Rogers — Orthogonal Trajectories of a Congruence of Curves. 109 



Tlieiofore 



cos ^ 1 

 pi p 



if) To find the. torsion of anij W-cwvc we must be given the rate of 

 variation of ^ ; and we can use the formula (1) on p. 96, which gives 



1 1 cl^ 



-= - + ^. 

 T T as 



This corresponds to the known formula for surfaces, (/> bemg the angle between 



the principal normal and the normal to the surface.* 



((/) TJie condition that the expression 



Idx + mdy -v ndz 



may be an exact differential or a multiple of one, is equivalent to any one of 



these geometrical conditions : 



(1) The mean torsion is everywhere zero. 



(2) The curves of limiting curvature coincide with those of zero torsion. 



(3) The curves of zero torsion intersect at right angles. 



(4) The Indicatrix of Form is a conic (see G, below). 



(A) In conformal representation in space (see p. 99), the difference between 

 the principal curvatures is divided by k where + k is the linear magnification, 

 and the relation between the principal curvatures for corresponding directions 

 is given by 



Hence also directions of princi-pal ciLrvature as well as directions of zero 

 torsion (p. 101) transform into each other hy inversion. These results follow 



from the theorem already proved that the torsion ( - 1 for any direction is 



transformed into — for the corresponding direction. 



(i) We have noticed that the two equivalent defining properties of a line 

 of curvature on surfaces (zero geodesic torsion and limiting curvature) become 

 separated when / = 0. Corresponding to this logical bifurcation, we have two 

 (jeneralizations of the surfcwe of centres. On a n -family, the directions of zero 

 torsion at P are those along which the normal {I, m, n) intersects the 

 ' consecutive ' normal, and they are given by their equation 



dx dy dz 

 I m n 



dl d'/n 



•■0. 



* Siiliiioh, £>;;. cit., p. 426. 



[15*] 



