384 



lines, and of which the product of the tangents of the semi- 

 angles at the base is constant, the locus of the vertex will be a 

 line of curvature. 



2. And if the ratio of the tangents of the aforesaid angles 

 be constant, the locus of the vertex will be a line of curvature 

 of the orthogonal system. 



3. As the arc 5 of a plane curve is expressed in polar co- 

 ordinates by the equation 



ds' = dp' + pMuiS 

 and the arc of a spherical curve by the equation 



ds^ = dp^ + sm^pduy^, 

 so let the arc of a curve on the surface of an ellipsoid, referred 

 to the geodetic distance (p) from one of the umbilics, and the 

 angle (w) made by p with the section containing the umbilics, 

 be given by the equation 



ds^ = dp^ + pVwS 



and, as Mr. Roberts has demonstrated, 



psinw 



will be the perpendicular distance of the point (p, w) from the 



plane of the umbilics. Hence, p', w' denoting the same things 



'for the contiguous umbilic, we have 



, . , P sinot' 



psiuoj = p sino) , or -y = 



p' sin w 



which may be regarded as an extension, to the surface of an 

 ellipsoid, of the fundamental property of plane and spherical 

 triangles, that the sides (or the sines of the sides) are propor- 

 tional to the sines of the opposite angles. 



4. Let w be a right angle, and the corresponding geodetic 

 vector will pass through the vertex of the mean axis, and its 

 length comprised between this point and the origin (the um- 

 bilic) will be equal to the quadrant of the elliptic section con- 

 taining the umbiHcs. The function p of this arc will be equal 

 to the mean semiaxis of the surface, in the same way as the 

 sine of the quadrant is the radius. 



