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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 are s of aplane curve is expressed in polar co- 
ordinates by the equation 
ds” = dp* + p°do”’, 
and the arc of a spherical curve by the equation 
ds? = dp* + sin*pdw”, 
so let the are 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” + P*dw’, 
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  sinw’ 
Psinw = P’sinw’, or — = 
P 
sinw’ 
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- 
bilie) will be equal to the quadrant of the elliptic section con- 
taining the umbilics. The function Pp of this are will be equal 
to the mean semiaxis of the surface, in the same way as the 
sine of the quadrant is the radius. 
