AND IMAGINARY ROOTS OR EQUATIONS. 
631 
the case before us, if the quintic have three equal roots, we may reduce it to the form 
ax 5 -j- 5 bxSj +10 czhf. 
Suppose now, if possible, an invariant of the degree m ; the weight of each term therein, 
say a r b*c\ in respect to x or y would be the same ^viz. so that we should have 
5r+4s+3£=^=s+2£, or 5s+3s+£=0, 
and therefore r=Q, s— 0, £=0, m= 0. So for a sextic with three equal roots reduced 
to the form ( a , b , e , 0, 0, 0]$+, y ) 6 . Supposing any term in one of its invariants to be 
a r b s c\ we should have 
6r+5s+4£=^=s+2£, or 6r+4s+2£=0, 
which is absurd, unless r— 0, 5 = 0, tf=0, m= 0, and so in general for a binary form of 
any degree. If in the above example for the degree m only three roots were equal 
inter se (the form assumed being (a, b, c, d, 0, 0, 0%x, y) s , any term in a supposed inva- 
riant being a r b s c t d u , where r+s+£+w=m, we should have 
and, as before, 
6r+5s+4£+3w=3»i=s+2£+3w, 
6r+4s+2£=0, r— 0, s=0, t— 0; 
no longer, however, m= 0, but m=u, which is left undetermined. 
(49) Before proceeding further it will be proper to consider under what circumstances 
a variation (in the coefficients of any equation) arbitrary, except that the coefficients are 
to remain real, can affect the character of the roots. 
Let F(#)=0 be any algebraical equation with real coefficients, and let ?5(F#) be the 
variation of F due to the variation of the coefficients, dF(x) the variation due to the 
change of x into x-\-dx. If, now, r be a root of F#=0, and r-\-dr the corresponding 
root of F(tf)+&F(tf)=0, we have 
Fr=0, F(r+c7r)+5F(r)=0, or SF(r)+^ F(r)dr+-L(^X¥r(dry+8cc.=0. 
civ I . L \ar / 
Hence, unless = 0, i. e. unless there are two equal roots r, we shall have 
dr- 
= a real quantity; so that the character of the root r-\-dr will be the 
same as that of r. 
But if 
?=0, *?=o, ... 
dr i dr 2 
F=0, 
so that there are i roots r, i being any integer greater than zero, then to find dr we 
have the equation 
WS+^f= 0 ' 
a 
Thus dr will have i distinct values ; of these, if i is odd, all but one will be imaginary, 
but if i is even they will be all imaginary, or only all but two imaginary and the remain- 
