a 

 1 



P 

 4 



7 

 1 



11 



48 



11 



13 



64 



13 



57 



320 



57 



64 G. H. KNIBBS. 



X. — Integral expressions for volume: three symmetrical sections. 



V= l -(aA + pB m +yC m ). 

 o~ 



Index p o- = 



2 or 3 6 



4 70 



5 90 



6 434 



The index 1 is satisfied with the others and of course : but no 

 other index : 2 and 3 are conjugate, and as shewn on the fi curve 

 on Fig. 2, the indices conjugate to 4 and 5 lie between 1 and 2, 

 and those conjugate to 6 to oo between 1 and 0. 



If n = 3, (57) becomes 



/3{(p + l)(l + 2v)-2.3v} +(p- 1)3^ = (59) 



which gives the following values for four sections : — 



XI. — Integral expressions for volume: four symmetrical sections. 

 V=±aA + /3B 1 + yC i + 8I>o) 



<T 



Index p cr a ft y 8 



2 or 3 8 1 3 3 1 



4 640 77 243 243 77 



5 60 8 27 27 8 



6 9296 1003 3645 3645 1003 



The indices and 1 are simultaneously satisfied with any one of 

 these : 2 and 3 are again conjugate. The indices greater than 3 

 are conjugate to indices less than 2 : the curve j3/a being similar 

 to the /3 curve in Fig. 2. It will be noticed that the four sections 

 satisfy only a cubic function, the coefficients being 1, 3, 3, 1. 



If n = 4, (57) becomes 

 /^{(^+1)(1 + 3p)-2.4 p j+/{(^+1)2.2 p -2.4p} + (^- 1)4^ = 0. ..(60) 

 solving which for either p = 2, or p = 3 gives 



y' = 2-i/3 (61). 



Hence there is a one-fold infinity of solutions, whenever five 

 symmetrical transverse sections are taken, if the indices are l, 2 

 and 3. Thus we may write out such solutions as the following, 

 in all cases doubling the value of 7' as it is a middle section. 



