the Analytical Investigation of a Formula, Sfc. 29 



and (^ r^Y = 0; 



and at last -^1^ = -rp • («) 



Now, if h and t be the houses and taxes of the medium 

 borough, the condition will be exactly satisfied ; and it is cer- 

 tain that h' and t' being the houses and taxes of any one of 

 the boroughs, we have nearly 



-T=-r' ^^) 



Therefore, compounding the ratios a, /3, we have in all the 



boroughs 



h! t^ 



-^ = -Tjr, nearly. 



Thus it appears that our general formula satisfies all the 

 conditions which our most strenuous opponent has required, 

 as far as the thing is possible. 



It has been admitted that a more powerful instrument of 

 analysis was employed in the investigation than was absolutely 

 necessary, (see p. 223 of the Number for March). Neither the 

 learned Doctor, nor the meii of Trinity, have however com- 

 plained of this. But the very same conclusions may be ob- 

 tained by the ordinary algebraic analysis, from the simple 

 common-sense principle, that " the importance of a borough 

 may be truly expressed by giving a certain numeral import- 

 ance to each house, and a certain numei'al importance to each 

 pound, paid in taxes;" just as we would estimate the share of 

 political influence due to the possessor of an estate from his 

 annual income, found by adding into one sum his rents derived 

 from land of different qualities, — say arable and pasture. Pro- 

 ceeding on this principle : 



Let X denote the numeral importance of a house, 

 y that of one pound paid in taxes. 

 Then, B, H, T, 6, li, t denoting the same as before, we have 



h — xh ■\- yt ^=. x\^1i -\- — t\\ 



B = xH + j/T = ^ (H + ^ T) ; 



In the second equation, x H and y T express the whole rela- 

 tive importance of the two elements, houses and taxes. We 

 may make any hypothesis we please concerning the power of 

 each separately to raise the importance of the borough. Let us 

 suppose that their powers are to each other as u given num- 



