MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
109 
Now the quantities U are expressed in terms of the quantities A ; and from the 
values of those quantities it at once follows that 
BA 
8A„, = + ^ + {qr - + {qs - tp) 
?q 
— ■^m.+ 1 
— A,;, 2 
^ / 8Ao 8A,„ 
\ d}) dq 
1 T 
2 
0)R > 
8A„ 3A„, \ 
dq dp ) 
and therefore in particular 
SAq = A^, 
8Ai = A 2 
SAo A^ — ^J Qj. 
Hence 
8U2=8(AoA 2-Ai^) 
— Ai-lAg A^^Ag -^AqJqj a ; 
and therefore 
V = Ag^Ag — 4 AqA;^A 2 d" ^A^^ — A^Hog -h AoAjJoi, 
an invariant proper to the rank 5. But 
Ug = Ao^Ag 
1 0 
3 d“ V 
is an invariant proper to the rank 5 ; hence 
Z = V Ug = ^ -I AgAjAg ^ AoUq2 + AqAjJqi 
is an invariant, and it is evidently proper to the rank 4. It must, therefore, be 
expressible in terms of the irreducible invariants within the rank 4 given by 
Q 55 Qg) Q?’ Qqj • • • j Qi 3 - The verification of this inference is as follows. 
32. We have 
Qi 3 — '^4.Pi3 j 
when the values of P^g and of Pg—viz. : 
(^i^Wjg + 12upi^u^ + SGWg^^) and ^ {uiU^ + 
respectively—are substituted, we at once have 
Qi3 ~ '^4'^13 '^8^ 
= A0A3 - (-Ai)2 = A0A2 - Ai^, 
thus identifying Qjg with Uo. 
