92 MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
C/ombinations of this kind, which are of uniform grade and are homogeneoiis, are 
Aq, AqAo — A^^Ag — /^AqA^Ao + AAj®, AqA.j, — m^A^^Ag + and so on. 
Vf hen these are substituted in A^/ = 0 and the coefficients Jc, I, in,. . . are determined 
so that the equation is satisfied, we find the following set of invariants :■— 
Uq = Aq, 
V, = AqAo - A^q 
Ug - Ao^Ag - J#AoA,A, + ^^Al^' 
U 4 , = AqA^, — SAjAg + ^-A.j", 
These combinations suggest an analogy with tlie coefficients of the principal irredu¬ 
cible covariants of a quantic. If we change the symbols by the relation 
A,„_3 — 111 \ {ni — 1 ) C.q„_.2. 
then, except as to numerical factors, the functions are 
OoCh - 
Co^Cg - sCoCjC^ + 20 ,\ 
CqC,. - lO^Cg -f say 
that is, they follow the same law of formation as the leading terms of the covariants 
referred to ; and they can therefore be expressed in terms of the quantities C and 
can thence be deduced in terms of the quantities A. 
All these functions satisfy the equation 
cv 
+ 2Ci 
0Co 
= 0 , 
that is, they satisfy the equation 
S {n + 2) {11 + 1) A 
dv_ 
0A„. 
= 0, 
hy means of which the numerical coefficients in U can be directly determined. 
It is evident from the form of U ;„_3 that the higliest order of differential coefficient 
which enters is the mth, that all the differential coefficients of the with order enter 
linearly and into only one set of terms, and that the remaining terms all involve 
coefficients of lower order of differentiation. 
