232 PROFESSOR A. G. GREENHILL ON THE 



dt* 



I(2t 



2 



2* x 2 ' 



l-l>+^) J /<,=P (i (l-X>0 -P+r) (8). 



The substitution ( p. ^ } will correspond to (v, 2v, 4-c) with a nonnalis- 



p 1 pi 



ing factor 



M = (MjMoM^) 5 =p(l-p) (y) f 



and 



P .(3-i>) _ . - 1 + + 3p- -p* __ r, _ + 3 

 M 6p(l p) "6 G 



'Math. Aim.,' 32, p. GG). 



> - - y __ - , , 



p-p}* W l~ P 



12P4_ -1 + l-2^-80/r + :54y> :i -21/. 1 +&/^-y> (; <9'4r (1 -;>)(! - 

 M a p 3 (l -/')' : M :i ' p 2 



while 



//-(I -|>)- M ' M 3 



unchanged by the substitution. 



Also 



p _ Vr + 92 1- + fr3 r + ^4 r _ ( - 1 + p - p : f , , ,, 



G '~ M^ - :^('-^) 2 ( } ' 



a quantity required in the transformation of the ninth order; and 



P(v)_5 + + 3pt-p< , } 



M "" 



P(2t/) L 

 "M 



P (4v) __ 7 + 18p 

 ~ 



