OF THE DIFFBEENCES OF THE EOOTS OF A GIVEN EQUATION. 
49 
and the terms in e to be added to this, in order to make it a seminvariant, are easily 
found to be 
— \^a^ce 
8. Hence, for the quartic equation («, 5, c, I)"*, the equation for (3, y)j 
is 0= 
a^X 
A 
fl- □ X 
A 
arx 
, A , 
^ □ X 
A 
□ X 
< o ’ 
1 or 
0 a 
r 1 
— 16 a“ce 
+ 18 erd^ 
+ 6 a¥c 
— 14 abed 
+ 4 ac^ 
+ 3 b^d 
- 1 6+2 
f 1 
— 8 a^d 
+ 4 abc 
-1 b^ 
\ 
+ 1 
For the quintic equation (a, b, c, cl, e,f%v, 1)®=0 ; 
9. We have 6 (3, y, ^), the roots being 
^j = 2^(f3, y, which for £=0 becomes 
^2~^(y5 11 ?? 
2 , a, 13), „ „ 
y)? 5? ?5 
7’ » » 
(3y^t^((3, y,'b), 
—yhal^iy, «) 5 
bci[3l^{h, a, f3) , 
— a(3yt^{K, (3, y), 
(3, y, b). 
10. The linear equation is ^a^+\/Z = 0; the quartic equation may be written 
U.^[d—6i)=0, for the determination of which see Annex No. 2. The two equations are 
. a® 
+^^-aVZ 
. a^lsie 
-\-Q . N^VZ 
+ 
= 0 
O.a^ 
+ x/Z 
= 0 ; 
and multiplying these together, the resulting equation is 
0^ . 
. 0 
. a®(Me— Z) 
. a^{J^e-\-M.)e\/Z 
+ 0 . (N+a^^yZ 
-(- 6^Z\/Z—0, 
