ON THE LINES OF CURVATURE 

 Hence, after a slight reduction, 



(& - c *) Wi 2 + ( c ' - a 2 ) ^ ^ - (a 2 - Z> 2 ) w, 

 j y 



+ ( 2 -c 2 K+(c 2 ->! = <) ......... (9), 



a quadratic in , of which the roots are real and of unlike signs. 



This is obvious, for u t , v lt are essentially positive, and a, Z>, c, 

 being in order of magnitude, the signs of b 2 c 2 and c 2 a 2 are 



opposite. Similarly, ^ is determined by a quadratic, whose roots 



are always real and of opposite signs. Thus two lines of cur- 

 vature pass through every point on the surface of the ellipsoid. 

 Let us now consider the envelope of the surfaces represented 



by (8). 



Differentiating (7) and (8) for/, g, h, we get 



-o (ii). 



I being an indeterminate factor, we may put 



7 _ (7,2 2\ U 1 _ f 2 2\ V 7 _ / 2 tf\ W . 



/* ' g*' K" 



whence, taking the values of /#, ^, to substitute them in (8), 

 we deduce 



As the signs of the radicals are independent, this represents 

 four planes ; but c 2 a 2 is negative. Hence the possible part of 

 these planes is their traces on the plane of xz, for which v = 0. 

 Thus we get the two straight lines 



V6 a c 2 "Ju + 'Jc? fr 2 *Jw (13), 



and for the points where they meet the ellipsoid, 



+ =! (14), 



whence 



