VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 65 



XII. — Integral expressions for volume: Jive symmetrical sections. 



Index p = 1, 2 and 3. 



o- a /3 y 8 e 



9 12 3 2 1 



12 1 4 2 4 1 



15 1 6 1 6 1 



18 1 8 8 1 



etc., etc. 



On solving for p = 4, we find 



y' = If - Tih-eP (62); 



and on combining this solution with (61): and putting y = 2y', the 

 solution for /3 and y becomes determinate, and for indices at any 

 rate as far as p = 1, 2, 3 and 4, 1 we have 



Consequently for those indices the series of coefficients are 



o- = 90, a = 7, /3 = 32, 7=12, 8 = 32, e = 7 

 as already given in Table IX. 



Again solving for p = 5, 6 and 7 we find 



p = 5 y' = if - iifhr/3 ■ (63) 



p = Q y'=W-im-/3 (64) 



p = 7 /=V - HV £ (65) 



By combining these results with (61) and (62) we obtain formula? 

 satisfying different indices. For example combining (63) with (61) 

 the resultant coefficients are again the same ; identical results 

 being given by the solutions for p = 2 or 3 and 4, p= 2 or 3 and 5, 

 or again for p = 4 and 5. Hence the series of weight-coefficients 

 last given satisfy a quintic function, or the formula 



V= _L a (7^ + 32 A + 12 C m + 32 A + 7^o) (66) 



is absolutely exact when the original function is 



4, = 4 + Bz + Cz 2 + Dz z + Ez 4 + Fz 5 

 these indices, viz. 2 to 5 are thus seen to be conjugate for the 

 indicated coefficients. 2 



1 It will be seen later that the solution is true also p — 5 ; that is a 

 five-section formula is true for a quintic function, the weights being as 

 shewn . 



2 This fact does not appear when the formula is deduced by finite 

 differences. 



E— June 6, 1900. 



