GENERATION OF FIGURES ON STRAIGHT OR TORTUOUS AXES. 325 



identical, and make an angle of \ir ± w with the plane of the curve 

 itself. In this case planes parallel to the plane of the curve, 

 intersect the surface in curves identical with the curved axis itself. 

 The plane, containing both the radius of the osculating circle at 

 any point in the curve, and the path of this point when generating 

 the oblique-axial-surface, determines unequivocally the abscissa of 

 the generatrix; and this plane, the radius and radius produced, 

 and the path of the point, constitute what may be called the 

 abscissa-plane and the axes of reference thereon. It is at once 

 obvious that at least condition (6) must be satisfied as the gener- 

 atrix moves along the axis, the planes of the osculating circles in 

 which p is measured being taken always parallel to the plane of 

 the curved-axis t, for each point in the plane of generatrix, and 

 for every position on the axis. This condition alone is, however, 

 inadequate. 



The surface generated by an element of the generatrix. (8s) 

 parallel to the radius of the osculating circle, is greater than that 

 generated by an element of equal length, parallel to the other 

 axis of the generatrix-plane, each being equidistant from the centre 

 of that circle. Consequently in order that the constant k, of 

 reduction for obliquity, shall be identical for every point on the 

 £-axis, the successive figures on the plane of the generatrix, as it 

 moves along the axis must be homothetic with respect to the axes 

 of that plane, 1 even when the inclination of such axes is identical. 

 This inclination however changes as the plane moves along the 

 axis, since it is a function of the angle a/ between the tangent to 

 the curved-axis t and the orthogonal projection, on the plane 

 thereof, of the line (making the angle \ir ± w with the plane) 

 which generates the oblique axial-surface. Let a>" denote the 

 excess or defect from 90° of the angle between the axes on the 

 abscissa-plane; 1 then evidently 



sin to" = sin to sin a/ (7) 



In this expression w is constant, but w', and therefore in" also, are 



1 So that the inclination of the XY axes, XPY say, would be \tt ± on , 

 or they may be similar and similarly oriented. 



