220 



ing any one of n quantities connected by m equations in terms of 

 any (w m) others of the same. 



Let there be any number of variables, say u, v, w, of which x, y, z, 

 are given functions, it is required to expand 



dx \dy \dz 



in terms of the partial differential coefficients of $, x, y, z in respect 

 of M, v, w. 



Form the determinant 



dx dx dx 



du dv' dw' 



dy dy dy 



du dv' dw' 



dz dz dz 



du dv dw' 

 which call J. 



The required expansion will contain in each term an integer nume- 



rical coefficient, a power of , one factor of the form 



J 



(d\p sd\* /rf\% 

 \du) \dv) \dwj 



and other factors of the form 







du dv dw 



du) \dv) (dw) y 



(ir (*Y (r,. 



\duj \dvj \dwj 



Let the latter class of factors be distinguished into two sets, those 

 where l-\-m + n = I, 



/li.il=Q m = n=0\ 

 I or /=0 m=l w=0j 

 \ or /=0 m=0 n = l/ 



which I shall call uni-differential factors, and those in which 

 l+m + n7 1, which I shall call pluri-differential factors. 



First, then, as to the form of the general term abstracting from 



