210 
[923 
923. 
NOTE ON A HYPERDETERMINANT IDENTITY. 
[From the Messenger of Mathematics, vol. xxi. (1892), pp. 131, 132.] 
The following is in effect a well-known theorem; but I am not sure whether 
it has been stated in a form at once so general and so precise. 
If 
o = (*)(«!, Vi) A (x 2 , y 2 ) B (x 3 , 2/ 3 ) c 0r 4 , y#> ... 
be a function separately homogeneous, and of the degrees A, B, G, D, ... in the sets 
of variables (x 1} y^), (x 2 , y 2 ), (x 3 , y 3 ), (x 4 , y 4 ), ... respectively; and if 
12 = ~ dxfy,, &c., 
then 
{A 23 + B 31 + C12) il = 0, 
when the variables (x 1} y x ), (x 2 , y 2 ), (x 3 , y 3 ), (¿r 4 , y 4 ). .... or only the variables (x 1} y 4 ), 
(x 2 , y 2 ), (x 3 , y 3 ) are therein severally replaced by (x, y). 
In fact, we have 
Ail = (x^j + y^) il, Pil = (x 2 % 2 + y 2 y 2 ), Cil = (x 3 t; 3 + y 3 y 3 ) il; 
thus the expression is 
= {Oi£ + ViVi) 23 + (x 2 % 2 + y 2 y 2 ) 31 + (x 3 ^ 3 + y 3V3 ) 12} il, 
and if we herein replace the variables (x 1 , y 2 ), (x 2 , y 2 ), (x 3 , y 3 ) in so far as they 
appear explicitly by (x, y), the expression becomes 
= {№ + yvi) 23 + (x% 2 + yi7a) 31 + (x% 3 + 2/773) 12} il, 
where the factor in { }, substituting for 23, 31, 12 their values ^ 2 y 3 — ^ 3 y 2 , i- 3 Vi —ZiV3> 
%iV2 — %2Vi> becomes identically =0. The value of the expression is thus =0; and of