562 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
If y 1 . . . . y n are given implicitly as functions of x x .... x n by the n 
equations 
.... x n y, • * * * Vn) 65 f 2 6 . . . . f n 0 , 
we put 
• • • • "t" ^nfn ^ 
and say that e(p, a ) is a given fictor function of p, a and the implicit rela- 
tion is e = 0. Thus 
0 = dc — (Sefy)|v. + Sc2oj' v'.) € 
Taking discriminants and remembering that [x(p] = x n [(p], and therefore 
that [ — (p] = ( — )”[ <p\ we have the well-known Jacobian theorem 
da 
_dp 
This seems to me a great improvement on the ordinary algebraic proof, 
and it must be remembered that (13) is a much more general statement 
than (14). 
I think the above fully justifies the importance of the notation that 
da I dp, or something very analogous, such as da : dp, should mean a fictor- 
linity. It strikes me with much surprise that quaternionists have not 
adopted this notation long ago. It seems to me immensely to decrease the 
complexities of change of variables in general. 
We now pass to integrations. The well-known theorem of change of 
variables from x v x 2 , .... x n to y x ... . y n in 
x 2 ... . )dx 1 dx 2 .... dx n 
at once suggests itself, but the integration implied here is of a very 
specialised type, and it is best to regard that theorem as a special case. 
Integration “along a path” in Thermodynamics, along curves and over 
surfaces in Physics, integrations connected with “ actual ” paths of 
dynamical systems and along “varied” paths thereof, suggest what we 
ought to regard as the general type of integration in our present system. 
Let p (and therefore or) be a function of (say four) scalar parameters u, v, 
w, z. We may think if we like of n — ^ other parameters e 1 ... . e n _ 4 , of 
which p and a are in general functions, but for the present e x ... . e n _ 4 are 
