[326 
neral form, and 
ie quantic £7 a 
nic roots of the 
be rx + sy, then 
thing, they are 
have 
326] 
BINARY QUANTIC OF AN ODD ORDER. 
105 
my, then 
3 
} 
cubic function 
quantic. More- 
— I", or writing 
may coexist, their resultant must vanish ; and conversely, when the resultant vanishes, 
the equations will have a common root. Now if the equation canont. (Xd x + Yd y f U = 0 
has a common root with the equation canont. £7 = 0, all its roots are roots of 
canont. U = 0 ; and, moreover, if rx + sy = 0 be the remaining root of canont. £7=0. 
then X : Y = s : — r, that is, we have 
canont. £7 (X, F in place of x, y) = 0 ; 
or the resultant in question can only vanish if the last-mentioned equation is satisfied. 
It follows that the resultant must be a power of the nilfactum of the equation; and 
observing that canont. £7 is of the form (a, ...) n+1 (x, y) n+1 , i.e. that it is of the degree 
n+1 as well in regard to the coefficients as in regard to the variables (x, y), it is 
easy to see that the resultant is of the degree 2n(n +1) as well in regard to the 
coefficients as in regard to (X, F) ; that is, we have 2n as the index of the power 
in question. 
3. In particular, if F = 0, the theorem is that the resultant of the functions 
canont. £7, canont. dfU is equal to the 2nth power of the first coefficient of canont. £7 
Thus for n — 1, that is, for the cubic function (a, b, c, dffx, yf, we have 
canont. £7 ^ 
canont. dJU— 
y\ 
-xy, 
a, 
b , 
b, 
c , 
y> 
-X | 
a, 
b j 
= (ac — b-, ad — be, bd — dXfx, yf, 
— ax + by ; 
and the resultant of the two functions is 
= (ac — b-, ad — be, bd — cX}]), — a) 2 
= — (ac — 6 2 ) 2 , 
which verifies the theorem. 
The theorems were, in fact, given to me in relation to the quantic £7 and the 
second differential coefficient d x 2 £7; but the introduction instead thereof of the second 
emanant (Xd x + Ydyf £7 presented no difficulty. 
2, Stone Buildings, W.G., February 16, 1863. 
their resultant 
C. V. 
14