OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. 537 
=> {(a—f)’ | yotyet+de, y+ Ste |} 
yoe 5 yo+ yet de 
and also = | 38P, , 4P,| =6P3;—12P,P, 
SP 5 p23 
that is 600 (d’—ce) =a’ {(a—B)’ (yo+ ye + Se—y + 5+. yde)} . 
The results for the equation of the fifth degree are 
100 (0? —ac) =a’.> {(a—B)’} 
600 (c? —bd) =a’. 2 {(a—B)’ (Qi QQ} 
600 (d’ — ce) =a’. 2 {(a—B)’(Q;—Q,Qs)} 
100 (e’ —df) =a’. 2 {(a—B)’(Q; — Q,Q,)} 
=~7.2%(a—6).yoe; , since Q,=—0, (5), 
where Q,,, is interpreted to be the sum of the products m at a time of all the 
roots of the given equation, other than those in the squared difference of roots 
with which it is associated, and Q,=1. 
: : n.nm—1 
Hence each quadratic element consists of —j-5— terms: and each term con- 
sists of two factors ;—the one being the square of the difference of a pair of 
roots, and the other a function of all the remaining roots. 
Let the symbol of roots (4) last discussed be denoted by (3), 3 being the 
dimensions of the second line of the second matrix ; then in an equation of the 
mth degree 
J (m) = 71 cg (eee oe 
n—m+1.P,.,, »—m.P,, 
and if 
ean tees M—M+t1 
‘4 Lge SS he 0 ; 
™m fe 
Pra et a” 
s 
iP, Sy? 
nm—m ¢ 
Pa+i=YnFi wet 
we have 
; s—rt 
J (m) =v’.n—m,.m—z ° 
Thus, when 2=6, the successive /’s have the numerical factors 180, 1800, 
3600, 1800, 180. 
