ON THE THEORY OF LINEAR TRANSFORMATIONS. 
Using the notation there employed, we have 
t 
11... O) 
mm ) 
a complete hyperdeterminant when n is even ; and when n is odd the functions 
t 
t 
r 11.-» ' 
J 
11... (n) 
2 2 
- 
2 2 
: : 
[ mm 
mm J 
are each of them incomplete hyperdeterminants. 
(J.) In the case of n = 2, the complete hyperdeterminant is simply the ordinary 
determinant 
11, 
12,. 
.. lm 
2i, 
2 2, . 
.. 2?/i 
ml, 
ml,. 
.. mm 
[13 
Stating the general conclusion as applied to this case, which is a very well known one, 
“ If the function 
U = XX (rs . x r y s ) 
be transformed into a similar function 
2X (VS . X r ys) ) 
by means of the substitutions 
x r = \y cc x + \ r 2 x. 2 ... + A r m x m , 
y$ = /Av' ÿi -f ÿ% ••• ~¥ H*™ ÿm, i 
then 
Ü, Ì2, ... 
= 
V, V,... 
H-2, • • • 
11, 12,... 
2Ì, 22, 
V, V, 
/v, y-f, 
: 
21, 22, 
so that the theorem is easily seen to amount to the following one—“ If the terms of 
a determinant of the m th order be of the form X,.X S (rs. x rp y S(T ), r, s extending as 
before, from 1 to m inclusively, the determinant itself is the product of three deter 
minants ; the first formed with the coefficients rs, the second with the quantities x, 
and the third with the quantities y."