OF THE ROOTS OF AN EQUATION AND ITS COEFFICIENTS. 543 
The left hand side of this equation, in terms of the roots, is 
=> tla 56" , ¥ i a Sek died Pewldy's 
@ SB POR ye ye ns 5 4 
a’, B, y' | By+Bo+y6 , 
Bys oe OE 8 
= La’ B'y?(a? ~ B’) (a? — y’) (2 —y’). (a % B) (a— Y) (8 a Y) 
hy 0 
C5. 
OF'O4 
OF0 
for instance, the equation got by omitting the line | A | is, after slight simplifica- 
tion, 
© {a3 B%_°5?(a2— B?) (a? —y)(B?—yy*) (a— B)(a—y)(B—y)} =PsPi |8 » 85» S|— Py? |s6 5 $5 583] (12), 
$4 , §3 , 82 nein 
Sos Sid Says Sq a0 
and that got by omitting the line | F is | 
2 { a'p*y'd"(a’ — B’)(a’—y")(B’ —y’)(a—B)(a—y)(B—y)} = Pil se, 95, Sa], 
Sz, 83 , Se 
Cry Sra 
or 2 {a—B)"(a—y)'(B—y).(at B)(aty)(B+y)} =| S65 85» S (13). 
84, 83 , So 
So ) sy )’ 4 
This last one is easily verified ; for, the determinant 
SG, ay polo, B 7 Oo 
a,B,y,8)a,B,y,8 
Tagua AR ce i 
=2 (7 —B.a—y. B’—7)(a—B.a—y.B—y)}- 
V. We may combine matrices of other forms. For example, in a biquad- 
ratic— 
a8 ,ay,ad, By, BS, 6 
“| Oia pie aie SP 
yo, Ba 5 By , a0 5 ay , of 
aByd , aByd , aByd , aByd , aBys , aByd 
uy ,| =86Pi—P;P, 
VOL. XXIX. PART II. 6 Y 
