164 Proceedings of Royal Society of Edinburgh. [sess. 
implies that any one of the latter is a function of the others. 
Functions of several variables are said to be mutually independent 
when no one of them is constant or can he expressed in terms of the 
rest. This is extended and made more definite by saying that if 
functions of x , x 1 , . . . x n involve also the quantities a, a x , a 2 , . . . 
the functions are said to he mutually independent with respect to 
the quantities x , x x , . . . , x n if no equation subsists between the 
functions and the quantities a , a x , « 2 , . . . these definitions will 
suffice to indicate the analogy above referred to, and the deduced 
propositions (pp. 325-327) need not he entered on. 
All this introductory matter having been disposed of, Jacobi 
proceeds (§ 5, p. 327) to deal with the subject proper, his starting- 
point being the fact that if there he n + 1 functions /, f \ , f , • • . , f n 
of the same number of variables x, x lt . . . , x n there arise in 
connection with these the \n-\- 1) 2 quantities 
V m 
dx k 
The determinant formed therefrom, viz., 
y + v Wi . . . . ¥. 
^ dx dx x ' ’ dx n 
he calls the ‘ determinant pertaining to the functions /, f x , ... , f n 
of the variables x f x lt ... t x nJ ’oT the £ determinant of the functions 
f, fu ... 5 f n with respect to the variables x, x 1 , ... , xj The 
case where n = 0 is then referred to in a line, after which cases are 
taken up where it is the functions that are specialised. The first 
of these is that in which 
fm+l = *^m+l j fm + 2 = *^m+ 2 5 • • * ) f n ~ ■> 
it being pointed out of course that the order of the determinant is 
then lowered, being equal to 
xry V * V\ Vm 
~cx dx x * dx m 
Another is that in which the functions f m+1 , / m+2 , ... § f n do 
not involve the variables x , x li ... , x m , the peculiarity then 
being that the determinant breaks up into two factors similar to 
itself, being equal to 
