and Lines connected with them. 231 



the equation to a pair of orthogonal straight lines, the pro- 

 jections, as is obvious from (8) or comparison with (6), of 

 the principal axes of the indicatrix. 



Thus in general through every summit, immit, and col two 

 ridge-lines pass, crossing each other orthogonally and touching 

 the lines of curvature, and therefore also the lines of slope ; 

 for at summits and immits all the lines of slope touch one of 

 the lines of curvature, except one, a limiting case, which 

 touches the other line of curvature*, and at a col there are 

 only two slope-lines coinciding with the lines of curvature. 



In the foregoing it is supposed that the point, summit, 

 immit, or col is an ordinary point on the surface ; but if 

 rt = s 2 , or s = and r = t, or r = s — 1 = 0, in which cases the 

 point is a parabolic point, umbilic, or point of no curvature 

 respectively, the equation (9) will vanish identically, and 

 terms of the third degree must be retained in (8). The 

 equation corresponding to (9) will then be of the third 

 degree, so that three ridge-lines will in general cross each 

 other at such a point ; but as two of these may be imaginary, 

 there may be in special cases only one ridge-line through the 

 point — for instance, where the surface is a conicoid and the 

 summit one of its umbilics. 



The case in which the summit or immit is an umbilic may 

 be examined. In this case s = and r=t ; hence, retaining 

 terms of the third degree in (2) , the surface near the point is 

 given by 



t=*+irie+f} + M<&+^ + W*+:*fh . (10) 

 and, by (7), this gives 



b?+(2c-a)Z 2 v -{2b-d)% v 2 -c V B = 0, . . (11) 



as the equation to the projection, for points near the summit 

 or immit, of the three ridge-lines which cross at the umbilic. 

 This is the same equation as that which gives the three lines 

 of curvature at the umbilic, so that the ridge-lines and lines of 

 curvature there coincide. At an umbilic there is obviously 

 no special connexion between the ridge- and slope-lines, for 

 the latter proceed in all directions from such a point. 



Other exceptional cases need only be mentioned. At par- 

 ticular points on a surface the indicatrix may have any form : 

 if at any such point the tangent plane is horizontal, there will 

 obviously be as many ridge-lines and lines of curvature 



* Cayley, I. c. ante* 

 R2 



or 



