110 
ON LINEAR TRANSFORMATIONS. 
[14 
Passing on to a more general property; let TJ l , f/ 2 , ... U p be functions of the orders 
/1, /2 .••/*>; and suppose 
U p ), = □t/ 2 ... 
a function of the degree f x in the variables : suppose that © ( U 2 ... f/^) contains the 
differential coefficients of the order r 2 for U 2 , r 3 for U 3 , &c., so that fi = (f 2 —r 2 ) + ...(f p —r p ). 
Consider the expression 
■fyltfi, ®(U 2 ...U P )}, 
which reduces itself in the first place to 
(12 + 13 ... + Ip) I I U Uo... Up, 
then to K l:/ 3_r3 ... lp v ~ rp □ U x U 2 ... U p ; 
where for shortness 
e. EÆ 
For if one of the indices were smaller another would be greater, for instance that of 
12: and the symbols £ 2 , r] 2 in 12' 3 3 would rise to an order higher than f 2 , or 
the term would vanish. Hence, writing 
and B'iUt.U,..., U p ) = n'U,U,...ü p , 
we have B^{U X , 0 (U 2 , ... U p )} = K& (U u U 2 ... U p ); 
i.e. the first side is a constant derivative of U 1} U 2 ...U P . 
Suppose 
U x 
[fiY' ^ a(!pcf + 
0(H 2 ,... 
Up) 
r yj/, +•••)> 
then 
K&{U X ... 
Up) 
f 
=*= a o^-/ l — Y +•••'> 
i.e. 
> 
II 
~&(U X ... 
aa 0 
U p ), 
U,...U P )..., 
or finally, 
0(H 2 , . 
U ) = — : — 
- p) [/]'* 
^X‘A 
an equation which holds good (changing, however, the numerical factor,) when several 
of the functions U x ... U p become identical. Hence the theorem: if U be a function 
given by • 
(a<jX f + a 1 x f ~ l y + ...);