80 
[13 
13. 
ON THE THEORY OF LINEAR TRANSFORMATIONS. 
[From the Cambridge Mathematical Journal, vol. iv. (1845), pp. 193—209.] 
The following investigations were suggested to me by a very elegant paper on 
the same subject, published in the Journal by Mr Boole. The following remarkable 
theorem is there arrived at. If a rational homogeneous function U, of the n th order, 
with the m variables x, y..., be transformed by linear substitutions into a function 
V of the new variables, £, if, moreover, 6U expresses the function of the 
coefficients of U, which, equated to zero, is the result of the elimination of the 
variables from the series of equations d x U = 0, d y U = 0, &c., and of course 6V the 
analogous function of the coefficients of V: then 0V=E na .0U > where E is the 
determinant formed by the coefficients of the equations which connect x, y... with 
£, r]..., x and a = (n— l)™' -1 . In attempting to demonstrate this very beautiful property, 
it occurred to me that it might be generalised by considering for the function U, not 
a homogeneous function of the n th order between m variables, but one of the same 
order, containing n sets of m variables, and the variables of each set entering 
linearly. The form which Mr Boole’s theorem thus assumes is 6V = Ef. E. a ... E n a . 6U. 
This it was easy to demonstrate would be true, if 6U satisfied a certain system of 
partial differential equations. I imagined at first that these would determine the function 
6U, (supposed, in analogy with Mr Boole’s function, to represent the result of the 
elimination of the variables from d Xl U=0, d y JJ = 0, ... d x JJ = 0, &c.) : this I afterwards 
found was not the case; and thus I was led to a class of functions, including as 
a particular case the function 0 U, all of them possessed of the same characteristic 
property. The system of partial differential equations was without difficulty replaced 
by a more fundamental system of equations, upon which, assumed as definitions, the 
theory appears to me naturally to depend; and it is this view of it which I intend 
partially to develope in the present paper. 
1 The value of a was left undetermined, but Mr Boole has since informed me, he was acquainted with 
it at the time his paper was written; and has given it in a subsequent paper.