130 



G. H. KNIBBS. 



V= f a f(z)dx = Az + lB** +ltiz* +lDz' m (2); 



*^ o 



that is 



V=±z(SA + 3Bz + 2Cz'> + liDz s ) = irz(A + ±A la +A') (3) 



J m denoting the middle area, or the area for z m = ^z; and A' the 

 area of the other terminal plane : which may be readily verified. 

 We thus see that this last expression (3), viz., the ' prismoidal 

 formula ,' is true for any solid whose sectional area is a cubic or 

 lower function of the distance from either of its terminal planes; a 

 proposition due really to Newton. 1 



2. The prismoidal formula applied to solids with ' ruled surfaces.' 



Consider further the 'ruled surface' formed by the motion of a 



straight line as generator, the terminals making a complete circuit 



of two parallel planes serving as directors; the areas of the latter 



being 



f(x.y) = A; f'(x'.y') = A'; 



and the velocities of the generator- 

 terminals, viz., their contact-points 

 on the directors, being unrestricted. 

 If now ds and ds' be corresponding- 

 infinitesimal elements of the director 

 curves, projected in Fig. 1 on the 

 one plane, and if g and g denote 

 the projections of successive posit- 

 ions of the generator on the same 

 plane; it is evident that any inter- 

 mediate parallel plane A i will cut 

 the generators and their projections 

 in a constant ratio; and that any infinitesimal increment of area 

 dA x say, see the shaded portion of the figure, may therefore always 

 be represented by a quadratic function of z. That is to say, if 

 dO be the infinitesimal angle between the projections of the 

 successive positions of the generator, the increment of area will 

 always be of the form 



dA^ia + bz + cz^dd (4), 



1 "Methodus differentialis," 1711. 



Fig'. 1. 



