OF THE DIEEEEENCES OF THE EOOTS OF A GIVEN EQUATION. 
47 
To apply the method above explained, write 7=0, and therefore also d—^', the roots 
thus become 
and we have the quadric and linear equations 
(^+«)(^-/3)=0, ^_(a_/3)=0, 
where (a, (3) are the roots of the equation 
{a,h,cjv,iy={). 
Hence, writing 
Z=4«!c— 
we have 
a— /3=^\/ — Z, 
and the two equations become 
Z— c=0, — Oj 
or multiplying the two equations together, 
5V-}-S^O+5(3«c— — Z=0, 
which is what the required equation becomes, on putting therein (Z=0; the coefficients 
of the complete equation are seminvariants, and the terms in d are to be inserted by 
means of this property. The coefficient ‘^ac—h^ is reduced to zero by the operator 
it is therefore a seminvariant, and remains unaltered. The coefficient C\/ — Z is what 
^ —n becomes ( □ being the discriminant of the cubic equation) on putting therein 
d=0, it is therefore to be changed into \/— □. Hence 
4. For the cubic equation («, b, c, dJJ^v, 1)® the equation for /3)j is 0= 
1)^ 
a?x 
'v/— n X 
A 
+ 1 
( ■> 
0 
+ 3 ac 
-1 b- 
f ■> 
+ 1 
For the quartic equation (a, b, c, d, e^v, 1)'‘=0 ; 
5. Here 
6z=—^^(u, /3, 7)=— (a— ^)(a-y)(/3— r), 
the roots are 
^2= — ^^(y, a). 
^3= a,/3), 
^ 4 =— 7), 
the signs being in this case (and indeed for an equation of any even order) alternately 
positive and negative; in fact, if the equation is represented by <pv=0, then the roots 
divided by ^’(a, /3, 7, ^) should respectively be ip'a, (p'j3, (p'y, (p'^, and this will be the case 
if the signs are taken as above. 
H 2 
