VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 69 



that is the powers of n are all identical with those in (77), and 

 only the coefficients differ. 



Similarly 



A;= J_.A p = ^™p (76) 



p + 1 p + 1 



A^-^-W (77) 



n(p + 2) p + 2 



Hence we may divide all the equations with even indices by (p + 1), 

 and all those with odd indices by n(i+ 1). The resulting quantities 

 K p /(p + 1), Ki/(i + 1), etc., may be conveniently distinguished by 

 accents, as in these five last equations. Let the even indices be 

 denoted by^>, r, t etc.; then commencing the series (72), as modified, 

 with p — 2, we obtain the equations 



B p /3 + C p7+ ...A p =0 ' 



B p+1 /3 + C p+1 y+...A p+1 = I .(78) 



etc. etc. etc. 



the factors of f3, y, etc., being of the type (75) and (75a): k will 

 be 1 for the term f3, 2 for y, 3 for 3 and so on, and we may write 

 r and r + 1, t and t + 1, etc., for the successive pairs of indices. 

 Then it will suffice to shew that values of /3, y, etc. which satisfy 

 the general equation for p, an even integer, will also satisfy it for 

 p + 1, an odd integer, provided p be not greater than n. In other 

 words it will then be evident that the m coefficients may be cal- 

 culated from either the m even indices commencing with 2, or the 

 m odd indices commencing with 3 ; the solution from the one 

 series satisfying the other. That a n ic function is satisfied in any 

 case, when the coefficients are symmetrical with respect to the 

 middle section, is shewn in the derivation of formulae, by the 

 method of finite differences. If therefore we write the general 

 equation by commencing with the sections nearest the middle 

 section when n is odd, or the middle section when n is even, we 

 have for n = i 



2 2n) +[ 2 + 2n) i+\) i\2~ 2nJ + \2 + 2?i) i+lJ (79) 

 continuing with terms 5/2n, 7/2n, etc.; and also for n=p 



