350 Frof. Gayley, On the Schwarzian [Mar, 8, 



functions of 5, derived from the polygon, the double pyramid and 

 the five regular solids, and which are called polyhedral functions. 



The Schwarzian derivative occurs implicitly in Jacobi's dif- 

 ferential equation of the third order for the modulus in the 

 transformation of an elliptic function {Fundamenta Nova, 1829, p. 

 79), and in Rummer's fundamental equation for the transforma- 

 tion of a hypergeometric series (1836), but it was first explicitly 

 considered and brought into notice in two memoirs of Schwarz, 

 1869 and 1873 ; the last of these (relating to the algebraic inte- 

 gration of the differential equation for the hypergeometric series) 

 is the fundamental memoir upon the subject, but the theory is in 

 some material points completed in the memoirs of Klein and 

 Brioschi. 



I propose in the present memoir to consider the whole theory 

 and in particular to give some additional developments in regard 

 to the polyhedral functions. 



I remark that Schwarz starts with the foregoing equation 



{.,^} = (a,b,c.-.|^, ^^, ^J, 



and he shows (by very refined reasoning founded on the theory of 

 conformable figures, and which is in part reproduced) that this 

 equation was in fact algebraically integrable for 16 different sets 

 of values of the coefiicients a, b, c. It may I think be taken to be 

 part of his theory, although not very clearly brought out by him, 

 that these integrals are some of them of the form, x = rational func- 

 tion of s, others of the form, rational function oi x= rational func- 

 tion of s, the rational functions of s being in fact the same in 

 these last as in the first set of solutions, and being quotients of 

 polyhedral functions. 



But as regards the second set of cases, the solution of these 

 (writinof for convenience a new variable z in place of x) may be 

 made to depend upon the solution in the form, w = rational func- 

 tion of z, of an equation of a somewhat similar form, but 

 involving two quadric functions of x and z respectively, viz. the 



equation 



, , /dwV f , Y 1 1 1 



dzj ■ V ' ' ' "X-c — a ' x — h' X 



1_ J_ 1_ 



z — a' z — h' z— c. 



\^\ 



and we have the theorem that the solution of this equation 

 depends upon the determination of P, Q, R, rational and integral 



