■PM 
14] 
ON LINEAR TRANSFORMATIONS 
The functions 0 may be transformed in the same way as the functions B have been. 
In fact 
0.(U, V, If) = 12 a_fc 23 3l a ~* Ok (U, V, If); 
if in particular &=1, then 
G X (U, V, W)= tf>° U- 1 if- 2 
F *0 ^,1 -pr,2 
W’° |f,2 
but in general 
£/ *7i p £/ ^ & Vs Ci {U, V, W), where p + p = a + a = t + t = 2a - 2, 
2a—2 2a—2 2a—2 
= G l {U'P F’- F’ T )=| if'P Z7’P +1 U'P +2 
y,<r y, <r+1 "P", p+2 
|f.r |y,r+l |y,p+2 
whence G a (U, V, IF) = 22 {(-)p+° 
[ff 
^,p F ,.-! Jp.Sa-p-^ 
, p+1 F , <t y, 3 a—p—<r—l 
JJ, p+2 y,<r+l jy, 3a—p—a 
where A/ = l '~- r ^ J ; i extends from 0 to a — 1; p, a-— 1, and 3a —p —o- —2 may have 
each of them any positive values not greater than 2a - 2. 
In particular 
G a (if, U, if) = 622 {(-)>+-1 i/’P if--" 1 tf'^-p 
( if'P+! if* 0 ' JJ,3a—p—<r-\ 
i JJ,p+2 JJ ,<j+1 JJ,3a—p—j 
where p, <x need only have such values that p < a — 1, <r— 1 < 3a — p — o- — 2. 
In particular the derivative aei —... + 15e 3 may be transformed into 
a, d, 
9 
-3 
a, e, 
f 
-3 
b, 
c, 
9 
+ 6 
b, d, 
f 
—15 
c, 
d, 
e 
b, e, 
h 
b, /, 
9 
c, 
d, 
h 
c, e, 
d, 
e, 
f 
c, f, 
i 
c, 9, 
h 
d, 
e, 
i 
d, f, 
h 
e, 
/> 
9 
in which form it is obviously a linear function of the determinants