300 ON THE THEORY OF INVOLUTION. [348 
must contain, instead of the factor (x — ay), the factor (x — ay) 2 . In order that this 
may be so, we must have 
I KV, © 
ByV, - a© ’ 
that is, (a& s I r + 8,,F) © divisible by (x — ay), that is, either a8 x V + 8 y V, or else ©, 
divisible by x — ay, or, what is the same thing, either V, or else ©, divisible by 
x — ay. But if V be divisible by x — ay, then U, V have the common factor x — ay, 
and we have the case first above considered. And again if © contain the factor 
x — ay, then we have 
JJ = — & X F+ (x — ay) 3 ©', 
and we have the case secondly above considered. Finally if the new critic centre be 
the distinct centre x — ¡3y = 0, then for x — ¡3y = 0 the equation 
-Ic : 1= 8 X U : 8 x V = $ y U : 8 y V 
should give k = k 1 ) but this will only happen if 8 x .(x — ay) 2 ©, 8 y . (x — ay) 2 © vanish 
for x — (3y= 0, that is, if © contains the factor (x — f3y) 2 ; and when this is so, 
17 = — k l V+(x — ay) 2 (x — /3y) 2 ©', 
or we have the case thirdly above considered. 
14. Hence the equation D=0 being satisfied if R = 0, or else if Q = 0, or else 
if P = 0, and in no other cases, the function □ must be made up of the factors 
R, Q, P, each taken the proper number of times, and knowing the degrees of the 
several functions, it follows that we must have 
□ = RQP 2 , 
in fact, considering the coefficients of either U or V, the comparison of the degrees 
gives 
2 (n — 1) (2n — 3) = n + 9 (n - 2) + 4 (n — 2) (n — 3), 
where the function on the right-hand side is 
= n 
+ 9w-18 
+ 4 n 2 — 20n + 24 
= 4ft 2 — 10 ft + 6, 
which is the value of the function on the left-hand side. 
15. In the very particular case n = 2, Q and P are each of them of the degree 
= 0; and we have simply □ = R, that is, the resultant of the two quadric functions 
U=(a, b, clfix, y) 2 , V=(a', b', c'fitx, y) 2 
= Disc 1 . (U + kV) 
= Disc 1 . (ac — b 2 , ac + tie — 2bb', a'c' — b' 2 $1, k) 2 , 
is