OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. 541 
Hence the four equations— 
atd {(a—B)?(a—y)*(B—y)? O|}=|—46, 12¢, —12d, 4e , Pali 
aU a, —4b,° 4x46¢, — 4d, @ 
1 ., ~4b, 12c,—12d, 4e 
3) 
Dee = 19. We. Ae 
or, more symmetrically, 
+—- + — + 
ame Bo OG5 Obey 6 gait. (8) 
Lia, so, de, d.,. 
as. b. 36, 3a ie 
—4|., a,3b,3c,d 
az {(a—B)(a—y)(B—y)’ 
the signs and numerical multipliers of each column and line on the right hand 
side being written respectively above and to the left of them. 
Write these four equations, for a moment, thus— 
Aa’ + BB’ + Cy’? + D&=A, 
Aa’ + BR’ + Cy’ +De=A, 
Aa +BB +Cy + Dd =A, 
A +B +C +D =A,, 
multiply them respectively by 42 , 6c , 4d , e , and add: then 
—a(Aa‘ + BB + Cy* + DS’) = J P= "40 SS Ge. = 4d, é | 
pAb Be PAG SY 4e, | 
a, — 4b, 6c, — 4d é | 
Pa AD Ie 190 Ae 
| oe 4a, —120, 12c, —4d 
that is “2 {(a—B)(a—y)(B—y)y’st}=| 12¢, -12d, 4e, , 
— 4b, 6c, — 4d, e 
— 40, 12¢,-12d, 4e 
—- + — + 
= 436, 0d 6, ot". : (9), 
[sD oC O,.. 
4\. 6, 3¢, 3d., € 
—4; a, 3b,3c,d 
thus completing the set of five equations for a biquadratic. 
