88 
MR. A. R. FORSYTH OH A CLASS OF FHNCTIOHAL IHVARIAHTS. 
Then we have 
Q? = + T2%X) 
{u^u^ + u^v^) + 12 ^ 4 “ — u.^Ui)] . 
But 
u^Ug + Ui>-f 7 = { 9 s — pi) {p^d — ?>p^qc + 'dpq^ — q^a) 
+ {q{qr — j^s) — {qs — pt)] {p\I — 2 pqc + q^b) 
= q {qr — ps) {phi — 2pqc 4- <fb) q{qs — pt) (— p“c + ^^qb — q^a) 
_ 1 _ i., T 
dq dq dj) ^ 
()“l ^01 > 
where Jq]^ denotes the Jacobian of Aq, with regard to and Wq;^. Similarly, 
so that 
up — upt^ = — ^’0 = ~ ; 
Q- = + 2AoJqi + 12Aq®Hq, 
Hq being’ the discriminant of Aq. 
For the modifi(;ation of the expression of Qg we have, on substituting for the 
quantities P in terms of the quantities u, 
ur 
Qg = + 1-8 + upp) + 18 “ (4^3+8 — upipig + Gupipij + Sw/wg). 
n-y up 
The modification of the second term has already been given ; for the third we have 
— Aupipig = 4t/-^^AjHQ ; 
upig + 2upi^ + upUg = {r [pd — qc) — 25 [pc — qb) + t [pb — qa:)] ; 
3 S^Ai _ . 9^^ ^4 , ^ 9lAil . 
^ ^ ^ [ dq^ dp" dp dq dp) dq dp" dq" J ’ 
— 14 , 
where Hq^ denotes the simultaneous Hessian of Aq and A^ with regard to and Mjq. 
Hence 
Qg = — Aj® — SAqA^Jq^ + 72 Aq^A^Hq — f Aq^Hqi . 
For Qg we have, after substitution for P., Pg, P,, Pg, the form 
Q 5 = -1 (^^6% - 
2 
+ {uph'^h + + upipu.r. + 2tlpUrj) 
'^1 
- A (4^3+/ - MpipiJ + 4 M 3 Q . 
