140 



G. H. KNIBBS. 



Fig. 5. 



Fig. 3, C B being \ w : and let also 

 the shaded figures CBG'E'C repre- 

 sent the vertical sections on OC 

 and OC l5 so that the point E is 

 vertically over C, and similarly the 

 point G vertically over the line OC. 

 Let y c denote an ordinate on the line 

 CoCi perpendicular to the plane 

 CgBi : if then the angle CoOCi be 

 denoted by 0, and dy /dd be constant, 

 the curved surface-line EqE^^ will be 

 a spiral with a uniform rise. Let 

 also y g be similarly the ordinate of 

 any point on the curved surface-line G G l3 measured from its 

 projection on the CB plane; then if dyjdd be also constant, the 

 surface-height at the centre line of a right section of the solid 

 between the terminal BO EG planes (see Fig. 3), the 'side height' 

 — or height where the uniformly sloping side of the road-cutting 1 

 meets the natural surface — and the horizontal projections of the 

 lines EG or BG will together change uniformly, 2 that is as a linear 

 function of 6, or of z the distance on the curved centre line, CoCi. 

 This is the analogue of the case already considered in § 7, § 8, etc.; 

 and since if the radius of curvature become infinite the surface 

 EoGoGxEi becomes an ordinary warped surface, we may call the 

 similar surface in the case now considered, a 'circularly warped 

 surface.' Such a solid as is here considered, viz., one whose longi- 

 tudinal axis is curved, and the terminals of whose cross-sectional 

 surface lines together change their height uniformly, may be 

 taken as defining a class of solids which often present themselves 

 in engineering practice, as in road and railways ; and to find its 

 volume it is necessary only to take account of the changing position 

 of the centre of gravity of its right section as it moves along the 

 longitudinal axis ; as will hereinafter be shewn. 



1 As before a similar statement applies to embankments. 



2 Tt is not essential that the slope (r) of the cutting should be constant 

 but it must change uniformly with $, i.e., dr~ 1 jdQ must be constant. 



