90 ON THE LINES OF CURVATURE 



gestions may, perhaps, show how we must determine the lines 

 of curvature which pass through an umbilicus, a problem dis- 

 tinct from that solved by Leroy of finding the directions of 

 curvature. 



Many curious properties may be deduced from the equations 

 we have arrived at. Thus, if we take on two concentric and 

 confocal ellipsoids, a series of pairs of corresponding points, 

 (such as are spoken of in the enunciation of Ivory's theorem,) 

 and if the locus of the points on one of the ellipsoids is a line of 

 curvature, then that of those on the other is so too. Again, the 

 traces on the tangent planes at the extremities of the three axes, 

 made by one of the cones represented by (8), are an ellipse and 

 two hyperbolas respectively. The areas of this ellipse, and of 

 the ellipses conjugate to the two hyperbolas, are so related that 

 their continual product is constant for the same ellipsoid, and 

 for all ellipsoids of the same volume. The method of demon- 

 strating these two theorems is so obvious, that it seems unneces- 

 sary to. enter more fully on either. 



It still remains to be shown how we pass from (8) to the 

 projections of the lines of curvature on the co-ordinate planes. 

 The symmetry of the problem is destroyed by the transition ; 

 but as it is in this shape that the results are commonly ex- 

 hibited, we shall dwell rather more upon it than would other- 

 wise have been necessary. 



Putting C=a z b*, B = <? cf, A = b* c 2 , and eliminating 

 u, v, w, successively between (4) and (8), there result 



A\ (B & = 



(17), 



(A B\ (C A\ A 



[-2 }V -y- -?] W = ~? 



\f gi U // / 



(B C\ (A B\ B 



r w ( ~? } u= 



\ff h) \f g) g 



Put 



C A . B C A B 



Then k-l = 



