Rogers — Orthogonal Trajectories of a Congruence of Curves. 103 



where — , — are the limiting curvatures at the point. These may be 

 Pi pz 



described as extreme curvatures. The curvature for any direction is 



proportional to the square of the corresponding radius vector of the 



indicatrix. If pi and p^ have opposite signs, the indicatrix will he a 



hyperbola and its conjugate. 



There are two inflexional directions at each point, for which = 0, 



P 

 corresponding to the asymptotes of the indicatrix, and the normal curves 



in these directions are inflected at the point. The inflexional curves form 



a congruence of curves in space, two passing through each point. 



Points in space may evidently be classified, with reference to a given 



family, as elliptic or synclastic and hyperbolic or synclastic, according as 



jOi|02 is positive or negative. 



If we use general axes, - may be expressed as a quadric function 



of a, /3, y, 



«.,, «2, a„ I (&3 + C2), I (ci + ffs), h («= + ^) I (n. /3, yY: 



P 



where 



dl dl , dvi 



«i = -J- , «= = -1- , Oi = -T— etc. 

 ax dy d.r 



We have thus a first generalization of lines of curvature, viz. the curves 

 of extreme curvature, the tangent-line at any point being a direction of 

 maximum or minimum curvature. These curves plainly form a cowjruciue, 

 since two can be drawn through each point iu space. They cut at right angles, 

 and they must be real ; but they do not, except for suiiaces, eoincide with the 

 directions of zero torsion, which, as we shall see, may be imaginary and may 

 cut at any angle. 



The differential equation satisfied by the lines of extreme curvature for 

 general rectangular axes is 



= 0, 



where 2/ = 63 + C2, 2g = ci + a,, 2h = a, + bi, and the directions are 

 determined by solving for dx:dy: dz from this equation and 



Idx + mdy + ndz = 0. 



dx 



dy 



dz 



I 



m 



n 



ttidx + hdy + gdz 



lidx + h-idAj + fdz 



gdx + fdy + c^dz 



