APPLICATION AND DEVELOPMENT OF PRISMOIDAL FORMULA. 143 



x denoting the abscissa of the centre of gravity on the initial plane, 

 i.e. for 1 = 0. As may easily be verified, this expression is identical 

 with 



r=H{A(i+|)+^ m (i +1^+^(1 +|)j < 36 ) 



in which x m is the abscissa of the centre of gravity of the 'mid- 

 section'; x is positive when it is on that side of the centre line, 

 or axis of reference, which is opposite the centre of rotation. 1 

 Since the triangle beneath the line AB, Fig. 3, is symmetrical, its 

 centre of gravity is in the line CD and its volume to be subtracted 

 from (36) say, is Iw^jir. The general result above indicated is, 

 that the prismoidal formula may be applied to circularly warped 

 solids of triangular section ; it being necessary only to multiply 

 their end and middle areas by the ratio p/R, p being the radius of 

 the centre of gravity, and R of the 'centre line,' since pjR = 1 + x/R. 

 By considering formulae (31) and (36) it is evident that the 

 latter is true for any 'circularly warped solid': for let the right 

 section of such a solid be divided into a series of triangles, whose 

 areas are A, A' etc., the abscissae of the centre of gravity of these 

 being x, x etc., measured from the curved axis, of radius R, of 

 the solid ; then for any section we have from those formulae 



2 { 4( l + |)-j=^{l + ^.^)} r ^(l + |) (37) 



x being the centre of gravity of the whole figure and HA its area. 

 Consequently formula (35) may be applied to any 'circularly 

 warped solid' of polygonal section. It is moreover obvious from 

 (33) that the formula is true for any circular solid the centre of 

 gravity of whose right section changes linearly with the z curved- 

 coordinate continually perpendicular to the section. 



16. Prismoidal formula not applicable with variable radius of 

 curvature. — It may however be shown from (33) that if R be not 

 constant, the prismoidal formula will not apply generally, since on 



1 In earth-work computation it is more often negative than, positive 

 in cuttings, since on hill sides the rising ground is usually on the same 

 side as the centre of the curve. The values of x/R can be obtained from 

 tables of double entry in practical calculation. 



