THEOREMS RELATING TO A GENERALIZATION OF BESSEL’S FUNCTION. 405 
We notice that in the expressions for sn, cn, dn, two arbitrary constants (orders 
of the functions) n, m appear : 
- Jacosr’s function Ta) ° Bs is related to the J functions as follows, 
2K(p-1) ,/2Kr\ f J(u) J(u) -z#f 32) J_(o) 
a Tf ) Se 2 CN _ i —e-* 2 Sin “in . 7 A(45) 
tralp ( es ) Fa) — In) dm(%) Tea) 
It is plain that Wererstrass’s functions c, (, ”, may be expressed by similar formule. 
For convenience of printing, the order of the functions will sometimes be expressed 
by n instead of [n]. The known formule of Jacosi’s functions will, it is evident, 
give rise to corresponding forms in the case of the J functions : for example 
sn? + cn? =] 
gives rise to 
- FlpwyH (rr) Hip) lip'o) 4, HC)I’C) 
sin? 2 -)-$___, — +k cos? =, —_ = ___, —_—* _ ; . (46) 
J,,(piu)d, (pir) J (tpiu)dS (iptv) 4g? J (w)J%(v) 
=~ Ps K 
=U VP ye y= UP 8 Ege CoRr 
p-l p-1 
Using (11) and (12) it is easily found by the method of § 10, p. 116, vol. xli., Trans. 
R.S.E., that 
© 
€ m—1 ,, a a ou n2 i a 
m=1 
By Fourirr’s theorem we write therefore 
21. 
il ao 
"De mile oe {H(-2 ‘as. 2pin-2) | - di ; 
p 5-7) zl ( ap cos «+ a®p*™*) f COS mar dic Ye (47) 
In ease «= 1, this reduces to 
PAs 
n2 1 1 Ka 
e Sie eg | co nr O(a . ee 
A. 
Funcrion J? (x). 
Forming in a series, according to powers of t, the product 
J pn(2t) X Jin (at *) 
we obtain 
IZ (a) + DS (- Or e-™)I Ze): _ (49) 
in which i 
M goimt ont 4r 
{2m + 2n + Ir} l{ 2m + Br }l{Qn + Ar} !{ Ir}! 
{2r}!=[2][4] ... . [27] 
(50) 
Jem(&) = 2S (= 
