MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
115 
the weights of its concomitants are estimated by assigning to q, i, g) • • • 
the weights 0, 1, 2, ... in succession, that is, the integers which represent the y-grade 
of these coefl&cients 
If we have a covariant of order m which is simultaneous to A„_ 2 , A„/_ 2 , A;;-/_o, 
. . . , and of degrees I, V, I", ... in their respective coefficients, and its leading term 
be Co9''“, then the weight of Cq is 
I {nl -b nV + n"l" — m), 
whicli is, therefore, the number representing the y-grade of Cg, considered as the 
leading term of a functional invariant. Since the grade of each of the coefficients 
of A ;(_3 is n, it follows that the grade of Cg, so far as it involves the coefficients of 
A „_3 is In, and, therefore, the grade of Cg is, in the aggregate, 
nl + nl' + n"l" . 
But the aggregate grade of Cg is the sum of the .-r-grade and the y-grade ; hence, the 
ic-grade of Cg is 
{nl + nV + n"l" + . . . + m). 
In order, then, to express 'F in terms of the quantities u, we should proceed to 
substitute for the coefficients r, s, t, . . . the values above obtained, and assuming 
= Cg2"' — + . . . , 
it is evident that the only term in ^ from which terms independent of y) can come is 
the first term. Moreover, since is expressible as a function of the quantities 
M alone, it follows that when these substitutions are carried out the terms involving 
p must disappear, for p is the only non-u quantity which enters into the expressions 
substituted ; and the value of is, therefore, the aggregate of terms which survive, 
that is, the aggregate of terms independent of p arising from Cgf/™. 
Now, in Cg this aggregate is obtained by replacing a coefficient, by a quantity 
±'ih u-^ ; since Cg is isobaric qua seminvariant, it is of uniform a;-grade qua part 
of functional invariant; and therefore the result of these substitutions is to give a 
function of u^, Wg, u^, , divided by a power of u-^ equal to the cc-grade of Cg, that 
is, divided by 
at \(nl + n'V + n"l" + . . . + jn) 
If, then, Fg denote this function of u^, u^, Uj^, . . . , we have 
or 
+ • - • —?«) -vjr — p 
Q 2 
