1879.] the Values of the Theta and Zeta Functions. 351 

 (" Fundamenta Nova," p. 184), so that 



J" xe-**K 1 j dx= 1 & w+1 ^ +1 K3 M+ i } ^ • (76) . 



Put %=1 and this formula gives 



|>^ H {K?)> =4 ^(*^) ! • • (77) - 



IX. " Values of the Theta and Zeta Functions for certain Values 

 of the Argument." By J. W. L. Glaisher, 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 

 G(iK), e(lK + ;K'), eQiK'), 9(K + J/K'), and ©^K + ^K'), which 

 ■were required in the evaluation of some of the integrals. This led me 

 to calculate a table of the values of the and H functions when the 

 arguments were of the form K-HuK' , for the values 0, -J, 1, f, of 

 n 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 

 formulas 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 



©(IK) =^0{i+ky, 



G(iK+fK') = q ->R.^(l-lcy(l + i), 

 e(^K') = r A^*(l-£)i, 



* I may here note that the value of cn (|K + ^K') should be — — instead 



» 1 — i 



