THEOREMS RELATING TO THE CANONIC ROOTS OF A BINARY 
QUANTIC OF AN ODD ORDER. 
[From the Philosophical Magazine, vol. xxv. (1863), pp. 206—208.] 
I CALL to mind Professor Sylvester’s theory of the canonical form of a binary 
quantic of an odd order ; viz., the quantic of the order 2n +1 may be expressed as 
a sum of a number n + 1 of (2n +1) th powers, the roots of which, or say the canonic 
roots of the quantic, are to constant multipliers près the factors of a certain covariant 
derivative of the order (n + 1), called the Canonizant. If, to fix the ideas, we take 
a quintic function, then we may write 
(a, b, c, d, e, f\x, yj = A (lx + my) 5 + A'(l'x + m'y) 5 + A"(l"x + m"y) 5 
(it would be allowable to put the coefficients A each equal to unity ; but there is 
a convenience in retaining them, and in considering that a canonic root lx + my is 
only given as regards the ratio l : to, the coefficients l, to remaining indeterminate) ; 
and then the canonic roots {lx + my), &c. are the factors of the Canonizant 
y 3 , -y% yx-, -x â . 
a, b , c , d 
b, c , d , e 
c, d , e , f 
It is to be observed that this reduction of the quantic to its canonical form, i.e. 
to a sum of a number n-\-1 of (2l)th powers, is a unique one, and that the 
quantic cannot be in any other manner a sum of a number n +1 of (2n +1) th poweis.