traced upon the Surface of the Sphere. 4 1 7 



able relations to the loxodrome. I mean the equi-subtangential curve on 

 the sphere, or as, for reasons which will be at once apparent, I shall denomi- 

 nate it, the Spherical Logarithmic Curve. Even this can be but slightly 

 noticed here, and that merely to ascertain its general form, leaving its invo- 

 lute, evolute, and locus of intersection of perpendicular upon the tangent, al- 

 together unnoticed. I also wished to add some speculations respecting the 

 circumstances of the curve when the sphere becomes infinite ; but these must 

 also be deferred till a future period, when I can complete my paper on the 

 singular points, etc. of Spherical Loci. 



On the Spherical Logarithmic. 



1. The subtangent of a certain curve intercepted on the equator by the 

 current meridian is constant, What is the nature of the curve ? 



Let QE = a be the given subtangent : then by (XVI. 2) we have 



tanESQ= ( ,"'" ^ ,orcotESQ= . 

 d(p m 



Also by right-angled triangles, 



cos </> tan a cot ESQ ................................. (2.) 



