1921-22.] Concomitants of Quadratic Differential Forms. 193 
then, except as to an irrelevant numerical factor, each of the quantities 
D%{u(P)}, 
for r — 1, 2, . . ., 6, and s — 0, 1, 2, . . ., r, is an invariant. There would 
apparently emerge twenty-seven such invariants (2-|-3-f4-f5 + 6-f7); but 
they are not independent of one another, and the independent members 
must be selected. We take them, in turn, from the successive sets; and 
shall use the notation 
Out of IF'a-j-yuA), we have 
j^ = N + 0 + Y, 
= n + 0 -f- V, 
two invariants ; they happen, each of them, to be zero ; but they occur as 
invariants of the system. 
Out of + /X A), we have 
oC^AD-fBE-hCF-h . . . +2RV 
as in § 31, 
il2 = DQ 
= aD + bE-}- -f-dA, 
and 
= ad + be -}- cf - 1 - . . . -f-2ru. 
When the values of d, . . ., a are substituted in , we find that 
2l2 = 3A, 
so that, because A has been retained as an invariant, the last gives no new 
member. Thus out of /ulA), we have the two invariants 
b)} 1^2’ 
in addition to A. 
Out of Igfi^-h^A), we have 
ol3 = .N^ + 03 + V3 + 3N(AI)+ ... ) + . . . 
ifs ^ 
= 3[n(AD+ . . . ) + o(BPH- . . . )-fv(CF-f ... ) 
+ 2(AD + MS-tLT)-t . . . ], 
^6A(AT-f04-V) 
= 6AQ, 
= 0 . 
VOL. XLII. 
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