316 



Mr. J. Griffiths. Abstract of some [Apr. 30, 



April 30, 1885. 



THE PRESIDENT in the Chair. 



The Presents received were laid on the table, and thanks ordered 

 for them. 



The following Papers were read : — 



I. " Abstract of some Results in Elliptic Functions. (Part II.)" 

 By John Griffiths, M.A. Communicated by Professor 

 G. G. Stokes, Sec. R.S. Received April 9, 1885. 



1. On the z(u) Function Complementary to Jacobi's 7i(u). 

 The double periodicity of the elliptic functions gives rise to an 

 interesting function of the form z(u) = a—— f u, where 



ftf d6 Co / 



u=\ /====, a=\ >/l-k*Bm*0 (J0=E(«O, J'=K'-E'. 

 Jov 1— & 2 sin 2 J 



By changing a, u, respectively, into (1) a + 2^J', u + 2iK', (2) <x+2E, 



«+2K, (3) q+E-^ sn ^ C -, u + K, it is easily seen that z(u) satis- 



fies the following relations, viz. : — 



z(u + 2iK')=z(u), 



z(u + 2K)-z(u)=~ (since KE' + K'E-KK' = ^\ 



2(lf + K)-2W=— 



2K' dmi 



2. Deduction of a ®(u) Function from z(u). 



Writing . /?t^L e fo zlu)du we can take the foregoing z (u) 



relations as equivalent to — 



®(u + 2iK') = -<S>(u), 



<P(u+2K)= ±e~K<t>(u), 



r 



