79 
If the moduli of the two functions which express the va- 
lues of s, + s, be complementary, the curve will coincide with 
the locus of the vertex of a spherical triangle whose base is 
given, and of which the product of the tangents of the semi- 
sides is constant, and less than the square of the tangent of 
the fourth part of the base. 
Equation (a) may be transformed into 
—sin 2X 
tan‘1 4p 2tan? 1 zp F(w) +S 
where F(w) simply is written for ee The class of plane 
Iya 
curves whose polar equation 
r— 2r°Fr(w) + c*= 0 
is analogous to the above, may be rectified by similar formulas. 
These are 
rer WV (AE) to 
iT a a (AE to *) de. 
Each of these integrals will be reducible to an elliptic function 
of the third order, provided that r(w) be a linear function of 
cos 2w. The following curves, among others, possess this 
property : first, the locus of a point, such that, tangents being 
drawn from it to two equal non-intersecting circles, their 
rectangle is constant, and less than the square of the tangent 
drawn to either from the point midway between their centres: 
secondly, the locus of the intersection of tangents to an 
ellipse which include a given angle. This curve is composed 
of two closed branches concentric with the ellipse, and satisfy- 
ing the given condition by angles which are supplemental. 
The well-known property of the lemniscate may be ex- 
tended to the locus of the orthogonal projections of the centre 
of an ellipse or hyperbola in general upon its tangents. First, 
in the case of the ellipse, the curve may be derived by taking 
