142 G H. KNIBBS. 



15. Prismoidal formula applicable to circularly warped solids. — 

 Since the projections of EF, FG, change linearly in proceeding 

 along the centre-line CoC^, see Fig. 3, not only in solids with 

 warped surfaces, but also when the surfaces are circularly warped, 

 as shewn in § 13, the relation 



x z = x Q + (x 1 -x ) I (32) 



always holds good, if x z denote the abscissa of the centre of gravity 

 on the right section, distant z, measured curvilinearly on the 

 centre-line OoC x Fig. 5, from section 0. The length I is the total 

 distance to section 1, also measured on the curve; and x and x x 

 are the abscissae of the centres of gravity on the initial and terminal 

 sections. 



Let R denote the constant radius of curvature of the centre-line 

 CoC^, p z that of the centre of gravity of section A z ; then since x z 

 may be written x + Xz, see (32), and remembering that the area 

 of a circularly warped figure is a quadratic function of z, and that 

 not p z d0 but Rd6 = dz; the volume of a solid, of the type illustrated 

 in Fig. 5, will be 



V=fA 2 p z de = ff l (A + Bz + Cz^(R + x + kz)j d ^ 



= z{A' + B'z + C'z* + D'z 3 } ...(33) 



in which the constants have the following values, viz., 



^'=i~ (34) 



that is, the volume is a quartic, and the area of the right-section 

 therefore a cubic function of z, 1 the length on the centre-line. 

 Consequently, see § 1, the prismoidal formula is rigorously applL 

 cable. Remembering that X — (x l - x )/l = Ax/l say, and that 

 the limits of z are and I, equation (33) may be written in full 

 in the form 

 V=i{a(1+ £+ l ^) + lBl(l + -£_ + $. *£) + 



+ i<^(l+Jr + f-^)} (35) 



1 Since A = dVjdz : or as is immediately obvious from (33) itself. 



