ON THE LINES OF CURVATURE ON AN 

 ELLIPSOID*. 



THE following investigation of the Lines of Curvature on an 

 Ellipsoid has the advantages of symmetry and of giving a 

 distinct geometrical conception. The artifice on which it de- 

 pends may, it is thought, be found useful on other occasions. 



The symmetrical equation to the lines of curvature is 

 d?-c*}xdydz+(c*-a*}ydzdx+(a*-tf)zdxdy = Q ...... (1), 



(see Mathematical Journal, Vol. I. p. 142), where xyz are con- 

 nected by the equation to the surface, 



j+f?*?- 1 ........................ 



= , !' = , J = w .................. (A). 



Then 



vv vw vuvw 



Hence, after the substitution and multiplying by 4 vw , (1) 

 becomes 



(J'-c 2 ) udvdw+ (c 2 -a 2 ) vdwdu+(a*-t>*) wdudv = 0...(3), 

 with the relation 



\-v + w \ (4). 



Difierentiate (3) ; then, since 



* Cambridge Mathematical Journal, No. IX. Vol. n. p. 133, May, 1840. 



