AND IMAGINARY ROOTS OR EQUATIONS. 
617 
are all zeros, the quadratic covariant becomes 
4( c 1 — l d) x 2 + 4 (d 2 — ce)vf. 
Supposing neither of those coefficients to vanish, and calling its two factors u and v, and 
making 
F(x, y)<P(u, v)=(a, (3, y, i, g, iXu, v ), 
it is clear that the minors of 
a (3 y 
y'bsi 
can no longer all be zero, since in that case we should have 
4(y 2 — (3tyu 2 + 4(§ 2 — ys)v 2 
containing u, v as factors. Consequently the canonizant of <E> must vanish under one 
or the other of those remaining suppositions which had been previously shown to con- 
duct to the form au 5 + bv 5 , or else to the case of three or more equal roots. When the 
quadratic covariant vanishes, we know that there must be four equal roots ; and when 
it becomes a perfect square but does not vanish, it will be found on examination that 
the equation has three equal roots. 
(36) Returning to the general case, where < P=u 5 -j-v 5 ~l-w 5 , and making "r+w+p 
. U V TV 
identically zero, and writing u’, v 1 , w’ for respectively, T> becomes ru l5 -{-sv' 5 -\-tw' 5 , 
or, if we please, rvJ’-\-sv‘-\-tw 5 , with the condition u-\-v-\-w=-Q. 
Moreover u , v, w will all three be factors of the canonizant of F. For taking the 
canonizant of F with respect to w, v, it becomes 
r—t 
-t 
-t 
-t 
1 
0 
0 
0 
-t 
-t 
—t 
—t 
or rx < 
-1 
-1 
-1 
-1 
—t 
-t 
—t 
s—t 
-1 
-1 
-1 
1 
v 3 
—v 2 u 
vu 2 
—u\ 
v 3 
—v 2 u 
vu 2 
-u 3 
or rst(uv 2 +w 2 ), i. e. —rst(uvw). 
Hence if x-\-ey, x-\-fy, x-\-gy are three distinct factors of the canonizant of F with 
respect to x, y, if we choose the ratios X : y* : v so that X+|U>+!>:=0, eX-\-fyj-\-gv=0, we 
may make u=X(x-\-ey) ; v=y,(x-j-fy) ; w—v{sc-\-gy) ; and shall then have 
F(^, y)—rv?-\-sv 5 -\-tw 5 , with the condition u-{-v-\-w—§, 
where r, s, t may be found from three equations obtained by identifying any three of 
the six terms in F with the corresponding terms ru 5 -\-sv 5 -\-tw 5 expressed as a function 
of x, y. These equations being linear, it follows that ru% sv 5 , tw 5 form a single and 
unigue system of functions of x, y. 
