331 


1879.J Definite Integrals involving Elliptic Functions . 

the bodies examined by them, and obtaining only negative results, and 
always a certain result with a crystal of the salt, they have insisted 
that this is the only nucleus. Others, again, have sought for an 
explanation in some catalytic or other mysterious force ; while a third 
set of observers have declared it to be a matter of uncertainty or 
hazard whether a foreign body acts as a nucleus or not. In reviewing* 
the subject and repeating my experiments in various ways, I see no 
reason for withdrawing from the theory which I had the honour of 
submitting to the Society eleven years ago, namely, that the action of 
nuclei is simply mechanical, and is capable of being expressed by the 
familiar word adhesion. 


VIII. 44 On Definite Integrals involving Elliptic Functions.” By 
J. W. L. Glaisher, M.A., F.R.S., Fellow of Trinity College, 
Cambridge. Received July 31, 1879. 

§ 1. The chief object of this paper is to apply to definite integrals 
involving elliptic functions certain special methods which have been 
employed for the evaluation of integrals of a similar kind involving 
circular functions. 

§ 2. One of the most elegant and direct investigations of the value 
of the integral 

I log sin x dx--=^7r log (J) 

J 0 

is afforded by the product 

. 7T . 277- . (n -1)7T / 

sm - sm —- . . . sm a—_—A_= n . 2 d{w 1; , 

• n n 2n 


for, taking the logarithm of both sides of this equation, and writing 
n 

7^(log sin h + log sin 2 /&...+ log sin A?r) = 7 r lim. -1 

n 

= “ i7rlog2. 

The same principle also gives the value of the integral 

7T 

log (1— 2&cos x-{-aF)dx . (1), 

0 

which =0 or 2?rlog a according as a < or > 1, and it is easy to see in 
general that if 


then [ log 0(ff)d#= b \im . n==(yi — b ^ t S.El . 

J« n V( °°) 


2 a 2 


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332 


Mr. J. W. L. Glaisher on 


[Nov, 20, 


For example, from tlie product 

r(a)r^a+i^ ... r^+2^^==r(m). (27^-%*-^, 
since, wlien n is great, 

T(na) = (27r) h (na) na ~ h e~ n % 
we derive the evaluation, 

Ca+l 

log r($)cfo?=41og (2?r) + <x(loga — 1). 

Ja 

The number of products that yield integrals of interest is not great, 
and those just noticed are all that I remember to have seen applied to 
this purpose. The transformation formulae in elliptic functions lead, 
however, in this manner, immediately to definite integrals, as will 
appear in the next section. 

§ 3. Writing sn, cn, dn in place of sin am, cos am, A am, we have 
(“ Fundamenta JSTova,” p. 46) :— 

{sn2wsn4a . . . sn(n-l)^== ^^ —^~J . . (2). 

{cn 2o> cn4w . . . cn (»—.(3). 

{dn2w dn4w .. . dn («-l)«} 2 =^V.(4). 


In the case of the first real transformation (n being an uneven 
prime) 

K 


K 


A = - 


Thus, when n is infinite, 
A=Jtt, 


but A'=log-, so that 
X 


M= 


nW 


2K 


A'=* 

M 


, ! ttK' 


A '=tn 


K 


A -in**! 

X — e k , 


j 7tK' j , 7 


and we therefore have 

,og (®)‘ = ^ 

log (|rj= i n log V — 2 log (i 7r ) > 




1879.] Definite Integrals involving Elliptic Functions . 
whence the products (2), (3), (4), give 

f K 

log snxdx= — JwK'—|Klog 1$ . . . 

Jo 

j* logcna?daj=--j7rK'+JKlog^-^-^. . . 

fg 

log dn xdx — iK log k' . 

Jo 

which are the analogues in elliptic functions of the integrals 
(V 

I log sina?doj=|-7rlog (|). 

Jo 

(V 

l logcosajda?=-|7rlog (J). 

Jo 

The other formulae, such as, ex. gr n 

snwsn {^ + 4<w} . . . sn 4(w—-l)iw}= ay sn (54 

do not lead to new integrals, for this product gives 

(*4K 

I logsn (u + x) dx=— 7rK'~ 2Klog/s, 

Jo 

in which the sign of sn ( u-\-x ) when negative is to he changed, so that 
the quantity under the integral sign should be written logsn 3 (u-\-x). 
This result, however, may be readily deduced from (5). 

The remaining formulae on page 51 of the “Fundamenta Nova” 
only produce the equation 


333 

( 5 ) , 

( 6 ) , 

( 7 ) , 

( 8 ) , 
(9). 


Vg(i-^)M Ki og(i- 

Jo \ snW Jo 


1<? sn 3 u sn 3 x)dx, 


which may be otherwise deduced from the formulae (44) and (45) of 

§ io. 

§ 4. In the second real transformation 
iK' 




, , ttK 

Xi =e ln w , 


so that we find 


fK' 

\ ^ log snHxdx= —log k . . . . 
Jo 

• • (io), 

j* logcnixdx=^7r’K-\-^K.' log^-^ . . 

• • (ii), 

f K ' 

1 log djiixdx—\7rK-\-\'K! log k r . . . 

• • (12), 







334 


Mr. J. W. L. Glaislier on 


[Nov. 20, 


but if, in virtue of the formulae 


. sn (a?, k ') 
cn (x, k'y 


cn (x, k) 

. dn (a?, 7s') 
cn (as, k ) 

these equations be written in a real form, they only reproduce (5), 
(6), and (7), as might have been anticipated a priori. 

§ 5. The double products (“ Fundamenta Nova,” p. 66) 

n gn 2 2wzK 4- 2m'^K'_(— 

2 2mK + 2m4K' / A? Y w * " 1) 


XI dn 2 ^ m ^h2m'£K „ yi{ n *-i) 
ni 


give rise to the remarkable integrals 


^log sn (x + iy)dxdy~ — KK' log 7s 


'k f K ' . (k!\ 

log cn (x + iy) dxdy =KK' log I — ) . 
JoJ-k \k J 


f log dn (x -4- iy)dxdy =KK' log k f . 


These may of course also be written 


log (sn (x + iy) sn ( x—iy)}dxdy = —KK' log k ; &c. 


§ 6. Using the values of the three integrals in (5), (6), (7), and 
observing: that 


log sn 2 xdx —JI log sn xdx, 


= 2 j l°g sn ®dx + ^-J log sn xdx , 


= i log sn $£&# + •§• logsn (2K— a?) da?. 


= log sn xdx , 

Jo 



1879.] Definite Integrals involving Elliptic Functions . 
we find, by means of tbe formula 


335 


sn 2a?: 


„ 2 sn a? cn 03 dn a? 
1— 7c 3 sn 4 a? ’ 


that 

Similarly 


K log (1 — 7c 3 sn 4 *)cte = — +Klogf. . (16). 

Jo \ k k / ' 


fK fK 

I ^log*cn 3 2a?£7a?= log 

Jo Jo 

1 “ 


cn xclx , 


log dn 2 xdx — log dn xdx, 


whence, from tbe formulas 


cn 2a?= 


1—2 sn 3 x + 7s 3 sn 4 x 


1—& 3 sn 4 a? 

dn 2a?—^ sn3 X ^ sn4 x 

1 — Zc 3 sn 4 a? ? 

by tbe use of (16), we find 

| 2 log (1 — 2 sn 3 x -f 7c 3 sn 4 a?) 3 c7a?= —\7rK! -f Klog (~y—^ 

| K log (1 —2& 2 sn. 2 »+7e 2 sn 4 x)dx = —Jn-K' + K log (-p-) • 

These two integrals may also be put respectiyely in tbe forms 
fit 

I 2 l°g (cn 3 a?—sn 3 a?dn 3 a?) 3 c7a?.(19), 

Jo 

f log (dn 3 a?—7? 3 sn 3 a? cn 3 x)dx .(20), 


(17) . 

( 18 ) . 


the former expression is written \ log ( ) 3 , as tbe quantity in brackets 
is negative from a?=-|K to a?=K. 

§ 7. Using tbe formulae 

1 —cn 2a?=2sn 3 a? dn 3 a? -f-(l—7s 3 sn 4 a?), 

1-f cn 2o?=2en 3 a? -r-(l—7c 3 sn 4 a?), 

1 —dn 2a? = 27i? 3 sn 3 a? cn 3 a? -f-(l —7c 3 sn 4 a?), 

1+ dn 2a?=2dn 3 a? -r-(l —7<? 3 sn 4 a?), 

cn 3 a?—(dn2a?-f cn2a?) -f-(l-f dn2a?), 

dn 3 a? = (7<?' 3 + dn 2a? + 7c 3 en 2a?) (1 -f dn 2a?), 

and, observing that 


’K j*2K 

0(2a?)(7a?=-|s (j){x)dx , 
Jo Jo 




336 


Mr. J. W. L. Glaisher on 


[Nov. 20, 


we have 


f2K 

log (l + cnx)dx=— IttK'—K log&.(21), 

f2K 

J log (1— dnx)dx= — fTrK'+K log h ..... (22), 

f2K 

log (1 + dnx)dx=z ±7rK' + Klog]c .(23), 

log (dnaj + cn 0 )c^=—^K' + Klog . . . (24), 


i ^ 

J log (k'z + dnx+Wcn^dx^TrK' + Klog (M/ 3 ) . (25). 

Since sn(2K— u) = sn u, dn(2K— u) = dnu, cn(2K— u) = — cn u 9 it 
follows that 


f2K CK 

(f)(snx)d%=2 0(s nx)dx .... 

Jo Jo 

(*2K fK 

I (fi(dnx)dx=2 \ (j)(dnx)dx .... 

Jo Jo 

f2K CK 

0(cn x)dx~ {0(cna?) + 0( — cnx)}dx 

Jo Jo 

and therefore from (22) and (23) we deduce that 


> • (26), 


*K 

Jo 


log (1 — dnx)dx= —f^K' + ^K log h 


(27) , 

(28) . 


j* log (Y-\~dx±x)dx= ^K'+^KlogZ? . . . 

But (21) only gives the value of 

f K 

{log (1 — cn x) + log (1 + cn x) }dx, 

Jo 

and is, therefore, equivalent to (5), while (24) and (25) are merely 
transformations of (5) and (28). 

We also see that in (24) and (25) the integrals may be written 

f2K 

I log (dna?-— cnx)dx ....... (29), 

Jo 

f2K 

j log (h'z + dnx — cnx)dx .(30), 

Applying (26) to (17) and (18) in the forms (19) and (20), we see 
that 

J 2 l°g (cnas + sn# dn xf'dx— — ^TrK'-fKlog • • • (31), 

j* log (dnx + Jc snoj cnx)dx = — ^7rK / + K log . . . (32). 





1879.] Definite Integrals involving E Hip tie Functions. 337 

§ 8. Since 

j* <p(x)dx—^ 0(K— x)dx .(33), 

it follows from (27) and (28) tliat 

log (dn x—k'^dx— —f ttK' + ^ r K log (Jett) . . . (34), 

log (dn x+ti)dx— JirK'+IKlog (kk f ) . * . (35). 


Jo 

j: 


It may be remarked that the transformation (33) applied to the 
functions sn, cn, dn does not suffice to give the values of the integrals 
in (5) and (6), although we thus immediately obtain (7) ; for 


*K PK fK pK 

log dn xdx= log dn (K —x)dx= log k'dx— log 
Jo Jo Jo Jo 


dn xdx , 


giving 

but 


'K 

Jo 


log dn xdx—\ K log k'; 


CK pK pK 

I log sn xdx— I log cn xdx— logdn xdx, 

Jo Jo Jo 

pK pK pK pK 

and logon xdx =\ logk'dx+\ log sn xdx— \ logdn xdx, 

Jo Jo Jo Jo 

only lead to one equation between the integrals of log sn x and log cn x , 
viz., 

pK pK 

I log cn xdx — log sn xdx = |K log k', 

Jo Jo 


9. Putting x=y in the formula 


0(xd-y)0(x—y) = ^^^- (l—k 2 sn 2 a? sn 3 y) 


it becomes 


whence 


0(2$) =^*-^(1 — k 2 sn 4 $) 
0 3 O 


(36) , 

(37) , 


f K f K f K 

J log 0(2$)cZ$:=4j log Qxdx—‘S 1 log QOdx \ log (1 — 1$ sn 4 x)dx. 

pK pK 

log Q(2x)dx=\ log( 

Jo Jo 


How 

so that the equation is 


Jo 
; Qxdx, 


pK pit 

ffig 0$db=Klog 00—1 log (1 — k 2 sn 4 $)d$, 
and substituting the value of the last-written integral from (16) and 




338 


Mr. J. W. L. Glaisker on 


[Nov. 20, 


putting for 90 its value, viz. — ) , this gives 

r log Gad®=**■£' + iK log(^) . 


. . (38). 


Also Bx= */k . sn xBoc, 

and integrating the logarithm of this equation, using (5) and (38), 
we have 


log Bxdx = ttK' + fK log 




Replace x, y in (36) by mx, nx and the formula becomes 

B(m-\-n)xB(m—n)x — ® ^ mx ^S nx ^ (1— W* sn 2 mx sn 3 nx), 
whence, since (p being any integer) 


log 0 (px) clx — log Bxdx, 


we have 


log (1—7c 3 sn 3 mx sn 3 nx)dx — 2^Klog 00 — log Bxdx ^ . (39), 


if m and n be different integers, and 


= 3^Klog 


00— log Bxdx 

Jo 


if m and n be equal integers. 
Thus we find 


log (1— W sn 3 mx sn 3 nx)dx— — ^K' + fK log^^ . (41). 


if m and n be different integers, and 


if m and n be equal integers. 
The relation 


:-^K' + Klogg') . (42), 


log (1 — W sn 3 mx sn 2 nx)dx =f log (1 — 1$ sn 4 mx )cZ; 


is curious. The value of the latter integral is of course immediately 
derivable from (16). 

§ 10. ( Lemma .) If 0(a?) be an even function of x, and if 0($-{-2K) 
z=z<fi(x), then 


<t>{x-\-y)dx -\-1 cf)(x — y)dx=2\ (£>(x)d,x, 



1879.] Definite Integrals involving Elliptic Functions . 339 

for, if u denote the left hand member of this equation, 

!r = [ ■y- ( t>(. x +y) dx + f ^-4>{x—y)da!, 

ay jo ay Jody 

= f -^(»+y)&>— f ~<fi(x—y)dx, 

Jo dx Jodx 

=(p(K+y')— 0(2/) — 0(K—1/) + 0(— y), 

= 0 , 

so that u is independent of y and the lemma is proved. Now from 
(36), 

fK pC (*K 

log0(a?-f-y)cZa? + I log 0(a?~ 2/)c7a?=2 log 0a?c7a? + 2K log 0y 
Jo Jo Jo 

*—2Klog 00+j* log (1 —W sn 2 a?sn 2 y)c7a?, 

and since 0a? satisfies the conditions supposed in the lemma, the left- 
hand member of this equation is equal to the first term on the right- 
hand side, so that we obtain the formula 

f log (1—7<? 3 sn 3 a? sn 3 y) dx =2K log—=K log( ^ --— \ —2K log Qy (44), 

Jo Qy \ 7 r J 

which we may write in the form 


f: 


log (1 — a 3 sn 3 x 


)dx=K log^^-2Klog0^sn- 1 |). 


In virtue of the formula 


sn (a?+ 2/) sn (a?— y) 


_ sn 9 a?—sn 3 y 
1 — & 2 sn 3 x sn 2 y 


cn 


z . n z \ 1—sn 2 a?—sn 2 y + 1ft sn 2 a? sn 2 y 

O + y) on 0-2/)=-,— JK— -5- J ~, 

1 — Ar sr x sn’ 4 y 


dn (a? + y) dn (x—y) 


_1—A? 2 sn 2 a?—& 3 sn 3 y -f- 7c 2 sn 2 x sn 2 y 

1 — 7c 2 sn 2 x sn 2 y ’ 


we can deduce by means of (44), since sn 2 a?, cn 2 a?, and dna? each 
satisfy the conditions of the lemma, that 

J ilog(sn 2 a?—-sn 2 2 /) 2 ^a?= —-|7rK'-f-Klog^?^-5^—2Klog 0y . (45), 

£ 


r log (1 — sn 2 a? — sn 3 y -f 1$ sn 2 a? sn 2 yfidx 


= -IttK' + K log 2K log Qy 


(46), 



340 


Mr. J. W. L. Glaisher on 


[Nov. 20, 


J log (1 — 1$ sn 3 x—k 2 sn 3 y + k 2 sn 3 x sn 3 y)dx 
o 

=K log —2K log Qy . (47). 

The last two integrals may also be written 


log (cn 3 x —dn 3 x sn 3 y) 2 dx^ 
log (dn 3 x — l b 3 cn 3 x sn 3 y)dx , 


and if these be transformed by the substitution of K— x for x , they 
become respectively 

fK [*K 

2 1°S“ ( sn3 x ~sn 3 y) 2 dx —2 log dn xdx +2K log Tc\ 

Jo Jo 

Ck Ck 

log (1— k 2 sn 3 x sn 3 y)dx — 2 log dn xdx + 2K log k ', 

Jo Jo 

so that of the four integrals (44), (45), (46), (47), the pair (44), 
(47) are convertible one with another, and also the pair (45), (46), by 
the substitution of K— x for x. 

Integrating (44), . . . (47) with regard to y between the limits K 
and 0, and using (38) we obtain the following evaluations :— 

j^ log (1— h? sn 3 x sn 2 y)dxdy = — Jjt-KK'+ fE 2 log(^rf) • • (48), 

•Jlog (sn 3 *— sn 1 yydxdy= — fjrKK' + fK^ogt^-l . . (49), 

0 \ k 2 / 

log (1 — sn 2 x — sn 3 y + k 2 sn 3 $ sn 3 ?/) 2 dxdy = — f 7rKK' + § K 2 

• • (80), 


o 

K fK 


0 JO 


KlK 

ojo 


log (1—k 2 sn 3 x—k 2 sn 3 y + k 2 sn 3 x sn 3 y)dxdy 

= — e ^KK'+f K 2 log (~p~) • • (51), 

§ 11. The lemma at the beginning of the last section may be de¬ 
duced directly from the definition of integration when y is real; for, 
in virtue of the equation 0(a)+ 2K) =0(a?), 0(a? + ?/) + 0(a)— y) can 
always be reduced to the form 0(a? + &) +<fi(x—a) where a< K; and 

Tk Ck 

<fi(x-\-a)dx-\- (j)(x—a)dx 
Jo Jo 

— A{0(&) -}- 0((Z 4" A) -J-0((X -|- 2/&) . . . -J-0(<x -}- hi) 

+.0(— c£) + 0(~+ 0(—^)+0(O)+0(&) . . . +0(K—&)}. 



341 


1879.] Definite Integrals involving Elliptic Functions . 

Since 0(K + a—/&) = $(2K~K~a-fA)=0(K+a + ft), &c., and0( — -Ji) 
=(£>(h), &c., tliis 

=/i'-[0(O) 4"0(7&) * * • +0(K—$) -|-0(K— cc-\-Ji) . . . -J-0(K) 

+0(0)+0W . . . +0(a)+0(a4-^) . . . +0(K)} 

fK 

=2 cf>(x)dx . 

Jo 

The lemma is thus evidently true for all real values of y , and from 
the proof in § 11 it would appear that in general it was probably true 
when y was imaginary or of the form a + bi. It is, however, certainly 
not true in the case of y=2^K', for 

0 (x + 2iK') — — eh K '- ix) e%. 


O—TT'/ 

so that log 0(^ + 2^')+log 0 (a?— 2£K') = —g- + 21og 0a?, 

and therefore 


i*K Pk Pk 

log0($ + 2 iK!)dx+\ log0(®-2tK / )d»=2wK' + 21 lo gOxdx. 

It is also evident that (44), . . . (47) cannot be universally true* for 
the left-hand members of these equations remain unaltered when y is 
increased by 2'iK', which is not the case with the right-hand members, 
since 0 is not periodic with respect to 2iK'. 

To determine the imaginary values of y for which the lemma and 
the theorems (44), . . . (47) deduced from it are true, consider the 
expansion of log 0 ( 0 ) in a series of cosines, viz.: 


log 00 =A— : 


2 q 


2 q* 

K 2 l-q i 


cos —— 


cos 


2t tx , 2 o 3 3? tx o /k 0 n 

(52)> 


where 



From this we obtain (38), viz.: 


i: 


since 


log Qxdx— AK, 


nirx 7 A 

cos- ax—0. 

) K 


How log 0(0 + a) -flog 0 ( 0 — a) 

=2A—2 / cos! Mr (* +a > 

n l—fi n l 

=2 A—2 1 4 ? B 


K 

6 mrx nm 

COS . COS —r 

n 1 ~-(f n K K 


+ cos 


mr(x—a) 
K 



342 


Mr. J. W. L. Glaisher on 


[Nov. 20, 

so that the lemma will necessarily hold good so long as the series 


1 4 q 1 


n 1- 


' mrX mrCb 
cos —— COS 

K K 


is convergent. 

Let a—mi K', then 


cos -1-<Z m; 0“ 

K 


_ii +tf m 


and the series 


, 2 1±2 2 “ »a- m) n*rx 

n l-q 2 ^ K 


which is convergent when m< 1. 

It follows therefore that the lemma and the theorems (44), . . . (4?) 
are true when y is of the form a + hi, where h lies between K' and 
—-K', and a is unrestricted. 

In the case of b=K! the series becomes neutral, but it is easy to see 
that the lemma and theorems are still true; for, transform (44) by 
putting y=i K' + 8, so that 

Jcsny=—, 

snz 

then log Qy =log 0 G'K'+ z) 

=logsnzOz} 

=-J log h + +log sn z 4- log Qz +log i — 


K 


2K 


Thus we have 

log(sn 2 sr - sn 2 ®) dx = - \ttK' + K log (?|2E) - 2K log 9« + tV(*—K). 

The imaginary term i7r(z — K) is due to the fact that sn 3 ^ —sn 3 a? 
changes sign when x—z, so that we might expect the term 
(K— 2 )log(—1) to appear in the equation. Writing, therefore, the 
integral in the real form 

J 2 ( sn3 2“ sn3 a?) 3 , 

and throwing away the imaginary term, w~e have (45). It may be ob¬ 
served that (46) and (47) are also connected by the same substitution 
of y—iK! -j-z. 

§ 12. In the formula (44), viz.: 

J log (1 —-7t 2 sn 2 x sn 2 y)dx=:2K{log 00—log Gy}, 
put y equal to *K, JK+^K', KK', K + ^'K', -JK + \i~K! re- 



343 


1879.] Definite Integrals involving Elliptic Functions . 

spectively; we thus find, on substituting for snits value in the 
different cases 


log{l-(l-A/) sn=2K{log00-log 0(iK)}, 
log{l-(l + 35/) s y?x}dx =2K{log 00—log 0(JK + iK')}, 


J log{l + Jfcsn*®}d® =2K{log 00-log 0(^K')}, 

j* log{l—7c sn 2 a?}cZa? =2K{log 00—log* 0(K+|7K f )}, 


log{1—7s (7s— iti ) sn»a>}da> =2K{log 00-log 0(£K-^K')}, 


j* log{l —&(&+$>') sn 2 a? }c7a?== 2 K{log00—log0(-JK+^K')}. 

To obtain the values of the thetas, put o?=^K, 2 /=JK in (36), and 
this equation gives 


whence 


e-‘QK)=™e 3 0GK, 


e(iK)=^V»(l+^) J 


From this, by taking a?=-|K in the formula 


we deduce that 


0 (a?+) = i i e~w~ sn x 0a?, 


0(|K+iKQ=eil fl . t*) Kl V (1 -y)« . . . (54). 
2M 


The value of 0(^K') may be deduced from (53) by putting. ®==^K 
in the formula 


^ A'KV 1 7 ,. 

0a3—| _— 1 e 4 KK'-0(^33, A; ), 

vsK / cna? 


and changing the modulus from 7s to h ': it is thus found that 


and thence that 


e(^K')=eil -7 C )‘.(55), 




e(K+itK') =eiSK —3— 7c*(l + 7c)i. 


Finally, from (36) by putting a^^K+^'K', TrK-b-^K', we find 
that 




344 


Mr. J. W. L. Glaislier on 


[Nov. 20, 


0*(iK+i»KO 


e s oe(K+;K') 


’rK'2K 2 7 ,,/ i /j/ . ... 
-e iK — -k , k\k +ik) 


(56). 


Substituting these values in the integrals and reducing, we have 
j Q !og{l — (1—&') sn 3 x}dt>=±K log j - 2 -^ j.(57), 

j^log{l-(l + ^su 2 *}%r==-ivK'+iKlog[^lJ . . (58), 

j" log{l + &sn 2 a;}c&B=—^ ttK'+-|K log { 2 (^ j. .... (59), 

J" log{l— Tcsn?x}d%= —-g-^K' + -g-Klog | ^ .... (60), 

j" logjl— h(le — ik!) sv? <e}dx= — ^n-K'+-|Klog {- —— j- ( 61 ), 

log{l— h(h+iW) sn*x}dx- -^K' + |K log 1 2yi y~^) j (62), 


In (58) as the integral is so written that its value may be real, the 
term involving i that enters from (54) has been rejected. 

By the addition of (57) and (58) we obtain (17), while (59) and 
(60) lead in a similar manner to (16), and (61) and (62) to (18). 
The reason why this happens is easily seen, for since 1 —2sn 3 a?-b7/sn 4 a? 
is the numerator of cn 2x, we have 


which = {1 — 7/ sn 3 x sn 3 (^K+ iK !) } {1—7s 3 sn 3 x sn 3 ^K}, 

= {1 —(l-f-7(f) sn 3 a?}{l— (1—7/) sn 3 a?}, 

and, similarly 

1 — 7/ sn 4 x= {1—& 3 sn 3 x sn 3 J7K'} {1 — 7/ sn 3 x sn 3 (K+^7K') } 
= {1 + h sn 3 x} {1 —h sn 3 a?}, 

and 


1 — 2& 3 sn 3 x +7/ sn 4 x 

= {1-7/ sn 3 a? sn 3 (-|K-^K')} {1-7/ sn 3 a? sn 3 (JK+^K')} 

= {1 — h(h — 77/) sn 3 x] {1— h(h +$/) sn 3 x }. 

§ 18. In any integral we may replace sna?, cna?, dna?, by sin a), cos a?, 

(1—& 3 sin 3 a?) J , if we also replace dx by-- tt—s —r> and make the 

(1 — 7/ sin 3 a?p 




345 


1879.] Definite Integrals involving Elliptic Functions . 


necessary changes in the limits. This is evident, since 

d am* 


dx 


=dnc 


and as am K=|7r, am 2K=7r, we see that to the limits K or 2K and 
correspond respectively the limits \tt or w and 0. 

Thus (5), (6), (7) maybe written 


f * 75 * log sin x 

o (1 — & 3 sin 2 a?y 

f|7T 


log COS X 


v dx— — JttK'—JK log Tc, 

^=-^K' + |Klog7<;, 


Jo (1—& 2 sim 

p-lo^l K J 

Jo (1—7s 2 sin 2 xfi 


while, for example, (57) becomes 


*"Tog{l-(l-&') sin 2 a?} 
o (1—7c 2 sin 2 at)* 



and the other integrals may be similarly transformed. 

In this form several of the above integrals have been obtained by 
Mr. William Roberts in'his papers “ Sur TEvaluation de qnelqnes 
Integrales Definies par les Eonctions Elliptiques 99 (“ Lionville’s 
Journal,” t. xi, 1846, pp. 157—173), “ Snr ITntegrale Definie 

f ’ (Id., pp. 471—476), and “ Note snr qnelqnes 

Jo (1—-7^sm 2 0)* 

Integrales Transcendantes ” (Id., t. xii, 1847, pp. 449—456), which 
contain evaluations equivalent to (5), (6), (7), (16), (38), (44), (48), 
(57), (59), (60). In the first of these papers the value of the integral 
in (7) is found by means of the transformation h r tan 0 tan ^=1, which 
is equivalent to the substitution of K —x for x ; and in the last paper 
the values of the integrals in (5) and (6) are found directly from the 
^-products, viz.: 



(1—2g 3 cos 2o? + g 4 )(l — 2g 4 cos 2o? + g 8 ) ... 

(1 — 2 q cos 2a? + g 3 ) (1 — 2g 3 cos 2a? + g~ 6 ) .. . 

(1 + 2g 3 cos 2a? + g 4 ) (1 + 2g 4 cos 2a? + g 8 ) . . . 
(1—2g cos 2a?-{-g 2 )(l —2g 3 cos 2a? + g 6 ) . . . 


by means of the integral (1), which gives 

(V 

J log(l—2g # cos 2x+q% n )dx—0. 

The equation (38) is also obtained directly from the g-product for 
0; this is of course practically equivalent to the use of the cosine 
series (52) for log Go? quoted in § 11. 

VOL. xxix. 2 B 



346 


Mr. J. W. L. Glaisher on 


[Nov. 20 ? 


The last two papers relate chiefly to the integral 

fK 

log (1 + n sn 3 x) dx, 

Jo 

and one of the methods indicated for obtaining its value depends upon 
the formula 

2Y amp+2Y am$=Y am (p + q) + Y am (p— q) —log (1—1$ sn 3 p sn 3 q), 

and is therefore the same in principle as that employed in § 10. 
Mr. Roberts generally uses the Legendrian notation and the function 
Y, for example the formula (44), viz.: 

f log (1—& 2 sn 3 x sn 3 y) dx =2K log — 

Jo Oy 

is written 

log(i_|WW0) #=B W[F( ^ 0)) 


but he remarks (“ Liouville,” t. xii, p. 458) that some of the results 
could have been obtained more readily by the use of 9 instead of Y. 
Mr. Roberts also gives the values of the integrals 

p*- log (1 + cot 3 6 sin 3 0) ^, 

Jo (1 — 7s 3 sin 3 0)^ ’ 

pTri 0 g {l_(l—fc'8gin 3 6 ) sin 3 0} , , 

Jo m30)* ^ 


V p* log (1 + cot a 6> sin 2 0) f , , 
Jo Jo (1 —7c 2 sin 3 0) 4 (1—7c' 3 sin 3 6) h 



l°g {1 ~ (1 ~ ■fc' 2 s™ 2 0 sin 2 0} M 

(l-7c 3 sin 2 0)4(1— 7^2 g in2 0 )i 


These can be derived from (44) by the substitution of iK!—iy and 
K + 7K 7 —iy in place of y, for 

lc sn(7K'— iy) =i Cn ^ 1 

sn (y, fe') 

& sn (K + 7K' —%) = dn (y, 7c'), 

7/iTr-i 7r 

9(^K r —%) = - t - -eiKK/ (Kf ~y )a sn (y, k')Q(y, lc'), 

R* 


e(K+tK'-iy)=A —A— e(7K'-iy), 


whence we find 


sn (y, A;') 

f iog[i+^^A^AAsii 2 *|(fe 

Jo l sn 3 (y, h ) J 

= ~ 2 gA K ’ — 2/) 2 + K log' ( 2 A)—2K log 9(y, 7c')—2Klog sn(y, 7/) (63), 



347 


1879.] Definite Integrals involving Elliptic Functions . 

fK 

log {1 — dn 3 (y, k') sn*x)dx 

Jo 

= -l^(K'- 2/ )3+Klog(^ , )-2Kloge( 2/ ,^) . . . (64), 


and integrating with regard to y between the limits K' and 0 these 
give 


17>{ i+ :wi^“^h=i' Ks -^ K ' ,+ * KK ' i “<T) - <65) ’ 

log{l —dn 3 (y,h') sn2 x}dx= -£rE?—£irK'»■+ AKK'log(V?) (66), 


which are in fact Mr. Roberts’s results (“ Liouville,” t. xii, p. 456). 
In the “ Philosophical Magazine,” for December, 1860,^ Professor 
Sylvester has given the values of two integrals which are equivalent 
to (5) and (28). The method depends upon an expansion in a series 
and is quite different to any employed in this paper. Professor 
Sylvester notices particularly the fact that the values of the integrals 
in (5) and (28) should be equal in magnitude. 

§ 14. Recently in the “ Mathematische Annalen” of Clebsch and 
Neumann (t. xi, 1877, pp. 567—570), Dr. Enneper has obtained the 
evaluation 

J dn 3 xlogdnxdx= JElog&' + -J(l +&' 3 )K — E 

by the substitution of K— x for x and further transformations: but the 
value of this integral had been previously found in a somewhat similar 
manner by Mr. Roberts, in a note in “ Liouville’s Journal ” for 1846 
(“ Extrait d’une Lettre addressee a M. Liouville,” t. xi, pp. 843, 344). 
Dr. Enneper, in his paper, by the use of (33) shows that 

I arc tan ( -— - - — -- \ dx 

Jo l dn (x + a) dn (x — a) J 

=J* arc tan { An ( x + a )^ (»:=g) j dx=fr K . . (67), 

and it may be observed that, by following the same method, we have 
at once 


* “ Notes to the Meditation on Poncelet’s Theorem, including a Valuation of the 
two new Definite Integrals 

jl log cos <p f| log{l+ V'l-a^COB „ 

J 0 — S 3 (costp) 3 J 0 \/ l — t 2 (cos<p) a 

“ Phil. Mag.,” ser. iv, t. xx, pp. 525—533 (1860). See also, by the same author, 
“ Note on certain Definite Integrals,” “ Quarterly Journal of Mathematics,” t. iv, 
pp. 319—324 (1861). 


2 b 2 



348 


Mr. J. W. L. Glaisher on 


[Nov. 20, 


[ 4 - (dn x\j f K , / y k \ 7 

arc tan I —— \dx= arc tanJ- )dx 

\ h * / Jo \dn x) 


+ arc tan 


^7rclx=^7 l 

quantity, we see 

dx _ f K da 
, dn^o? Jo-! , I 


Also, n being any quantity, we see from (33) that 
f K dx f K dx 


o -i . dn n x Jot , h* n 

-l~t~ . -■ >;■ ■■ - 1+-- 

h* n dn n x 

fK/ ym 


dn w aj V 

0 'ln _|_ x J 


so that 


o \h' hl + dn n x lc' bl + dn 


dx __i K 

bl + d.n w x 2 h’ h 

dn w x 7 i -r 


and, similarly, 


U11 *—dx=jK . . 

J o h' bl + dn w x 

f__* 

J o h' n + dn n (x—a) dn n (x + a) 
f K dn w (x — a) dn. n (x + cb) 

J o h' n + dn n (x—a) dn^ (xd-co) 


Further, if 0 be an uneven function and n any quantity, 
f K , / dn x Aj'* \ { 1 .dn n x\j 

J„ 0 h?r-5^) lo n 1+ T»r 

f K , /dn x h’ h \, h' ln \ 7 

= “J» H^“3=)M 1+ <E^r- 


whence 

J>G 


0^ log (]d bl + dn w ' x) dx 
\ Id* dnav 


and of course there are other formula involving —and 

h 4 

dn(x —JjQcfaOf. jlfO which may be proved in a similar manner. 






1879.J Definite Integrals involving Elliptic Functions . 349 

§ 15. It was shown in § 5 that 

KfK' 

log cn ( x + iy) dxdy —KK' log —-; 

h 

the value of this integral when the limits with regard to y are K' and 
0 may be found as follows. 

In the formula 

rriu 0('M + K+lK , ) ) 

cn u——fe 2 K —-- 1 

¥ Qu 

put u~x-{-iy —K— !K.\ 

and, taking logarithms, we find 

log cn (*+iy—K— %K ')=i log —i-g- + 2 (x+iy—JL—iK') 


Now 


0 jo 


-j-log Q(x-\-iy > ) — log © (x~\~iy —K—£K/), . . .. (72) 

/ ’ TeTk/ 

(j)(x-{-iy— 1 K.—i’K!)dxdy—\ | <fi(x+iy)dxdy 


o Jo 


if 0 be an even function, so that (72) gives 

K fK' 7/ 

log cn (x + iy)dxdy= JKK' log _-JttK* 
o Jo h 


+¥ C (i K3K/ +^KK'2 - K 2 K' - iKK'2) 




=iKK'log-|—JwrKK' 


( 78 ). 


§ 16. I conclude with the determination of the values of the in¬ 
tegrals 


when n is a positive integer. 

0^5?^=1 — 2$ cos 2x+ 2 <fi cos 4x—2q 9 cos 6x + &c., 


dx 


whence 


= e"** — 2qe x * cos 2ax+ 2q 4> e~ x * cos 4±ax— &c. 


\ 7 T J 


Now 

so that we have 


j; 


e x2 cos 2bxdx=jf7r h e~ 


“\- x °e(‘^-)dx=^(l-2r I e-“ 2 + 2r/e~^-%fe~' 1 ‘‘'* +■ &c.) 

0 \ 7T / 



350 


Mr. J. W. L. Glaisher on 


[Nov. 20, 


n being a positive integer, whence 

■Kt)' 

and 

J o e-^e | j &=l^(l-22» +1 -224te+i) + 2 2 9te+D- & c .) 

—i ^74 +1 K. +iy 

= (^^ y .(74), 

where ¥ n+1 , K w+1 are what h and K become when q is replaced by q n+1 , 
so that h u+l is the modulus obtained by the first real transformation of 
the (^-fl) th order. 

As a particular case put n— 1, and the formula (74) gives 


> 12a(lE) 4 ) dx= (IK//*) 4 


■ (75). 


To evaluate the other integral, multiply the equation 


=2q k sin ax —2 q% sin 3a$4- 2 q* s sin 5an e— &c. 


by e * a and integrate, replacing the integrals on the right hand side 
by their values from the formula 


xe x * sin 2bxdx=^7r h be'~ i *, 


we thus find 


xe^H — 3q%e~% a * + 5— &c.) 


Put, as before, < 

w being a positive integer, then 


e~ a *=q n 7 




and the equation becomes 


®e-* a Hj2s ^y( 2 ite+J)_ 3 2 |(»+i) + 5 2 V(»+d _ &c .) 

Lt 2 2 *-6^+102V- &c.= 17^—J) 




351 


1879.] the Values of the Theta and Zeta Functions. 

( u Fundamenta Nova,” p. 184), so that 

J" xe-*’H 1 ) dx= 1 lc„ +1 ¥n +1 KVi} 5 

Put n~l and this formula gives 


(76). 


(77). 


IX. a Values of the Theta and Zeta Functions for certain Values 
of the Argument.” By J. W. L. G-laisher, M.A., F.R.S., 
Fellow of Trinity College, Cambridge. Received July 31, 
1879. 


§ 1. In § 12 of the preceding paper “ On Definite Integrals involv¬ 
ing Elliptic Functions/’ it was necessary to determine the values of 
0(iK), e(pl-KK'), e(^K'), 0(K+±iK'), and e^iK+^K'), which 
were required in the evaluation of some of the integrals. This led me 
to calculate a table of the values of the 0 and H functions when the 
arguments were of the form , for the values 0 , 1 , f, of 

m and n , and the results are contained in this paper. For the sake of 
completeness the corresponding values of the Z function are also 
given; and some remarks connected with the g- series to which the 
formulae lead are added. 

A table of the values of the sn, cn, dn for the above-mentioned 
arguments is given by Professor Cayley on page 74 of his “ Elemen¬ 
tary Treatise on Elliptic Functions ” (1876) : this table* is so useful 
that it seemed desirable to supplement it by a similar one for the 0, 
H, and Z functions. 

§ 2. The values found in § 13 are 

e(*K) +V)\ 

e(p:+;K') = g -iJ^'i(i-Wi+;)> 

G(iiK') 

7T P - 


* I may here note that the value 


of 


1 — i s/k' 

V^V*" 


of cn (^K + i^K') should be 


1 — i \Z&' 

f/2 f/T 


instead 



