312 



Mr DA VIES on the Equations of Loci 



XXIII. 

 LEXELL'S THEOREM*. 



On the surface of the sphere, the line in which are situated the ver- 

 tices of all the triangles, having the same base and the same surface, 



is a less circle of the sphere. 



By resolving (III. 3.) supposing a, a {3 m the variable co-ordinates of the 

 point in question, and the other quantities engaged in the expression (viz. 



V 



a, (3,, a,, /3,, and cos ^) as constants, we shall find an equation to the circle : 



but as this reduction is somewhat operose, and the resulting expression ra- 

 ther complex, we shall take a shorter course, as follows : 



Let EQ, the base, be bisected in S, and EP, SP, QP, drawn to the 



FIG. 17. 



pole of EQ. Take P and PS as origin of < and 0. Put ES = 7, SR = 6, 

 and PM = (p. Puta Iso, for the present, % = ^ (j> MR. Let the 



spherical excesses of the triangles EMR, RMQ, and EMQ be respectively 

 e, (, and E. Then, since the angles at R are right, we have 



* Acta Petrop. torn. v. Dr BREWSTER'S Translation of LEOENDRE'S Geometry, 

 p. 266, and other places. 



