ELLIPTIC FUNCTIONS
INTRODUCTORY CHAPTER *
The first step taken in the theory of Elliptic Functions
was the determination of a relation between the amplitudes of
three functions of either order, such that there should exist an
algebraic relation between the three functions themselves of
which these were the amplitudes. It is one of the most re-
markable discoveries which science owes to Euler. In 1761 he
gave to the world the complete integration of an equation of
two terms, each an elliptic function of the first or second order,
not separately integrable.
This integration introduced an arbitrary constant in the
form of a third function, related to the first two by a given
equation between the amplitudes of the three.
In 1775 Landen, an English mathematician, published his
celebrated theorem showing that any arc of a hyperbola may
be measured by two arcs of an ellipse, an important element
of the theory of Elliptic Functions, but then an isolated result.
The great problem of comparison of Elliptic Functions of dif-
ferent moduli remained unsolved, though Euler, in a measure,
exhausted the comparison of functions of the same modulus.
It was completed in 1784 by Lagrange, and for the computation
•Condensed from an article by Rev. Henry Moseley, M A., F.R.S..Prof, of
Nat. Phil, and Ast., King's College, London.
2
ELLIPTIC FUNCTIONS.
of numerical results leaves little to be desired. The value of
a function may be determined by it, in terms of increasing or
diminishing moduli, until at length it depends upon a function
having a modulus of zero, or unity. .
For all practical purposes this was sufficient. The enor-
mous task of calculating tables was undertaken by Legendre.
His labors did not end here, however. There is none of the
discoveries of his predecessors which has not received some
perfection at his hands; and it was he who first supplied to
the whole that connection and arrangement which have made
it an independent science.
The theory of Elliptic Integrals remained at a standstill
from 1786, the year when Legendre took it up, until the year
1827, when the second volume of his Traite des Fonctions
Elliptiqnes appeared. Scarcely so, however, when there ap-
peared the researches of Jacobi, a Professor of Mathematics in
Konigsberg, in the 123d number of the Journal of Schumacher,
and those of Abel, Professor of Mathematics at Christiania, in
the 3d number of Crclle’s Journal for 1827.
These publications put the theory of Elliptic Functions
upon an entirely new basis. The researches of Jacobi have for
their principal object the development of that general relation
of functions of the first order having different moduli, of which
the scales of Legrange and Legendre are particular cases.
It was to Abel that the idea first occurred of treating the
Elliptic Integral as a function of its amplitude. Proceeding
from this new point of view, he embraced in his speculations
all the principal results of Jacobi. Having undertaken to de-
velop the principle upon which rests the fundamental proposi-
tion of Euler establishing an algebraic relation between three
functions which have the same moduli, dependent upon a cer-
tain relation of their amplitudes, he has extended it from three
to an indefinite number of functions; and from Elliptic Func-
tions to an infinite number of other functions embraced under
an indefinite number of classes, of which that of Elliptic Func-
J.VTROD UCTOR У CHAPTER.	3
tions is but one ; and each class having a division analogous to
that of Elliptic Functions into three orders having common
properties.
The discovery of Abel is of infinite moment as presenting
the first step of approach towards a more complete theory of
the infinite class of ultra elliptic functions, destined probably
ere long to constitute one of the most important of the
branches of transcendental analysis, and to include among the
integrals of which it effects the solution some of those which at
present arrest the researches of the philosopher in the very
elements of physics.
CHAPTER I.
ELLIPTIC INTEGRALS.
The integration of irrational expressions of the form
or
Xdx VA 4* Ex 4~ Cod,
Xdx
1C4 4- Ax 4- Cx' ’
X being a rational function of x, is fully illustrated in most ele-
mentary works on Integral Calculus, and shown to depend upon
the transcendentals known as logarithms and circular functions,
which can be calculated by the proper logarithmic and trigono-
metric tables.
When, however, we undertake to integrate irrational expres-
sions containing higher powers of x than the square, we meet
with insurmountable difficulties. This arises from the fact that
the integral sought depends upon a new set of transcendentals,
to which has been given the name of elliptic functions, and
whose characteristics we will learn hereafter.
The name of Elliptic Integrals has been given to the simple
integral forms to which can be reduced all integrals of the form
(i)
where A(A" R) designates a rational function of x and R, and
R represents a radical of the form
R = \CAx' 4- Ex' 4- Cx‘ 4- Dx 4- E,
4
ELLIPTIC INTEGRALS.
5
where А, В, C, D, E indicate constant coefficients.
We will show presently that all cases of Eq. (l) can be
reduced to the three typical forms
(2)
x'dx
- «1 - ^ж*) ’
Г*, dx
(*’ + д) Я1 * -	— zv) ’
which are called elliptic integrals of the first, second, and third
order.
Why they arc called Elliptic Integrals we will learn further
on. The transcendental functions which depend upon these
integrals, and which will be discussed in Chapter IV, are called
Elliptic Functions.
The most general form of Eq. (i) is
(3)
A^BR,
C + DRdx'
where А, В, C, and D stand for rational integral functions of x.
A + BR
C + DR
can be written
A +-BR AC- BDR1 (AD - CB)R'
C + DR~ C‘ — D'R C* - D'R3
i
R
ELLIPTIC FUNCTIONS.
yVand P being rational integral functions of x. Whence Eq.
(3) becomes
(4)
V=
Eq. (4) shows that the most general form of V can be made
to depend upon the expressions
(S)
_ jFdx
~ f R ’
and
/VzZr.
This last form is rational, and needs no discussion here.
We can write
_ G. J- GS +	+ •  • + (g. + GS + •  >
Multiplying both numerator and denominator by
H, + H,x' + Htx' + . . . - (Z7, 4- H,x> + Яъх* + . .
re have a new Jiumerotnr which contains only powers of x°.
Uhe result takes the following form :
_ < + <жа + ЛГл*4- ... 4-« 4- M3x' +
лг„ 4- л>’ 4-	4-  • •
- ф(х’) 4-	. х.
Equation (5) thus becomes
(6)
. x .dx
R
V =
ELLIPTIC INTEGRALS.
7
We shall see presently that R can always be assumed to be
of the form
X(i - л-’Xi - k'x').
Therefore, putting x‘ = z, the second integral in Eq. (6)
takes the form
£
2
l'(i - ,Xi -^s) ’
which can be integrated by the well-known methods of Integral
Calculus, resulting in logarithmic and circular transcendentals.
There remains, therefore, only the form
&(x')dx
R
to be determined.
We will now show that R can always be assumed to be in
the form
4z(i — x~(i — Л’х’).
We have
R =	-|- 8х* + Cx‘ + Dx + E
= \fG{x — «)(л- — b)(x —	“ d)>
a, b, c, and d being the roots of the polynomial of the fourth
degree, and G any number, real or imaginary, depending upon
the coefficients in the given polynomial.
Substituting in equation (i)
x~ i+y’
we have
(7)
V =f ^У’ №y’
8
EI.1JPTIC FUNCTIONS.
p designating the radical
p= vg[/>—[>-ад(?- ад [/—^+(?-ад. • • •
In order that the odd powers of у under the radical may
disappear we must have their coefficients equal to zero; i.e.,
whence
and
(8)
(/*—«) (? - ^) 4- о—ад?—*)= o,
{p — c) (q - d) + (/ - d)(q — z) = о ;
2pq _ (p q^a _|_ /_>) _|_ 2ab = O,
W — О + ?)0	2«f = O, -
ab(c 4- d} — cd(a 4- b)
№ ~ a -|- b — [c -j- d) '
2ab — 2td
a b — (c + d}‘
Equation (8) shows that / and q are real quantities, whether
the roots a, b, c, and d are real or imaginary; a, b, and c, d being
the conjugate pairs.
Hence equation (i) can always be reduced to the form of
equation (7), which contains only the second and fourth powers
of the variable.
This transformation seems to fail when a 4-^ —- (c-|-<Z) — o;
but in that case we have
R =	— (л4~ ад 4-«^]l>1 — («4-ад4-^]>
and substituting
a 4- b
х=У------—
will cause the odd powers of у to disappear as before.
If the radical should have the form
— ri)(^ — b'){x — r),
ELLIPTIC INTEGRALS.
9
placing x = f 4- a, we get
V = f Ф{у, P)<ty,
P= \'G(f + a — b}{f 4- a — c).
<p designating a rational function of j and p.
Thus all integrals of the form contained in equation (i), in
which A stands for a quadratic surd of the third or fourth
-degree, can be reduced to the form
<9)
У= f R}dx,
R designating a radical of the form
+z(7(i 4“ i + nx'\
»/ and я designating constants.
It is evident that if we put
x' — x V — k1 —-------»
m
we can reduce the radical to the form
/(i r- .r’Xi - ~*’)-
We shall see later on that the quantity R, to which has
been given the name modulus, can always be considered real
and less than unity.
Combining these results with equation (6), we see that the
integration of equation (i) depends finally upon the integration
of the expression
(10)
У(|
/(p^x^dx
R ’
IO
ELLIPTIC FUNCTIONS.
The most general form of is
_ <+^’4-«h
Ф\^) — ,.V _1_ /Vr“ -I- .Vx* -k
L
(Д-4 + «Г
Hence
(U)
ГГ" __ > P d* 1 V/ / ________________________
V ) R +^ J (x' + dyR
But
——— depends upon
dx ,
R •‘nd
x'dx , . ,
—77-, which can
Л
be shown as follows:
Differentiating Rxm~\ we have
У«4-A-^’+
= (2m — ^x^'^dx У a + fix* -}- yx*
xim~\fix 4- 2yx')dx
Integrating and collecting, we get
Rx”*'^ = (’w — з)«
x^-'dx
R
4~ (2»Z —
xmdx
~R~
(12) =a / ~ X» '
x^-'dx
xMdx
~”R~
ELLIPTIC INTEGRALS.
II
Whence we get, by taking m = 2,
Оз)
/x2mdx
which shows that the general expression I ~~r~ can be found
by successive calculations, when we are able to integrate the
expressions
and
the first and second of equation (2).
We will now consider the second class of terms in eq (l 1),
Ldx
VIZ., . s ] ч
(л-1 -}- a)nR
This second term is as follows:
Each of these terms can be shown to depend ultimately
upon terms of the form
x'dx dx j dx
~R’ and (x3 + a)R
The two former will be recognized as the two ultimate forms
already discussed, the first and second of equation (2). The
third is the third one of equation (2).
This dependence of equation (14) can be shown as follows:
12
ELLIPTIC FUNCTIONS.
We have
xR
(ж,4-д)я-(^77?4-7?г7.г)-2ж1/?(я+1)(.г,4-д)"-,<7.г
(x* -|- a)(xdR -|- Rdx) — 2л^/?(лг — i) dx
Substituting the value of
R = V a -f- fix* + yx' and dR = {fix 4~ 2ухг) -g-,
we get
(л-,14-д)(/?х*4-2ул'4-^д-|-/Зх°-|-ух*)—2х2(я — 1)(д4-/3х84-улг*) dx
ЗГ
— 2\tl — l)y
+ ЗаУ
- 2 (я - I)/?
+ 2«/f ж’ + аа
“Г а
— 2(я — l)rt
dx
’ ~R
5)ух‘ - (гя - 4)/? I л-‘
+ 3«Г ।
— (2я — з)« х3 4" аа
—2а fl
dx
d
or, by substituting in the numerator Xs = z — a,
- (2й- 5)г^+(2я- 5)зд/
— (2« — 4)А
+ 3«Г
z1— (2я— 5)3«’y
+ (2« — 4)2«/?
— 6afy
— (2Я — 3>
+ 2Л/?
—(2Я-4)й’/?
+ 3«V
+ (2я — з)д«
-2^fl
-|- аа
dx
ELLIPTIC INTEGRALS.
13
or, after resubstituting s = x’ + a, and integrating,
л-А .	,
XM^=-<2?z-5)r
dx
— (2Л — 4XA
— (2H — з)(з«“у — 2a fl J- «)
dx
dx
dx
(x* -|-
C dx t л (' dx . Г dx
~a>J (xa + д)и-’А3-г	+	(x‘ + «)"”/?
, A C dx
'J (х’ + дХА*
Making » = 2, we have
. л-А	f (x* 4- d)dx . й Cdx Г dx
<l6) (x’ 4- ff)-« - a'J A’ + A + У' J (r’ +
Equation (16) shows that
depends upon the three forms
xfdx

dx
14
ELLIPTIC FUNCTIONS.
the three types of equation (2), and equation (15) shows that
the general form
Г dx
J (P + dfR
depends ultimately upon the same three types.
We have now discussed every form which the general equa-
tion (1) can assume, and shown that they all depend ultimately
upon one or more of the three types contained in equation (2).
These three types are called the three Elliptic Integrals of
the first, second, and third kind, respectively.
Legendre puts x — sin ф, and reduces the three integrals
to the following forms:
/’* tty
(i7)	F(A0) = /
l i — k sin ф
Vi—sin’ ф. tty;
<l8> П{п, k, Ф) - /-------------------—_____________;
JQ (i — n sin’ Ф) Vi — k* sin’ ф
the first being Legendre's integral of the first kind; the form
09)
sin’ ф  tty
being the integral of the second kind; and the third one being
the integral of the third kind.
The form of the integral of the second kind shows why they
are called Elliptic Integrals, the arc of an elliptic quadrant
being equal to
ff
a I Vi — f* sin’ ф. tty,
Ф being the complement of the eccentric angle.
gLUPTiC INTEGRALS.
15
By easy substitutions, we get from Eqs. (17), (18), and (19)
the following solutions:
C* sin* ф F — E
I ~d~	= ~7F~’
cos’ ф E — (1 — F)F
—-Аф=	;
** tan’ Ф ,. A tan ф — E
—j- Фф- r_kt
/**sec’ ф ,,	4 tan ф + (i — F}F — E
/ =;
<z 0
Jr* I , I {r F sin Ф cos 0\
^ФФ —	-	3	/’
D
sin* ф 1 (E — (1 — F)F sin ф cos
—	-----F-------J--!•
f)
Г* cos* ф F — E , sin Ф cos ф
/	= ~fc*-+--J---*
CHAPTER IL
ELLIPTIC FUNCTIONS.
0* is called the amplitude corresponding to the argument u,
and is written
ф — am (u, A) = am u.
The quantity A is called the modulus, and the expression
|T — A3 sin'J Ф is written *
lzi — A‘sin“ ф = Л лт и — Лф,
and is called the delta function of the amplitude o. u, or delta
of ф, or simply delta Ф.
и can be written
и = F(k, Ф).
The following abbreviations are used :
sin ф = sin am и = sn j- и;
cos ф = cos am и = cn f и;
Лф = J am и = dn + и — Ли;
tan ф — tan am и — tn u.
Let ф and ф be any two arbitrary angles, and put
ф = am и;
ф — am v.
* Legendre.
f (iudermann, in his “ Thcoric der Mudularfunctionen ” : Crelle's Journal,
Bd. 18.
16
ELLIPTIC FUNCTIONS.
J7
In the spherical triangle ABC we have from
Trigonometry, c and C being constant,
r
<?Ф . </>/• _o
cos В ‘ cos A
Since C and c are constant, denoting by k an arbitrary con-
stant, we have
О)
But
sin c
sin /<
 Л . , sin 2? sin C ...
sm A — sin ф -.— • = sin ф —.----------— k sm
r sin 90 sm p
Whence
cos A = V1 — sin* A — Vi — A? sin* ф.
In the same manner
cos В = Vi — sin’ В = Vi — Z’ sin’ ф.
Substituting these values, we get
Аф
~	“ •___— — O.
Vi — /•’ sin’ ф Vi — Z* sin’ ф
Integrating this, there results
When ф = o, we have ф — j.i, and therefore the constant
must be of the form
_____^ф_______
Vi — Z’ sin’ ф’
18	ELLIPTIC FUNCTIONS.
whence
, x Г лф , Г <ц> Л
' Ja Vi — P sin* </> J, *'i sin* 0 J, Vi — P sin* </>’
or
и + r = m;
and evidently the amplitudes ф, ф, and p. can be considered as
the three sides of aspherical triangle, and the relations between
the sides of this spherical triangle will be the same as those
between ф, ф, and ju.
But the sides of this triangle have imposed upon them the
condition
sin C .
—— = k;
Sin /4
and since k < 1. we must have ju > C, which requires that one
of the angles of the triangle shall be obtuse and the other two
acute.
In the figure, let C be an acute angle of the triangle ABC,
and PQ the equatorial great circle of which C is the pole.
The arc PQ will be the measure of the an-
gle C.
Let AG and AH be the arcs of two great
circles perpendicular respectively to CQ and
CP. They will of course be shorter than PQ.
Hence AB = /л must intersect CQ in points
between CG and HQ, since )i>{C — PQ). In
any case either A or В will be obtuse according
as В falls between QH or CG respectively; and
the other angle will be acute.
In the case where C is an obtuse angle, it will be easily seen
that the angle at A must be acute, since the great circle AD,
perpendicular to CP, intersects PQ in D,PD being a quadrant.
The same remarks apply to the angle Б. Hence, m eithet
ELLIPTIC P UNCTIONS
19
case, one of the angles of the triangle is	c
obtuse and the other two arc acute, as
a result of the condition	/i[\	\\X\
sin C	,	/ I V- 
----- = £>< I.	I \	\	T'4
Sill	к \	\	1-И
From Trigonometry we have	p—I
cos p = cos </> cos ф sin ф sin cos С; X,	\	\ /
and since the‘angle C is obtuse,
cos C — — Vi — sin’ C — — Ti — k'‘ sin” /<,
and
(5) cos p — cos ф cos ф — sin ф sin ф У i — fi‘ sin’ /z,
the relation sought.
The spherical triangle likewise gives the following relations
between the sides:
Icos ф = cos /г cos ф sin /z sin ф Vi — fi‘ sin’ </>;
______________
cos ф = cos /< cos ф -|- sin p sin ф i 1 — A’’ sin” ф.
These give, by eliminating cos
cos’ ф — cos’ ф
Sin /Z —	cog ф J ф _ sjn ф COs ф J 0 >
which, after multiplying by the sum of the terms in the de-
nominator and substituting cos’ — 1 — sin’, can be written
_ (sin’ ф — sin’ r/?)(sin ф cos ф А ф -J~ sin V’ Cos ф J ф
1	sin’ ф cos* ф А' ф — sin’ ф cos” ф Д' ф
Since the denominator can be written
(sin* ф — sin’ 0)(l — k' sin’ ф sin”»/»),
. sin ф cos ф Д ф -4- sin Ф cos ф J ф
 7	1 — k sin sm
In a similar manner we get
<6)*
cos ф cos ф — sin ф sin ф Д ф Д ф
I — k' sin* 0 sin’ ф ’
Д ф Л ф — k' sin ф sin ф cos ф cos ф
I — Z'1 sin’ ф sin’ Ф
20
ELLIPTIC FUNCTIONS.
These equations can also be written as follows:
sin am и cos am v 4 am v ± sin am v cos am и Л am и
sin am (и ± v) —	i — s[n aill K sjni am у	*
(7)'
,	cos am и cos am v T sin am « sin am v A am « J am v
cosam(« ± r) =  -—J^siii* am“«Tii? am v	'
A am и Л am V T A*sin am « sin am v cosam и cos am v
zi am (« ± И —	"	i _ sin-' am и sin* am v
v
or
(8)
sn w cn v dn v ± sn r cn si dn и
sn (и ± v) _	J _ bn5 u sn' v	’
cn я cn v T sn и sn v dn и dn v
cn (« ± »') —	p_ p sns u sn* у	»
dn и dn r T sn и sn v cn и cn v
dn (a ± r) —	j snl — snM у
Making « = v, we get from the upper sign
sn 2SI
(9)
cn 211
dn 2SI
2 sn st cn st dn si
I — sn’ « ’
cn’ и — sn" я dn" U I — 2 sn’ SI -|- X"* sn* я
I — X?" sn’ «	I — X'" sn’ к ’
dn’ si — k* sn" SI cn* SI I — 2X* sn* я -|- k' sn’ и
I — k' sn’ w — I — k‘ sn' и
From these
(io)
I	— cn	2« =	2 cns st dn* st I — A’ sn’ и ’
I	+ cn	2SI —	2 СП* Я
			I — X’* sn’ st ’
I	— dn	u =	2X’’ sn’ и cn* и
			I — k" sn’ st
	-f- dn		2 dn’ st
I		и —	i — X’* sn’ и '
ELLIPTIC FUNCTIONS.
21
and therefore
	sn* и —	I — СП 2Я
		I 4- dn 2« ’
(II)	СП* и —	dn 2U + cn 2U I -|- dn 2U ’
	dn“ и —	I —	+ dn 2М + № СП 2й
		I 4- dn 2U	’
and by analogy
		и	/1 — СП и
	sn	2-\/	I -|- dn и ’
(12)		U ж i	/cn « + dn и
	СП	2 V	1 + dn и ’
	dn	и	/1 —	+ dn и -|- Л*3 сп и
		2-V	I + dn и
In equations (7) making и = v, and taking the lower sign,
we have
' sn 0 = 0;
-	cn О = 1;
dn о - - I.
Likewise, we get by making и = о,
" sn ( — «) = — sn и;
-	cn ( - я) = cn я;
dn (— и) — dn и.
PERIODICITY OP THE FUNCTIONS.
CHAPTER III.
WHEN the elliptic integral
Vi
Лф
№ sin’ ф
has for its amplitude , it is called, following the notation of
Legendre, the complete function, and is indicated by K, thus:
с1ф
— A1 sin2 ф
When k becomes the complementary modulus, k', (see eq. 4,
Chap. IV,) the corresponding complete function is indicated by
K', thus:
zr= p—&—.
Ja Vi — k! “ sin” ф
From these, evidently,
am (A, A) = J
am =
(0
fsn (л; A) = i;
-I cn (A' A) — о;
I dn (Ar, k) — k'.
32
PERIODICITY OF THE FUNCTIONS.
23
From eqs. (7), (8), and (9), Chap. II, we have, by the sub-
stitution of the values of sn (A') = 1, cn (A”) — o, dn (AT) = K,
	' sn 2K = 0 ;
(2)	cn 2K = —l ; dn 2K = I.
These equations, by means of (1), (2), and (3) of Chap. II,
give	sn (й! 2АГ) — — sn и ;
(3)	-I cn (a -j- 2K) = — cn и ; [ dn (w 4- 2K) — dn и;
and these, by changing и into « 4- 2K, give
(4)	’ sn («	4A') = sn и ; cn (и 4~	= cn w; dn (u 4- 4AT) = dn u.
From these equations it is seen that the elliptic functions
sn, cn, dn, are periodic functions having for their period 4K.
Unlike the period of trigonometric functions, this period is not
a fixed one, but depends upon the value of k, the modulus.
From the Integral Calculus we have
= nK’
Дф
from which we see that
я- = am (nK);
24
ELLIPTIC FUNCTIONS.
or. since — = am K.
2
am (&K) = n . am K,
and
mt — am {2nK},
and also	nit = 2n am K.
In the case of an Elliptic Integral with the arbitrary angle a,
we can put
o' у = nit ± /9,
where /9 is an angle between о and —, the upper or the lower
sign being taken according as — is contained in a an even or
an uneven number of times.
In the first case we have
or, putting Ф,— — nir,
In the second case
or, putting ф1 = mt — ф.
Фф,
^Ф,
PERIODICITY OF THE FUNCTIONS.
25
or in either case,
= 2«K ±
* Лф,
4<t>;
Thus we see that the Integral with the general amplitude
a can be made to depend upon the complete integral К and
an Integral whose amplitude lies between о and
Put now
This gives
fm±6 <1ф
/	~2ф = 2nK ± *
ь/ 0
or	am (211K ± «) = nn ± /5
(5)	= n?t ± am «
(6)	= 2n . am К ± am «;
or, since . am (— г) = — am s,
am (w ± 2nK) = am и ± пя
- am и ± in . am K.
Taking the sine and cosine of both sides, we have
sn (w inK') — ± sn и ;
cn (и + inK} — ± cn и;
the upper or the lower sign being taken according as n is even
or odd. By giving the proper values to n we can get the same
results as in equations (3) and (4).
Putting я = 1 in eq. (5), we have
(7)
sn (2ЛГ — «) = sin я cn и — cos я- sn «
= sn u.
2б
ELLIPTIC FUNCTIONS.
Elliptic functions also have an imaginary period. In order
to show this we will, in the integral
assume the amplitude as imaginary. Put
sin ф = i tan ф.
From this we get
cos	;
cos p
(8)
Аф - V1 ~ sin'	;
cos ф	cos ф
.	. dp
Фф — г------.
cos ф
From these, since ф and ф vanish simultaneously, we easily get
dp
— г
dp
Дф
Put
whence
dp _
4{p, k") ~ U
and
dp
= ги
4ф
and these substituted in Eq. (8) give
and
Ф = am (iii);
sn iu = i tn (a, k') ;
(9)
1
СП ги — -------------;
cn (?/, k )
dn iu
dn (u, k')
cn {u, k') ’
ф = am (u, /'),
PERIODICITY OF THE FUNCTIONS.

By assuming
Л	, f* <1ф
J, Щ>,А)	J,
we get
sn (— w) = i tn {iu, A’),
cn (— «) =------у-—/7, ,
v	J	cn	{ги,	A)
dn	{iu,	A')
dn (— u) - ------p----tk ;
4	'	cn	{iu,	A)
or, from eq. (14), Chap. II,
sn
и — — i tn {iu, A1):
(10)
cn
dn
U cn {iu, A') *
dn {iu, A')
W cn {iu, A')'
From eqs. (7), Chap. Il, making v = K, we get, since
sn К — I, cn К — О, dn К — A1',
cn и dn и cn и
sn {u ± K) — ±-----75—;— = zb 3— >
v ’	1 — я sn и dn и
T sn и dn uA’ _ A' sn и
dn и '
cn (« ± K) —
dn“a
A'
dn {и ± 70 = 4- 3— .
4	'	1 dn и
In these equations, changing и into iu, we get, by means of
eqs. (9),
Sn(/«± ^)=±-d-2-^;
,.	iA' sn (w, A')
cn {iu ± ЛГ) = T —;—7—77Г-;
	1 dn {u, A )
dn (ги±^) = +-п-_.—.
28
ELLIPTIC FUNCTIONS.
Putting now in eqs. (9) и ± K' instead of u, and making use
of eqs. (io), and interchanging k and k', we have
,.	i cn (и, Л')
sn (tu ± iK ) — — --;
	1 k Stl (it, k) ’
,r„ dn (u, /')
dn(,«±«-)=Ts^pj.
Substituting in these — iu in place of u, we get, by
means of eqs. (9) and eqs. (14) of Chap. II,
(i4)
sn (u ± iK') = j—-— ;
v ' k sn и
,	i dn w
cn (и ± гК ) = T 7--------;
' k sn и
dn (« ± zA'') = i cot am u.
In these equations, putting и -j- К in place of u, we get
05)
sn («+Л7± ,^ = -1--^;
41	7	1 К cn и
ik’
cn (и -4- К ± НС) = т т-;
7 k cn и
dn (« + A' ± 1 A') = ± ik' tn u.
Whence for и — о we get
sn {К ± ?7C') = j;
A
(16)
,	i^r
cn (A ± iK) = т -J ;
dn (Л" ± «А") = o.
FEHiOVICITY OF THE FUNCTIONS.	2p
If in eqs. (14) we put и — о, we see that as и approaches
zero, the expressions
sn (i iK"), cn (± iK'), dn (± iK7)
approach infinity.
We see from what has preceded that Elliptic Functions
have two periods, one a real period, and one an imaginary
period.
In the former characteristic they resemble Trigonometric
Functions, and in the latter Logarithmic Functions.
On account of these two periods they are often called
Doubly Periodic Functions. Some authors make this double
periodicity the starting-point of their investigations. This
method of investigation gives some very beautiful results and
processes, but not of a kind adapted for an elementary work.
It will be noticed that the Elliptic Functions sn u, cn w, and
dn и have a very close analogy to trigonometric functions, in
which, however, the independent variable и is not an angle, as
it is in the case of trigonometric functions.
Like Trigonometric Functions, these Elliptic Functions can
be arranged in tables. These tables, however, require a double
argument, viz., и and k. In Chap. IX these functions arc de-
veloped into scries, from which their values may be computed
and tables formed.
No complete tables have yet been published, though they
arc in process of computation.
CHAPTER IV.
LANDEN'S TRANSFORMATION.
Let A В be the diameter of a circle,
with the centre at 0, the radius AO = r,
and C a fixed point situated upon OB,
and ОС = kQr. Denote the angle PBC
by ф, and the angle PCO by Ф,. Let
P' be a point indefinitely near to P.
PP' sin POP' _ sin PCP
~PC ~ sin PPC ~ cos OP’C
But PP* = ътФф, and sin PCP' — PCP* =	;
therefore
2п1ф <1фх
~PC~ ~ cos OP'C
But
PC* — P + r’Z'/ + гг’Х.'о cos 2ф
= (r 4- rkrf cos’ ф + (r — sin" ф;
also	P cos’ OP'C = P -P sin’ OP'C
= P — r’X'c’ sin* 0,.
Therefore
2 <70	_________<70,______
4z(r 4- r^y cos’ 0-j- (r — r^,) sin* ф VP — Pk° sin’ 0/
which can be written
2	(1ф	_I <70,
J 4^P . . ~	sin’0/
FI—.. , Ta Sin 0
30
LA .V.DEN' S TRANSFORMA TION.
31
Putting
4V’
(r +
d<f>
/1 — P sin’ ф
i + A, Л- _
2	V i — k* sin’ Ф, ’
no constant being added because Ф and ф, vanish simulta-
neously ; ф and ф, being connected by the equation
<3)
sin OPC _ sin (рф — 0,) _ rk„ _
sin OCP sin </>] r
From the value of k" we have
and therefore
k' is called the complementary modulus, and is evidently the
minimum value of Дф, the value of Дф when ф — 90° :
VF-Z’ = z<
From eq. (1) we evidently have £ > к0, for, putting eq. (1)
in the form
P __	4
+ 2Aj2 + Al ’
we see that if = 1, then k — but as ka < 1, always, as is
evident from the figure, k must be greater than kt.
It is also evident, from the figure, that ф, > ф. Or it may
be deduced directly from eq. (3).
Since k < 1, we can write
k — sin 0,
k' =	—~k‘ = cos в.
32	ELLIPTIC FUNCTIONS.
Substituting in eq. (5), we have
k, =	= tan" 10,
I -f- Л
and we can write
k„ = sin ₽0, A' = Vi —A’ = cos 0t.
From eq. (5) we have
Substituting the value of k, in that for Л/, we get
k.-^L
 -1 + f •
We also have
2ф — 0, = Ф — (<Д — Ф)
Ф, = Ф+{Ф, — Ф),
and, eq. (3), sn (2Ф — ф,) = A sin ф,,
becomes
sin ф cos (Ф, — Ф) — cos Ф sin (0, — Ф) —
A sin Ф cos (ф, — </>) + A cos Ф sin (<A “
or
tan ф — tan (0, — Ф) = A tan 0 4" A lan (Ф» ~ Ф)>
or
tan (ф, — Ф) = tan &
— k' tan ф.
Collecting these results, we have
,	2 tZA • a
(6)
j __Д'
(7)	A = fqr£ = sin = tan* ;
LANDEN'S 7'EANSFOKMA TIQN.	33
2 V~kf
(8)	A' - r+T= COSA;
(o)	k' = —г-7* — cos 0;
™	i + A
, .	1,2	2+<A	A' 1
(IO)	i + A = —г“г/ = — l =	— —stz ;
v '	1 4	14-Z’ k	cos’40’
(u)	sin (20 — 0,) — Z’, sin 0,;
(12)	tan (0, — 0) = k' tan 0;
Г - *+A Г dK
3}	AAO 2 J ° AA.A)’
(14)	Z = v'i — Z’’, A = 4zi — Z’\
Upon examination it will easily appear that k and Za, and 0
and 0O, are the first two terms of a decreasing series of moduli
and angles; k' and Z’/, and 0 and 0,, of an increasing series;
the law connecting the different terms of the series being de-
duced from eqs. (6) to (12).
By repeated applications of these equations we would get
the following scries of moduli and amplitudes:
Ae — О(и-ы|
А.	А'	Фг
kt	At	ф1
k	A	0
The upper limit of the one series of moduli is i, and the
lower limit of the other series is o, as is indicated, k and k',
34
ELLIPTIC FUNCTIONS.
which are bound by the relation k' -|- Z'a = I, arc called the
primitives of the series.
Note.—It will be noticed chat the successive terms of a decreasing series
are indicated by the sub-accents o, oo, 03, 04, . . . ои; and the successive terms
of an increasing scries by the sub-accents 1, 2, 3, . . . n.
Again, by application of these equations, we can form a new
series running up from k, viz., k,, klt .,. .	: and
also a new series running down from k', viz., kJ, k'*, . . . k„„ —
O(n.=J. So also with ф.
Collecting these series, we have
k,a = O	kn' =	I	фя
kJ	0,
k,	kJ	0,
k......................................0
k,	kJ	ф„
kt	k'„	ф„
k„= I	kj„ =	о	0O« =	о
Note.—In practice it will be found that generally я will not need to be very
large in order to reach the limiting values of the terms, often only two or three
terms being needed.
Applying eqs. (7), (12), (13), and (14) repeatedly, we get
(i4.)
k = sin 6,
kt = 73-1 = tan" fl? = sin
1 -f- л
A» = tan“ = sin
k„ — tan* = sin fi„,
k' = cos
kJ = cos ;
kJ = cos 0CO;
kJ = cos ;
k„ — tan“ = sin ()„,
kj — cos
J^t.VDE.V'S TJU.VSFOXMst TION.
35
(M,)
(14,)
' tan (0, — ф) = k' tan ф;
tan (0, — 0,) — tan 0,;
tan (0, — 0,) = А/ tan 0,;
. tan (0Я — 0„_,) = tan 0Я_,.
^’,0) = Ц^Ж, «/>.);
<м-
Multiplying these latter equations together, member by
member, we have
(15)	0) = (I-+ ло)(1 + Лоо) . . . (I + 4»)	:
Д, &<><>> etc-, and 0,, 0,, etc., being determined from the pre-
ceding equations.
From eqs. (9) and (10) we get
1 +	- cos’ |0’	1 + *°° “ cos’ iff,’
Substituting these in eq. (15), we get
(	16)	0)	=	 £	q---------------(f-
•	»	•	1	t>«
cos	- COS	— ...	cos	—
2	2	2

Зб
ELLIPTIC FUNCTIONS.
From eqs. (15) and (10) we get
—к--------------
And this with equations (8) and (9) gives
, .	/cos tia cos . cos’ 8tn фк)
(i7)	Ш,ф} = у/--------------------------и--.
Applying equation (13) to (Zr, , 0„), (£,, <Aj, etc., we get
Л(/’>» *A>) = “3~ etc-:
but since, eq. (10),
these become
^,0) = (1 + т,Л);
, 0«) = 0 +	;
ЛА.-.. ^о(н-п) = (1 + А„)ЛАи ;
whence
(18)	F(£, </>) = (!+ 4,'Xi + Z-'J - . . (1 +	фт),
in which Z>'„, etc., , Z',, etc., ф,, ф„, etc., are deter-
mined as follows:
Let
From eq. (10),
Л — sin в,
Z\ = sin 0,.
2 Vk . .	2 t'sin 6
LANDEN'S 7'NANSJ-OEMA T70N.
37
Solving this equation for sin 0, we get
sin 0 = tan’ .
Hence we can write
	' k — sin в = tan* ;
(’8.)	Л, = sin 0, = tair ;
	kK = sin 0n.
From equation (12) we get
	sin	— фу — k si n ф ; *
(18,)	sin (2<^M	—	s>n Фо;
	. sin (2ф„п	—	= A.-t sin Ф^.к-о-
* When sin ф = I nearly, ф is best determined as foliotvs: From eq. (is)
we have
tan (ф — ф») = io tan ф,
= io’ tan ф nearly;
whence
ф — ф, = /lit,' tan ф nearly,
L‘ being the radian in seconds, viz. 2об2б4".8о6, and log R = 5.3144251.
Substituting the approximate value of фо , we can get a new approximation.
£хатр&.	— $2° 30'	i'»o = log 1 5.B757219
tan 62“ 30'	10.6805709
Л'ло	5.8757219
Л	5.314-1251
	2.0707179	U7".634 = i'.96i4
Фо — Фп — l'-96l4
фоя = 8s0 гв'.озБб	1st approximation.
38	ELLIPTIC FUNCTIONS.
To determine k’t k'm, etc., we have
' k'	—	sin rj,	k — cos r);
К	=	fqrj = tan’ t? = sin	,	k, = cos %;
^.o	=	tan’ iq, = sin %.,	A, = cos г/и;
etc. etc.	etc.
Or, since i r^,' = —y-j-, i + A'6t) = —J-r—, etc.,
cos in 1	‘° cos	*
we can put eq. (l8) in the following form :
(J9) Ф) — cos> cos’ . cos’ ’ ^«)-
From equation (13) we have
(19)*	=
whence
By repeated applications this gives, after combining,
2	2	2
= Г+?' Г+А  • , +3— W
This value gives
0. — <Z'oo = ii7".i675 = 1'.95279
.'. 0oo = 82° 28'.04721	2d approximation.
This value gives
0o — 0„ = Ч7".1бд8 = i'-95283
0oo = 83" 28’.04717	3d approximation.
LANVEN’S TEANSEOEAfA T!CW.
39
(20)	1Ц, ф) =	. Щ,я ,	.
k,, Z\ , etc., being determined by repeated applications of
or by equations (i8,).
In equation (19)* let us change Z’, and ф„ into k' and Ф re-
spectively, so that the first member may have for its complete
function
K' = I\k', Ф).
Upon examination of eq. (19)* we see that the modulus in
the second member must be the one next less than the one in
the first member, that is, Z’„'; and likewise that the amplitude
must be the one next greater than the amplitude in the first
member, viz., ф1; hence we get
I _l_ b ’
Щ’,Ф) =
Indicating the complete functions by K' and KJ, we have,
7Г
since ф = — when Ф, — ?r (see Chap. V),
^ = (i+Z-X/;
and in the same manner,
A7 = (x+^„)^,
/CM = (i + Z''0X'«,»
KJi*-ti = (T + ^,Я)К'„;
40
ELLIPTIC FUNCTIONS.
whence
A? = (I + £/)(i + k’J . . . (I +
Since
JC1	7t
' (1ф — — ,	(я = limit,)
2
о
we have
(20)* (I + £,'Xl +	• • • (I +
From eq. (19)* we have, since [eq. (10), Chap IV]
1 4~ * _	1
2	1	Z*,' *
<1ф.
е1ф
whence also, since for Фо = ~,ф = тг,
(I +ZV) л; = 2К,
(i + *'«) a; = 2Kt,
(1 + z;.) a; = 2Kn_,,
and
(I + £/)(! + kJ)... (I + Z'„)A; = 2яК;
or
a; ____________к__________
2" ~ (i + £.'Xi + ^ce) • •.
(я = oo)
(21)
“za;
Z.iA'ZJZ'tV'J 7ЛАЛ S/WtMA TIOW.
41
Let us find the limiting value of F(kaK , 0„) in eq. (15). In
the equation tan (ф„ — <?>„_,) = knl tan </>„_,, we sec that when
Л-, reaches the limit J, then фа — <Д,_, = </>„_, or фя = 2фя_1.
Therefore
Ф_„ _ 2фя.1 _ Фи-t .
2" “	2"	~ 2я’1 ’
constant, whatever т may be
Therefore eq. (1$) becomes
(21)* F{k, ф) = (1 + A,)(i +Z-..) . . . (1 + Л„)
ф
n being whatever number will carry and —- to their limiting
values.
In the same way, eqs. (16) and (17) become
(22) ДД ф) = e J----------------------• |-
cos - cos — . . . cos —
2	2	2
_	/ cos &q cos . . . COS8 . Ф^
^*3)	cos 0	2я’
n — i being the number which makes = I.
In these last three equation i,, are determined by eqs.
(14,); 0,, Ф.,, etc., by eqs. (14J * ; ft, 0,, etc., by eqs. (14,); and
k', k.', k,, etc., for use in eq. (14,) by eqs. (14,).
* Taking for <pi — Ф, etc., not always the least angle given by the tables,
but that which is nearest to ф.
42
ELLIP TIC FUNCTIONS.
BISECTED AMPLITUDES.
We have identically
Therefore
и = F[k, am u) = am
и
— 2" . am - ,	(я — limit.’
я	4
am — being determined by repeated applications of eq. (12) of
Chap. II, as follows:
. я i - cn a 2 sin’ i am и
sn - = —;—i— — -------г j------;
2 i -|- dn и i -|- dn «
. am и
• i	sin-----
и __ sin 4 am a	2
2	^/i l-cos/i COS ’
-	2
fl being an angle determined by the equation
(25)	cos fl = dn и = у,i — sn“ 11,
and я being the number which makes
к
2" am - — constant.
n
u .
am — is found by repeated applications of eq. (24).
MA'D£X'S TRAWSfORMA TION
43
Indicating the amplitudes as follows:
am и = Ф,
и
am - = 0„,
n
amT =
am| = <j\e,
и
am- = «V—
(26)
0) = 2"0И»;
n being the limiting value.
In eq. (18), when k„ reaches its limit I, we have

cos ф„н
- log. tan (45° +
and eqs. (18) and (19) become
(27) ДЛ, ф) = (1 + /’/)( 1 + Л'м)... (i +	log. tan(45°+J0ol,)
I __________________
cos'	cos’ f % . . . cos’ £г/оя
log. tan(45°-H0<,-)
“ cos’ i?? cos’ b?.. • • cos’	’ Af loStan(45°+i0.n);
n being the number which renders kn = I.
Eq. (20) becomes
44
ELLIPTIC FUNCTIONS.
(29) F{k, ф) = л	]oge tan (450 _]_ ^ф^
\f	я
= л M -P7-1 log tan (45° + W
у я M
_ . Icos	c°5 ?»» • • • cos 'Л>« .	log tan (450 4- |0ОЯ).
у	cos т/	AT
In these equations /•/, i'№, etc., are determined by eqs. (18J;
V, v., etc., by eqs. (18,); ф„, ф„, etc., by eqs. (18,); klt etc.,
by eqs. (18,).
Substituting in eq. (27) from eq. (20)*, we have
2K'
log< tan (450
2K'
<3O)	== log tan (450 + 4^,).
CHAPTER V.
COMPLETE FUNCTIONS.
Indicate by К the complete integral
(0
Фф
V I — /•’ sin3 ф *
and by K, the complete integral
(2)
Лф,_______ .
<zi — Ло* sin’
and in a similar manner Ko<), K„a, etc.
From eq. (12), Chap. IV, we have
tan (<Д — ф) = k' tan Ф
tan Ф, — tan ф
I -|- tan 0, tan ф ’
whence
tan 0, =
(1 + Z,r) tan ф
I — k' tan’ ф
1 -|- k'
—--------k' tan ф
tan ф
45
46
ELLIPTIC FUNCTIONS.
From this equation we see that when ф = -, Ф1 — я. This
same result might also have been deduced from Fig. I, Chap.
IV, or from the equation
(3)	0i = 20 — sin 20 -J- sin 40 — etc.,
this last being the well-known trigonometrical formula
tan x — n tan y,
i — n . l/l -«V . i[i -nV .
x = v — —:— sin 2y -I —:—) sin 4 у — ( —:— I sin pi' -I- etc.
S I-J-Я y 1 2\l+«/	3\l+«'
Since
;
AW °’
we must have
’ <?Ф,
AW
These values substituted in eq. (13), Chap. IV, give succes-
sively
(4)	К — (i +	,
AL = (i+We,
^Ч1(н -1>   ( I "T" OM »
whence
(5)	K= (1 + 4>)(i + U ••(!+
Since the limit of kaK is о, Kan becomes
!rZ0=-,
COMPLETE EVICTIONS.
47
and we have
<6)
(7)
______________i*____________.
~ cos’ cos’ AW, . . . cos’ ’
Л„ etc., and Wo, etc., being found by eqs. (14,) of Chap.
IV.
From the formulae in these two chapters we can compute
the values of и for all values of ф and k and arrange them in
tables. These are Legendre's Tables of Elliptic Integrals.
CHAPTER VI
EVALUATION FOR ф.
TO FIND ф, If AND X’ BEING GIVEN.
From eqs. (21) and (23), Chap. IV, we have (fl having the
value which makes cos = i)
2" и	_ 2”m Vcos t)
(') Ф« (1-|-£„)(i.. . (1 + fcos . . . cos’
from which ф„ can be calculated, kB, X',.,, etc., being found by
means of equations (14,). Chap. IV.
Then, having ф„,	, Х’„, etc., we can find ф by means of
the following equations:
sin (20M.X - ф^ = sin ф„,
sin (2фн-? Фн-i) — sin фп_19
sin (20 —0,) = A, sin 0,;
whence we can get the angle 0.
When k> 11 the following formulae will generally be found
to work more rapidly :
From eq. (29), Chap. IV, we have
(2)
log tan (450 + |0on) =
uM
46
EVALUATION FO!t ф.
49
from which we can get фик; ks, Д,, etc., being calculated from
eqs. (l8,), Chap. IV, and ф being calculated from the follow-
ing equations:
tan (</><«_,) — <£„) = kn tan 0O„,
tan (0O - 0OO) = Д, tan ,
tan (</> — </>,) = k tan ф„;
whence we get ф.
This gives a method of solving the equation
Гф = nF<+>,
where n and ф and the moduli are known, and ф is the
required quantity, n and ф give F</\ and then ф can be deter-
mined by the foregoing methods.
When k = i nearly, equation (2) takes a special form,—
Iе. When tan ф is very much less than In this case
(1ф
Vcos“ Ф k'* sin’ Ф
d-ф
4z(l + kn tan“ ф)с.оь‘ф
J'^ф = 10g tan ^45° +	’
whence we can find ф.
2°. When tan ф and approach somewhat the same value,
A
and k' tan ф cannot be neglected, Ф) must be transposed
into another where k' shall be much smaller, so that k! tan Ф
can be neglected.
50
ELLIPTIC FUNCTIONS.
These methods for finding ф apply only when 0< that
is, и < К. In the opposite case (a > K) put
и = 2tlK ± V,
the upper or the lower sign being taken according as A" is con-
tinued in и an even or an odd number of times. In either case
r < K, and we can find r by the preceding methods.
Having found v, we have from eq. (5), Chap. Ill,
am и — am (2nK ± v)
= яп ± am v.
CHAPTER VII.
DEVELOPMENT OF ELLIPTIC FUNCTIONS INTO FACTORS.
FROM eq. (12), Chap. IV, we readily get
sin (2фс — ф) = k sin ф;
sin 20
Sin Ф = ,	------.===
Vi + & + 2^ cos 2фя
________________________
V(i -|- £*) — 4k sin’ vt
_ i + kJ sin 20,
“	2 Vi — kJ sin’ ф„
(л	2	ч
since —=kt and i+^— ecls- (Q and (10), Chap. IV1;
and thence
<0
sin ф =
(1 + £.') sin ф0 cos ф,
А.Ф.,6,)
From eq. (13), Chap. IV, we have
• (!Ф.	_ i + k Г* Лф
> A) - ’ 2 / ^{Ф, k):
0	'
and from eq. (4), Chap. V, passing up the scale of moduli
one step,
, ,	A',
'+^K’
51
52
ELLIPTIC FUNCTIONS.
whence

Put
whence
, 4) = “i ап<1 A) — u>
Furthermore,
ф — am (u, Z);
0, = am («,, Z’,) = am
Substituting these values in eq. (l), we have
sn
sn (и, Z) = (i + 49 —
But from eq. (i i), Chap. Ill, we have
cn (v, 4)
dn (r, Z’,)
or
cn I -?>«, 4
\2Л
= sn
dn a,
= sn
DEVELOPMENT INTO PACTOPS.
53
whence
(2) sn (», k) = (1 + Z.') sn sn
(Mod. = Z,.)
From this equation, evidently, we have generally
(2)* sn (v, Z„) = (1 + Z„'(M H>) sn ^"+-’ v sn	+ 2-K.fJ.
(Mod. = Z„ + 1.)
Applying this general formula to the two factors of eq. (2),
we have
(K,u \	K,« Г Kt (Kji VI
sn СТ’ 4 =(l + k sn 2ЛТ CT •sn Ict ct +2A J J
(Mod. ZJ
= (I + Z'„) sn sn (« + 4/0; (Mod. Z,;)
<3) sn TK' + 2K = + Л'“) snV^ (“ + 2K^
• sn 2/<D& + 2A') + 2A']‘	(Mod- •)
The last argument in this equation is equal to
Й-(» + 6ЛГ);
and since, eq. (7), Chap. Ill,
sn (w, Z,) = sn (zKt — u, Z4),
f The analogous formula in Trigonometry is
sin = J sin sin Цф 4- ff).
54	ELLIPTIC FUNCTIONS.
we can put in place of this,
2Л; - ^ (« + 6A' =	(2A - «);
whence eq. (3) becomes
sn [ S +2A'>- *•] =(I + s" Й (2A'+ »>
 sn 2^ ^2A" ~ “)•	(Mod. £a.)
Substituting these values in eq. (2), we have
(4)	sn («, k) = (1 4- ЛД1 + sn
' Sn ^2K ± sn ^A" + (Mod- M
in which the double sign indicates two separate factors which
are to be multiplied together.
By the application of the general equation (2)* we find that
the arguments in the second member of eq. (4) will each give
rise to two new arguments, as follows:
A"a« K,u
2Vf g,ves PA’
and
+ 2A'’) “ Й + 87Г);
(2K ± u) gives (2/f ± «),
DEVELOPMENT INTO FACTORS.
55
and
^-Г^-(2ЛГ±«)+2л;“|=^(1оЛ-±й), . .(*)
2A3 L.2 A	_I z л
2^(4AT + «) gives ^(4Z<+«),
and
sx Й +u}+2Ю = й (12*+K)- • •
Subtracting («) and (Z>) from 2АГ,. by which the sine of the
amplitudes will not be changed [eq. (7), Chap. I1IJ, and since
our new modulus is A,, we have for the expressions (r?) and (ty,
^(6ZCTW);..........(«')
..........................(*')
Substituting these values in eq. (4), and remembering the
factor (I 4~^„s) introduced by each application of eq. (2)*, we
have
Ku
sn (u, k) = (1 4- £.')(i 4- i 4- Л'„)‘ sn ~K-,
. sn (2K ± u) sn ~ (4ЯГ ± u) ;
. sn (6K ± u) sn	(8K 4- u).	(Mod. A,.)
2 A	2 A
5б
ELLIPTIC FUNCTIONS.
From this the law governing the arguments is clear, and we
can write for the general equation
(S)	sn («, Л) = (I + 0(1 + ZOXi +	• • • (I + £'.У'
, ...
•Sn^sn^>(2^±")
’ ЗП Й ± SI1 (бАГ ±
• ‘ ‘ Sn 2^A'^2"“ ~'K * "J
.sn^(2"TC+«).	(Mod. О
Indicate the continued product of the binomial factors by
A', and we have
л - (i +<>0+*;Л1 +*;,y(i+^y...
Since the limit of A’/> ctc-> *s ^ero.it is evident that
these factors converge toward the value unity. It can be shown
that the functional factors also converge toward the value
unity. Thus the argument of the last factor can be written
From eq. (n), Chap. Ill, we get then
(Mod. o
But since i„ at its limit is equal to unity, cn = dn ; whence
the last factor of eq. (5) is unity.
DF.VELOPMENr INTO FACTORS.	57
From eq. (21), Chap. IV, we have
Kn 2Я
Therefore for n = w, eq. (5) becomes
sn (a, Z) = A' sn —sn —777 (2K ± u)
'	'	2K 2K
7Г	7t
. sn <AK ± a) sn (6A' ± a), . . . .
(Mod. i.)
or
тга Г ” ~1 ?r
(6)	sn (a, k) = A' sn |_ Л* J sn ^77 (2/1К ± a),
(Mod. 1,)
where the sign [77] indicates the continued product in the same
manner as 2; indicates the continued sum.
When k = 1, / V = J whence and sin ф =	^(ф, k) becomes Л	„1!	1 4 J COS Ф	I - I + sin ф *—	:	, I — sin 0 — I f* - £22 _ rfiv _L т	—I— v
58
ELLIPTIC P UNCTIONS.
Hence in equation (6)
. mi faA" - e
s,n 2/C = ™-----------ПЕ;
Jtv/C пн	kttK itti
i
SO 9 i"	±	АжЛ" ttu	Air A'	WH *
em z,±^“+ e ~ *' ef^'
Put
irA''	_wA'
(6)*	? = *" K , i' = e K'f
and the last expression becomes
sn —УУ7 (2/1K ± tl} =
qT h e* lh' — e* aA
, -A ± lA-i , A
4 е +<?
sn 2K' (2/zj^+ “) bn ^K‘ ^2kK ~
1TM	ITW	gw	irw
?'-лГзА"-/\^
ПЫ	Я-W	«Й	gtf
i 2,Kf I t b — йАГ/ f л	I	J9
У e -\-^e q e 4- q e
(gw	g« \
~7c 1	— к * 1
_____________e J e ) _
(ITU	gw \
г*' + ЛА')
From plane trigonometry we have the equations
f*— e~-r ...
---:----= — г tan гх, - e~^— 2 cos гх;
4- e~x
DEVELOPMENT INTO FACTORS.
59
where » =	— i: which gives
пи . теги
sn' —777 — — г tan —777
2K'	2K
(Mod. i;)
sn 7^77 (zkK 4-«) sn 2^y (zkK— U)
У	У аЛ — 2 COS
q' -aA 4“ /lA + 2 COS -^7
, . rniu , , ,
I — 2q aACoS -re 4" Q ‘A
A
, . mu ‘
I 4- 2? зЛ COS 4- 4*
From eq. (io), Chap. Ill, we have
sn (u, A) = — i tn (iu, k').
Substituting these values in eq. (6), we have
tn (iu, A') = A' tan

, . TtiU . .
I — 2q 1K cos 4- 7
, * mu
I 4“ 2? COS -777'
Now in place of the series of moduli A', ka' and the cor-
responding complete integral K', we are at liberty to substitute
the parallel series of moduli A, A‘a and the corresponding com-
plete integral К; calling the new integral u, we have
(8)
I — 2^ COS -гл +
7ZU	Л
tn (u, A) = A tan n-------------------—--------
I + 2^ cos 4- q^
бо
ELI.!РTIC FUNCTIONS.
rrw
I — 2<? cos -y? 4- q
z 4.	A	1	2
= A tan ---------------------------
IK	7ГК.
14-2? cos ^ + (7
I— 2/cos 7f+?”
I 4- 2Q COS +<?
7ГМ
I — 2q" cos4' ?ia
JTW
14-2? cos к 4-?
where
(9) A = (1 4- Z’.)(I +	4- W4- *..)• • • -
Now in equation (6) put и 4* К for и, and we have, since
cn и
[eq. (11), Chap. Ill] sn (к + A ) =
=A' *<	[я] « V.2k + .)/r+ «]
.sn — [(2Л- i)A7-a].
(Mod. 1.)
Now from 2J1 — 1 and 2Л 4“ 1 we have the following scries of
numbers respectively:
2/z — I ; I, 3, 5, 7, 9, etc.
2Л + 1 :	3, 5, 7, 9, etc.
It will be observed that the factor outside of the sign [If],
viz., sin am	would, if placed under the sign [П], supply
DEVELOPMENT INTO FACTORS.
6l
the missing first term of the second series. Hence, placing this
factor within the sign, we have
(IO) dn7z ~ A' И sn i K2/z -
.sn —777L(2^ — i)AT— к].	(Mod. i.)
Comparing this with equation (7), we sec that the factors
herein differ from those in equation (7) only in having 2/; — I
in place of 2/z; hence we have
sn ^L(2^ -	«] sn "" »)A'- «]
I — 2Q COS -777 +	**
=. (Mod. 1.)
I 20 aA-1 COS j£>’4" ? +*”a
From eqs. (10), Chap. Ill, wc have
cn (и, Z) _	1
dn (?z, k) dn (iu, Z') ’
whence eq. (10) becomes
, . niu . , i
1 — 20 lA-1 cos 4- 0
(”) dn (iu £4= A P'	яТм 	’
'	*- J J 4. =a-i cos —у 4*-»
A
and when in place of iu, k', K', q', A', wc substitute «, k, K,
q and A, and invert the equation, we have
I + 204*-1 cos 4- 0<A-S
(12) dn («, Z) = z [И]--------------- —-n— •
L J I — 20=*-1 COS -д, 4~04А-а
б2
ELLIPTIC FUNCTIONS.
Bearing in mind the remarkable property (Chap. Ill, p. 29)
that the functions sn и and dn и approach infinity for the same
value of m, we see that both these functions, except as to the
factor independent of u, must have the same denominator.
Furthermore, since sn и and tn и disappear for the same value
of u, they must, except for the independent factor, have the
same numerator. Hence, indicating by 7? a new quantity,
dependent upon k but independent of к, we have
1 — 24^ cos -v>- + y**
<13) sn («, Л) = В sin
I — 2y2A_I COS -vy -|-
and since
cn и —
sn и
tn и'
we also have, from eqs. (8) and (13),
<I4)
cn (?f, k) =
В 7ГИ
A cos Ik
7TU
I — 2y14-’COS
П
Collecting these results, we have the following equations :
пи г -] 1 -2?3Acos —+
(15) sn (г<, Л) = В sin |_7lJ-------------------- ------,
j I — 2^*-' COS
, mt
r>	r- 1 + C0S + 4*
(гб) cn («, Л) = z cos [л]------------------------------
1 — 20=A~1 COS	4-
DEVELOPMENT INTO FACTORS.
63
т I 4-2^- COS -4-0*-*
<.7) dn («,Л) = Л [л]------------------—---------
‘	I — 2y’*-1 COS -дг 4-
To ascertain the values of A and Б, we proceed as follows:
In eq. (17) we make и = о, whence, by eq. (13), Chap. II,
we have
whence
<18)
I —
Ц _ ?0Л-.
In equation (17), making и = К, we get, by equation (1),
Chap. Ill,
09)
) = л-
•4=^-
We have identically
=4=-»з=я^
A A
whence
A
ЛГЯ
To calculate B, put = r; if we change —=? into
2A
64
ELLIPTIC FUNCTIONS.
пи , iifK' ... ,	. .
v will change into v
2K 1 * 2Л
У7, and sn и will become, by
eq. (14), Chap. Ill,
sn (« + г^') = т—-
ft- Э11 «
Now, replacing sin and cos by their exponential val-
ues, and observing that
I — 2?- cos -J- <?” = (l —
we have
_ В v — r- [7T](i -
Changing и into « iK', and consequently v into v tfq, we
have
1______В
k sn U ~ 2
v Vq — I'-’ iV7’ [Я](1 —
Multiplying these equations together, member by member,
and observing that
v — v~' =
we get
1 _	1 ~ 4V'. V(1 _ v.^.
A 4* v\lq	[П](1—
DEVELOPMENT INTO FACTORS.
<>5
S'
4*V
(l — jv’)r(l — r"=)
(I - g^‘)(i - /И)
(i —	— <?’*'”)
. 0 — g^~*Xi —	•  •
(I _	_ ^V-’) ...
7?’ j_
4 '
... В =	•
whence
!k'
Л = 2^У^’
Substituting these values in eqs. (15), (16), and (17), we
have
(20)
sn (w, Л) =
2*7
. 7111Г rr
sm Ж|_Л
I — 2^ cos -)- q*
.	71 U
I — 2/*-1 COS -g -|- ^*-2
I \ I 1Л 2	^9
(21) сп(»,Л) =-----
T k
cos St77]
1 + 2/* cos + <7^
nil ’
I — 2qth * 1 COS -p. -f-
_ I + V*“’ cos к +?**-3
(22) dn (u, Z-) - УУ nj------------------- ------
L J 1 — 2/*'1 cos + «т4* 1
CHAPTER VIII.
THE 9 FUNCTION.
We will indicate the denominator in eq, (20), Chap. VII, by
thus:
(1)	ф(м) = [77](i — 2?1*- cos 4-
We will now develop this into a series consisting of the cosines
ttw	it и	,
of the multiples of . Put ^7 = xt whence
2 cos -i, = (e** + г-**);
Л
but
1 _ 2<7k~1 co-s + y*-a = (I —
and therefore
(2)	0(«) = (1 - ^“Xl -	•  •
(1 —	— yV 2 jr)(i — qbe~3")....
Putting now и 4- 2гЛГ' instead of u, wc have
7г(м4"2гЛт/) niK'
27fK'
2ixl = 2гх------;
66
ТЛЕ в FUNCTION.
67
and
?
From these we have
ф(и + 2iK') = — ^ e~-,JI (1 —	— 7*^) • . .
(1 — $tf"I,jr)(i —	. ..;
whence
ф(и -|- 2г'Л") = — e~‘“ 0(a),
or
ir/«
(3)	0(« + 2tK')= — q-'e *
Now put
4	.	. . „	2mi , -inrw ,
(4)	0(a) = A + л cos + C cos + D cos 4- etc.
Since
cos = i(eI,x + f™),
this becomes
(5)	0(a) = A + iBe** +	+ iL>cJ,Le + . • .
4- |Z?c+ iCe ^ix -r	;
whence
(6) — - e~*" 0(a) — — — c-”* —-------- e>!x — e*ix — . .
x ’ q v / q 2q 2q 2q
B	 C « D	и
____p-VX ____.т-OJJf _ _ .,-WJf _
2q 2q 2q
68
ELLIPTIC FUNCTIONS.
Now in equation (5) put и -|- 2tK’ in place of г/, remember-
ing that e*ix and f3i* are thereby changed respectively into
and q 3e and we have
(7)	+ 21Я") = A ~r e** +	e*‘* + -%-	4- . . .
^2	2
В	c
+ a“ + —i e-*ix + . . .
2?	-2Q	'
Since equations (6) and (7) are equal, we have
2(7
£ _
2q~ 2 *
B> Cq'
2q~ 2 ’’
В = — 2qA ;
£>= -2/Л;
whence
[77](I — zq^-' cos
(8)
пи ,	, тли , \nu
= л(1 — 2<y cos -f- 2q cos — 2q cos
+ 2g cos
A
The series in the second member has been designated by
Jacobi and subsequent writers by <У(«), thus:
(9)
nu	2nu
<9(и) = i — 2g cos -77 2q cos
CHAPTER IX.
THE 9 AND H FUNCTIONS.
In equation (20), Chap. VII, viz.,
.	2 Vq . nu
sn («, Z-) =-- sin —
V 7 Vk 2A
I — 2^’ACOS-^
I — 2<?гА"' cos -.J. -j-q^~3
the numerator and the denominator have been considered sep-
arately by Jacobi, who gave them a special notation and de-
veloped from them a theory second only in importance to the
elliptic functions themselves.
Put [see equation (8), Chap. VIII
(l)	0(w) = 1	— 2?51-1 cos -f- ^-a).
! 7	. iru Г П	пи
(2)	Я(«) = 2 J Sin |J7JO - 2<? cos yf + <r) ;
A being a constant whose value is to be determined later.
From these we have
Sn (w, A) = —p .	.
V 7 Vk ®(«)
(3)
6g
ELLIPTIC FUNCTIONS.
The functions sn и and cn и can also be expressed in terms
of the new functions; thus we have
(4) cn (», £) — у . 2 Vq cos
к	2K
 пи
I + iq21' cos „
nu
I—2<7sb-'COS T>-
or, since sin -r = cos
and cos ж — — cos
?Kx
and putting и = ——,
Replacing —— by its value, u, we have
(5)
cn (u, ty —
r# +^)
k &(ti)
Furthermore,
(6)
dn («, k) — Vk' £Tzj
I 4- 2q>!,~1 cos + t4*-’
I - 2^-* COS
THE ® AND H FUNCT/ONS.
71
gives in the same manner
	<’n—’
or
(/)	dn (a, Z) — Vk	.
If we put
(8)	Я(» + А’)=Я1(«),
(9)	<9(w + A') = ©,(«),
the three elliptic functions can be expressed by the following
formulas:
(10)	, Al	1 sn (а, я) = —7=. • —J—. 4 7 Vk <У(") fk' fkht)
(«0	cn {и, к) - у k	,
(12)	dn(«,Z-)= VZ'
These functions 0 and H can be expressed in terms of
each other. By definition,
ELLIPTIC FUNCTIONS.
72
but
and consequently
(13)	H{u)—C^qe ,K	AJb - A‘ Д1-^ \
Now, changing и into и 4" iK', and remembering that
-*k-
e A = q, we have
(14)	/7(« + :7T)
—iriu	/ жйЛ f	win\ /	iriu\f	irr*A
= Cq *<гаЛГ y~li— qe Kl/\I — qe Д1— <?VAД1— q'e
and reuniting the factors two by two, this becomes
(15)	H(« + /A-)
—C^—xq «ЛД1 — 2y cos	2^’cos^r 4-0*
and finally, according to equation (1),
(16)
H{u 4- iK') -
le
THE 0 ЯЛ7/? П EUHC77OHS.
73
In the same manner, we can get
(17)	(У(«4~ iK) =	>? *e
Substituting и — 2K for и in equations (1) and (2), we get
(18)	©(к + 2K) = ©(«),
(I9)	/7(« + 2АЭ=-Я(«),
7t ,	,,	ПИ . It ,	. %l<
since cos [и + 2Л) = cosand sin ^>(«-|-2A )=—sin
The comparison of these four equations with equations
(10), (11), and (12) shows the periodicity of the elliptic func-
tions. For example, comparing eqs. (10) and (16) and (17), we
see that changing и into и -f- iK' simply multiplies the nu-
merator and denominator of the second member of cq. (IO) by
the same number, and does not change their ratio.
The addition of 2K changes the sign of the function, but
not its value.
We will define 0, and Я, as follows:
(20)
(21)
0,(x) = O(x + A');
Я,(ж) = H(x + A').
Hence we get, from equation (17),
О,(л- -f- iK) = 0(x + iK' + A') =	-J- A'+ iK')
---'0x4-aA'-r<A")
^ iH(x + K)e ‘A
--^.0x + .A")	,__
= /Я.(^*А	(—^=4),
rr
since	e 1 = cos------/—~I sin - = — У — i ;
2	2
74	ELLIPTIC FUNCTIONS.
whence
-	iJP)
(22)	©.(x-HZC) = Ht(x)e <A
In a similar manner we get
(22)*	//,(* +/A") = <y,(x> <A
2Кл
In eq. (9), Chap. VIII, put и =	, and we get
(23)	=1 — 2? cos 2z 4- 2/ cos 4s — . . .
Now, in this equation, changing z into s + ~ , and observing
eq. (20), we get
(24)	6>,	= 1 + 2? cos 2z + 2/ cos 4г + • • •
Applying eq. (22) to this, we have
THE G A.VD -H EUHCTJOHS.
= W [I +	+ •  •]
= [I + ?’^£1 + ?'f1" +•••
+ <f-^ + ^-^_|_...]
= S'1 I?" +	+••
+ ^ + ?v-3*4-A_i“+-- •]
= 2gi [cos z + q' cos 3-s + <7* cos 5-® + •  •] J
whence
(25)	H, (-^77 = 2 +V cos s + 2 *У cos 3* + 2 cos 5^4- ...
я
In this equation, changing z into z — -, and applying eq.
(21), we get
(26)	//(—) = 2 sin z — 2 sin iz + 2 ^’sin sz — ... ,
We will now determine the constant A of eq. (8), Chap.
VIII, and eqs. (1) and (2) of this chapter. Denote A by/(?),
and we have
(26)*	[Я](1 - 2^*- cos у +	= №)©(«)
Kr
Substituting herein и — о and » = --, wc have
[H](i -?’*-)’= Ж«(о);
[Я](1 + ^1)=Л?)<у(7).
ELLIPTIC FUNCTIONS.
7$
From eq. (9), Chap. VIII, we get
(27)	®(o) - I — 2q + 2<?‘ — 2tf’ 4- 2q'r' —
(28)	= I - 2q' 4- 2?1’ - 2?” 4- 2/’ - . . .;
from which we sec that 0(o) is changed into 0^—J when we
put q' in place of q.
Whence
and therefore
А?) _ ГгЯ 1 -1 ^~a
(29)	= H(i -/*-->)( 1-*
Now, the expressions 4/z — 2, 8/z — 4, and 8/z give the
following series of numbers:
4/z — 2,	2, 6, 10, 14, 18, 22, 26, 30, 34 ;
8Л—4,	4,	12,	20,,	28,	36;
8/z,	8,	i6,	24,	32.
Hence, the three expressions taken together contain all the
even numbers, and
Therefore, multiplying eq. (29) by
we have
THE & ЛЛ’£> H FUHCT/OHS.
77
Now in this equation, by successive substitutions of g4 for qr
we get
Л/) _ [>/11 -g^
ЛГ) “ L Ji -
ЛИ _ Ггг~11 - <r*
/(g“) - L"J I -	’
ЛИ _ r.rrl1 -
ЛИ) - L - 4^’
Now q being less than i, q” tends towards the limit о as n
increases, and consequently i — g" tends towards the limit I.
Also, from eq. (8), Chap. VIII, we see that До) = i. Hence,
multiplying the above equations together member by member,
we have
(30)	Л?) = Hrzpi»
or
(3°	A = (i-?’Xi-«?')(*-?“)••• ’
Substituting this value in equation (8), Chap. VIII, we have,
after making и = о,
(I - g) (I - ?*) (I g)---(^g’Xl-g’Xl-g0)...
__________6(0)________
“(r -/X1 —/X* -g’)--.'
(See equation (9), Chap. VI11.)
Transposing one of the series of products from the left-
hand member, we get
,	v „	6(0)
°	?K*	n-------(i-?Xi-g‘Xi-g’)(i-g‘)---‘
/8
F.Ll.IP I'lC FUNCTIONS.
Introducing on both sides of the equation the factors I —$*,
I — у4, I — q“', etc., we get
(I - ?)(I - /)(! - tf’Xl -?')••
' 7 I — q 1 — q I — q I — q
= ©(oX1 f <?)(’ + ^X1 + /)-••;
whence
(i — q)i I — tf’Yi — q')
<32>	= (T+l, 4x7+7)-
Resuming equation (20), Chap. VII, and dividing both
members of the equation by 11, we have
sn »	2 t'q
~u ~
. як ,
I — 2q^ cos
This, for « — o, since the limiting value of —— for и = o is 1,
Till
sm
and of ---— for x — О is —-, becomes
_ П (I ~ g?(l ~ ?У(1 ~ gV   •
1 - 7Л- • К • (I - q}\I - ?T(I - qy . .
УТгК _ Г(1 — 4)(i — /X1 —?*)•• -T
n \'q 41 - Q )(« - S’1/!.- <y}---------1 ’
Further, from equation (21), Chap. VII, for и = 0, we have
_/z; _ _ rei	-1-^(1 ± Л • • -T
2^'i'q Ц1 -?)(l “«’’К*—!?’)---I ’
THE <9 AND П FUNCTIONS.
79
The quotient of these two equations gives
2 SPA'_ F(i i-^(» -?•).--T.
n ’ I-(I +/)(! + ?‘)(i +«?")-. J ’
or, substituting the value of SP from eqs. (l8) and (19), Chap.
VII,
(36)
2££= rO-gXi-g’Xi -ga)---T
Ц*+«?)(!	J-
Comparing this with equation (32), we easily get
<37)
/ k‘' K"
From equation (9), Chap. VIII, making и = К, we get
(38)	0(A') - I -J-2? + 2?’+ 2/+ 2/e + . . .
Making s = 0 in equation (24), Chap. IX, we have
<39)	©.(o) = I 4"2? + 2/ + 2/ + . . .
This might also have been derived from eq. (38) by observ
ing that
0/o + A’) = 0,(о) = 0(A).
Knowing 0(o), it is easy to deduce 0(A") and if(AT)
From equation (7) we have
,	£)(г< -J- K)
dn « =	_^J.
Making и — о, we have, since dn (o) = 1,
<40)
т Л’
So
ELLIPTIC FUNCTIONS.
From equation (5) we get, in the same manner,
(4i)
ВД =
From eq. (12), Chap. IX, we have
.------------------------------- ------ 0 («)
dn к — V I — A sin ф = 1 k ... , ;
(4 0*
TtU
and putting x = wc have
dn в I + 2g cos 2X -|- 2g* COS 4X -I- 2g* COS 6jr 4-   
^4-)	— I — 2q cos 2X -j- 2g* COS 4-Г — 2g*’ cos 6x -J- . . .’
Putting
dn и
(42)*	= cot y,
V к
we have
cot у — I	, „	.	cos 2.Г4-/(4 cos3 2X— 3 COS 2T)+ ...
c-^7+i=t“(4S ~r}=2^---------------------------------------
whence
tan (45° -	+ g’(4 cos3 2x - 2)]
(43)	COS 2X = --------------------------------
— g'(4 cos’ 2л: — 3 COS 2x) — ...
and approximately,
tan <45° - y)
(44J	cos 2'r =----~Tq-----'
From equations (37) and (40), Chap. IX, we have
(45)	x =	:
whence
(46)
и = л-@3(Ал).
CHAPTER X.
ELLIPTIC INTEGRALS OF THE SECOND ORDER.
FROM Chap. I, equation (19), wc have
£(Л, 0) = / 4zi — k' sin’ ф . d(j> = f Аф . с!ф.
From this we have
£(0) + £(0) Лф . <1ф +£*Лф . <1ф.
Put
(1)	Еф-\-Еф = 8.
Differentiating, we get
(2)	ДФ .	+ Лф . Дф = dS.
But we have, Chap. II, equation (2),
4ф г/0
4Ф "+ Лф ~ °’
or
(3)	4ф . $ф Ц- 4ф . dty = о.
Adding equations (2) and (3), we get
(4)	(4ф + 4ф№Ф + Лф) = 4$.
8t
82
ELLIPTIC FUNCTIONS.
Substituting cos ju from eq. (5), in cq. (5)*, Chap. II, we get
’	, sin 0 cos 0^ju -T cos ф sin ф
Дф — —	-------------
sin /<
(5)	sin ф cos 0zJ/z -I- ccs Ф sin Ф
sin j.i
whence
(6)	J0 ± J0 = sin (0 ± 0).
Substituting in equation (4), wc have
= S*n +
(7)	=-^T7I</cos^+^
Integrating equation (7), we have
Еф + Еф = К - cos (0 + 0)].
The constant of integration, C, is determined by making
0 = o; in this case ф = ц, Еф — о, Еф = £/<, and ,S = Ец;
whence
z?	+ 1	ч
<r-COS/‘>’
and by subtraction,
£0-|- Еф — Ец = I(cos ju — cos 0cos 04~ s’n Ф s'n ^)-
But, Chap. II, eq. (5),
cos ju — cos 0 cos ф = — sin 0 sin ф4ц;
ELLIPTIC INTEGRALS OF THE SECOND. ORDER. 83
whence
_ , . _,	_ I — 4'p .	,
ЕФ + ЕФ — Ep — s-n — sin </> sin ф
whence
(8)	Е,ф + Еф = Ep-\-k2 sin ф sin ф sin p.
When ф — ф, we have
{9)	Ер = 2ЕФ — £2 sin’ ф sin p.
But in that case
(10)
whence
<n)
cos p = cos’ Ф — sin’ ф4р;
Let ф, ф', ф^, etc., be such values as will satisfy the equa-
tions <12)	Еф = 2Ефь — k2 sin1 фк sin ф, Еф1 = 2Еф^ — k2 sin’ sin ф* , etc.	etc.
Assume an auxiliary angle y, such that
(13)	sin у — k sin ф;
whence
Аф ~ cos y,
and Chap. IV, eq. (24),
. .	. , sin 1Ф
{14)	sin ф} —----v—.
x '	cos Лу
84
ELLIPTIC FUNCTIONS.
Applying eqs. (13) and (14) successively, we get
	sin фь =	sm &Ф 	i , sin yi cos iy	r	= k sin 04;
	sin </>j —	sin Л01 		sm yt cos	'1	— k sin 04 ;
(15)	J			
		sin	
		2" -1	
	sin фт_—	cos Ay  ’	
		2“'«	
whence
(16) Еф = 2пЕф2_ — (sin ф sin’ yj + 2 sin 0t sin’ /4
««
4- 2’ sin ф~. sin’ y± + • • • 2”'T sin sin* у r
i4-
To find the limiting value, Еф\ , we have, by the Binomial
N
Theorem, since sin ф — I — т— r.-etc.,
Z?0 = (l — sin’ ф)Ъ
Whence
Екф^ — /’*” Иф^ф
а» Jo j”
(17)
ELLIPTIC INTEGRALS OF THE SECOND ORDER. 8$
Substituting in eq. (16) the numerical values derived from
equations (15) and (17), we are enabled to determine the value
of Еф.
Landen’s Transformation can also be applied to Elliptic
Integrals of this class.
I'rom eq. (11), Chap IV, wc get, by easy transformation,
(18) sin* 2Ф — sin’ ф1 (1 4-Z’o + 2/„ cos 2ф).
From this wc easily get
2/, cos 2ф sin* Ф, — sin’ 2ф — sin’ 0, — i' sin’ ф,
— I — cos* 20 — sin’ 0, — sin’ 0,
= zl’^0, — sin’ 0, — cos’ 2ф;
whence
cos’ 20	2/0 sin* 0, cos 20 = zl’X’,,0, — sin* 0i J
and from this,
cos 20 = — Z’o sin’ 0, ± УЛД — sin’ 0, + /0* sin* 0,
(19)	— cos 0, J/,0, — /0 sin’ 0,;
whence, also,
I—cos’ 20= 1—cos’ 0,0’0,4-2/ sin’ 0, cos 0,0/,.,0,—// sin’ 0,
= sin’ 0, (1 -|-// cos’ 0,4-2/, cos 0t J/,0,— sin’ 0,)
and
(20)	sin 20 = sin 0, (0/o0, -|- A> cos 0,).
Differentiating equation (19), we get
00	. , (A, cos 0, 4- ^/„01У
2 Sin 20 -j— = sin 0,  -----=—----------.
00,	1	^/.0!
Dividing this by equation (20), we have
200 k, cos 0, -|- 0£„0,
00,	0/„0,
86
ELLIPTIC FUNCTIONS
But from (19), and eq. (6), Chap. IV,
Л3 sin’ 0 —
k‘ (1 — cos 20)
2
2 A,
{i + kf sin’ 0, — cos 0, ^^0, J;
whence
л t	0, Ч- cos 0j
=-------—7-------,
and
... d<t> (A, cos 0, + JAt0,)*
(1+^.0,	'
and
<!ф МФ =	. fe5°»O.+/W
W	Zj£o0, 2(1 +^0)
This gives immediately, by integration,
= 5^) /ад И. “» * +	!
= Й, Iti ['+ 2Z'-“s Ъ‘иЖ ~ A"!
. .	£A„0, , A, Sill 0,	LXr-L,
<*0 =,+?;+-,адг-«'-«гал-
Thus the value of Ek$ is made to depend upon £AO0,
(containing a smaller modulus and a larger amplitude), and
upon the integral of the first class, Ek„<pt; ku, 0,, etc., being
determined by the formulae (6) to (12) of Chap. IV.
By successive applications of equation (21), Ek<p may be
made to depend ultimately upon Е/г,вфя, where A0B approxi-
mates to zero and Ek,w<t>„ to 0„.
Or, by reversing, it may be made to depend upon Екпфю>
where kn approximates to unity and Е/г„фаи to — cos 0ON.
ELLIPTIC INTEGRALS OF THE SECOND ORDER. §7
To facilitate this, assume
GW = Ek</> — Fk$.
Subtracting from equation (21) the equation
I - I- k
Fk$ = - ' -/56,0, (see eq. (13), Chap. IV),
2
we have
Gk$ = ~-r (G^W, + К sin 0i —	•
1 I
Repeated applications of this give
Gkjt\ — , i~T (g^o0, + s>n 03 - ^Ло0»),
* “T= Л410
GAtln.^„., =	+ k>* sin 0« “
1 l" Ло"
Whence
(22) GW = 2И‘
^0»r(sin Fk(ln$*) ( 1 Gkttiфн
Й(1+Ъ) J №+An)*
к	Я
But since (compare eq. (13), Chap. IV)
^ОЯ0В[П](1+^)
FW =------=4-----,
or
(23)
/^„0,2"Fk<i>
й(1+и ило+^У
88
ELLIPTIC FUNCTIONS.
and since, also, (compare.eq. (6), Chap. IV,)
^Л(я-1)
Awi
2
we have
2"Л„К
(24) -r^----------=
[i<](i+ W
kt
г;
2й
2" k,
2" L„_
X-
k


Substituting these values in equation (22), and neglect-
ing the term containing Gkm<p„ since, carried to its limiting
value,
= <Д> — Фп = О, (n = limiting value,)
we have
f k sin ф„ [jyj A,,.,, - Z-*[77] Z’0(M_n )
(25)	б7ф=2’„’л —----------=____________2_____ r
I	2H	I
= k »in A + sin	и sin ф. + . . .J
- 7 D + 2 + t +	+••]:
whence
(26)	w = w[i-^(l+| + ^+...)]
+	k sin	sin + -1®^” sin 0, + . . Л .
2
ELLIPTIC INTEGRALS OF THE SECOND ORDER. 89
From eq. (3), Chap. V, we see that when <f> =
фп = 2”-'я.
Substituting these values in equation (26), we have for a
complete Elliptic Integral of the second class,
In a similar manner we could have found the formula for
E{k, <•>) in terms of an increasing modulus, viz.,
Cl 2	2* 2s
•+Ц1+д+цтщ+ •  
r 2"-’	2"-’
+ z-д . . . A., “Z’A . . .
Г	2	2’
— k ^sin ф + -pj sin 0, + -r-— sin 0, + . . .
2м “1	2”
+"^7 2	' —Г— 5‘П	~ HTZ -^~T= S‘n ’
4/	b	A-	£•	—I
CHAPTER XI.
ELLIPTIC INTEGRALS OF THE THIRD ORDER.
THE Elliptic Integral of the third order is
/"♦ d-Ф
(i)	Z7(«, k, 0) -J (l+„sin’0)J0-
Put
(2)	П(0) + Л(0) = 5;
whence we have immediately
(1Ф	. dty_______________
(3)	dS =	_|_ n sin, ф^ф + я sin’ 0) J0’
But, eq. (2), Chap. II,
(4)
whence
(5)
dtp .dtp
J0 + J0 = O;
( I __________________I .r/0
4 + я sin’ 0	1 -|- n sin’ 0/ J0
«(sin1 0 — sin’ 0)	_ . t/0
(1 + n sin’ 0)(i + n sin’ 0) J0‘
From equation (8), Chap. X, we get by differentiation, since
<r (or ju) is constant,
J0 . Дф J0 . </0 = k1 sin <r <Z(sin 0 sin 0),
or, from equation (3),
(sin1 0 - sin’ 0)
= sin <Ti7(sin 0 sin 0).
91
ELLIPTIC INTEGRALS OF THE THIRD ORDER.
This, introduced into equation (5), gives
n sin <r<7(sin ф sin ф)
— I + « (sin* ф -|- sin’ ф) я’ sin* ф sin’ ф’
Put
sin ф sin ф = q, sin’ ф -|~ sin* ф — p',
whence
n sin <rdq
dS = 7~i—
47	I -|- n/ + я /
Prom equation (5), Chap. II, we have
cos <r = cos ф cos ф — sin ф sin фА&,
from which wc easily get
(cos cr + ^Jcr)’ = cos’ 0 cos’ ф)
= (1 — sin’ 0)(i — sin* ф)
= i -/ + ?’>
and thence
p — 1 +	— (cos a + ?^cr)’
= sin* <r — 2 cos crJcr^ + Л* sin* o’ . q*.
This, substituted in eq. (6), gives
n sin cdq
dS=T±7t sin’ <r — 2n cos <T.4aq -f- n{n -|- k' sin’ <r)j*
я sin adq
~ A— 2Bq + б?* ’
where
A = 1 + K sin* o’,
В — n cos <tJcf,
С = «X’’ sin’ <r
92
ELLIPTIC FUNCTIONS.
From this we get
Л' = n sin <r / —---------—
J A — 2Bq + Cq
f- Const.
In order to determine the constant of integration wc must
observe that for Ф = o, r/- = <7 and q — о ; whence
—2^+ty + C°nSL:
whence
5 = Пег -|- n sin
or
(7) Пф + Пф = Пег + n sin
But we have
Vq
A — 2Bq + Cq*’
J, A — 2 Bq -}- C"/
_ CMdq
~ AC- (Q - B?
CM	dq
~~AC—B*'	/ Cq-B у
1 + \ VAC — ВЧ
Cdq
M	VAC- B7"
~ Vac —h ' , / Q у
1 + \VAC-B'l
where M = n sin <r.
The integral of the second member is
M	Cq-B
=- tan*1—--— ;
VAC-ВГ VAC-Bi
ELLIPTIC INTEGRALS OF THE THIRD ORDER.
93
whence
A	M Г Cq -.В ,	В -1
VAC-1?\- VAC-B* VAC-B'A
or, since
tan ’ x + tan ' у = tan 1--,
1	i — xy
sl=-FAs==tm-C!A^.
IvlC—Z A—Bq
Substituting the values of А, В, C and M, we have
AC — В* = w(i + n — J’er)(i + n sin’ <г) — л’ cos’ erJ’er
= m(i -J- « — A'er -4- a(i -|- я) sin’ <r — ftA^a)
= я(1 -|- «)(i — A’<r + n sin" <r)
= a(i 4-	+ n) sin’ er;
and putting
(i 4-	4- a) _
—- Ллг»
П
we have
VAC— B1 =h Vn sin ст-
Substituting these values in eq. (/), we have
ll{n, k, ф] 4~	k, ф) — k, er) = S,
I	n t'Ti sin ф sin ф sin a
~ VX2 i 4- w s‘n’ — « sin ф sin ф cos о A er
CHAPTER ХИ.
NUMERICAL CALCULATIONS, q.
CALCULATION OF THE VALUE OF q.
FROM eq. (7), Chap. IX, we have
dn „ _ wlT -«J
whence, eq. (9), Chap. IV, eqs. (27) and (39), Chap. IX,
<0
Ceos & =
I — 2g -|- 2/ — 2g’ + 2gie —
I 4- 2g + 2/ + 2g- + 2g"* +
= i — 4g + 8g’ - 16g’ + 32g* — 56g6 + . . .
The first five terms of this scries can be represented by
From this we get
(2)
Ceos В =
I — 2g
J + 2?'
I I — Ceos t)
2 I + Ceos tf ’
which is exact up to the term containing g’.
Or we can deduce a more exact formula as follows; From
<’4- (0,
1 -|- Ceos t) _ Ci 4~ tan’	+ Ci — tan’l#
I — Ceos В Ci 4- tan’	— Ci — tan’
1 + 2?‘ + 2?“.+  • •.
2g 4- 2g" 4- 2g"S 4- • ♦ • ’
94
NUMEK/CAL CALCULATIONS.	95
whence, by the method of indeterminate coefficients,
(З)	? — itan’ “ + iV tan’ " + tan” 2 "Ь tan'* 2 +   •»
or
8
log q = 2 log tan - — log 4 +
log (i + i tan’ tan’ f tan” f •  •)
ft
(4)	= 2 log tan - - log 4 +
fl	fl	Д
tan* - -|- t’A tan” — -|- tjVt tan” - + . . .),
£	£
M being the modulus of the common system of logarithms.
Put
4?	ft ,
(5)	log q = 2 log tan - + 9.397940 + a tan - +
ft „ в
b tan” —I- c tan —
2 1	2 1
in which
log a - 9.0357243;
log b = 8.64452 ;
log c = 8.41518 ;
log d = 8.25283.
Example. Let k' — cos io° 23' 46". To find q.
4 log tan - = 5.835	2 log tan - = 7.9176842
2	2
log a — 9.036	9-39794OO
4.871	----------
log? = 7.3156316
8
a tan* - = 0.0000074
9б
ELLIPTIC FUNCTIONS.
When 6 approaches 90°, tan — differs little from unity, and
the scries in eq. (5) is not very converging, but у can be cal-
culated by means of eq. (6), Chap. VII, viz.,
wK'
~ A~ K'
q—e ,	q = e .
By comparing these equations with eqs. (6) and (9), Chap.
IV, we sec that if
?=Л*) =Ж
then
/=Л^)=.Л9О0-^
Therefore, having 0, we can from its complement, 90° — 0,
find q' by eq. (5), and thence q by the following process. We
have
j trA”'	j	it A*
— = €~К\	; =	*
<1 ?
whence
log log	= 1.8615228,
(6)	log log -|- log log —, = 0.2698684,
by which we can deduce q from q'.
Example. Let ti = 79° 36' 14". To find q.
90° — 0 = IO° 23' 46".
By eq. (5) we get
log / — 7.3156316, log = 2.6843684,
and log log = .4288421;
NUMERICAL CALCULATIONS. q.
97
and by eq. (6),
	log log = 9.8410263 ;
whence	log? = i.3065321.
When k’ =	k — cos 45° — J Vz, eq. (6) becomes
(7) whence	log = M« = 1.3643763 ;	(£ = kr;) log q = 2.6356237, q = O.O432138.	(k = k'.}
Example.	Given 0 = IO° 23' 46". Find q. Ans. log?= 7.315631.6.
Example.	Given в = 82° 45'. Find q. Ans. log q = 9.37919.
NUMERICAL CALCULATIONS. A'.
CHAPTER XIII
CALCULATION OF THE VALUE OF A”.
We have already found from eq. (37), Chap. IX,
(I)
and from eq. (40), same chapter,
6»<o)	/2Л'
ew=w=V^’
But, eqs. (38) and (27), Chap. IX,
®(A’) = 1 4- 2<? + 2?* + 2/ 4- 2<?” + . . .
©(o) = 1 — 2q -|- 2/ — 2/ 4- 2^‘ — .. .
whence, eq. (2),
(3)	К = J (! + 2<1 + 2?‘ + 2?’ + • • •)’•
By adding eqs. (1) and (2) we get
0(0) 4- 6VQ =	(14- ^);
whence
к_ ?(®(o) 4- y>y
NUMERICAL CALCULATIONS. A'.
99
(4)
7Г t 2	\*
= 2 (i + i<t7 (I + 2?* + 2?” +  •
Example. Let k = sin Q = sin 190 30'. Required K.
First Method. By eq. (3).
By e4- (5)» Chap. XII, we find log q — 8.6356236. Apply-
ing eq. (3), using only two terms of the scries, we have
I 2q = 1.0147662
(1 + 2?) = 0.0063660
2 log (l + 2q) = 0.0127320
log = 0.1961199
log К = 0.2088519
К — 1.615101
Second Method. By eq. (4).
Equation (4) may be written, neglecting q',
+ ^cos
— 2 \	2 i
whence
and
log cos 8 = 9.9743466,
log 4 cos = 9.9871733,
1 4- Vcos 61 = 1.9708973,
I -+ Vcos 6
-----------= O.98544865 ;
log К = 0.2088519,
К = 1.615101,
the same result as above.
TOO
ELLIPTIC FUNCTION'S.
Third Method. By eq. (7). Chap. V.
ft - 19° 30'
= 90 45'
log tan	= 9.235103
log cos	— 9.993681
log tan1 ) o f-
. t. > — 8.470206
log Sin ft, )
0,= Г 41'31''.!
= i° 41' 31".!
= o° 50' 45".$
log cos ie„ = 9.999953
log cos* = 9.987362
log cos’ |(9(i = 9.999906
9.987268
log = 0.196120
log AT = 0.208852
fiai) is not calculated, as it is evident that its cosine will be 1.
Example. Given h = sin 75е. Find A”.
By eq. (7), Chap. V.
From eqs. (14J, Chap. IV, we find
k = sin ft — sin 750
log = 9.9849438
( tun’ ift	=	tan’	370	30'	(
( sin 0°	—	sin	360	4'	i6".47	)
( tan’ I#. =	tan*	18°	2'	8".235	)
t sin вп	=	sin	6°	5'	9".з8	1
(tan*|0„= tan* 30 2'З4".б9 )
( sin ft„a = sin 9' 42' .90 )
9.7699610
9.0253880
7-451I672
NUM Eli/САГ. CAI.CULAT/ONS. Л".
IOI
COS	=	cos	37°	30'
cos	—	cos	180	2'.I3725
cos	=	cos	3°	2'. 57817
cos	~	cos	z4'.8575
log
9.8994667
9.9781184
9.9993873
9.9999995
2 log
97989334
9.9562368
9.9987746
9.9999990
a. c. 2 log
О.2ОЮ666
00437632
0.0012254
O.OOOOOIO
0.2460562
.1961199
log A' =
0.4421761
К = 2.768064 Ans.
Example. Given k = sin 45°. Find K.
Method of eq. (7), Chap. V.
From eqs. (14,), Chap. IV, we have
_ ( tan’ =tan*22° 30'	)
0 — 1	sin t),	—	sin	90	52'75683	I
_ j	tan’	=	tan’	40	5<5'-3784i	I
- 1	sin	=	sin	25'.679	f
_ j	tarf	-	tan’	I2'.3395	)
I sin — sin o'.o5	1
log
9.2344486
7.8733009
5-1445523
a. c. log cos’ if)
a. c. log cos'
a. c. log cos’
, rr
Ios?
0.0687694
0.0032320
0.0000060
0.1961199
log К — 0.2681273
К = 1.8540747 Ans.
Example. Given в = 630 30'. Find K.
Ans. log К = 0.3539686.
Example. Given f) = 34е 30'. Find AT.
Ans. K= 1.72627.
CHAPTER XIV.
NUMERICAL CALCULATIONS. «.
CALCULATION OF THE VALUE OF U.
When = siii’V' < 45°
Example. Let ф = 30°, k  sin 45°. Find a.
First Method. Eq. (23), Chap. IV, and eqs. (14,), (14,),
(14,), Chap. IV
By equations (14,),
log tan - = 9.6172243 ;
e
log tan’ q = 9.2344486 = log k„ = log sin 6,;
6'o= 9°52'45"4i;
log tan - = 8.9366506 ;
log tan5 = 7.8733012 = log h„ — log sin 0„ ;
= o° 25' 40".; ;
log tan* = 5.144552 = log .
102
NUMERICAL CALCULATIONS. U.
ЮЗ
By equations (14J,
0 — 3o°
log tan Ф — 9.761439
log cos в — 9.849485
log tan (<Д — Ф) = 9.610924
<pt — ф = 22° 12' 27". 56
Ф. = 52° 12' 27".56
log tan ф, — 0.110438
log cos 6u — 9.993512
log tan (0S — Ф,) — 0.103949
0, — 0. = 5Г<> 47' 32"-59
0, = 104° o' o".i5
log tan <f>, — 0.603228
log cos = 9 909988
log tan (0, — </>„) = 0 603216
0, - 0. = >04° o' i"-5
Ф, — 208° o' 1 ".65
Since — = 26° o' o".O4 and = 26° o' o".2i,
4	s
we need not calculate 04.
= 936oo".2i.
О
Reducing this to radians, we have
Ф
log g = 9.656852.
104
ELLIPTIC FUNCTIONS.
Substituting in eq. (23), Chap. IV, we have, since cos 0M= 1,
a. c. log cos $ = o. 150515
log cos fl, = 9.993512
log cos = 9.999988
0.144014
. /cos 6 cos fL
0.072007 = 108^.. cosg
<L>
log g- = 9.656852
log и = 9 728859
я = 0.535623, Ans.
When в = sin 1 k > 45°.
Example.
Given k = sin 750,
tan
First Fiet hod. Bisected Amplitudes.
By equations (24) and (25), Chap. IV, we get
Ф = 47° 3' 3°"9i.
<4 = 25° Зб' 5"-б4,
04 =13° б' зо"-98,
<’4 = 6° 35' 4°"-74>
0Л= 3° iS' S”-75,
0Л = J° 39' 7"-43>
fi =45°;
= 24° 40" io".94;
A<,= 12° 39' 15".8з;
A, = 6° 22' 8".4O;
A.=
Substituting in equation (26), Chap. IV, we have
Ф) = 32 X i° 39' 7"-43
= 520 51' 58".оз
= 0.9226878. Ans.
XUMEXICAL CALCULATIONS. U.	105
Second Method. Equation (29), Chap. IV.
From equations (iSj, Chap. IV, we have
log
k =	cos 7	=	cos	15 °	o'	o".oo	9.9849438
k' —	sin jj	=	sin	150	o'	o".OO	9.4129962
k' =	(tan’	=	ta,1‘	7°	30'	o".oo)	8.2388582
( sin	70	—	sin	О	59	35' .25	)
К =	cos	=	cos	o°	59'	3 5".25	9-9999348
4a
tan’I T), = tan’ O° 29' 47".б2 ) c о7„7,10
sin = sin o" o' IS"49 ( S '5' 9
Л, = cos = cos o° o' I5".49 0.0000000
1-1493838
From equations (i8a), Chap. IV, we get
Ф = 47° 3' 3°"-95 ;
— Ф — 45° ;
Фи = 46° i' 45"-475;
Ф„ = 46° i' 29".4i ;
<A,S = 460 1' 29".41 ;
45° + i Ф> = 68° °' 44"-7O5-
Substituting these values in eq. (29), Chap. IV, we get

log tan 68° o' 44".705
= 0.9226877. Ans.
Third Method. Equation (23)*, Chap. IV.
юб
ELLIPTIC FUNCTIONS.
From equations (14,), Chap. IV, wc have
k = sin 0	= sin 75° 0' 0" log = 9.9849438		
= cos в	= cos 75°		9.4129962
A,= ]	, tan’ i 0 — tan’ 37° 30'	1 . *	 у n J у f f _ 1	>	9.7699610
1	' sin	— sin 36 4 16 .47 j	
Л/ = cos		9.9075648
=!	j tan’4 0, = tan’ 18° 2'8'^235!	!	9.0253880
OS	I	I sin	= sin 6 59 .38 1	1
Л/ = cos		9.9975452
£ — J Л03 	 1	i tan’i^ = tan’ 3° 2' 34"-б9 1	r 74511672
	1 sin	— sin	9 42 -90	I
k\ — cos
= (i
k; =
9.9999982
4.3002761
0.0000000
From equations (14,), Chap. IV, wc have
Ф = 47° 3'3O"-94;
ф} — 62° 36' з".ю;
Ф, = i>9° 55' 47"-67:
d>, = 240° о' о". 19;
Ф, — 480° о' о".
ф Фг
Therefore the limit of ф, , or
2	4
30°
it
= 6'
Substituting these values in eq. (23)*, Chap. IV, we have
Z’,' kQ 'A, '//
it
6
= 0.9226874. Ans.
Example. Given Ф — 30°, k = sin 89°. Find u.
Method of cq. (28), Chap. IV.
NUMERICAL CALCULATIONS. U.	Ю/
From eqs. (18,) we find
Л, — sin 0, and tan’ | = k — sin 6,
from which we find that Л, = 1 as far as seven decimal places.
From eqs. (i8a) we have
sin 0 = 9.6989700
i — 9.9999338
sin (20, — 0) = 9.6989038
20c — 0 = 29е 59'-б97зз
20. = 59° 59-69733
45° + i 0.* - 59° 59-92433
log (45° + 40») = 0.2385385
From eqs. (l8s), Chap. IV, we have
k — cos 7 = cos 1 °, i 7 — 30'.
Substituting in eq. (28), Chap. IV, we have
a. c. log cos 177	0.0000330
• log log (45° + 4 0,) 9-3775 5^5
a. c. log M 0.3622157
log F(k, 0) = 9.7398072
F\k, 0) — 0.549297. Ans.
Example. Given 0 = 790, k = 0.25882. Find u.
Ans. 11 = 0.39947.
Example. Given 0 = 370, k = 0.86603. Find u.
Ans. и = 0.68141.
* Since kt — 1, 0»» = 0», and we need not carry the calculation further.
CHAPTER XV.
NUMERICAL CALCULATIONS. 0.
Example. Given и = 1.368407, V = 38° Find ф.
First Method. From eqs. (46) and (41}*, Chap. IX, we have
и = х<Э’(А'),
From equations (5), Chap. XII, and (38), Chap. IX, we
have
log? = 8.4734187
log &\K) - 0.0501955
log и = 0.1362153
log x = 0.0860198
x = 69° 50' 46". 12
From equations (23) and (24), Chap. IX, we get
log &,(x} = 9.9798368
log 6(jt) — 0.0192687
9.9605681
log Vk’ = 9.9482661
log = 9.9088342 = log sin A
But
F sin’ ф = I — Л'ф,
k sin ф = cos A;
10З
NUMERICAL CALCULATIONS. ф.	IO9
whence
log cos A = 9.7675483
log k — 9.7893420
log sin ф — 9.9782063
ф = 72°. Ans.
Second Method. From eq. (1), Chap. VI
From eqs. (14J Chap IV, we find
.	\ tan’| ft — tan’ 19°	) .
= sin 6° 4S'.5456g f |OS = W3«3«
cos ft,	9.9969260
_ ( tan’ il'ft,, = tan’ 3° 24'.2784 ) ” “ ( sin	=	12'. 16659) cos ft0D _ ( tan’ Aft,, = tan’	6г.о8з29 j ’',s — ( sin ftCi	f cos ft„	7.5488952 9-9999974 44957316 0.0000000
Substituting these values in eq. (1), Chap. VI, we have
log cos ft, 99969260
log cos ft„„ 9-9999974
__________	9.9969234
log Vcos ft, cos ffqa 9.9984617
a. c. log “	“	0.0015383
log?* .1362153
log Vcos^ 9.9482660
log 2s .9030900*
a. c. log V'cos cos 0.0015383
0.9891096
2.2418773
log ф* 2.7472323
0, 55s° 46'. MO
* я is taken equal to 3, because cosos =
no
ELLIP TIC FUNCTIONS.
Whence, by equations (i)* of Chap. VI, we get
Ь» log = 4.4957316
sin 0,	9.5075232.
sin (20, - 0.)	4 0032548.
20, — 0, - — o'.O0346
0, = 279е 23'.o6827
4» 1ои = 7-5488952
sin 0,	9.9941484.
sin (20, - 0,)	7-543O436-
20, — 0, = — I2'.O039
0. = 139° 35'-532i
]og = 9-O739438
sin 0,	9.8117249
sin (20 — 0,) 8.8856687
20 — 0, = 40 24:467
0 = 710 59:9999
= 720. Ans.
Example. Given и = 2.41569, 0 = 8o°. Find 0.
A ns. ф = 82°.
Example. Given и = 1.62530, k — £. Find 0.
Ans. 0 = 870.
CHAPTER XVI.
NUMERICAL CALCULATIONS. E(k, ф).
First Method. By Chap. X, eqs. (15), (16), and (17).
Example. Given k — 0.9327, Ф = 8o°. Find E{&, ф).
By eq. (15), Chap. X,
ф =80°:
= 5o° 43%
= 270 48'.5,
= 14° 16'7,
7° ”'-3»
Ф-h = 3° Зб'.о,
Ф& = 0.062831.
/. ф* < O.OOOOOOI.
Whence, by eq. (17),
Г = 670 44'-
П = 4б° 4O'-4
— 26° o'. I
П =13° 24'-o
ГЛ = 6° 45'-2
log sin = 8.77094;
0л) = 0.062794
sin ф sin’ =0.52116
2 sin	sin8	=0.29757
4 sin	sin8 yj =0.10023
8 sin	sin8	= 0.02728
16 sin Ф& sin8	= 0.00697
0.95321
Hence, by eq. (16),
£(^, Ф) = $2E(k, 0A) - 0.95321
= 2.0094 — 0.9532 = 1.0562.
XII
112
EI.I.1E TIC FUNCTIONS.
Second Method. By Chap. X, eq. (26}.
Example.
Given k — sin 75°,
Find
E{k, Ф).
From eqs. (14,), Chap. IV, wc have
h — sin в — sin 750 0' 0"	log = 9.9849438	
E = cos tf = cos 75°		9.4129962
_ | tan’ $0 = tan’ 37° ° — 1 sin	= sin 36°	30'	) 4' 16". 47 f	9.7699610
Л/ = cos &c		9.9075648
_ J tan’ № = tan’ 18° ~~ ( sin	— sin 6°	2' 8".235 ) 5' 9"-3& f	9.0253880
л; = cos		9.9975452
_ | tan’ |0(1D = tan’ 30 0B { sin 0ot = sin	2' 34"-69 ) 9' 42".9O f	7.4511672
— cos 6?DS		9.9999982
		4.3002761
		0.0000000
From eqs. (14,), Chap. IV, wc have		
Ф = 47°	3' зо".94;	
ф1 = 62°	36' з".ю;	
Ф. = 119°	55' 47"6?;	
Ф, = 240° o' o".i9-
NUMERIC.-!L CALCULATIONS. E(k. <>).
113
Applying eq. (26), Chap. X, we have
Z’7 log = 9.9698876
a. c. 2	9.69S9700
A a. c. 2	9.6688 576 9.7699610 9.6989700	.4665064
A. a. c. 2	9.1377886 9.0253880 9.6989700	•1373373
a. c. 2	7.8621466 7.4511672 9.6989700	.0072802
5.0132838	.0000103
.6111342
1 — .6111342 = 0.3888658.
From eq. (23)*, Chap. IV, we find ф) = 0.9226874.
Hence
ЛА — |-(i + + • • -)J = 0.3588016
& Vi
2 * sin ф1 =	0.3290186
i . .	o
-------sin Ф* —	0.0522872
4
k Vi i i
----sin Ф, = - 0.0013888
i Vi(,  . . i,t _ o.oooQQiQ
0.3799180
114
ELLIPTIC FUNCTIONS.
Whence
E{k, — 0.3588016 + 0.3799180 = 0.7387196. Ans.
Example. Given к = sin 750. Find E^Z=, -
From Example 2, Chap. XIII, we find
log = 0.4421761
log 0.3888658 = 1.5897998
log 7) = 0 0319757
I/’, —) — 1.076405. Ans.
\ 2 *
Example.
Example.
Example.
Example.
Given k = sin 30°, Ф — 81 °. Find E(k, 0).
Ans. Eik, 0) = 1.33124.
Find £(sin 80°, 550).
Find E^sin 27°,
Find E(sin 19° 270).
Ans. 0.82417.
Ans. 148642.
Ans. 0.46946.
CHAPTER XVII.
APPLICATIONS.
RECTIFICATION OF THE LEMNISCATE.
The polar equation of the Lemniscate is r = a Vcos 20,
referred to the centre as the origin. Prom this we get
dr a sin 20
dS~ ~	’
whence the length of the arc measured from the vertex to any
point whose co-ordinates are r and 0
<=/w
de
fZCOS 20
Let cos 20 = cos’ Ф, whence
sin ф ФФ
У1 — COS* ф
** Фф _ a С* Фф
1T -|- cos’ Ф <2 Vi — 4 sin’ 0
115
иб
EL1JPT1C FUNCTIONS.
Since r = a Vcos 20 — a cos ф, the angle ф can be easily-
constructed by describing upon the axis д of the Lcmniscate a
semicircle, and then revolving the radius vector until it cuts
this semicircle. In the right-angled triangle of which this is one
side, and the axis the hypotenuse, Ф is evidently the angle be-
tween the axis and the revolved position of the radius vector.
RECTIFICATION OF THE ELLIPSE.
у
Since the equation of the ellipse is —+ y= I, wc can
assume x = a sin ф,у — b cos ф, so that ф is the complement
of the eccentric angle. Hence
s =J tfdx* -|- dy* — aJ е1ф Ti — e* sin’ ф
— a£\e, ф),
in which e, the eccentricity of the ellipse, is the modulus of the
Elliptic Integral.
The length of the Elliptic Quadrant is

s' = aE
%* y* -
EXAMPLE. The equation of an ellipse is	= i;
required the length of an arc whose abscissas are 1.061162 and
4.100000 : of the quadrantal arc. Ans. 5.18912; 6.36189.
RECTIFICATION OF THE HYPERBOLA.
On the curve of the hyperbola, construct a straight line
perpendicular to the axis x, and at a distance from the centre
equal to the projection of b, the transverse axis, upon the

asymptote, i.e. equal to
Join the projection of the
APPLICA TIONS.
"7
given point of the hyperbola on this line with the centre. The
angle which this joining line makes with the axis of x we will
call ф. If j is the ordinate of the point on the hyperbola, then
evidently
P tan ф
and
e’ sin* Ф a
a' +	cos ф
whence
Г 1/ i—T-!	^ф
s = I ¥ax -|-	= - I ---------- —
cos’ ф I i — sin’ ф
Фф
cos’ ф Vi — Z* sin’ ф'
But
^tan ф Vi — Z* sin’ Ф) = <1ф Vi — /•' sin1-^-}- Дф у-—
cos’ ф Vi — e* sin’ ф
Consequently
(1ф
cos’ ф Vi — f? sin* ф
— у F{k, ф) — cE{k, ф) -|- c tan ф A(k, 0)
-|- ae tan ф4\^-, ф
118	ELLIPTIC FUNCTIONS.
Example. Find the length of the arc of the hyperbola
x* У
-----------= i from the vertex to the point whose ordinate
20.25	400	r
4° л
is —— tan 15.	Ans. 5.231184.
Example. Find the length of the arc of the hyperbola
x* y1
-----77— = 100 from the vertex to the point whose ordinate
144 81	1
is 0.6.	Ans. 0.6582.
Elliptic Functions
An Elementary Text-Book for
Students of Mathematics.
BY
ARTHUR L. BAKER, C.E., Ph.D.,
I’ROFffSSOR СР MATHEMATIC* ГМ THE STEVENS SCHOOL OF THE STEVENS INSTITUTE OF
TECHNOLOGY, Ноплким. N J.; FORMERLY PhofKSsOR IN THE Pardmh
Scikntihic: Department» Lafayette College, Eastof, Pa.
t
sni am и = —f
<4 tA«)
NEW YORK:
JOHN WILEY & SONS,
63 East Tenth Street.
i8<jo.
Copyright, 1890,
XIV
Arthur L. Baker.
Robert Drumxoxp,
FGfpptfr,
411 А Ш Fear] Ktrect,
Now York.
FKttRLS DKQB,,
PrinttTS,
S&4 Priori Strectr
Kw York.
PREFACE.
In the works of Abel, Euler, Jacobi, Legendre, and others,
the student of Mathematics has a most abundant supply of
material for the study of the subject of Elliptic Functions.
These works, however, arc not accessible to the general
student, and, in addition to being very technical in their treat-
ment of the subject, are moreover in a foreign language.
It is in the hope of smoothing the road to this interesting
and increasingly important branch of Mathematics, and of
putting within reach of the English student a tolerably com-
plete outline of the subject, clothed in simple mathematical
language and methods, that the present work has been com-
piled.
New or original methods of treatment are not to be looked
for. The most that can be expected will be the simplifying of
methods and the reduction of them to such as will be intelligi-
ble to the average student of Higher Mathematics.
I have endeavored throughout to use only such methods as
are familiar to the ordinary student of Calculus, avoiding those
methods of discussion dependent upon the properties of double
periodicity, and also those depending upon Functions of Com-
plex Variables. For the same reason I have not carried the
discussion of the & and II functions further.
IV	PEEPACE.
Among the minor helps to simplicity is the use of zero
subscripts to indicate decreasing series in the Landen Trans-
formation, and of numerical subscripts to indicate increasing
scries. I have adopted the notation of Gudermann, as being
more simple than that of Jacobi.
I have made free use of the following works: Jacobi’s
Fundamenta Nova Theorise Func. Ellip.; HoUEL'S Calcul
Infinitesimal; Legendre’s Traite des Fonctions Elliptiques;
Du REGE'S Theorie de r Elliptischcn Functionen; Hermite’S
ТЬёоп’е des Fonctions Elliptiques; VERHULST’s Theorie des
Functions Elliptiques; Bertrand’s Calcul Integral; Lau-
rent's Thtorie des Fonctions Elliptiques ; Cayley's Elliptic
Functions; BYERLY’S Integral Calculus; SCHLOMILCH'S Die
Hoheren Analysis; Briot et Bouquet’s Fonctions Ellip.
tiques.
I have refrained from any reference to the Gudermann or
Wcierstrass functions as not within the scope of this work,
though the Gudermannians might have been interesting
examples of verification formulae. The arithmetico-gcometrical
mean, the march of the functions, and other interesting investi-
gations have been left out for want of room.
CONTENTS
RACE
Introductory Chapter, .	.	.	.	.	.	. r
Chap. I. Elliptic Integrals, ......	4
II.	Elliptic Functions, ......	ц,
III.	Periodicity of the Functions, ....	22
IV.	Landen's Transformation, .....	30
V.	Complete Functions, ......	45
VI.	Evaluation for <!>,	.	.	.	.	.	.	4З
VII.	Factorization of Eiliptic Functions, .	.	•	51
VIII.	The. & Function, ......	60
IX.	The & and H Functions, .....	tig
X.	Elliptic Integrals of the Second Order,	.	, Sr
XI.	Elliptic Integrals of the Third Order, .	.	.go
XII.	Numerical Calculations, f, .	.	.	.94
XIII.	Numerical Calculations. /Г, .	.	.	.	96
XIV.	Numerical Calculations, .	.	.	’	,	.102
XV.	Numerical Calculations, Ф. .	.	.	.	.108
XVI.	Numerical Calculations, £{i.	<!>),.	.	.	.111
XVII.	Applications, ....... ik