328 G. H. KNIBBS. 



consecutive normal planes, infinitesimally separated, cuts the plane 

 of osculation to which it is perpendicular. It follows therefore 

 that the perpendiculars to the osculating planes at two consecutive 

 points on a tortuous curve, are analogues of the parallel perpen- 

 diculars to the plane of a plane curve. That is to say, they will 

 be everywhere equidistant from the plane perpendicular to the 

 tangent of the curve 1 at a point midway between the points. 

 Orthogonally projected on that plane however they will make an 

 angle equal to d<j> the angle of torsion for the length dt of the 

 curve. 2 



As before, let p denote the radius of the osculating circle, that 

 is the radius oj circular curvature. Hence we shall have 



dt = P d6 (11) 



where dO is the angle of contingence, or angle in the plane of 

 osculation between two consecutive tangents at points dt apart. 



If also we have 



dt = o-d^ (12) 



or is what may be called the radius of torsion, dcf> being, as before 



mentioned, the angle of torsion. 3 



If the generatrix be essentially one-dimensional, i.e. if it be a 



line, the surface generated will depend upon the angle it makes 



with the axes on the abscissa-plane, and upon its position on that 



plane. 



1 Containing therefore the radius of the osculating circle. 



2 These propositions will perhaps be more obvious when it is remem- 

 bered that a tortuous curve is misdescribed when called a curve of double 

 curvature. Its essential character is better understood by regarding it 

 as a curve of circular curvature, the plane of which however, twists about 

 the tangent to the curve at each point, instead of remaining constant as 

 in a plane curve. 



3 The rotation of the axis formed by the line of intersection of the 

 binormal axial-surface, and the normal plane, or, what is the same thing, 

 the rotation of the plane of osculation, is dcf> in the distance dt. This is 

 equal to the angle of contingence of the edge of regression of the polar 

 developable for the corresponding points. The radius T, of what may be 



jailed the complex curvature is given by 



1/T 2 = 1/p 2 + l/o-% 

 consequently ^ = ^ 



if d\b be the angle of complex curvature. Further 

 (d^) 2 = {dO} 2 + (dcf> 2 ) 



