14] 
ON LINEAR TRANSFORMATIONS. 
or if U be any function whatever of the variables x, y... which is transformed by 
the linear substitutions above into U, then 
or the function 
UÜ = Ef&f y ... 
DU 
is by the above definition a hyperdeterminant derivative. The symbol □ may be called 
“ symbol of hyperdeterminant derivation,” or simply “ hyperdeterminant symbol.” 
Let A, B,... represent the different quantities of the series ||il||,—A\ B\... those of 
the series ||iT|[, &c. ..., then □ may be reduced to a single term, and we may write 
□ = A«B? ... A'« BP ... 
Also U may be supposed of the form 
U=®<S> 
c. 
where 0, are functions of the variables of one of the sets x, y,..., of one of the 
sets x'', y y , .... &c., thus 0 is of the form 
y u . 
and so on. The functions 0, may be the same or different. It may be supposed 
after the differentiations that several of the sets x, y, ... or of the sets x\ y', ... 
become identical: in such cases it will always be assumed that the functions @,... 
into which these sets of variables enter, are similar; so that they become absolutely 
identical, when the variables they contain are made so. Thus the general expression 
of a hyperdeterminant is 
□ U = A*B* ... ... 04> ... 
in which, after the differentiations, any number of the sets of variables are made equal. 
For instance, if all the sets x, y ... and all the sets oc, y ... are made equal, the 
hyperdeterminant refers to a single function F(x, y...x\ y ...). In any other case it 
refers not to a single function but to several. 
What precedes, is the general theory: it might perhaps have been made clearer 
by confining it to a particular case: and by doing this from the beginning it will 
be seen that it presents no real difficulties. Passing at present to some developments, 
to do this, I neglect entirely the sets y x ... and I assume that the number m of 
variables in each of the sets x, y ... reduces itself to two; so that I consider functions 
of two variables x, y only. The functions 0, <!>, &c. reduce themselves to functions 
V 1} V 2 ... V p of the variables x 1} y lf or x 2 , y. 
- tin1= 12, &c. 
the symbols A, B... reduce themselves to 12, 13.... Hence for functions of twc 
variables, there results the following still tolerably general form 
□ H = 12“ 13 ß 14\..23 ß 24 v ... 34 r