rBEATTY] AN OUTLINE OF ANALYTICAL SPHERICAL GEOMETRY 263 



„, , . tan d)' — tan d)" tan x 



Ihat IS rr, r^n + = f>. 



tan d' - tan 6" tan y 



^ tan 0' — tan 0" _ tan ^ — tan ^' 



tan ^' -tan ^ ^ tan /9 - tan ^' 



since the three points are on a great circle. 



Therefore tan x (tan d — tan d') + tan y (tan cfi — tan (/>') = o. 



Or tan a; tan 6 + tan ,y tan <p = tan'^ r. 



XXI. The following theorems could be proven from this general 

 equation in precisely the same manner as the corresponding ones are 

 in plane analytical geometry: 



(a) If P lies on the polar of Q, then Q lies on the polar of P. 



(b) If a point P moves along a great circle, its polar turns 



about a point. 



(c) This is a special case of the previous one. The pole of PQ 



is the intersection of the polars of P and Q. 

 XXI T. In Uke manner the theorem for finding the tangent to 

 any curve could be used to find the tangent to a small circle with 

 centre {a, ^) and radius r. The equation of such a circle is 



(1 + tan- d + tan-^ cf)) (1 + tan' a + tan^ /? 

 ^ + ^^' '^ (1 + tan ^ tan a + tan cf) tan /?)' 



Let {d, (/)) be the point where the tangent is required, to this circle. 

 For convenience in manipulation write for tan d, the symbol d. 

 This givesus 



(1 + r^) (I + da + cf) ^) (a + ^ ^) = (1 + «^ + ^) (6 + cf> -^) 



defy ^ 6 (1 + œz + ^•^) - ( 1 + r^) g {\ + Q a + cf> ^) 

 ^^ d 6 {\ +r) ^ {\ + d a + cf) ^) - cf> {\ + a' + ^■') 



That is 



(1 + r^) (.r a + ^ /?) ( 1 + ^ a + </) /3) + {6' + cf)') (1 + «^ + /?2) = 

 {\ + 7^) {6 a + cf) ^) {\ + e a ^- cf) ^) -\- {xd + ycf)) {I + a' + ^'). 



But this becomes 



(1 +r2) (l+xa +2//?) (l+^a+c/»/?) -(1+^^ +.y0)'(l + «^ + /?') = 

 (1 + r^) (1 + ^ a + <^ /?) '- (1 + 6' + ^') (1 + œ + /5^) = o, 

 which is therefore the equation of the tangent. Writing it out in full 

 the equation is 



,, , , ,, , (1 + tan 6 tan a + tan cf) tan B 

 <' + '"" *•' (1 + tan- « + ta ??) - 



(1 + tan 6 tan x + tan cf> tan y ) 



(1 + tan a tan ^f^ + tan ^ tan :y) 



