48 
ME. A. CAYLEY ON THE EQUATION EOE A FUNCTION 
6. Putting now S = 0 (and therefore e=0) the roots become 
^,=(3y((3-y), 
4=ya(y— a), 
^3=a/3(a— f3), 
^4= — y)r 
where (a, /3, y) are the roots of (a, h, c, d'Xv, 1)^=0. Let Z denote the discriminant 
of the cubic function, then (3, — Z, and we have thus the linear equation; 
the cubic equation is 
n3{^-/3y(/3— y)}=0, 
the coefficients of which can be calculated by the method of symmetric functions (see 
Annex No. 1). 
7. The cubic equation being thus obtained, we have the two equations 
-\-6 . — ^c^d?-\-^dbcdj—lfd 
= 0 
+ v/^ 
= 0; 
and multiplying these together, the resultant equation is 
+^h0 
iahcd —¥d-\-'L) 
. {—%a^d-\-^abc—¥)d \/ — Z 
-(^^Z=0, 
where the coefficients have to be completed by adding the terms which contain e. We 
have \/ □ in the place of d\/ — Z, and □ in the place of —d^Z. The coefficient 
— — Jf is a semin variant, and requires no alteration. The coefficient 
— %a^d^-\- ^.abcd— Jfd-^-T^ 
is 
— ^abcd —Ifd 
— X^Lohcd 
+ 4ac® 
+ U^d 
- Wc\ 
that is. 
