109 
' FOE AN EQUATION OF ANY OEDEE. 
The form of the preceding results may be modified by writing we have 
thus the equations for 
thus for example, in the case of the cubic {a, h, c, djx, yf, 
the equation for '^{=ci\ot,—^)\x—yyY] is 
0=(1, 18H, 81ff, 27nU^I^, l)^ 
This equation may be written 
(^+9H)^^+27nU^=0; 
or putting we have 
v^+mv+vj-27n = 0 , 
an equation the roots of which are 
a(a—^)(x—yy), a{(5 — y)(x—ccy), a(y—u}(x—(5y), 
and which leads to the formula, given in my Fifth Memoir on Quantics, for the solution 
of a cubic equation. But this decomposition of the equation in is peculiar to the 
cubic. 
The equation of differences for an equation of any order may be found by the follow- 
ing entirely distinct method. Let the proposed equation 1)“=0, be for shortness 
represented by (pij=0, and let w, y be any two distinct roots ; we have not only 
^.r=0, <pv=0, but also (px-\-(py=^, *^^3^=0. Writing 0=[x—yf, s—x-\-y, we have 
X y 
values which are to be substituted for x, y in the equations 
<px+<py—^, 
We have thus two equations rational in s and and the elimination between them of 
the quantity s leads to the required equation in But it is proper to modify the form 
of the system ; in fact the two equations are, as regards s, the first of them of the degree w, 
the second of the degree n — 1 ; but if we write 
then each of the equations will be of the same degree n — 1 in s. 
For instance, let pv=(a, b, c, d^y, 1}®, then x=^{s~\-^d), y=^(s+^^); the equations 
px-\-py-0, are 
s^a-\-Ss^2b-\-3s(4:C-i-ad)-\-8d-\-6bd=0, 
3sV<+3s4J+ 12c-j-a^ =0; 
and multiplying the first equation by 3 and the second by — s, adding and dividing by 2, 
we have an equation 
s^3b-j-s(12c-i-4a^)^12d-j-9b^=0. 
Q 
MDCCCLX. 
