96 
MR. A. R. FORSYTH OH A CLASS OF FUHCTIOHAL INVARIANTS. 
From this Table it at once follows that 
^oQi3 ~ 1-^Qi3 “ ^iQi 35 
^oQl:; “ ~ ^iQi 2 ’ 
^oQii ~ 3*^Qii — ^iQin 
"oQlO “ ^'^QlU “ '^iQiOj 
^oQ'J ~ 54 Q,j = -O^Qg. 
Hence, Qg, Q^q, Qn, Qis, Q 13 cire invarictnfs of indices 18, 15, 12, 9, 6 respectively ; 
and every invariant which involves no ddferenticd coefficient oj order liujher than the 
fourth can he expressed as a function ff u^, Q^, Qg, Q-, Qg, Qio? Qn? Qi3> Qia- 
General Inferences. 
19. And if, among tlie sets of irreducible invariants thus obtained, those invariants 
which involve the partial differential coefficients of the rAh order as the highest that 
occur, and which are linear in those partial differential coefficients of highest order, 
are called irreducible invariants to the rank n, then we have the following 
propositions relating to the complete aggregate of invariants :— 
(i) The irreducible invariants can be ranged in sets, each set being proper to a 
particular rank ; 
(ii) There is no irreducible invariant proper to the rank unity; 
(hi) There is a single irreducible invariant (= u^= Aq) proper to the rank 2 : 
(iv) There are three irreducible invariants (= Qg, Qg, Q^) proper to the rank 3 ; 
(v) For every value of n greater tlian 3, there are n ffi 1 irreducible invariants 
proper to the rank n, and they can be so chosen as to be linear in the 
ditlerential coefficients of order n ; 
(vi) Every invariant can be expressed as a function of the irreducible invariants ; 
and, if such an invariant liave ditferential coefficients of order r as those of 
highest order occurring in it, the functional equivalent involves some or all 
of the aggregate of irreducible invariants proper to ranks not greater than r; 
it involves some of thi 3 irreducible invariants proper to the rank r, but no 
irreducible invariant proper to a rank greater than r. 
Simidtaneous 1 nvariants of Two Functions. 
20 . Hitherto we have considered invariants of only a single dependent variable 
which is a function of the two independent variables ; but we may consider a second 
dependent variable, say %, which is also a function of x and y. The two quantities 
