58 
ME. A. CAYLEY ON THE EQUATION FOE A FUNCTION 
\A’e have 
Next, 
M = 
+ 96 aV 
—60 c?bde 
- 40 aVe 
+ 27 a^cdr 
+ 47 aH^ce 
— 18 abc^d 
+ 4 ad^ 
- 9 ¥e 
+ 4 b^cd 
- 1 b-c^ 
'SjA^3 = 7 , X — y^a^^(y, K «) x «, f3) 
= 2,(Dy\l3 — 'y)((5 — h){y—'^) X —ySa(y—§)(7 — «)(§ — «) X icc(3{l—u)(^—(3)(cc~ /3) 
= a"f3y ^-(a— /3)(a— yX« - — y)(/3 - S)(y — §)2,a(a — ^)(«— y)(a _ 
= (3, y, l)l^^ci{ci—^){(x, — y){a — l) 
= — Jsx/ZSXa— /3)(a— yXo5— 
and observing that 
{a, h, c, d, e'Jy, iy=a{v—ci){v—j3){v—y){v—l), 
and therefore 
iav^-{-hv'‘-\-2cv-Yd=^a{v~^){v—y)[v--l)-\- &c., 
which, putting r)=a, gives 
4««^+35a^+2ea+d=a(a — /3)(a — y)(a — §), 
we have 
2^a(K—(3)(c^—y)(ci — ^)=^(4«2o5^+3i2«*d-2c2a*4-<^2a) 
where for a moment a is put equal to 
— 4(4)+ 3J(3)+2c(2) +<?(!), 
1. This is calculated by 
e 
-16 = 
-16 
bd 
+ 16 -9 -1 = 
+ 6 
<? 
+ 8 -4 = 
+ 4 
b’^c 
-16 +9 +2 = 
— 5 
¥ 
+ 4-3 = 
+ 1 
or restoring the powers of «, and putting 
_ 
16 
a^e 
+ 
6 
a^bd 
"h 
4 
— 
5 
aWc 
+ 
1 
b^ 
