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.