the relative Importance of a certain Number of Boroughs. 221 



the prefix y being put to denote, generally, a function of the 

 quantities x and y. By our third principle we must have also 



Therefore, 



f{x,y) +f{x\y>) =/{(x + x'), {y + y')) (1) 



We may consider x and y as two variable quantities which 

 are independent of each other, and susceptible of all degrees 

 of magnitude by increase or decrease, the symbols x and a^ 

 denoting succeeding values of one of the variables, a.nA y, y' 

 succeeding values ot the other. 



Our analytic expression (1) may now be regarded as an 

 identical equation which must be satisfied, whatever values be 

 given to the quantities x, x', y, y' which compose it. 



For the sake of brevity, let us put 



f{ix+x'),iy + y')}=:n. 

 Equation (1) will now stand thus: 

 f{x,y)+f{x',y')=u. 



Hence, by differentiation, on the hypothesis that .r, y, x',y' are 

 in succession variable, we get 



df (x, y) _du df(x,y) du 



dx dx' dy dy' 



df(x',y') _ dji_ df(x,y) _ rfu 



dx' dx" dy' dy'' 



Now from the peculiar form of the function tc, the differen- 

 tial coefficient found on the hypothesis that x is variable is 

 exactly the same as that found on the supposition that a/ is 

 variable ; that is, 



d 11 du 



dx dx' 



For a like reason, 



du du 



dy dy' 

 Hence it appears that 



dx dx' ' 



df(x,y) _ dfix'.yj 

 dy dy' 



If 



