the relative Importance of a certain Number of Boroughs. 223 



the borough. If the Legislature shall be of ophiion that the 

 importance of a borough depends more on one of the two 

 kinds of importance above defined than on the other, and shall 

 declare that an assigned numeral value of tax-importance is 

 equal to an unit of house-importance, then m and n will be- 

 come known ; at least their ratio, which is all we want. Un- 

 til this is done, we may make any supposition we please. Lieu- 

 tenant Drummond assumed that an unit of house-importance 

 had the same iioeight as an unit of tax-importance ; on this sup- 

 position TO and n are equal, and the formula may be expressed 

 thus: 



This is just the analytic expression of his rule. But whatever 

 assumption be made, the importance of a borough can be ex- 

 pressed correctly in no other way than by the formula 



It perhaps may be supposed that so refined an analysis was 

 not required to establish so simple a truth. To this we answer, 

 that a truth may be sufficiently obvious, and yet may require 

 the most refined mathematical theory to establish it on the 

 basis of demonstration. The calculus of variations is wanted 

 to determine by analysis the nature of the shortest line that 

 can be drawn between two given points in space ; yet none re- 

 quire to be told that a straight line alone can satisfy the pro- 

 posed condition. 



The assertion, that the importance of a borough can be ex- 

 pressed by a mean proportional between the number of houses 

 and the amount paid in taxes may be proved to be erroneous by 

 a very simple process indeed. Were it true, then, putting h and 

 t for the houses and taxes of one borough, and h' and t' for 

 those of another, we should have, by our third principle, 

 ^{(^h+h'){t + t')l = ^rt+ VhTi': 



h' t' 1 



This can only be true when -^ = — , that is, when the towns 



are constituted exactly alike in respect to power to pay taxes. 

 In this particular case, either the houses alone or the taxes 

 alone would serve to indicate the relative importance of the 

 two boroughs. ^- ^" 



XXIX. Notices 



