of a Formula fur the relative Importance of the Boroughs. 27 



tion, whether there could be more than one formula deduci- 

 ble from the data ? which were the houses and taxes of the 

 boroughs. 



The writer of this article had satisfied himself as well as 

 others, by the dictates of common sense, thatLieutenantDrum- 

 mond's rule was correct. He, however, judged that it might 

 be proper to discuss the subject upon mathematical principles 

 that should be beyond dispute, and he chose the very simple 

 axiom, " That 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." From this he formed a functional equa- 

 tion, and employing the Differential Calculus (not the Calculus 

 of Variations, as Dr. M'Intyre supposes)*, he determined the 

 form of the function, and proved beyond dispute that it could 

 have but one form, which is that of formula (1) of this article. 



LieutenantDrummond has not explained the views by which 

 he was led to his practical rules, given in the parliamentary 

 paper (see page 219 of the Phil. Mag. and Annals for March). 

 It is easy to see, however, that by making m' = n' in the gene- 

 ral formula (1), so that it becomes 



i = ?«'B 1^ + -7^1 , 



we have immediately Mr. Drummond's rule. It is true the 

 constant factor ?«' here put down, was left out in the former 

 communication, because it did not in the least affect the posi- 

 tion of the boroughs in the scale of comparative importance, 

 and the number B was supposed to be 1000000, to avoid 

 fractions in the results : but it was not expected that any one 

 would lay hold of so trivial a matter, in order to attempt to 

 show that Lieutenant Drummond's rule really involved the 

 absurdity that two is equal to one. It appears, however, that 

 the candour of his opponents was overrated : the objection of 

 Dr. M'Intyre is a fair specimen of it. The Doctor stands 

 exposed to some sharp remarks : but at present this purely 

 scientific speculation shall not be contaminated with any thing 

 personal. 



We shall now proceed to show, by carrying the analysis a 

 step further, that the Doctor's objection does not apply to the 

 general formula 



Since this must hold true whatever be the magnitude of a 



* Dr. M'Intyrc's paper will be iouiiil in Phil. Mag. and Annals, vol. xi. 

 p. 360.— Edit. 



E2 



