122 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



Let us now consider the function 



2 ...x n , z l9 z z ... z m ) - 



By means of the equations (i) we may insert the values of 

 the i/'s in terms of the a?'s and ^'s, so that G is explicitly given as 

 a function of the variables x^...x n and z^...z m . On the other 

 hand let us solve the equations ( I ) for the #'s obtaining 



#1 = & fyi> 2/2 ... 2/n, z lt z* ... s m ) 



(2) ....................................... 



#n = 0n (2/1, 2/2 2/n, *i, ^2 ... *,,), 



and by means of the latter let us insert in G the values of the a?'s 

 in terms of the y's and z'a. Let the function in this form, that 

 is, explicitly given as a function of y^...y n and z . . . z m be de- 

 noted by G. Then for all values of z*s and of #'s and ys com- 

 patible with the equations (i) or (2), we have identically 



G(x l ...sc n , z lf ..z m )= G(y l ...y n) ^ ... z m ). 



Differentiating both 6^ and G totally by varying all the 

 variables that occur, we have 



n flf m dF n 



= 2 x- dx s + S ^- dz 8 - 



1 O x s I VZs 1 1 



but as these are identically equal, we get by transposing, 



In this equation there appear 2n + m differentials, only n + m 

 of which are independent, in virtue of the equations ( i ), or their 

 equivalents (2). The equation (3) assumes importance when we 

 define the functions y in a particular way, namely as the partial 

 derivatives of the original function F with regard to the 

 variables x, 



dF 



