Canonic Roots of a Binary Quantic of an Odd Order, 207 



In fact, considering, as before, the quinticU = (a, b,c i d i e > f'$x i y) b , 

 we have 



V=A{lx + my) 5 + A'{l r x + 7n'y) b + A!'{l' ! x + m"y) 5 , 



and thence 



(Xd a + Td^V = B(lx + my) 3 + W(l , x + m , y) 3 + 'B' , {r l x + m n y) 3 J 



if for shortness B = 6 . 5(/X + wY) 2 A, &c. 

 Suppose (X^ + YB ) 2 U has in common with U the canonic 

 root lx-\-my, then 



(X.-d x + Yd/\J = C(Ix + my) 3 +C l (px + qy) 3 > 



and thence 



B'{l'x + m'yf + B"(l"x + m"y) 3 = (C - B) (Ix + my) 3 + C'( px + ?y) 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', Vx + m'y=px + qy, B" = 0, C = B, 

 and then we have 



(X^ + YB 2/ ) 2 U = B(^ + m 2 /) 3 + B'(^ + m'z / ) 3 ; 



so that all the canonic roots of the emanant are canonic roots of 

 the quantic. Moreover, the condition B" = gives /"X + ?/i"Y=0, 

 that is, X : Y = m" : —I", or writing rx + sy instead of l"x + m"y, 

 X : Y = s : — r ; and the system is 



U = A(Ix + myf + A!(Vx + m'y) 5 + A {rx + sy) b , 



(s-d x -r?)/V = B(lx + my) 3 + B'(l>x + m'y) 3 , 



which proves the theorem. 



2. The two functions, canont. U, canont. (X^^- Yd ) 2 U, have 

 for their resultant {canont. U(X, Y in place of x, y)} 2n y if 2n+ 1 

 be the order of U. 



In fact, in order that the equations 



canont. U = 0, canont. (Xd, + YB y ) 2 U = 0, 



may coexist, their resultant must vanish ; and conversely, when 

 the resultant vanishes, the equations will have a common root. 

 Now if the equation canont. (XB X + Y^) 2 U = has a common 

 root with the equation canont. U = 0, all its roots are roots of 

 canont. U = 0; and, moreover, if rx + sy = be the remaining 

 root of canont. U = 0, then X : Y = s : — r, that is, we have 



canont. U(X, Y in place of x, y) = ; 



or the resultant in question can only vanish if the last-mentioned 

 equation is satisfied. It follows that the resultant must be a 



