Dr. Booth on a Theorem in Analytic Geometry. 179 



nerating line is perpendicular to one of the coordinate planes, 

 the line drawn from the centre to the point where this line in- 

 tersects the surface is a conjugate diameter to this plane, a 

 result which might be obtained from geometrical considera- 

 tions. 



We may, as a simple consequence from the preceding de- 

 monstration, obtain a theorem in spherical trigonometry ap- 

 parently new. 



Let a, b, c be the sides of a spherical triangle, and P", P', P 

 the arcs of three great circles drawn from the vertices A, B, C 

 of the spherical triangle, through a point S assumed on the sur- 

 face of the sphere to the opposite sides, p" p 1 p the segments of 

 those arcs between the point S and the sides a, b, c, we shall 

 have 



tsinp sinp' sinp"~]*__ 

 sm~P + shTF + suTFj "" 



S m^sin^sin^J^ (1 _ cos ^ S J^ n 



sinPsinP / sinP"L s in^ v ' €va.p' y ' ' s»jr" J 



To show this, through the point O let a right line be drawn 



parallel to P Q, meeting the surface of the sphere in S, and 



let the sides of the spherical triangle, opposite the angles A, /*, y, 



be a,b,c; then in the triangle PCD: PD:PC::sinPCD 



: sin P D C : sin p : sin P. 



Since P D is parallel to O Z, and P C parallel to O S, 



, z sin p 



hence — = -. — £. 



c sin P 



c.- -i i * sin»" y sinp' . . . , . 



Similarly, — = ^j^,, -|- = ^— ^,; making these substitu- 

 tions in (2.), after some obvious simplifications we find 



tsinj9 siny sinj9"~l 2 _ 

 imP + imP 7 + imF J ~ ' 



+ Z^^sin/TsinP (i_ cos A + ^ (1 -cos b)+™L^ (1 _ cos a)~| 

 T sinPsinP , sinP"Lsini> v J ^sinp ,K ' sin/>" V ; J 



When the triangle becomes plane, the sines are changed into 

 the corresponding arcs, and cos a, cos b, cos c are each equal 

 to unity, and we thus derive the known theorem in plane geo- 

 metry, 



P , P* 



j i 



P 



+ p/ + pw — *• 



N2 



