1887.] Theta functions in ascending powers of the argument. 97 
C1, = 2" 8 4 Q” ih? p?, 
n $ 1/28 3d i 
= 4 ¢ eos ai i? fia 
(= (20+ 8)"; 
D, = 2" p#Q" pt 
— An 27/2 ad bn 
= 4" kp" ap") 
= (20 + &,)”; 
NV, = 2"h'"2 -4 Q” Kk? p 4 
epi 3% 1A 2)" 
=4" (i MF pk Sk 0) 
P dh 
= (20+ R,)"; 
where Deals, lo al, 
AKT AKG AKE 
Ee, = 1 ? R, = W ? R, 7 1 ? 
and ( is an operator defined by the equation * 
Q (R,? Re uy) =(B8+y) Res Rf RY+ (y+a) Ri, 1B Sigg 
te (GO) IRE IBF IRE eRe TE JPET TR Je 
ui G eee Ra 
The following symbolic forms for the Theta functions are 
deducible from the above values of the coefficients : 
s’ (u) =f 2 ® cos (20h! ly? a :) ht hi p 3, 
c(u)= hp cos (2ah > hh’? p = i p2, 
holt 
Q 
1 al = pul dz 
d (wu) = 2 cos (2an' 4 ~) Za 
. c Pane 
os tl 1 ‘il, d2 1 
n(u)= h'* pcos (2ahh ns ] hit 02; 
: Pal? 
s' (uw) = cos (21! h’3 p# = hi h’s pa ta), 
* The operator Q=2q" = 2hh/p? = The quantities I, G, H have been con- 
sidered in Vol. yv. pp. 184—208, 232—250. 
