im. A. R. FORSYTH ON A CLASS OP FUNCTIONAL INVARIANTS. 
98 
Invariants in the Fortrth Order. 
17. To obtain the irreducible invariants which involve no clilferential coefficient of 
order higher than four, we may proceed as in ^ 11 by forming the irreducible functions 
which satisfy the differential equations ; mnong these functions the invai'iants already 
obtained will occur. 
Foi\convenience, let the differential coefficients of the fourth order be denoted by 
e,f, g, h,i{ = z^.^, respectively). Then, beginning as before with A,/ = 0, 
the subsidiary equations additional to those already (§ 12) considered are 
de df dg dh d.i 
of which the irreducible independent integrals are 
Ro = L 
^bo — 
'hi = P~^ — + vV. 
- pH — ?>ppq], -P 8/9r/"q — q^f, 
— nff — Ap^qh + (yp'^q^g — Apq''^_f q^e \ 
and any solution of 0 is expressible as a function of ;q, %, . . ., 
The remainder of the analysis is very similar to that which has been used for the 
earlier question, and so it is not here reproduced ; the following are the results : — 
(i) The functional combinations of the thirteen quantities w which satisfy 0 
(and which are, therefore, the irreducible simultaneous solutions of = 0 = 
are 
^1, u^, u^, Rq ; 
Vf. = upiQ + 
Vj = upij - SWg-', 
= upi^ + Gupc ^; 
Uio = wp-Uo + 3M3W3, 
+ 3?/o?b, 
^13 = + 9 iqR 3 !q — 12^3''^, 
Vjg = + I2,upiptf^ + S6u^"u^. 
(ii) The functional combinations of these twelve quantities which satisfy Aof = 0 (and 
wdiich are therefore the irreducible simultaneous solutions of A^f= Aif= — d) 
are 
