84 ON THE THEORY OE LINEAR TRANSFORMATIONS. [13 
where the 2’s refer successively to f g, h,, denoting summations from 1 to m 
inclusively. Having this equation, it is perhaps as well to retain 
u' = LpMp№ ...u (B, 1), 
instead of (B, 2), that form being principally useful in showing the relation of the 
function u to the theory of the transformation of functions. 
It may immediately be seen, that in the equations (B), (C) we may, if we please, 
omit any number of the marks of variation (•), omitting at the same time the 
corresponding signs 2, and the corresponding factors of the series L, M, N ... 
Also, if u be such as only to satisfy some of the equations (A); then, if in the 
same formulae we omit the corresponding marks (•), summatory signs, and terms of the 
series L, M, N... , the resulting equations are still true. 
From the formulae (A) we may obtain the partial differential equations 
22 ... i^ast ... 
22 - { rat -diwz) u = 0 ’ 01 pu ’ 
according as a is not equal, or is equal, to /3; 
and so on : the summatory signs referring in every case to those of the series r, s, £,..., 
which are left variable, and extending from 1 to m inclusively. 
To demonstrate this, consider the general form of u, as given by the first of the 
equations (A). This is evidently composed of a series of terms, each of the form 
cPQR ... (p factors), 
d 
dfist ... 
u = 0, or pu, (D), 
in which 
, Is'/t'/..., 
... (m terms) 
«/*/■•• 
, W/'C-» 
h’X- 
Q, R, &c. being of the same form ; and we have 
22 
-./4... u=cQR - ■ ast - 
cL 
and 
22 ... [ast... 
d 
d/3st... 
P = 
dfist... 
Isftf ..., 1 , ... (m terms) 
«/*/-, ... 
P 4- &c. + &c., 
= 0; 
as;t;...