VOLUMES OF SOLIDS AS KELATED TO TRANSVERSE SECTIONS. 49 



function of the distance along its axis ; the same is true also for 

 an area, where the ordinate is similarly a cubic function. 1 If A x 

 and B x denote the values of the ordinates or sections at 0-2113249 

 and 0*7886751, then whenever 



A Z = A+Bz+Cz 2 + Dz 3 

 we shall have 



r=iz(A 1 + B 1 ) (25). 



11. Asymmetrical positions of two sections. — Since the number 

 of fractions is the same as the number of indices they cannot both 

 be arbitrarily taken : Prop, (a) § 2. Let the weight unity be 

 assigned to the section at a, so that the weight of that at b will 

 be relative thereto : we shall then have from (5) 



a? + pb*= ] +/3 , and a * + /ft* = Ji±£ (26) 



1 + p 1 + q 



so that the condition to be satisfied is 



(l+?)(a p + j86 p )= (l+q){a* + pb*) (27); 



as might be anticipated this does not lead to simple relations. The 



only cases that appear to be worth consideration are a = 0, and 



6 = 1, the former involving the determination of the value and 



weight of b : the latter the value of a and weight either of a or b. 



If a = Q, then its powers are also zero, putting its weight =1, 

 we have at once from (27) 



b = (lll^i^; or logarithmically log b J°«( U P) Z lo g( 1 +«) (28) 



1 + q) q-p 



and when b is obtained 



^ = 6 y (l+p)-l " ¥([ +q)-l ^ 



Putting A for the initial ordinate or area, and B x for that at b, 

 the formula for the area or volume will be 



V=-± r pZ(A, + pB 1 ) (30) 



and the integral solutions up to the fourth power are contained 

 in the table hereunder : — 



III. — Position and coefficients of second section, the first being 

 the initial section. 



l A less general proof of this is given by T. U. Taylor, op. cit., pp. 99 - 100 

 D— June 6, 1900. 



