1880.] Prof. Cayley, On the Schwarzian Derivative, &c. .3i9 



Monday, March 8, 1880. 



Peofessoe Newton, President, in the chair, 



A. Vinter, M.A., Gonville and Cains College, was ballotted for 

 and duly elected a Fellow of the Society. 



The following communications were made to the Society : — 



(1) Professor Cayley : On the Schwarzian Derivative and the 

 Polyhedral Functions. 



(Abstract.) 



The quotient s of any two solutions of a linear partial 

 differential equation of the second order 



d"ii dy 



is determined by a differential equation of the third order 

 d^s d^s 



ds 2 

 dx 



:=-i(/+^|-%). 



where the function on the left hand is what I call the Schwarzian 

 derivative, or say this derivative is 



where the accents denote differentiations in regard to the second 

 variable x of the symbol. 



Writing in general (a, b, c :^X, F, Zf to denote a quadric 

 function, 



(a, b, c, 1 (a - b - c), 1 (- a + b - c), 1 (- a - b + g)\X, Y, Zf, 



then if the equation of the second order be that of the hypergeo- 

 metric series, generalised by a homographic transformation upon the 

 variable x, the resulting differential equation of the third order 

 is of the form 



{.,«.} = (a, b,c.-.][-l 



a x — h' X — c 



and, presenting themselves in connection with the algebraically 

 integrable cases of this equation, we have rational and integral 



