ON LINEAR TRANSFORMATIONS. 
96 
[14 
□ =EfE y f" ... □; 
p K at least as great as m\ and so on. Let the analogous variables cc, y... be con 
nected with these by the equations 
x = Xx + y y +..., 
y =\'x + yy +..., 
sc~ = A' x + y y s + ..., 
y' = V'a'+/'y' + ..., 
where x, y,... stand for x lf yi,... or x 2) y 2 ,... or x p , y p ,...\x, y\... stand for x^, ... 
or X2, y%>... or x' p , y s p ', &c.... The coefficients A, y,..., A', y\... &c.; \\ y,..., X'\ y',... 
remain the same in all these systems. Suppose next, 
£ = 8*, 17 = ^, 
i.e. ^1 = ^, ^1 = S 2/1 ,... ,^1 = 83;,,... 
(where §3, 8^... are the symbols of differentiation relative to x, y, &c.). Then evidently 
£ = A£ + A'77 + .. •, 
y = y^ + yy + ■■■ , 
with similar equations for i~\ y,... Suppose 
£l> 
•• 1 : p 
, «il'n= 
fc\. 
Vi> 
■ 
V2, • 
.. y p 
Vi, v»\ 
.. 7/V 
that is to say ||il|| is the series of determinants formed by choosing any m vertical 
columns to compose a determinant, and similarly ||0'||, &c. Suppose, besides, 
E = 
A, y,... 
, E = 
A , fi , ... 
A,, fl, ... 
'V A A 
A , H , ... 
Then, by the known properties of determinants, 
Pll=^||ii||, Ifii'll ||il'|| &c., 
i.e. the terms on the one side are respectively equal to the terms on the other. Hence if 
D = F( ||n|K ||ft'F',...), 
i.e. □ a rational and integral function, homogeneous of the order f in the quantities 
of the series ||ft||, homogeneous of the order /' in the quantities of the series ||iT||, 
&c., we have immediately