VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 63 



which, if we make j3 = 2, satisfies these last mentioned values of p. 1 

 This fact, viz., that certain values of the weight-coefficients may- 

 satisfy other indices than those which are used to determine them, 

 will be found to have a wider applicability than is immediately 

 evident: in general the indices other than and 1, which are 

 satisfied by any system of weight-coefficients may be called 

 conjugate, as hereinbefore. 



For brevity let 

 *P k = n*xl (k/ny + (n-ky/n» i = k j k? + (n - kf I (55) 



and similarly in regard to $ k , i£ k , etc.; the capital letter corres- 

 ponding to that denoting the index, while the subscript k is to be 

 the same integer as k. Then the equations to be satisfied are 



2(#cP) - 2n^K/(p + l)=0 (56) 



k having the values a, /3, y, etc., and the limits for kP being to 

 l(ri+l) when n is odd, and to J (n + 2) when n is even. 

 Remembering that the number of sections is independent of the 

 number of indices in any series, and that the solution does not 

 lose generality by making a= 1, we have from (48) and the last 

 two equations, 



P{(p+l)P 1 -2nP}+y{(p + l)P 2 -2nP}+... + (p-l)nP = (57) 



and similar expressions in which q, Q iq} etc. are substituted for 

 p,P, v ', the number of terms in addition to the last or absolute 

 term being now the same as the number of weight-coefficients to 

 be evaluated, viz. n/2 if n be even, (n - l)/2 if n be odd. By means 

 of these last equations, viz. (57), any case can be readily solved. 



17. Examples of the application of the general formula. — For 

 w=2, that is for a middle and terminal sections, (57) becomes, on 

 dividing each quantity by 2, 



i 8'(p+l-2 p ) + (^-l)2 p - 1 = (58) 



which gives the following series of formulae, remembering that the 

 coefficient must be 2/5' as already pointed out. 



1 It has already been pointed out in connection with (48) that for even 

 values of n, the coefficient is half its proper value : (54) is of course one 

 half of (37). That the values p = 0, p = 1 hold, may be verified by consider- 

 ing the limits. 



