VOLUMES OF SOLIDS AS BELATED TO TRANSVERSE SECTIONS. 



43 



idex. 



a 



Index 







•3678794 



2 



i 



•4095999 



3 



A 



•4218751 



4 



* 



•4444444 



5 





•4647581 



6 



1 



•5000000 



7 



i* 



•5428836 



8 



oo 



idex. 



a 



9 



•7742636 



10 



•7867934 



11 



•7977974 



12 



•8075536 



13 



•8162746 



14 



•8241257 



15 



•8312380 



loga= - llogQo+1) (17a) 1 



V 

 from which the following numerical values for a, with the argu- 

 ment p, are calculated and shown by curves No. 1. Fig. 1. 



I. — Position of section of axis, for a " one-term" formula. 



a 



•5773502 

 •6299605 

 •6687430 

 •6988271 

 •7230203 

 •7429969 

 •7598357 

 1.0000000 

 Note. — a is the distance along the axis ( = 1) from the initial end. 



A formula for area or volume depending upon the ordinate or 



sectional area (16) at some particular point on the axis (z) may 



be called a " one-term formula" that is, V denoting the area or 



volume, 



V=zA 1 (18) 



A 1 being the ordinate or cross-sectional area at the point deter- 

 mined by (17). 



8. Positions of two sections. — Let the function of z be 



Az = A + Bz v +Cz* (19) 



and first let us suppose the weights of the selected values of the 

 function to be equal. Then from (7) and (8) we have the sym- 

 metrical equations : — 



9 



a p +6 p 



p+l 



and a q + 6 q 



q+ 1 



(20) 



1 The limits are somewhat peculiar. Suppose p to be very small, then 



l0ge (1+J?) 



(p 



etc.) 



*l+*:-eto.) = - (1 -.*-+*- 



p p 2 3 2 2 



that is - 1 when p is 0. Hence for the zero value of p, logio a = — -43429448, 

 or 1*56570552, that is a — '3678794. Again when p is very large the value 

 of - — log p is required, since the unit will be negligible in relation to p. 



V 



Put 



Hence obviously 



: p = p so that P - log e p. Then since e> 1, if P = oo , p - oo . 



P 



l0gei> 



and the reciprocal — sili: =0, that is a. 



