VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 45 



The following computed values of a are shewn by Curve .No. 2, 

 Fig. 1, on which they are indicated by dots. 



Fig. 1. 



sra 



£M7l 47345^0rdinates - a, Abscissae = /> Values of ~p 



Curve 1. — The ordinates indicate the distance from the initial line or 

 plane at right angles to the axis, to the point thereon where the line or 

 plane has a mean value corresponding to the index p ; the total length of 

 the axis being considered as unity. 



Curve 2. — Graph of the function aP - (1 -a)P - 2/(p + 1) = ; p being 

 the independent and a the dependent variable : a will be the distance 

 from the terminals of the axis of two lines or planes satisfying the corres- 

 ponding indices, which constitute the abscissae of the graph. 



Curve 4 — Middle and termia al sections, the indices 2 and 4 being made 

 conjugate by suitable coefficients. 



Curve 5. — Shewing the relation between index and the position of an 

 intermediate section, when it and the terminal sections have equal weight. 



9. Limiting positions of two symmetrically situated sections. — 

 Let a=f(p), in which p, or u in (5), is to be regarded as the inde- 

 pendent variable; then for symmetrically situated sections, a real 

 and positive value for a may be found by suitable methods 1 from 



1 For very Jarge values of p, we may put, at any rate for a first 

 approximation, 



a= 1 - log -1 1 — 



\p 



log 



p + 1 



