VOLUMES OF SOLIDS AS RELATED TO TRANSVERSE SECTIONS. 55 



From the nature of the curve we see that, in general, three 

 indices may be satisfied, when a middle and two other sections 

 equidistant therefrom are taken, each having equal weight, two 

 of these indices however will be conjugate. For the same index- 

 values the sections are nearer to the terminals than in the case of 

 two symmetrically situated sections. The result may be summed 

 up in the two following propositions : — 



Prop. (I.) When a middle section and two others equidistant 

 therefrom, all of equal weight are taken, the latter can never be at a 

 greater distance than '14-69624 of the length of the axis from its 

 terminals: at that distance the only indices that can be satisfied are 

 1 and about 2'44-9- 



Remembering that the indices 2 and 3 are conjugate, we have 

 also the second proposition : — 



Prop, (m.) For sections nearer the terminals of the axis than 

 this limiting value, the index 1 and two conjugate indices may be 

 satisfied the one greater and the other less than 2' '449; and if the 

 distances from the terminals be '1464466 the conjugate indices 

 satisfied, together with 1, will be 2 und 3. 



The formula for area of volume satisfying the function 

 A z = A + Bz + Cz n + Dz\ 

 u and v being conjugate, is 



V=iz(A 1 + B m + C l ) (41). 



14. Two terminal sections and one intermediate section. — Equa- 

 tion (5) in this case reduces to 



'-JU^T) '< 42) 



The values of a, /3 and y are all at our disposal, hence there is a 

 three-fold infinity of solutions for b v . Since the solution loses no 

 generality by making f3 unity, inasmuch as a and y merely become 

 the ratios of the weight-coefficients of the terminal sections to the 

 intermediate section, which is all that is required, the above 

 equation becomes simply 



i» = 1+ i "-yy (42a) 



1 + p 



