APPLICATION AND DEVELOPMENT OF PRISMOIDAL FORMULA. 139 



on a iortyous .curve, 1 be denoted by p; and let the tangent to this 

 circle at the middle point be called the tangent to the curve. If 

 then the centre of gravity of any plane figure, as generator, move 

 along this curve in such a manner that the plane is continually 

 perpendicular to the tangent of the curve as defined, the volume 

 of the generated solid will be equal to the length of the curve 

 intercepted between the terminal planes, multiplied by the area 

 of the generator, provided that it does not return on itself ; and 

 that the centre of the osculating circle as defined, is always outside 

 the boundary of the generator. 2 For if ds denote an element of 

 the curve, subtending at the centre of the osculating circle the 

 angle dd, the volume will be 



V = fApd9 =/Ads = As (27). 



With reference to the curve itself,- the plane generator may be 

 regarded as not rotating, if the radius p of the osculating circle 

 does not change its position with reference to the axes thereof : as 

 in the preceding section however, the rotation of the generating* 

 plane does not affect the volume of the generated solid, since the 

 relations expressed in (27) still hold good. 



13. Solids of curved longitudinal section with circularly warped 

 surfaces. — In Fig. 5, let O be the centre of the concentric arcs 

 BoBi, C C 1} representing in plan a half-width of roadway as in 



1 That is a curve of ' double curvature ' or one that will not lie in a 

 plane. Let O > Oi, O2 he the centres of circles of which the very small 

 arcs PP ,PoPi,PiP2 form part. Then if Oi be in the plane P0 Po 

 the curve is a plane curve, but if not it is tortuous. Let the radius P0O0 

 rotate through the angle d</> about the point P in a plane perpendicular 

 to that of the circle containing PP OJ and then change its length to the 

 radius of the curve P0P1 by the point o moving to Oi : and similarly 

 let P1P1 rotate laterally d$ x -> and the point Oi extend to 2 : these 

 rotations may be made to measure the tortuosity: the curve O0O1O2 

 contains the centres of the osculating circles as defined, and denoting the 

 arcs when infinitesimal by ds , dsi etc., and the radii Po, O1P1, etc.* 

 byPoiPi etc., we have p d0 o = ds ,p 1 dO x = ds x etc. It is obvious that 

 in a curved so developed the tangent at P is common to the arcs dso, dsi 

 and so on : d$/ds measure the curvature, d^ids the tortuosity. 



2 This condition is necessary, otherwise negative volumes would have 

 to be considered. 



