220 Analytical Investigafion of a Formula "which shall express 



1. Boroughs composed of the same number of houses, and 

 paying the same sum in taxes, must be considered as equal in 

 importance. 



2. If one borough contain double the number of houses in 

 another borough, and pay twice as much in taxes, the import- 

 ance of the former must be reckoned double that of the latter. 

 And, in general, if the number of houses in one borough be 

 any multiple of the number of houses in another borough, and 

 the taxes paid by the first the same multiple of the taxes paid 

 by the second ; the importance of the first must be the same 

 multiple of the importance of the second. 



3. If a town contain as many houses as are in any number 

 of boroughs, and pay as much in taxes as they all pay, its 

 importance will be equal to the united importance of all these 

 boroughs. 



We are now to determine by the application of these prin- 

 ciples the comparative importance of the boroughs, that is, 

 speaking mathematically, the ratio which the importance of 

 any one borough has to that of any other borough : now this 

 will depend on, and must in some way be expressible by the 

 ratio of the number of houses in the one borough to the 

 number of houses in the other borough, and the ratio of the 

 sum paid in taxes by the one to the sum paid in taxes by 

 the other. 



Let H denote the number of houses in a town or borough, 

 T the sum it pays in assessed taxes, 



B a numeral measure of its importance. 



Also, let h, t, b, and h', t', U denote the same things of any 

 other two boroughs. The mathematical problem to be resolved 



is to find a formula which shall express the ratio ^ by the 



h t 



two ratios n"j 7?r. For the sake of brevity, let us put 

 A— J_— ]L— ' il— ' 



In the language of analysis, -jt 's a function of x and y ; and 



the same function of .r* and y\ We are now to inquire, what 

 is the form of this function ? 



The tlependence of -^o\\ x and 3/, and that of „ o" 3.^ and 

 5/' may be expressed thus : 



-|=/(,-y), ^=/Cr',,/'), 



the 



