PROFESSOR CAYLEY’S NINTH MEMOIR ON QUANTICS. 
47 
355. Every covariant whatever is a form or sum of forms such as 
1218^3’... 
writing in regard to any such expression 
2 inch l=i, X inch 2 —j, . . . 
(viz. i is the sum of all those indices a, (3, &c. which belong to a term containing the 
symbolic number l,j the sum of all the indices a, y, &c. which belong to a term con- 
taining the symbolic number 2, and so on) then each of the numbers i,j, ... is at most 
=n, that is n—i, n—j, . . . may he any of them =0, but they cannot be any of them 
negative; the degree of the function is =m, and its order is =mn—i—j. ... It is to 
be further observed that the form is a function of the differential coefficients of f of the 
orders n—i, n—j, &c. respectively. It follows that if n — i, n—j , . . . are none of them 
= 0, the form in question may be obtained from a like form belonging to a quantic 
f of the next inferior order n— 1 by replacing therein the coefficients a', V, ... 
by ax Achy, bx-\-cy , &c. respectively: for example, if f denote the cubic function 
(a, b, c, djjc, y ) 3 , then the Hessian hereof is 12’// ; the like form in regard to the 
quadric f—(a!, V, c'jjc, y) 2 is 12//, which is —aid — V 2 ; and substituting herein 
ax -[-by, bx-\-cy , cx-\-dy for a’, V , d respectively, we have the Hessian 12// of the 
cubic. A co variant of f derivable in this manner from a covariant of the next inferior 
quantic/' is said to be a special covariant. 
356. Reverting to the form 
I2“l3' , 23'... 
if, as before, n—i, n—j, &c. are each of them > 0 ; if there is at least one index i which 
is = or < -In (that is, for which n—i>\n ), and if the order mn—i—j. . . be > n, then 
the form, or any sum of such forms, is said to be a form or covariant W. Every covariant 
W is thus a special covariant, but not conversely. In the particular case m= 2, the 
form is 
which will be a form W if n — a>\n, or, what is the same thing, 2n — 2a, >n, that is if 
the order be > n. Hence, as already mentioned, the covariants T of the degree 2 are 
W, or else y, according as the order is greater than n, or as it is equal to or less than n. 
357. Theorem B. — If any covariant T be expressible as the sum of a form W and of 
earlier T’s than itself, then forming the derivative (T f) k , either this is not a form T, or 
being a form T, it is expressible as the sum of a form W and of earlier T’s than itself; 
or, what is the same thing, (T f) k , if it be a form T, is (like the original T) the sum of a 
form W and of earlier T’s than itself. 
Hence also every form T is the sum of a form W, and of forms derived from the 
functions sa Y 
T=W+P X , 
or, what is the same thing, every co variant whatever is of the form W+P x . 
