222 Analytical hwestigation of a Formula niohich shall express 



If y, y be other values of the variables expressed generally 

 by X and y, we must have, in like manner, 



df{x\y) _ d{x",y") _ df(x',y>) _ df(x",y ")^ 

 dx' dx" ' dy' dy" ' 



and so on. Hence it follows that the functions 

 dfix,y) df{x,y) 



dx ' dy 



are constant quantities ; let these be m and n, so that 



dfjx.y) _ df(x,y) 



~dx ""' dy -"' 



we have now 



^JS^dx J^^JJ^^dy = mdx+ ndy ; 

 dx dy 



and by integration, 



f{x,y) =zmx + ny, 



or — = tn — +71 — : 



or g H ^ T 



The integral requires no arbitrary constant, because when 

 ?i = and ^ = 0, then we must have b = 0. 



We have now obtained a function which expresses the im- 

 portance of a borough, viz. 



=«{"-l+4}- 



We have put B for the importance of a town that contains 

 H houses, and pays T pounds in taxes. Let us now suppose 

 that this town is as large as all the boroughs taken together, 

 and that it pays as much in taxes as they all pay; then h, h', h", 

 &c. denoting the houses in the boroughs, and /, t', ^', &c. the 

 sums they pay in taxes, we have 



li = h + h' + h", &c. Tz=t + t' + i'', &c. 



Thus H and T are known quantities : as to B, it may be any 

 number we please; we shall assume that B = 1000000. 

 There are yet two quantities, m and 7i, but concerning these 



we can pronounce nothing : we may regai'd the fraction rj- as 



the index of that part of the importance of the borough which 

 depends on houses, which we shall call its house-importance; 



and the fraction t^ as the index of its importance resulting 



from the payment of taxes, which v;ill therefore be its tax- 

 importance ; these by their union constitute the importance of 



the 



