79 



If the moduli of the two functions which express the va- 

 lues of Si ± *2 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 



, ^ 1 — sin2X ^ 

 tan*|p - 2tan^i^ F(^) + y-j-^j^^ = 0, 



where f(w) simply is written for , . . V^. . The class of plane 

 ^^ ^•' l-|-sin2A 



curves whose polar equation 



r*— 2r2F(a>) + 0*= 

 is analogous to the above, may be rectified by similar formulas. 

 These are 



Each of these integrals will be reducible to an elliptic function 

 of the third order, provided that f(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 



