923] 
NOTE ON A HYPERDETERMINANT IDENTITY. 
211 
course it remains =0 when, consequently upon the foregoing change (x u y^, (x 2 , y 2 ), 
(x 3 , y 3 ) each into (x, y), we also change (f, Vl ), (&, % ), (f„ Va ) each into (£ v ); 
and if we also change (a? 4 , 3/4), &c., into (x, y). 
It is clear that O may denote the covariant symbol 
il = 23“ 31^ 12? 14 s 24 e 34^... U 1 V 2 W z T i ..., 
where U, V, W, T, ... denote qualities 
(a, ...\x, yf\ (a', y) n , ... 
of the degrees m, n, p, q, ... respectively, and U lt V 2 , &c., are the corresponding 
functions (a, ..ffx u y 1 ) m , (a', y 2 ) n , &c., the values of A, B, C being here 
A =m — ¡3 — y — S — ..., 
B — n — y — a — € — , 
G =p -a — /3 - £ - ...; 
the theorem expresses the property that the covariants 120, 230, 310, are linearly 
connected together ; or, writing it in the form (A 12 — C13 + B 23) 0 = 0, we have the 
proper linear combination A 120 - C130 of the two covariants 120 and 130, equal to 
— B 230, a determinate multiple of 230. Speaking roughly, we say that the difference 
of the covariants 120 and 130 is equal to 230. 
27—2