■■nHn 
ON THE THEORY OF LINEAR TRANSFORMATIONS. 
There is no complete hyperdeterminant (i.e. for p — 1), and the incomplete ones are 
ah — bg — cf+de = u / , suppose, 
ah — de — bg + cf= u /t , 
ah — cf — de + bg = u //r 
Thus, suppose the transforming equations are 
= V X x + V X 2 , 
X 2 ~~~ ^*2^ X x X 2 ) 
yi = Pi Vi + Pi ¿/2. 
2/2 = P-2 y l + P2 2/2 ; 
Zi = V\ Z x + vi ¿2 , 
= vi ¿i + v 2 2 ¿2; 
u, = (/¿a 1 /¿2 2 - Pipe) (vi vi - Vi vi) u x , where y, z are changed 
u „ = (V za 2 2 — vi vi) (V \ 2 — V V) , „ 2, a? 
U tn = (V V - v V) (pi pi- pi pi) U un 
We might also have assumed 
u t —ad — be, or eh — gf 
u /t = af — be, or ch — dg, 
u ni — ag — ce, or bh — df, 
but these are ordinary determinants. 
(G) n = 4, m = 2. 
[13 
y 
a= 1111, 
i =1112, 
b = 2111, 
j = 2112, 
c = 1211, 
k = 1212, 
d = 2211, 
/ = 2212. 
e = 1121, 
m = 1122, 
/=2121, 
n=2122, 
g — 2211, 
0 =2212, 
h = 2221, 
^= 2222. 
U = a x x y x z x w x + 6 x 2 y 2 z x w x + c x x y 2 z x w x + d x 2 y 2 z l w l 
+ e x x y x z 2 w x + fx 2 y x z 2 w x + g x 2 y 2 z x w x + h x 2 y 2 z 2 w x 
+ iX x y x Z x W 2 +j X 2 y x Z x W 2 4- k X x y 2 Z x W 2 + l X 2 y 2 z x w 2 
+ w X x y x Z 2 W 2 + n X 2 y x Z 2 W 2 + 0 X 2 y 2 Z x W 2 + p X 2 y 2 Z 2 W. x , 
= ap — bo —cn + dm — el +fk + gj — hi;