1895.] Mr Baker, On a certain automorphic function. 827 



which are easy to prove, we infer 



• c = l/@(l(^^• + ^^), 



which proves the theorem as originally stated. 

 For the case ^ = 1, putting 



^-B 1 , 7n-B 



V = 



log 1^—1' "^0 = ^^ 



27ri ^ ^-A' ' 27Ti ^ m-A 



we find from the definitions, 



E, .J. ._B-Asm'7r(v-Vo)^l- 2q^' cos 27r (v - v,) + g*'" 

 Zi sm TTu sm ttVq i=i (1 — q^^f 



- ^(B A) . ^"''"7°' ©(i^-Vo + i + JT) ^ 

 ^ sin TTv sin ttuo II (1 — q^'^f 



where q = | Vy"' | and e'^*'' = q, so that ^^ = 1 ; and 



^ (?, «0 - ^_ ,^,^ ^^^^ |1 + 4 ( ) J _ 2^2. cos 27r (u - u„) + W ' 



where /ij = 1 or 2. 



For example when Aj = 1 



1 ©'(i + lr) . , , ., , ©(v-Uo + i) 



= «; 7^T1^ sm TT (y - uo) e-'^*("-"'>>,=ry— ^ \ ^\ . , 



^-m 7r0(|) ^ "^ ©(u-i;o + i + iT)' 



and in virtue of the fact that 



®' (i + i-r) = 27rm (1 - g2i)3^ 

 1 



these agree with the formula found in the general case. 



Reply to a Paper hy Mr Bryan. By A. B. Basset, M.A., F.R.S. 



My attention has recently been drawn to a note by Mr Bryan 

 on page 51 of the present volume of the Proceedings; and I wish 

 in the first place to state that until October 5, 1895, I had never 

 seen the note in question, and was ignorant of its having been 

 written. It was therefore impossible for me to have made any 

 remarks upon this note ; and the paragraph at the end of page 53 

 has consequently been inserted without my knowledge, authority 

 or consent. 



