traced upon the Surface of the Sphere. 319 



We remark, in closing, also, another simplification of the general for- 

 mula, viz. when cos 2 i = 1. Here we have (3.) converted into 



1 



cot0 = - {sin a. cos 6 8, sin a a cos 6 8,,} ... (2.) 



cos a, H- cos a a l 



which represent two great circles perpendicular to one another, a result 

 which possesses claims to attention, on account of some consequences deriv- 

 able from it, as well as from its own geometrical beauty. 



The chief modifications which (3.) can undergo are easily traced, and 

 will all be found to depend upon the constants employed becoming, sepa- 

 rately, or in connexion, of the form -~-, where n is a whole number, But 



as they are easily followed out, it is unnecessary here to discuss the question 

 at greater length. 



XXVI. 



Given the perimeter of a spherical triangle, together with the mag- 

 nitude and position of its vertical angle, to find the curve to which its 

 base is always a tangent. 



Let EPQ be the given vertical angle, and PS, the great circle bisect- 

 ing it, be taken for the origin of 6, whilst P is the origin of <p. Put 



FIG. 21. 



EPQ 2, and perimeter = 2 a. Let the sides of any one of the tri- 

 angles so constituted be denoted by , , respectively. Then, by a well 

 known theorem, 



s s 2 



