa
y=
c */0,
in
=
= ds sin
= c tan
d
;
c log sec 0,
and thus the Cartesian equation y/c
dy
is
log sec x/c,
which the axis of x
is
the tangent at the lowest point.
Prob. 95. Using the same data as in Art. 3i find the parameter and the depth of the lowest point. (The equation x/c = gd s/c which, by putting i$/e = z, becomes IQ/C = gd i$/t gives c
9
HYPERBOLIC FUNCTIONS.
50
From
If the graph it is seen that z is nearly 1.8. f z, then, from the tables of the gudermanian at the
%dz~$z.
= gd z
f(z)
end of
this chapter,
= + .0432, 2 667 = + .0072, /(i.po) 1.2739 1.2881 1.3000 = .0119, 7(1.95) = whence, by interpolation, z 1.9189 and c= 7.8170. Again, y/c = log* sec x/c but x/c = lo/V = 1.2793; and 1.2793 radians = 73 '7' 55"; hence j= 7.8170 X .54153X2.3026 = 9.7472, the /(i.8o)
= = =
1.2000
1.2432
~
i
;
required depth.) Prob. 96. Find the inclination of the terminal tangent.
Show
Prob. 97.
oo,
that the curve has
two
Prob. 98. Prove that the law of the tension T, and of the section at a distance s, measured from the lowest point along the
curve,
is
T=& = cobh ,
H and show that is
vertical asymptotes.
in the
3.48 times the
Prob.
99.
s c
;
above numerical example the terminal section
minimum
section.
Prove that the radius of curvature is given by Also that the weight of the arc s is given by in which s is measured from the vertex.
= c cosh s/c. W = H sinh s/c,
ART.
An
33.
THE ELASTIC CATENARY.
uniform section and density in its natususpended from two points. Find its equation of
elastic string of
ral state is
equilibrium. Let the element d
=
+
hence the weight of the stretched element gpaods/(i
+ IT).
d(Ts\r\ 0)
T cos
and hence in
which
#/(tan /*
ds,
=^
Accordingly, as before,
=gpvds/(
= H = gpooc, sec 0), 0) = ds/(i + I*
stands for A/f, the extension at the lowest point
;
THE TRACTORY.
= <:(sec'
ds
therefore
= tan +
j/V
ju(sec
51 *
1 -j" A
sec 0)d?0
+ gd"
tan
1
[prob. 20, p. 37
0),
the intrinsic equation of the curve, and reduces to that The coordinates x, y of the common catenary when yu o.
which
is
=
may ting
be expressed
in
terms of the single parameter
= ds cos = r(sec + = ^(sec* + dy = ds sin
dx
=
1
/*
8
}*
sec 0) sin
Whence
*/0.
-4-
/*
tan 1 0.
These equations are more convenient than the eliminating 0, which is somewhat complicated.
AKT.
To
34.
by put-
0X0,
= sec
tan 0,
gd"
sec*
result of
THE TRACTORY.*
the equation of the curve which possesses the property that the length of the tangent from the point of contact to the axis of x is confind
stant.
Let
PT
9
secutive
P'T' be two
tangents
PT=P'T' = = /; draw TS
and
c,
to/"r'; then is
let
conthat
OT
perpendicular if
that
evident
such
PP' = ds, it ST' differs
M
from ds by an infinitesimal of a higher order. Let /Tmake with OA, the axis of y\ then (to the first order of an angle
= TS = TT cos 0; that = cos t = c gd~'0, sin 0), y c sin 0, = ^(gd~
infinitesimals) PTd
ctt
x
=/
This
is
;
and
c cos 0.
a convenient single-parameter form, which gives
*This curve p. 242)
is,
in
is used in Schiele's anti-friction pivot (Minchln's Statics, Vol. I, the theory of the skew circular arch, the horizontal projection
of the joints being a tractory. (See "Arch/' Encyclopaedia Britannica.) gd t/c furnishes a convenient method of plotting the curve. equation
=
all
The
HYPERBOLIC FUNCTIONS.
5
values of x>
pressed in the same form, ds
= ST' = dt sin
is
=
=
At the point ^4, o, Cartesian equation, obtained
= gd*
=
.r
^
0^/0,
= o,
j
o,
= log, sec 0. * = o, j=.
ex-
s,
<:
by eliminating
*
/
1
-\ ^/
=
0,
cosh"
1
The
is
-
A
y
/i
?-.
\
c
//
.2658^,
x
=
/
=
1.0360^.
Show
Prob. 101.
=
=
=
,
Prob. 100. Given
7
c tan
sin [cos'
-)
v
value of
be put for //, and be taken as independent variable, // tanh //, j//^: sech w, 5/^ gd log cosh //. ;r/
If
==
[cos"
The
i/T.
found from the relation
1
1
o to
increases from
as
y
show that At what point
= 74 = c?
2C,
35', s
=
1.3249^,
is /
that the evolute of the tractory
is
the catenary.
(See Prob. 92.) o'
Prob. 102. Find the radius of curvature of the tractory in terms derive the intrinsic equation of the involute.
0; and
ART.
!n
THE LOXODROME.
35.
On the surface of a sphere a curve starts from the equator a given direction and cuts all the meridians at the same angle.
To
find its
in latitude-and
ordinates
equation longitude co-
:
Let the loxodrome cross
two consecutive meridians
AM, AN'm
^
MN= dx, RQ = dy,
all in
let
PR be
radian measure
;
the points/3 Q\ ,
a parallel of lati-
and
let
the angle
MOP=RPQ = a\ then tan a = RQfPR, but PR = MN cos MP* hence dx tan a = dy sec/, and x tan a = gd~ y, there !
no integration-constant since y vanishes with x quired equation
is
y *
;
=
gd (x tan
a).
Jones, Trigonometry (Ithaca, 1890), p. 185.
being thus the re-
COMBINED FLEXURE AND TENSION.
To
OP:
find the length of the arc
= dy esc
ds
To
whence
of,
Integrate the equation
= y esc a.
s
suppose a ship
illustrate numerically,
53
sails
northeast,
from a point on the equator, until her difference of longitude 45, find her latitude and distance
is
:
Here tan a
=
andj/
I,
= y 1/2 =
radians: s
= gd
1.0114
= gd \n =
x
radii.
The
gd(.;854)
= .7152
latitude in degrees
is
40.980.
the ship set out from latitude y^ the formula must be modified as follows: Integrating the above differential equaIf
and (x^y^ gives
tion between the limits (x^ j,) (*,
- *,) tan
a
=
gd" >,
- gd-'j/,;
hence gd" /a = gd~'j, -f- (X, x^ tan <*, from which the final latitude can be found when the initial latitude and the differ!
The
ence of longitude are given.
~7i)
(/a
a
csc
radii,
Mercator's Chart. parallel straight lines, line y'
= x tan
distance sailed
a radius being 60
X
is
equal to
i8o/7r nautical miles.
In this projection the meridians are and the loxodrome becomes the straight
hence the relations between the coordinates of
ar,
corresponding points on the plane and sphere are x'
y
= gd~ y.
is
tabulated under the
y"
;
Thus the
the values of
latitude
name
y and
y
of
"
= x,
l
magnified into gd y, which " meridional part for latitude is
of y' being given in minutes.
A chart
constructed accurately from the tables can be used to furnish graphical solutions of problems like the one proposed above. Prob. 103. Find the distance on a
(30
E) and (30
N, 20
ART.
A beam other,
and
36.
that is
plied at the
assumed by
S,
40
rhumb
line
between the points
E).
COMBINED FLEXURE AND TENSION. is
b'ult-in at
one end
carries a load
P at
the
Q
ap-
also subjected to a horizontal tensile force
same point; its
to find the equation of the curve
neutral surface: Let x>
y
be any point of the
HYPERBOLIC FUNCTIONS.
54
end as
elastic curve, referred to the free
origin, then the
Px. ing moment for this point is Qy notation of the theory of flexure,*
A cosh nx + B sinh nx y = mx + A cosh nx + /? sinh
whence that
u =.
is,
The
9
B
arbitrary constants A,
At
terminal conditions.
must be
Hence, with the usual
mx = u, anddy/t/x* = d*u/dx*>
which, on putting/
zero,
becomes
[probs. 28, 30 #.
are to be determined by the
= o, y = o
x
the free end
;
hence
A
and
= #/;r /? sinh nx -J- = /# 4- nB cosh # y
-f-
y
ax but at the fixed end,
x=
and dy/dx
/,
5= y^ =
hence
--
/,
f^ sinh
/^r
;
=. o,
cosh
w/
and accordingly
To
bend-
nx
-
cosh nl?
.
obtain the deflection of the loaded end, find the ordinate
of the fixed
end by putting x deflection
Prob. 104.
Compute
=
giving
/,
= m(l -- tanh nl).
the deflection of a cast-iron beam,
2X2
inches section, and 6 feet span, built-in at one end and carrying a load of 100 pounds at the other end, the beam being subjected to a horizontal tension of 8000 pounds.
E=
15
X
deflection
*
io
6 ,
Q = 8000, P =
= sV(7 2 "" 5
tan ^
I
100
-44)
;
/ = 4/3, = 1/50, m = 1/80, 44-^9) = -34 inches.]
[In this case
hence n
= A(7 2
Menriman, Mechanics of Materials (New York,
1
1895), pp. 70-77, 267-269,
ALTERNATING CURRENTS.
55
Prob. 105. If the load be uniformly distributed over the beam, say per linear unit, prove that the differential equation is
w
=
or
and that the solution is>'= A cosh nx
Show
how
also
-f
v
-///*,
B sinh nx + MX* +
n
determine the arbitrary constants.
to
ART.
37.
ALTERNATING CURRENTS.*
In the general problem treated the cable or wire
is
regarded
as having resistance, distributed capacity, self-induction,
leakage
and
although some of these may be zero in special line will also be considered to feed into a receiver
;
cases.
The
circuit
of
in
and the general solution will inwhich the receiving end is either
The
electromotive force may, without
any description
clude the particular cases
grounded or insulated.
;
be taken as a simple harmonic function of the time, because any periodic function can be expressed in a Fourier series of simple harmonics.f The E.M.F. and the
loss of generality,
which
may differ in phase by any angle, will be have given values at the terminals of the receiver supposed to circuit; and the problem then is to determine the E.M.F.
current,
and current that must be kept up at the generator terminals and also to express the values of these quantities at any intermediate point, distant x from the receiving end the four
;
;
line-constants being supposed
known,
viz.:
= resistance, in ohms per mile, = coefficient of self-induction, in henrys per mile, c = capacity, in farads per mile, g = coefficient of leakage, in mhos per mile. J r
/
It is
shown
in
standard works
* See references in footnote, Art. 27. t
This
that
if
any simple harmonic
t Byerly,
Harmonic Functions. on the Application
article follows the notation of Kennelly's Treatise
of Hyperbolic Functions to Electrical Engineering Problems, p. 70.
Thomson and Sound, Vol.
I.
Tait, Natural Philosophy, Vol.
p. 20; Bedell
I.
p. 40; Raleigh,
and Crehore, Alternating Currents,
Theory
p. 214.
of
HYPERBOLIC FUNCTIONS. function a sin
a and angle 27T/G0,
be represented by a vector of length then two simple harmonics of the same period -f- 0)
(<*>
6,
but having different values of the phase-angle
combined by E.M.F. and the current
adding their representative vectors.
any point of the from the receiving end, are of the form e
in
=e
which the
are
all
at
sin (&?/ -f #)
t
maximum
functions of x.
sented by the vectors are the
=
*
*i
snl (<&*
can be
0,
Now
the
circuit, distant
+
x
(64)
#')>
and the phase-angles 0, ff' These simple harmonics will be reprevalues /
*\,
1
e /&,
iJO
t
9
whose numerical measures
;
complexes e (cos tt -{-/sin #)*, i, (cos ff -\- j sin #'), be denoted by e, L The relations between e and i may be obtained from the ordinary equations f
which
l
will
c
for, since de/dt
will
=
de
de
di
w^
cos
(o>/
-'
+ 0) =
=n + w^
di l
sin (ut
>
+ + ^
TT),
then
be represented by the vector otfi/0+ \K\ and
by the sum
two vectors ge /6, cwe^/Q -f- ^x whose numerical measures are the complexes ge, juce\ and similarly for de/dx in the second equation thus the relations between of the
l
;
;
the complexes
jx
e, i
=
are
(g
g = + ;0i.
+ ;0&
(f
(66)J
i. symbol/ is used, instead of i, for V and Crehore, Alternating Currents, p. 181. The sign of dx is changed, because x is measured from the receiving end. The coefficient of leakage, g, is usually taken zero, but is here retained for generality and sym-
* In electrical theory the f Bedell
metry. t
These relations have the advantage of not involving the time. Steinmetz them from first principles without using the variable /. For instance,
derives
he regards to
i,
with
r
+jwl
as a generalized resistance-coefficient,
i.
dielectric
which, when applied
phase with /, and part in quadrature Kennelly calls r -f jul the conductor impedance; and g -f- juc the admittance; the reciprocal of which is the dielectric impedance.
gives an E.M.F., part of which
is
in
ALTERNATING CURRENTS
57
Differentiating and substituting give
are similar functions of x to be distinguished terminal values. only by their It is now convenient to define two constants z by the * equations
and thus
I
e,
9
,
cf
and the
=
+
(r
j o)l) (g
+ juc}
differential equations
ZQ
,
e
=A
(68)
;
-"
<w
j
+ JB sinh a^,
cosh ax
jV)
then be written
may
-* the solutions of which are
= a/ (g +
I
=
;
^4
cosh
+ B' sinh ojc,
ajc
wherein only two of the four constants are arbitrary; for substituting in either of the equations (66), and equating coefficients, give
(g
whence
Next
+ MM = B B' = A/z <*
=
(S
A = B/zQ f
Q,
.
the assigned terminal values of e, i, at the reo gives E A, by E, / then putting x f the thus soB and whence B z 7, A', general /2 let
ceiver be denoted
I
-
'>
lution
=
;
=
=
=
;
is
e
f
= E cosh a# + z I sinh a#, Q
= / cosh ojc H
isinh a^, ^o
* Professor Kennelly calls
impedance of the
a
and Art.
27, foot-note.
(7)
r J
the attenuation-constant, and
line.
t See Art. 14, Probs. 28-30;
]
z
the
surge-
HYPERBOLIC FUNCTIONS'.
68
these expressions could be thrown into the ordir nary complex form -}-jY', by putting for the let-\-jY, ters their complex values, and applying the addition-theorems If desired,
X
X
X
The
% Y, X' 9 quantities Y' would then be expressed as functions of x and the repre where e? X*-\-Y*, sentative vectors of e, i, would be ^/0,
for the hyperbolic sine
and cosine.
;
i? = X" +
Y'\ tan 8
=
2,^0',
=
7 Y/X, tanT
= ~Y'/X'.
For purposes of numerical computation, however, the formulas (70) are the most convenient, when either a chart,* or a table,! of cosh
,
sinh
,
is
available, for
complex values of
u.
=
Prob. io6.J Given the four line-constants: r 2ohms per i == 20 millihenrys per mile, c microfarad 1/2 per mile, o; and given w, the angular velocity of E.M.F. to be 2000
=
mile,
g
=
radians per second; then
= 40 ohms, conductor reactance per mile; = + /w/ 2 + 407 ohms, conductor impedance per mile; we = .001 mho, dielectric susceptance per mile; g + /we = .001; mho, dielectric admittance per mile; = iooo/ ohms, dielectric impedance per mile; (g + ;w)"~ a = (r+ /w/) (g + /we) = .04 + .oo2/, which is the measure w/
r
l
2
of .04005/177
a = measure
8';
therefore X
=
+
.0050 .2000;, an abstract coefficient per mile, of dimensions [length]" l ,
So
of .2001/88
= <x/(g + /we) =
200
-
34
s/
ohms.
Next let the assigned terminal conditions at the receiver be: I = o (line insulated); and E = iooo volts, whose phase may be taken as the standard (or zero) phase; then at any distance x,
by
(70),
e
= E cosh ax,
I
=
E*
sinh
ax
t
2o
in
which ax
is ai>
Suppose
it is
must be kept up
abstract complex.
required to find the E.M.F. and current that at a generator 100 miles away; then
* Art. 30, foot-note. J
The data
p. 38).
for
this
t See Table II.
example are
taken from
Kennelly's artidf
(1.
c.
ALTERNATING CURRENTS.
59
+ 207), I = 200(40 /)"' sinh + 207), = cosh (.5 + 2o/ 6nj) (.5 + 207) = cosh + 1.157) = .4600 + -475 /
e .= 1000 cosh (.5
(.5
but, by page 44, cosh .
(.5
obtained from Table II, by interpolation between cosh and cosh (.5 -f 1.27); hence
(.5
+
!!/)
~ 460 + 4757 =
l
and hence '
=
(*
+ /)( 2126 +
I.028/)
=
=
= "^('495 = (cos 0' + /sin 0'), = 1495 sec #'/t6oi = 45', i,
f
where log tan
10.7427,
e
let it
=
79
/,
and magnitude of required current.
5.25 amperes, the phase
Next
ff'
=
be required to find e at x
1000 cosh (.04
by subtracting tween sinh (o
^JT/,
+
Similarly
07")
+
1.67")
10007* sinh (.04
and applying page 44. and sinh (o -|- .if) gives
sinh (o
+
037)
sinh
+
.037)
(.1
Interpolation between the sinh (.04
Hence/ =7(40.02
=
8; then
last
+
-f-
.037),
Interpolation be-
= ooooo + 02 99S/ = .10004 + .030047'. '
two gives
3/)
+ 29.997)=
= .04002 + .029997. 29.99+40.027" =^,(cos
0+j sin
#)>
where log tan
= .12530, 0=
126
51',*,
=
29.99 sec I2 6
51'
= 5-i
volts.
Again,
let it
be required to find
e at
x
=
16; here
= 1000 cosh (.08 + .067"), 3-27") = 1.0020 -f .0067; = but cosh (o + .067") .9970 + o/, cosh (.1 + .o6/) cosh (.08 + -o6/) = i.ooio +.00487", hence e= iooi+4.8/= ^,(cos 6^+/sin ^), and = 180 17', t = 1001 volts. Thus at a distance of about where e
1000 cosh (.08 H-
t
16 miles the E.M.F.
is
the
same
as at the receiver, but in opposite
HYPERBOLIC FUNCTIONS.
60 Since e
phase.
proportional to cosh (.005
is
+ .2j)x
9
the value of
which the phase is exactly 180 is n/.2 = 15.7. Similarly the phase of the E.M.F. at x = 7.85 is 90. There is agreement in phase at any two points whose distance apart is 31.4 miles.
x
for
In conclusion take the more general terminal conditions in line feeds into a receiver circuit, and suppose the current in advance of the elecis to be kept at 50 amperes, in a phase 40 sin 40) tromotive force; then / 5o(cos 40 38.30 32.14/2
which the
=
+/
and
+
substituting the constants in (70) gives
e=
+ ,2j)x + (7821 + 62367)
icoo cosh (.005
=
460-] 4757
+ .2j)x
sinh (.005
4288+98417 ^(cos #+7
-4748+93667=
sin #),
where 0= 113 33',
If
i
\ 2 ii
/
where n 2
=
= o, g = o, / = o; = wrc/2, w = r/2wc;
then a
=
and the
2
t
(i
+ j)n,
solution
2
is
2nx
+ cos 2nx,
tan 6
= tan nx tanh nx,
E I/cosh 2nx
cos 2nx,
tan 0'
=
E ^cosh
2Wi
tan nx coth nx.
1 08. If self-induction and capacity be zero, and the rebe insulated, show that the graph of the electromotive end ceiving force is a catenary if g j* o, a line if g = o. Prob. 109. Neglecting leakage and capacity, prove that the
Prob.
solution of equations (66)
= /,
is I
E+ r
e
(r
+ jul)Ix.
Prob. no. If x be measured from the sending end, show equations (65), (66) are to be modified; and prove that
= E cosh ax zjo sinh ax, I = I where E I refer to the sending end.
cosh ax
e
-- E
how
sinh ax,
z
.
c
ART. I.
The
38.
MISCELLANEOUS APPLICATIONS.
length of the arc of the logarithmic curve
y= M(cos\\ //+logtanh The
2.
s
=
ture
+ 2), where sinh u =
In the hyperbola x*/tt is
p=
a
(#
sinh
8
measure of the sector 4. In
M= I/log
;/
+
b*
A OP,
f/b*
is
0.
=
i
cosh* u)*/a6
i.e.
= a* = r = ati is
y u
a, sinh
length of arc of the spiral of Archimedes
#(sinh 2u 3.
which
#), in
cosh u
the radius of curva;
in
= x/a,
which sinh u
//
is
the
=y/b.
an oblate spheroid, the superficial area of the zone
MISCELLANEOUS APPLICATIONS.
61
between the equator and a parallel plane at a distance y is 5 = irP(sinh 2u 2u)/2e wherein b is the axial radius, e eccen-
+
tricity, sinh u
J
= ey/p
and
/ parameter of generating ellipse. length of the arc of the parabola 2px, measured from the vertex of the curve, is /= ^/(siiih 2u in which 2//), t
y=
The
5.
sinh u
~y/p
+
= tan
where
0,
is
the inclination of the termi-
nal tangent to the initial one. 6.
The
centre of gravity of this arc
given by
is
1
3/^r
nr/^cosh u
i),
64/;7
and the surface of a paraboloid 7.
The moment
minal ordinate
of revolution
of inertia of the
/=
is
p* (sinh 4;*
2x\
^[xKx
same -f-
is
5= 2n yL
arc about
wP*N\
where
its ter,
p
is
the mass of unit length, and
yV= 8.
The
J-
sinh 2u
in its
which
= jr/f
is
2u -f
^(sinli ;/
s ' nn
6.
centre of gravity of the arc of a catenary measured
from the lowest point 4/j/
4+ iV
sinh
;
/ = /^ rV (
2#),
/JF= c\u sinh
and the moment
terminal abscissa b
given by
cosh u
-f- i),
of inertia of this arc
about
is
sin ^ 3 W
~H t sin ' 1
;/
~~
u cos ' 1
7
')'
Applications to the vibrations of bars are given in RayVol. I, art. 170; to the torsion of leigh, Theory of Sound, in Love, Elasticity, pp. 166-74; to the flow of heat prisms 9.
Byerly, Fourier Series, pp. 75-81; to wave motion in fluids in Rayleigh, Vol. I, Appendix, p. 477, and in
and
electricity in
Bassett,
Hydrodynamics,
arts.
120,
384; to the
theory of
in Maxwell, Electricity, arts. potential in Byerly p. 135, and Non-Euclidian to geometry and many other subjects '172-4;
Gunther, Hyperbelfunktionen, Chaps. V and VI. Several numerical examples are worked out in Laxsant, Essai sur lea
in
fonctions hyperboliques.
HYPERBOLIC FUNCTIONS.
(&
ART. In Table sinh
,
cosh
I
#,
EXPLANATION OF TABLES.
39.
the numerical values of the hyperbolic functions tanh u are tabulated for values of u increasing
from o to 4 at intervals of may be used.
in
When
.02.
u exceeds
4,
Table IV
Table II gives hyperbolic functions of complex arguments, which cosh (x
=a
iy)
and the values of
sinh (x
t& 9
#, b, c,
d
if)
are tabulated
ranging separately from o to
and of
y
When
interpolation
is
necessary
1.5
=c
wf,
values
for
may be performed
it
of
at intervals of in
x .1.
three
For example, to find cosh (.82-}- 1.341): First find stages. cosh (.82+ l-30 by keeping/ at 1.3 and interpolating between the entries under x
=
.8
=
and*
.9
;
next find cosh
(.82 -f l-4*)
by keeping y at 1.4 and interpolating between the entries under x = .8 and jr = .9, as before then by interpolation between cosh (.82 1.41) find cosh( .82 i-34*) 1.31) and cosh (.82 ;
+
in
which x
of j/,
however
sinh (x It
great,
.82.
+ 2** = sinh^r,
occurs
in practice.
x
is
= cosh
cosh (x-\- 2tn)
)
does not apply when
dom
+
+
The table is available for all values means of the formulas on page 44: by
kept at
is
greater than
1.5,
x, etc.
but this case
This table can also be used as a
sel-
com
plex table of circular functions, for cos (y
ix)
= a ^p ib,
sin
(y
and, moreover, the exponential function
exp
(
x
i
=^
^r
ix) is
=
given
i(*
by
^/),
which the signs of c and */are to be taken the same as the sign of jr, and the sign of i on the right is to be the product of in
the signs of
x and
of i on the
left.
(See Appendix, C.)
Table III gives the values of v= gd and of the guder0= 180 v/n> as u changes from o to I at inter,
manian angle
EXPLANATION OF TABLES. from
vals of .02,
intervals of
%
6iJ
to 2 at intervals of .05, and from 2 to 4 at
I
i.
IV are given the values of gd log sinh u, log u as increases from to 6 at of intervals u, .1, from 6 to 4 of intervals and from at to of at intervals .2, 7 .5. 7 9 In Table
,
cosh
In the rare cases in which sary, reference
Glaisher,
may
made
be
more extensive
tables are neces-
to the tables* of
Gudermann, Gudermanthe independent variable, and
and Geipel and Kilgour.
In the
first
the
ian angle (written k) is taken as increases from o to roo grades at intervals of .01, the corresponding value of u (written Lk) being tabulated. In the usual
which the table
case, in
entered with the value of
is
//,
it
gives
by interpolation the value of the gudermanian angle, whose circular functions would then give the hyperbolic functions of
When
//.
u
is
large, this angle
is
so nearly right that inter-
To remedy
this inconvenience Gupolation dermann's second table gives directly log sinh u, log cosh //, log tanh u, to nine figures, for values of u varying by .001 from 2 to 5, and by .01 from 5 to 12. is
not reliable.
and c~* to nine sigfrom o to i, by .01 from O from o to $00. From these
Glaisher has tabulated the values of
x
nificant figures, as
to
2,
by
.1
from o to
varies 10,
by .001 and by I
the values of cosh x, sinh
x
e*
t
.
are easily obtained.
Geipel imd Kilgour's handbook gives the values of cosh;t, sinh r, to seven figures, as x varies by .01 from o to 4. There are also extensive tables by Forti, Gronau, Vassal, and there are four-place tables in Byerly's Callet, and Houel Fourier Series, and in Wheeler's Trigonometry, (See Ap;
pendix, C.) In the following tables a dash over a that the number has been increased. Gudermann
final digit indicates
in Crelle's Journal, vols. 6-9, 1831-2 (published separately Theorie der hyperbolischen Functionen, Berlin, 1833). Glaisher in Cambridge Phil. Trans., vol. 13, 1881. Geipel and Kilgour's Electrical Handbook.
under the
title
HYPERBOLIC FUNCTIONS. TABLE
I.
HYPERBOLIC FUNCTIONS.
TABLES. TABLE
I.
HYPERBOLIC FUNCTIONS.
65
HYPERBOLIC FUNCTIONS.
66 TABLE
II.
VALUES OF COSH
(JT
+ iy)
AND SINH
(x -f
iy).
TABLES.
TABLE
II.
VALUES OF COSH
(x
+ fy)
6?
AND SINH
(x
+
iy).
HYPERBOLIC FUNCTIONS.
68 TABLE
II.
VALUES OF COSH
(x
+ iy)
AND SINK (x
+
iy).
69
TABLES. TABLE
II.
VALUES OF COSH(*
+
iy)
AND SINH(*
+
iy.)
70
HYPERBOLIC FUNCTIONS. TABLE
III.
TABLE
IV.
APPENDIX.
A.
HISTORICAL AND BIBLIOGRAPHICAL.
What is probably the earliest suggestion of the analogy between the sector of the circle and that of the hyperbola is found in Newton's Principia (Bk. 2, prop. 8 et seq.) in connection with the solution of a
On
dynamical problem.
the analytical side, the
first
hint of the modi-
and cosine is seen in Roger Cotes' Harmonica Mensurarum where he suggests the possibility of modifying the expression
fied sine
(1722), for the area of the prolate spheroid so as to give that of the oblate one,
by a
certain use of the operator
\/i. The
actual inventor of the
hyperbolic trigonometry was Vincenzo Riccati, SJ. (Opuscula ad res Phys. et Math, pertinens, Bononiae, 1757). He adopted the notation Sh.<, Ch.< for the hyperbolic functions, and Sc.<, Cc.0 for the circular ones.
He
proved the addition theorem geometrically and derived Soon after, Daviet
a construction for the solution of a cubic equation.
de Foncenex showed tions
the
the use of
by work
resting
how
to interchange circular
and hyperbolic func-
V
i, and gave the analogue of De Moivre's theorem, more on analogy, however, than on clear definition
(Reflex, sur les quant, imag., Miscel.
Turin
Soc.,
Tom.
Heinrich Lambert systematized the subject, and gave the
i).
Johann
serial devel-
opments and the exponential expressions. He adopted the notation sinh ft, etc., and introduced the transcendent angle, now called the gudermanian, using it in computation and in the construction of tables (1.
c.
page
30).
The important
history of the subject
is
place occupied by indicated on page 30.
Gudermann
in the
of the circular and hyperbolic trigonometry naturally considerable a part in the controversy regarding the doctrine played
The analogy
of imaginaries, which occupied so tury,
and which gave
much
birtU to the
attention in the eighteenth cen-
modern theory
of functions of the
HYPERBOLIC FUNCTIONS.
72
In the growth of the general complex theory, the
complex variable. importance of the
"
singly periodic functions"
was gradually developed by such log. et trig., Florence, 1782);
became
still
clearer,
and
writers as Ferroni (Magnit. expon.
Dirksen (Organon der tran. Anal, Ber-
lin, 1845); Schellbach (Die einfach. period, funkt., Crelle, 1854); Ohm (Versuch eines volk. conseq. Syst. der Math., Nurnberg, 1855); Hoiiel
(Theor. des quant, complex, Paris, 1870). in systematizing
helped
them
and tabulating these
Many
other writers have
functions,
and
in
adapting
The
following works may be espeGronau mentioned: (Tafeln, 1862, Theor. und Anwend., 1865); cially Forti (Tavoli e teoria, 1870); Laisant (Essai, 1874); Gunther (Die to a variety of applications.
Lehre
.
.
.
,
1881).
The last-named work
and bibliography with numerous
contains a very full history Professor A. G. Greenapplications.
various places in his writings, has shown the importance of both and inverse hyperbolic functions, and has done much to popularize their use (see Did. and Int. Calc., 1891). The following articles
hill, in
the direct
on fundamental conceptions should be noticed: Macfarlane, On the definitions of the trigonometric functions (Papers on Space Analysis,
N.
Y., 1894); Ilaskell,
functions (Bull.
On
N. Y. M.
the introduction of the notion of hyperbolic Attention has been called in Soc., 1895).
30 and 37 to the work of Arthur E. Kennelly in applying the hyperbolic complex theory to the plane vectors which present them-
Arts.
selves in the theory of alternating currents; and his chart has been described on page 44 as a useful substitute for a numerical complex It may be worth mentioning in this table (Proc. A. I. E. E., 1895).
connection that the present writer's complex table in Art. 39 is believed to be the earliest of its kind for any function of the general argument
x
+ iy. B.
(See Appendix, C.)
EXPONENTIAL EXPRESSIONS AS DEFINITIONS.
For those who wish to start with the exponential expressions as the u and cosh u, as indicated on page 25, it is here pro-
definitions of sinh
posed to show
how
these definitions can be easily brought into direct
geometrical relation with the hyperbolic sector in the form #/a=cosh 2 2 S/Kj y/b sinh S/K, by making use of the identity cosh u sinh w= i,"
and the differential relations d cosh tt=sinh u du d sinh w==cosh u du, which are themselves immediate consequences of those exponential Let 0-4, the initial radius of the hyperbolic sector, be definitions. y
EXPONENTIAL EXPRESSIONS AS DEFINITIONS. taken as axis of #, and
OB=b,
its
AOB=w,
angle
73
conjugate radius OB as axis of y\ let OA = a, and area of triangle AOB=K, then JRT
Ja&sinw. Let the coordinates of a point P on the hyperbola be x and y, then x2/a2 yi/b2 =i. Comparison of this equation with the 2 sinh 2 u=i permits the two assumptions #/a=cosh u identity cosh
and y/b=smh
u,
wherein u
is
a single auxiliary variable; and
remains to give a geometrical interpretation to u=S/K, wherein S is the area of the sector OAP. of a second point
POQ POQ
Q
be
x+dx
,
and
it
now
to prove that
Let the coordinates
and
y-\-Ay^ then the area of the triangle is, by analytic geometry, %(xJy ydx)sin aj. Now the sector a ratio whose limit is unity, hence the bears to the triangle
POQ
differential of the
Ja&sin w(cosh
u=S/K,
2
S may be written dS=$(xdy y
sector
sinh2
the sectorial measure (p. 10)
geometrical relations
C.
;
this establishes the
fundamental
#/a=cosh S/K, ;y/6=sinh S/K.
RECENT TABLES AND APPLICATIONS.
The most
extensive
tables
of
hyperbolic
functions of real
arguments are those published by the Smithsonian Institution, prepared by G. F. Becker and C, E. Van Orstrand (IQOQ). For complex arguments the most elaborate tables are those of Professor A. E. Kennelly: "Tables of Complex Hyperbolic " Circular Functions (Harvard University Press, 1914).
Three-digit
tables of sinh
and cosh
of
x+iy, up to
and
x= i and
by W.
E. Miller in a paper, .01, given by steps y=i " Formulae, Constants, and Hyperbolic Functions for Transmission" in the General Electric Review Supplement, Schenline Problems are
of
ectady, N. Y.,
May,
1910.
There are interesting applications and an extensive bibliography " The Application of Hyperbolic in Professor Kennelly 's treatise on " Functions to Electrical Engineering Problems (University of
London
Press, 1912).
It should
be noted that this author uses the term
"
hyperbolic hyperbolic sectorial measure," the analogy being due " " for the circle and ellipse sectorial measure to the fact that the "
Bangle
is
"
for
an actual angle
(p.
n).
been suggested by Professor
The convenient term "angloid" has S.
Epsteen.
INDEX.
Addition-theorems, pages 16, 40.
Admittance of
dielectric, 56.
Algebraic identity, 41. Alternating currents, 38, 46, 55. Ambiguity of value, 13, 16, 45.
Amplitude, hyperbolic, 31. of
complex number, 46.
Anti-gudermanian,
Complementary triangles, Complex numbers, 38-46.
Applications of, 55-60. Tables, 62, 66.
Conductor resistance and impedance, 58.
Construction for gudermanian, 30. of charts, 43. of graphs, 32.
28, 30, 47, 51, 52.
Anti-hyperbolic functions, 16, 22, 25, 29,
Convergence, 23, 25. Conversion-formulas,
35. 45-
Applications, 46
et seq.
18.
Corresponding points on conies,
Arch, 48, 51.
sectors
Areas, 8, 9, 14, 36, 37, 60.
Argand diagram,
10.
and
7,
28.
triangles, 9, 28.
Currents, alternating, 55.
43, 58.
Curvature, 50, 52, 60. Bassett's
Hydrodynamics, 61.
Cotes, reference to, 71.
Beams, flexure of, 54. Becker and Van Orstrand, 73. Bedell and Crehore, 38, 56.
Deflection of beams, 54.
Byerly's Fourier Series,
Difference formula, 16.
etc., 61, 63.
Derived functions,
20, 22, 30.
Differential equation, 21, 25, 47, 49, 51, Callet's Tables, 63.
52, 57-
Capacity of conductor, 55. Catenary, 47. of uniform strength, 49. Cayley's Elliptic Functions, 30, 31.
Electromotive force, 55, 58. Elimination of constants, 21.
Center of gravity, 61.
Ellipses, chart of confocal, 43.
Characteristic ratios, 10.
Elliptic functions, 7, 30, 31.
Elastic, 48.
*
Dirksen's Organon, 71. Distributed load, 55.
Chart of hyperbolic functions, 44, 58.
integrals, 7, 31.
Mercator's, 53. Circular functions, 7,
sectors, 7, 31.
u,
14, 18, 21, 24,
29, 35> 4i, 43-
of complex numbers, 39, 41, 42. of gudermanian, 28
Equations, Differential (see).
Numerical, 35, 48, 50. Evolute of tractory, 52.
Expansion
in series, 23, 25, 31.
76
INDEX.
Exponential expressions, 24, 25, 72.
Hyperbolic functions of complex
num-
bers, 38 et seq.
Ferroni, reference
relations
to, 71.
among, 12. gudermanian,
relations to
Flexure of beams, 53.
Foncenex, reference to, 71. Forti's Tavoli e teoria, 63, 71.
29. relations to circular functions, 29, 42. tables of, 64 et seq.
Fourier
variation of, 20.
series, 55, 61.
Function, anti-gudermanian (see). anti-hyperbolic (see). circular (see).
Imaginary, see complex.
Impedance,
34.
Integrals, 35. elliptic (see).
gudermanian
Interchange of hyperbolic and circular (see).
hyperbolic, defined, n. of complex numbers, 38. of pure imaginaries, 41.
of
sum and
difference, 16.
functions, 42.
Interpolation, 30, 48, 50, 59, 62. Intrinsic equation, 38, 47, 49, 51.
Involute of catenary, 48. of tractory, 50.
periodic, 44.
Jones' Trigonometiy, 52.
Geipel and Kilgour's Electrical Hand-
book, 63.
Kcnnelly on alternating currents, 38, 58. Kennelly's chart, 46, 58; treatise, 73.
Generalization, 41.
Geometrical interpretation, 37. treatment of hyperbolic
Liisant's Essai,
etc., 61, 71.
Lambert's notation, 30.
functions, jet seq., 16.
place in the history, 70. of conductor, 55.
Glaisher's exponential tables, 63.
Leakage
Graphs, 32.
Limiting ratios, 19, 23, 32. Logarithmic curve, 60.
Greenhill's Calculus, 72.
expressions, 27, 32.
Elliptic Functions, 7.
Love's
Gronau's Tafeln, 63, 72. Theor. und An wend.,
Gudermann's
72.
notation, 30.
Gudermanian, angle,
Loxodrome,
function, 28, 31, 34,47, 53, 63, 70. etc., 61, 71.
et seq.,
30, 37, 44, 60.
Hyperbolic functions, defined, addition-theorems for, 16. applications of, 46
W.
Modulus,
Moment
E., Tables, etc., 73
31, 46.
of inertia, 61.
Multiple values, 13, 16, 45.
n.
et seq.
derivatives of, 20.
expansions
definitions, 72.
Maxwell's Electricity, 61. Measure, defined, 8; of sector, 9 Mercator's chart, 53. Miller,
Haskell on fundamental notions, 72. HoiiePs notation, etc., 30, 31, 71.
Hyperbola, 7
52.
Macfarlane on
29.
Gunther's Die Lehre,
elasticity, 61.
of, 23.
Newton, reference
to, 71.
Numbers, complex, 38
Ohm,
of, 32.
integrals involving, 35.
seq.
reference to, 71.
Operators, generalized, 39, 56.
exponential expressions for, 24.
graphs
et
Parabola, 38, 61. Periodicity, 44, 62.
et seq.
INDEX. Permanence of equivalence, Phase angle, 56, 59.
41.
Physical problems, 21, 38, 47 Potential theory, 61.
Self-induction of conductor, 55. Series, 23, 31.
et seq.
Product -series, 43. Pure imaginary, 41.
Spheroid, area of oblate, 58 Spiral of Archimedes, 60.
Steinmetz on alternating currents, 38. Susceptance of dielectric, 58.
Ratios, characteristic, 10.
Tables, 62, 73. limiting, 19.
of Sound, 61.
Rayleigh's Theory Reactance of conductor, 58.
Reduction formula, 37, 38. Relations
among
functions, 12, 29, 42.
Resistance of conductor. 56.
Rhumb
line, 53. Riccati's place in the history, 71.
Schellbach, reference
Terminal conditions, 54,
58, 60.
Tractory, 48, 5 1.
Van Orstrand, C. E., Tables, 73. Variation of hyperbolic functions, 14* Vassal's Tables, 63. Vectors, 38, 56. Vibrations of bars, 61,
to, 71.
Sectors of conies, 9, 28.
Wheeler's Trigonometry, 6l.
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