96 Mr J. W. L. Glaisher, On the expansions of the [May 30, 
May 30, 1887. 
Mr TROTTER, PRESIDENT, IN THE CHAIR. 
The following Communications were made: 
(1) On the expansions of the Theta functions in ascending 
powers of the argument. By J. W. L. GuatsHEr, M.A., F.RS. 
The author laid before the Society certain symbolic formule 
which he had obtained for the coefficients in the general terms in 
the expansions of the Theta functions, and for the Theta functions 
themselves. 
r 
2 2K. 
Put p= and wa" = pe; 
also let ©, (u), ©, (u), ©, (uv), © (w) denote the four functions 
H(u), Hiwt+k), Ou+kK), O (wu), 
and 
eg w) O,(u _ 9@) 
Suppose s(w), e(u), &e. ae in PEN powers of a, 
and let S, C,, D,, 1, be the coefficients of the nth terms in these 
n? n? 
expansions, SO that 
sw =—>- (—)’ ae 
ai "(2n +1)!’ 
sW= egy °C)= 
e(u)= Xo (-)*C, 
d(u)= 52 (~)"D on 
n(u)= 0 (-)"¥, aay - 
The symbolic expressions found for S,,, C,, &c. were as follows: 
n? 
S, = 2" ee ae: p? Mie 2 ky’? p: 
n 3 a 4d Bs nr 3 n 
a (% A gh © ih ) 
=(20+ Rh, +R, + R.)"; 
