148 
[745 
745. 
ON THE SCHWAEZIAN DEEIVATIVE, AND THE POLYHEDEAL 
FUNCTIONS. 
[From the Transactions of the Cambridge Philosophical Society, vol. xm. Part I. (1881), 
pp. 5—68. Read March 8, 1880.] 
The quotient s of any two solutions of a linear partial differential equation of 
the second order, ¿¿J* + P + TU — 0> is determined by a differential equation of the 
third order 
d 3 s 
dec 3 
ds ~ \ ds 
dx 
(r+zf x -*q), 
where the function on the left-hand is what I call the Schwarzian Derivative; or 
say this derivative is 
1-1.-£-*©■. 
where the accents denote differentiations in regard to the second variable x of the 
symbol. 
Writing in general (a, b, c Y, Z) 2 to denote a quadric function 
(a, b, c, ^(a-b-c), ¿(-a + b-c), £ (- a - b + c)$X, F, Z)\ 
then, if the equation of the second order be that of the hypergeometric series, 
generalised by a homographic transformation upon the variable x, the resulting differ 
ential equation of the third order is of the form 
{s, æ) = (a, b, c .•. 
x — a ’ x — b ’ x — cj ’