MR. A. CAYLEY’S INTRODUCTORY MEMOIR UPON QUANTICS. 
257 
(i. e. when ^=0) md is necessarily even*; so that a qnantic of an odd order admits 
only of invariants of an even degree. But there is an important distinction between 
the cases of md — g evenly even and oddly even. In the former case the covariant 
remains unaltered by the substitution of (y, x), («', b\ .. b, a) for (x,y), (a, b , .. b\ a) ; 
in the latter case the effect of the substitution is to change the sign of the covariant. 
The covariant may in the former case be called a symmetric covariant, and in the 
latter case a skew covariant. It may be noticed in passing, that the simplest skew 
invariant is M. Hermite’s invariant of the 18th degree of a quantic of the 5th order. 
23. There is another very simple condition which is satisfied by every covariant of 
the quantic 
(a, b..b\ a'Xx, y) m , 
viz. if we consider the facients {x,y) as being respectively of the weights — i, and 
the coefficients (a, b..b\ a') as being respectively of the weights —\m, — \m-\- 1, 
..\m — 1, \ m, then the weight of each term of the covariant will be zero. This is the 
most elegant statement of the law, but to avoid negative quantities, the statement 
may be modified as follows: — if the facients ( x , y) are considered as being of the 
weights 1, 0 respectively, and the coefficients (a, b..b\ a') as being of the weights 
0, 1..7W—1, m respectively, then the weight of each term of the covariant will be 
24. The preceding laws as to the form of a covariant have been stated here by 
way of anticipation, principally for the sake of the remark, that they so far define the 
form of a covariant as to render it in very many cases practicable with a moderate 
amount of labour to complete the investigations by means of the operations {.rB^} — xb 9 
and {yb x }—yb x . In fact, for finding the covariants of a given order, and of a given 
degree in the coefficients, we may form the most general function of the proper order 
and degree in the coefficients, satisfying the prescribed conditions as to symmetry and 
weight: such function, if reduced to zero by one of the operations in question, will, 
on account of the symmetry, be reduced to zero by the other of the operations in 
question ; it is therefore only necessary to effect upon it, e.g. the operation {.tB,,} — xb^ 
and to determine if possible the indeterminate coefficients in such manner as to 
render the result identically zero: of course when this cannot be done there is not 
any covariant of the form in question. It is moreover proper to remark, as regards 
invariants, that if an invariant be expanded in a series of ascending powers of the 
first coefficient a, and the first term of the expansion is known, all the remaining 
terms can be at once deduced by mere differentiations. There is one very important 
case in which the value of such first term (/. e. the value of the invariant when a is 
put equal to 0) can be deduced from the corresponding invariant of a quantic of the 
* I may remark that it was only M. Hermite’s important discovery of an invariant of the degree IS of a 
quantic of the order 5, which removed an erroneous impression which I had been under from the commence- 
ment of the subject, that m0 was of necessity evenly even. 
2 L 
MDCCCLIV. 
