APPLICATION AND DEVELOPMENT OF PRISMOIDAL FORMULA. 131 



and this property depends merely upon the fact that the generator 

 is cut in a constant ratio. * And since the total difference of area 

 between any two planes is made up of terms similar to (4), that 

 is differing only with respect to the value of the constants, it 

 follows that the xy sectional-area, in a solid whose mantle is a 

 1 ruled surface] is merely a quadratic function of the z coordinate. 

 Hence without restriction, the volume of such a solid is also 

 expressed by the prismoidal formula, as shewn in § 1 . 



3. Skew, warped, or ruled quadric surfaces. — If the directors are 

 dissimilar polygons whose sides are neither necessarily parallel nor 

 equal in number, the prismoidal formula still applies, as the last 

 section, § 2, demonstrates; and the middle- area may readily be 

 found, provided the scheme of generating the '"mantle'' 1 be specified. 

 If the generator-terminals move simultaneously with uniform 

 velocities over any two sides S and *S" of the polygons, starting from 

 the initial and reaching the terminal points of those sides at the 

 same instant ; then if the sides are parallel a plane is generated, 

 but if not, the 'ruled surface' formed is a 'skew,' 'warped' or 'ruled 

 quadric surface'; 3 that is a surface upon which a straight line will 

 lie wholly on the surface in two directions, and only in two direc- 

 tions. From projection it is evident that the corresponding side 

 of the middle-section is the straight-line 

 S m = i(S+S') 

 if #and S' are parallel, 4 and similarly in regard to the coordinates 

 of the terminals, but without that restriction : that is, 

 x m = ^ (x + x) etc (5). 



1 In Fig. 1 let arcs be drawn from the points A, A it A', and from B, B if 

 B' with O as centre. It is evident that in the limit, the triangular areas 

 on opposite sides of the curves, of which these arcs, the curves, and the 

 radii form the boundaries, become equal, both approaching zero as B B' 

 approaches A A '; thus every elementary area is expressible by a quadratic 

 function (4). 



2 The ' ruled surface ' bounding the solid between its terminal planes 

 may be called its ' mantle/ 



3 Throughout this article ' warped* means skew or plane- warped unless 

 otherwise specified. A skew surface is one on which the successive 

 positions of the generator do not intersect. 



4 Added 21 Sept. This restriction was erroneously omitted in the paper 

 as read. 



