98 
ON LINEAR TRANSFORMATIONS. 
[14 
The functions V 1 , V 2 ... may be the same or different: but they will be supposed the 
same whenever the corresponding variables are made equal. This equality will be 
denoted by writing, for instance, 
□ VV'VV... 
to represent the value assumed by 
UViV. 2 V 3 V 4 ... 
when after the differentiations 
*1, y 1 = «3, y-i = &T y4. = as, y ; 
x-2, 2/2 = ve', y' ; 
&C. 
It is easy to determine the general term of □ 17. To do this, writing for shortness 
a + /3 + y ... =fi, 
a +/3' +y ... =/ 2 , 
(3 + P + y". 
&c. 
= i_\r+s+t-• • -t 
M r W W ••• OT M r ■” [t'j 
f 
y^rfV or 8 x f- r 8 y r V= V> r or V’ r , 
the general term is 
A 
N V x ,r+8+t ■ • ■ y , a—r+s’+t' • • • y^, /3—S+P'— s'+t' ■ ■ ■ 
Avhere r, s, i, ...s', i', ... i",... extend from 0 to a, /3, y.../3\ y',...y"... respectively. It 
would be easy to change this general term in a way similar to that which will be 
employed presently for the particular case of □ T 7 iF 2 T 7 3 . 
If several of the functions become identical, and for these some of the letters 
f are equivalent, it is clear that the derivative □t/’ refers to a certain number of 
functions V 1 , F 2 ... the same or different, of the variables x, y\ x, y'\... and besides 
that this derivative is homogeneous, of the degrees 0 lt 0/,... with respect to the 
differential coefficients of the orders f lt fi, ... &c. of V u (consequently homogeneous of 
the order 9 1 + 6 1 '+... with respect to these differential coefficients collectively), homo 
geneous and of the degrees 6. 2 , 0 2 ',... with respect to the differential coefficients of 
the orders / 2 , //... of V 2 , (consequently of the order 0 2 + 0 2 '... with respect to these 
collectively), and so on. The degree with respect to all the functions is of course 
0i T 0 2 ... + 6. 2 + 6 2 -f ...,=p suppose. In general, only a single function will be considered,