579 



the tangent plane at any point on a line of curvature, and 

 the tangent planes at two umbilici, is constant ; or, as the 

 tangent planes at the umbilici are parallel to the planes of 

 circular section, we have the following elegant theorem : 



" The sum or difference of the angles between the tan- 

 gent plane at any point along a line of curvature on an ellip- 

 soid, and the two planes of circular section, is constant." 



The proposition just mentioned was, it is believed, first 

 published by Sir William Hamilton, in the Dublin University 

 Review, part (3) ; the short article which contains it being 

 dated June, 1833. It has also been published by Dr. 

 Joachirasthal, in a paper printed in the 26th vol. of Crelle's 

 Journal, and dated January, 1842, where that geometer 

 claims it as " novum neque inelegans." 



The reciprocal of the line of curvature has for its equa- 

 tion 



— -5— x'H zr-Y^ + — -o— =0, 



in which if we make 



^ 1 y 

 x= -, and Y = -, 



z 



we shall get the equation of the cone, whose generatrices 

 are parallel to the normals along the reciprocal of the line of 

 curvature, and whose vertex is at the centre of the ellipsoid. 

 After this substitution the last equation becomes 



from the form of which it is evident that the cone passes 

 through the intersection of the given ellipsoid, and a con- 

 centric sphere having k for its radius. The properties of 

 this cone, and its relation to the lines of curvature, were first 

 noticed by Professor Mac CuUagh.* 



* Froceedinga of the Academy, vol. ii. p. 499. 



