GENERATION OF FIGURES ON STRAIGHT OR TORTUOUS AXES. 327 



independent of the inclination of the axes, but taking equivalence 

 of area in rectangular units into account; hence 



8A t =f(x,y,o>"); 2(8A t .Ap) = (10) 



indicates the conditions. 



The case of curved axes in the generatrix leads to more complex 

 conditions, and may be dismissed without consideration, as of 

 little practical moment. 



We now consider the case where the £-axis does not lie in a 

 plane. 



7. Tortuosity of the principal axis. — Let the axis be a tortuous 

 instead of a plane curve; and the surface formed by the sheaf of 

 lines drawn, through every point of the curve, perpendicular to 

 the plane of the osculating circle thereat, be called the binomial 

 axial-surf ace f and let also the plane, perpendicular to the osculat- 

 ing circle and containing its radius, be called the normal plane; 

 then the line of intersection of 'normal plane' and the 'binormal 

 axial-surface,' and the radius and radius produced of the osculating 

 circle, are rectangular axes, at the intersection of which the 

 tortuous curve 2 is everywhere perpendicular. The normal plane, 

 and the axes thereon, are consequently the analogues respectively 

 of the plane perpendicular to a rectilinear axis, and the rectangular 

 axes thereon; and are also respectively analogous to the plane 

 continually perpendicular to the tangent of a plane curve, and 

 axes thereon ; one of which is the radius of the osculating circle, 

 and the other the vertical to the plane of the curve. We shall 

 therefore, as before, call it the abscissa-plane. 



The circular curvature uniquely defines the curvature of tortuous 

 curves, inasmuch as the planes of osculation pass through any 

 three consecutive points, infinitesimally separated. And again, 

 the locus of the centres of circulation curvature, are denned by 

 the point, where the line of intersection (or polar line 3 ) of two 



1 This will of course, not be identical with the 'rectifying developable/ 



2 Or its tangent. 



3 The locus of the polar lines is the polar developable, i.e. the centres of 

 circular curvature lie on the polar developable. 



