MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS, 
105 
M being an absolute invariant. Then, by means of all the eductive operators associ¬ 
ated with absolute invariants of successive ranks, we can obtain from I the set of 
educed invariants 
T; 
I. 
I.', 
f (7 3 
I(/3 ^ 6 } I/' 3 
1.3 1/3 . . .3 
all proper to the rank n 1 •, and there is a similar set from each of the other 
invariants J, K, . . . 
This number, however, can be at once reduced ; for, if be any educed invariant 
other than 1 ' and la, we have 
and therefore 
r, 
1/73 
P 
dA dA 
- 3 
d// oj: 
OiAi an 
di/ ’ a^c 
IM' = TM, + 
which shows that can be expressed in terms of I' and and of invariants proper 
to lower ranks if M be different from J, K, . . and that, if M coincide with one of the 
invariants J, K, . , the invariant can be expressed in terms of the set I', J', . . ., 
the set la, Ja, . . ., and of invariants proper to lower ranks. 
it therefore follows that the invariants, educed from the absolute irreducible 
invariants I, J, K, , . , proper to the rank n, can be expressed in terms of I', J', K', . . . ; 
la, Ja, Ka, . . . pi’oper to the rank n 1, and of invariants proper to lower ranks. 
All these educed invariants are, if n be greater than 3, linear in the partial differential 
coefficients, which are of order n 1, and so determine tiie rank of the invariants. 
We know that, for values of n greater than 3, the number of iiTeducible invariants 
proper to the rank n is n 1, all of which can be made absolute on division by an 
appropriate power of Aq ; hence, the number of invariants educed as above is 2 (w + 1), 
which must all be expressible in terms of the n -f- 2 irreducible invariants proper to 
the rank w + 1. But so far there is nothing to indicate which of them, or how many 
of them, are equivalent to irreducible invariants proper to the rank to which they 
belong. 
28. Again, we have seen that there are four simultaneous invariants of two func¬ 
tions proper to the rank 2, and that there is a single invariant proper to the rank 1 ; 
so that 
MDCCCLXXXIX.-A. 
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