[beatty] an outline OF ANALYTICAL SPHERICAL GEOMETRY 263 



XVII. The solution may be had, however, in the case where R 

 becomes infinite, and we see it is then a straight Hne 



-^ = cot a 

 ax 



or integrating (y — y') — cot a . {x — x') . 



XVIII. The next question we shall consider is to derive a 

 general formula applicable for finding the general equation of the 

 tangent to the curve oîf {6, ^) = o. The tangent at a point P is 

 the limiting great circle of all great circles, which pass through P and 

 Q, as Q moves up to P along the curve. Let {d, (j)) be the co-ordi- 

 nates of the point P. Suppose we choose the running co-ordinates, 

 to avoid confusion, to be (a?, y). The equation of the great circle 



through {6, (})) and (d -\- A ^, </> + A (?S) is tan x | tan («^ + A 0) - 

 tan j 4- tan y I tan (^ + A ^) - tan l9 | -f | tan (^ + A ^) • tan 



4> — tan (<^ -}" A ^) t an ^ > = o. 

 That is 



tan a; I A -^ tan (;6 4- ( ( A 0') ) | - tan y / A ^ — - tan ^ -f- 



( ( A^') ) I + I A ^ tan -^ tan ^ - A <^ tan d -r-r tan <^ > = o. 



Therefore 



(tan X — tan 6) A 4> -j~t ^^'^ 4* ~ (^^^ V ~ tan 0) A <9 -j-^ tan 6 



+ ((A ^'0 ) + ( (A ^ A </.) ) + ( (A c/»^) ) = 0. 

 Therefore 



A sec _ (tan y - tan 0) + ( ( A ^) ) + ( ( A 0) ) 

 A ^ sec» 6 (tan ic - tan (9) + ( (A ^) ) + ( (A 0) ) 



But the limit of ^ as each term of the ratio approaches zero is 



by definition -t-~ The limit of the left side is given by the cir- 

 cumstance that the right side approaches the definite limit, 



(tan y — tan 0) , . , . , , . , , 



T- ^,. which, therefore, gives us the equation 



^tan X — tan U) 



d tan _ (tan y — tan 0) 

 d tan d (tan x — tan 6) 



The value of -, ^ can be found from the curve / (0 6) = o, which, 



d tan 



