13] ON THE THEORY OF LINEAR TRANSFORMATIONS. 89 
so that, with the same sets of transforming equations as above, and the additional one, 
w 1 = p^ + p-?w 2 , 
w n = p^w x + p 2 2 w 2 , 
we have u = (X^V — X^Xf) (p ftp ft — pftpft) (v-fvf — v^vfj {pipft — p*Pi) • w, 
this is important when viewed in reference to a result which will presently be obtained. 
If we take the symmetrical case, we have 
U — oi oft + 4/3 ofty + 67 ofty 2 + 4Bay 3 + e y 4, ; 
which is transformed into 
U' = a'x'* + 4/3 'x' 3 y' ft- 67 'x' 2 y' 2 + 4 h'x'y' 3 ft- e / y /4 , 
by means of 
then, if 
we have 
x = X x' ft- p y-, 
y = X/ + p/; 
u = a e — 4/8 S + 37 2 , 
w' = aV — 4/3'3' + 37' 2 , 
u' = (Xp / — X,p) 4 . u. 
II. Where p = 2, m = 2, n is odd. 
The expression 
m = 
1111 ... (n) 
1222 
1111 ... (w) 
2222 
t 
1111 
2222 
+ 
2111 
2222 
(n) 
(n) 
is a complete hyperdeterminant; and that over whichever row the mark (*|*) of nonper 
mutation is placed. The different expressions so obtained are not, however, all of them 
independent functions: thus, in the following example, where n= 3, the three functions 
are absolutely identical. 
{A). n — 3, notation as in I. (B). 
u — aftJi 2 + b 2 g 2 + c 2 f 2 ft- d 2 e 2 — 2ahbg — 2 ahcf— 2ahde — 2bgcf— 2bgde — 2 cfde + 4 adfg + ftbech, 
and then u = (X/Xo 2 — X 2 1 X 1 2 ) 2 {pftpft — Pvpi) 3 (vivft — v 2 v 2 ) 2 u. 
This is in many respects an interesting example. We see that the function u 
may be expressed in the three following forms: 
u = {ah — bg — cf + de) 2 + 4 {ad — be) {fg — eh) (1), 
u = {ah — bg — deft- cf) 2 ft- 4 {af— be) {dg —ch) (2), 
u = {ah-cf-deft- bg) 2 + 4 (ag - ce) {df - bh) (3),