RoGEU.s — Ovthocjonal Trqfectories of a Congruence of Curves. 107 



Let T\ be the extreme radius of torsion for ^ = - . Then, since 



4 



this implies 



1 1 1 1 „ 



= = Jj, say. 



Ti 72 pi pi 



It may be of interest to notice that the differential properties of tho 

 normal curves, torsion, and curvature are determined at a point by means 

 of the n-tangent plane, and the three quantities ^7, ^J, and ^D (i.e. the 

 mean torsion, mean curvature, and deviation), provided a particular direction 

 be associated, say, with p^. For then the principal radii are given by 



1 



J+D 



2 ' 



1 J- D 



p. 2 



1 



I+D 



2 ' 



I I-D 



71 



r. 2 ' 



and thus - and - can be expressed in terms of Q, I, J, D by equations (3) 



P '■ 



and (4), which become 



- = J" + Z> cos 2ff, - = I + D sin 2d. 



P T 



E. — Various Formulae, Geometrical Expressions of Conditions of 



Integrabilitv. 



The following results may easily be proved : — 



(a) The angles which the two intiexional directions make with a 

 direction of principal curvature are given by 



tan»0=_e:^ = ^^, 

 Pi JJ - 'I 



and the angles they make with a direction of principal torsion satisfy 



1^ 1 



sin 2^ = ^^ = ^ = - cos 20. 



pi Pz 

 K.I A. PROC, V(lL. X.XIX., SECT, A. . [15] 



