358 



Mr. J. W. L. Glaisher on 





[Nov. 20 



then 















J - L V2 JV /" — \ J=L 2 JV >' 





±J 



e(jK+*KO = 





H(iK + iK') 



= <r* 



A'.i(l + {) 



eQ-K+iK') = 





H(fK + *K') 





A'.i(l-i) 





grA.B 



H(itK') 



— *2 1 



. JJ 



ii- T 2 ^ ) — 



^ ^ . A. 





= gr~£ 



.A 



G(f^K') 



T 9 6 . B 







0(K + -|zKO = 





H(K + f*K') 





.A 



Thus 











whence 



A'=0(iK)=e(|K) 



== g* (1 - 1) H (JK + *K') = 2* (1 + H (|K + *K') 

 B'^H(JK)=H(fK) 



^2*(l-t)0(iK+tKO 

 A=^e(K+ifK / ) 



B=^e(JiK0 



^(1+00(^ + ^0 

 gr*e(K + fiK') 

 ^AHCK + fiK') 

 ■q&Q (fiK') 



g _ H (jK) _ H (f-K) _ e (|g + j KQ _ (f K + iK ') 

 A' e(jK) G (f K) HQK+tK') H(fK + ;K') 



B_ e(JiK') _ e(^K0 _ , H(JiK') _ , H(f*K') 



A e(K + ^K') e(K + |*K') H(K+^K') 

 equations which admit of immediate verification. 

 Also, if C=—kWK 



then (IK + j*K') H ( JK + fc"K') = 2 -»0 (1 + , 



(}K + \iK!) H (fK + J*K') = gr*C CI— 0. 

 0(iK+^KOH(iK + ^KO = -^C(l-O 5 

 0(lK+|iK')HaK+|«KO = -gr*0(l+0. 

 § 8. The g-series for and H are 



Gu=l—2q cos 7 ^ + 22 4 cos 2g 9 cos &c, 

 K K K 



H«=22* sin ~ -2qi sin ^ + 2 2 V sin !£f- &c, 

 2K 2K K 



and comparing tne ^-series for 0(-|K), &c, with their values A', &c, 

 given at the beginning of § 7, we have the following equations : — 



