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ROYAL SOCIETY OF CANADA 



as a rule, appears in the form F (tan d, tan ^) = o 

 That is 



d F d F d Vàiï 6 



+ -,~ 7 . —- \ = O. 



d tan 6 d tan ' d tan d 



XIX. This could readily be applied to finding the tangent to 

 the circle at a point {6, (j)) on it. 



tan' r = tan'^ 6 + tan* 



d tan tan d _ (tan y — tan (f)) 



d tan d 



tan (tan a? — tan l9) 



Therefore 



tan 6 tan .r + tan <j> tan y = tan^ d + tan' <^ = tan* r. 

 The analogy is plainly discernible between this and the equation to 

 the tangent to a circle in plane geometry. Furthermore, the tangent 

 at {6 (j)) whose equation we see is 



tan d tan x + tan tan y = tan* r 



can be proven to be perpendicular to the spherical radius of the small 

 circle. The equation of the radius to the point {d <f>) is 



tan y tan d — tan x tan (f) = o. 



The perpendicularity follows since 



tan d (— tan 0) + tan ^ tan 6 = o. 



The general theorem dealing with the question of perpendicularity of 

 great circles will be given later, now that its use has been rendered 

 necessary, 



XX. Moreover, all the theorems dealing with poles and polars 

 in plane geometry may be here demonstrated. The polar of a point 

 {6, 4>) is defined to be the locus of the intersection of tangents 



drawn from the two points, where a 

 great circle through {d, 0) cuts the small 

 circle whose equation has been found. 

 The locus of L is arrived at thus — 

 Let P be the point {d, (f>). Let a great 

 circle through P cut the small circle in 

 two points {d', 4>') and {6" <t>"). The 

 point //, where the tangents at {d' (f)') 

 and {6" (j)") intersect satisfies buth of the 

 equations 



tan X tan 6' + tan y tan <f)' — tan' r. 

 tan X tan d' + tan y tan 0" = tan' r. 



