VOLUMES OF SOLIDS AS BELATED TO TRANSVERSE SECTIONS. 67 



XIV. — Integral expressions for volume: six symmetrical sections. 

 Indices p - 1,2, and 3. 



<r 



a or 



c 



f3 or e 



y or 8 



24 



1 





8 



3 



48 



1 





19 



4 



72 



1 





30 



5 



96 



1 





41 



6 



etc., etc. 

 So also expressions can be deduced satisfying p = 1, 2, 3, 6; 1, 2, 3, 7; 

 etc.; 1, 4, 6 ; 1, 4, 7; 1, 5, 6; 1, 5, 7; etc., etc.; that is to say the 

 weight coefficients may be so determined as to make these indices 

 excepting 1 conjugate. 



This will be a sufficient indication of the application of the 

 general formula. The identity of the formulae may be shewn 

 graphically by treating p as the independent variable, and writing 

 y instead of in (57), and in the particular formulae deduced 

 therefrom. For example the curve represented by (60) and that 

 represented by (67), are plotted with identical parameters and 

 shewn in Fig. 3, Curves 10 and 11. The numerical results areas 

 follows : — 



p= J 1 11 2 21 3 31 4 41 5 6 



(60) = - -0268 +-0058 - -0068 + -0140 - -0721 +10.67 

 (67)- - -0247 + -4058 - -0063 +0163 - -1128 +73.33 



18. On the number of indices satisfied by a given number of 

 symmetrical sections. — Let, in (57), and in the similar expressions 

 in which p is replaced by q, r, etc., the absolute or final terms be 

 denoted by A with suffixes corresponding to the indices. Express- 

 ions of that type may then be briefly written 

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

 B q/ S + C q7 +... +A q = ol (72) 

 etc. etc. etc. 



the number of unknowns, viz. ft, y, etc., being as already pointed 

 out, \n when n is even, or %(n- 1) when n is odd. We proceed 

 to shew that in a system of equations of this type, viz. (72), A, B, C, 

 etc., being the particular functions of p, q, etc., indicated in (55) 



