98 MR. A. R. FORSYTH OH A CLASS OF FUHCTIOHAL IHVARIAHTS. 
Taking the eqaations in the order adopted before and beginning with (Ag + A'g) i/; = 0, 
we have the set of snbsidiarv equations 
cliJ clq dr ds dt dj)' dq' dr' ds' dt' 
q ~ 77 ~ 2.S ~ ~ (7 ~ q^ ~ ~ Ys' ^ V ~ 0 ’ 
nine in number. It is necessary to obtain nine independent integrals of the set; and 
we may take these in the form 
= q I Wj = M — 9.^' ~~ lY I % — ^0 
u\ = q \ u'z = i — q^ — lY J = A'o 
bearing in mind the property above proved. By the theory of linear partial 
differential equations of the first order, it follows that every solution of the equation 
(Ag + A'g) iff = 0 can be expressed as a functional combination of u\, u^, 
u^, u\, Thus, 
qs — 
UyU 3 — 
St' — s't = 
qh'' — 2pqs' + = 
q'Y — 2p)qs + pH = 
upb 3 — U 
U.lC 
U'' 1 
U^lb'^ — hb'pipb^ — u' 
u'-^u^ + 2ibpb'^ib^ — upb^ 
and so on. 
To obtain the most general solution of ®gr// = 0 = it will be sufficient to form 
the irreducible combinations of . . . , Wg which satisfy ©pjj = 0. Now, 
0pq =z 0, ©1^7 = 0 ; 
©pq = 0, ®iu'^ =: 0 ; 
©1^4 = 0, ©pfi^ = 0 ; 
and 
so that 
©pq 
= qq, ©lAg = q'q ; 
©1 (?/,3 - u^) = 0 ; 
^ 5 
and, therefore, the irreducible combinations which satisfy ®pp = 0 are u^, u'-^, u^, 
%, %, u'^, u^, where 
Ug = Mg — Mg = qs — q's — pt' + p't. 
To obtain the most general solution of ©gi); = 0 = @pp = ©pj/, it will be sufficient 
to form the irreducible combinations of the preceding eight quantities which satisfy 
(tdpjj = 0. Now 
