VOLUMES OF SOLIDS AS BELATED TO TRANSVERSE SECTIONS. 47 



From these results, and the figure representing them, it is 

 evident that : — 



Prop. (/). Two symmetrically situated sections cannot he at a 

 greater distance from the terminals of the axis than about 0-2123179, 

 the length of the axis being regarded as unity; at that distance p 

 and q are respectively 1 and about 2-Jfll, and no other values can 

 be satisfied. 



Since the curve in Fig. 1 is intersected by the ordinate for p = 1, 

 at the distance of about 0*1997088 from the axis of abscissae; for 

 values of a greater than this, and less than the maximum, two 

 indices greater than unity can be satisfied, together with unity 



Rejecting the second and higher powers of p, since it is extremely small, 

 as inappreciable, and transposing we have 



op = l-p(2- »-HlL-^- etc.) 



Taking logarithms, and again rejecting the 2nd and higher powers of p 



a 2 

 p loge a = -p (2 - a - — - etc.) 

 2 



Dividing both sides by - p, applying the operator log -1 , transposing, and 

 dividing the numerator and denominator of the left hand number by a, 



we have l^+l+ a + a 2 + etc. = log7 2 =logw 2 fi = 7-3890561 



a 

 ix being the modulus of common logarithms ; so that 



— + a + a 2 + etc. = 6-3890561 

 a 



from which a is found by suitable methods of approximation to be '1613782. 



5 In general a is of course indeterminate and may have any value 

 whatsoever. The curve studied has a limiting value for p-\, which may 

 be found by putting 1 + h for the index. Rejecting powers of h 



a l+h + h l+h _ O ( l + k logo ) + 6 (i + Ti, log 6) = 2/(2+ h) = 1-^h 



from which after putting b — 1- a, remembering that 



t ,, \ a a 2 a 3 , 



log(l-a)= ---_-_- etc. 



dividing by ah, and arranging the terms in the order of their numerical 

 magnitude, we obtain 



1 ,, , a , a 2 , a 3 , a n ,, .. 



+ log e a + + + — + ... — ■ + etc. = 1 



2a 1.2 2.3 3.4 n(n+L) 



from which by suitable methods the limiting value of a may he found. 



The convergency of the first three terms is very slight, consequently in 



practical computation it is advantageous to tabulate the sum of these 



three at least. See § 16 hereinafter. 



6 Both roots are i ± ^ V3. 1 \ ± ^(V^ - f). 8|i V(Vif - - i)- 



