104 THEOREMS RELATING TO THE CANONIC ROOTS OF A [326 
Professor Sylvester communicated to me, under a slightly less general form, and 
has permitted me to publish the following theorems: 
1. If the second emanant (Xd x + Yd y f U has in common with the quantic U a 
single canonic root, then all the canonic roots of the emanant are canonic roots of the 
quantic; and, moreover, if the remaining canonic root of the quantic be rx + sy, then 
(X, Y), the facients of emanation, are = (s, —r), or, what is the same thing, they are 
given by the equation 
canont. U (X, Y in place of x, y) = 0. 
In fact, considering, as before, the quintic U = (a, b, c, d, e, f\x, yf, we have 
U = A (lx + my) 5 + A' (l'x + m'y) 5 + A" (l"x + m'y) 5 , 
and thence 
(Xd x + Ydyf U = B (lx + my) s + B' (l'x + m'y) 3 + B" (l"x + m"y) s , 
if for shortness 
-B = 6.5 (IX + mYf A, &c. 
Suppose (Xd x + Yd y f U has in common with U the canonic root lx + my, then 
(Xd x + Ydyf U=G(lx+ my) 3 + C' (px + qy) 3 , 
and thence 
B' (l'x + m'y) 3 + B" (V'x + m'y) 3 = (C — B) (lx + my) 3 + C(px + qy) 3 , 
which must be an identity; for otherwise we should have the same cubic function 
expressed in two different canonical forms. And we may write 
B' = C', l'x + m'y —px + qy, B" = 0, C = B, 
and then we have 
(Xd x + Ydyf TJ — B (lx + my) 3 + B' (l'x + my) 3 ; 
so that all the canonic roots of the emanant are canonic roots of the quantic. More 
over, the condition B" = 0 gives l"X + m"Y=0, that is, X : Y = m" : —l", or writing 
rx + sy instead of l"x + m”y, X : Y = s : — r; and the system is 
U — A (lx + myf + A' (l'x + m'y) 5 + A (rx + sy) 5 , 
(sd x — rdyf TJ = B (lx + my) 3 + B'(l'x + myf, 
which proves the theorem. 
2. The two functions, canont.?/, canont. (Xd x + YdyfU, have for their resultant 
{canont. U (X, Y in place of x, y)} m , if 2n +1 be the order of TJ. 
In fact, in order that the equations 
canont. U = 0, canont. (Xd x + Yd y fU = 0,