1880.] Derivative and the Polyhedral Functions. 351 



functions of z (containing, each of them multiple factors), which 

 are such that P + Q-{- R = Q'. (using accents to denote differentiation 

 in regard to z this imphes P'+ Q'+ R = 0, and consequently 



QR' -Q'R = RF -R'P = PQ'- P'Q) : 



and are further such that the equal functions QR' — Q'R, 

 RP' — R'P, PQ' — P Q contain only factors which are factors of 

 P, Q or R. In fact writing f, g, h = h — c, c — a, a — b, the required 

 relation between oc, z is then expressed in the symmetrical form 



f{x-a): g{x-h) : h{x-c)=P : Q : R. 



The last mentioned differential equation is considered by Klein 

 and Brioschi : the solutions in 13 cases, or such of them as had 

 not been given by Schwarz, were obtained by Brioschi : and those 

 of the remaining three cases (subject to a correction in one of 

 them) were afterwards obtained by Klein. 



The first part of the present memoir relates, say to the forego- 

 ing equation 



{5, a;} = (a, b, c .•.{) , ., -) , 



^ V A^ — (^ X — x — cj 



although the other form in [x, z] may equally well be regarded as 

 the fundamental form : and we consider in the theory 



A. The derivative [s, x], meaning as above explained, 



B. Quadric functions of any three or more inverts - 



C. Rational and integral functions P, Q, R having a 

 sum = 0, and which are such that 



QR' - Q'R = RP' - R'P = PQ' - P'Q 



contains only the factors of P, Q, R. 



D. The differential equation of the third order. 



E. The Schwarzian theory in regard to conformable 

 figures and the corresponding values of the imaginary variables 

 s and X. 



F. Connection with the differential equation for the hyper- 

 geometric series. 



The second part of the memoir relates to the polyhedral func- 

 tions. 



VOL. III. PT. VIII. 26 



