14] 
ON LINEAR TRANSFORMATIONS. 
[From the Cambridge and Dublin Mathematical Journal, vol. i. (1846), pp. 104—122.] 
In continuing my researches on the present subject, I have been led to a new 
manner of considering the question, which, at the same time that it is much more 
general, has the advantage of applying directly to the only case which one can 
possibly hope to develope with any degree of completeness, that of functions of two 
variables. In fact the question may be proposed, “ To find all the derivatives of any 
number of functions, which have the property of preserving their form unaltered after 
any linear transformations of the variables.” By Derivative I understand a function 
deduced in any manner whatever from the given functions, and I give the name of 
Hyperdeterminant Derivative, or simply of Hyperdeterminant, to those derivatives which 
have the property just enunciated. These derivatives may easily be expressed explicitly, 
by means of the known method of the separation of symbols. We thus obtain the 
most general expression of a hyperdeterminant. But there remains a question to be 
resolved, which appears to present very great difficulties, that of determining the 
independent derivatives, and the relation between these and the remaining ones. I 
have only succeeded in treating a very particular case of this question, which shows 
however in what way the general problem is to be attacked. 
Imagine p series each of m variables 
#i, 2/x, — &c. x% y y%) • • • &c. Xp, yp, ... &c., 
where p is at least as great as m. 
Similarly p s series each of m variables 
X\ ■> y\, • • • &c. x<i, y%,..., &C.