1881.] 77 



March 21, 1881. 



Professor Newton, President, in the Chair. 



Dr Armistead. Dr G. M. Bacon, Mr H. Baxter, Mr R. Bowes, 

 Mr Bumpstead, Mr J. Carter, Mr A. Deck, Mr H. Gotobed, 

 Mr A. Graham, Mr Marshall (Ely), Mr W. E. Pain, Mr A. Schuster, 

 and Mr W. W. Smith, were balloted for and re-elected associates 

 of the Society. 



The President stated that it was with very great pleasure 

 that he was enabled to place upon the table the charter of the 

 Society, which had been out of their possession since 1852 ; and 

 he was glad to think that the remarks made by him at the Annual 

 Meeting on October 25 had led to its recovery. They were much 

 indebted to Professor Paget and Messrs. S. and W. Peed for the 

 assistance they had rendered, and the Council had passed a cordial 

 vote of thanks to them for their services. 



The following communications were made to the Society : 



(1) On Conjugate Functions of Cartesians and other Quartics. 

 By A. G. Greenhill, M.A. 



1. The well-known case of the conjugate functions of confocal 

 ellipses and hyperbolas in which x + iy is a trigonometrical func- 

 tion of £ + in may be considered as derived from the integral 



h 



dz 



= lu, 



\J(z.\-z) 



from which z = sin 2 w, 



or x + iy = sin 2 (£ + in), 



and the foci of the system are at z = and z=\. 



If r, r denote the distances of a point from the foci, then 

 r 2 = (x + iy)(x-iy), 

 = sin 2 (f- + in) siir (f - irj), 

 or r = sin (£ + in) sin (£ — it)) ; 



and r" = (as — 1 + iy) {x — 1 - iy) ; 



r — cos (£ + in) cos (£ — in) ; 



