[beatty] an outline of analytical spherical geometry 2 71 



The limit of the right side being zero indicates that ultimately the 



angle at R is 90°. Therefore the limit of !E_^ = the limit of sin 



sin /\d 



(r + A »-) = sin r = the limit of 

 tânQB 



QR 



Ad- 



Besides tan tp = limit 



sin PR 

 Therefore 



since R has 90° for its limit. 



tan (p = limit 

 = limit 



= s m r 



QR 

 PR 



QR 



Ad ■ 



dd 

 d r 



Ad 

 PR 



XXXV. To apply this to any cm've, / (r, d) = o we need only 



equate tan (p cosec r to the value of 



dd 



dd 

 dr 



dj_ 

 dr 



dd 

 dr 



found from the relation 



This serves to give tan </-. The angle ^ can 



be obtained from the relation cos <p> = 

 cos d cos ip — sin d sin ^ cos r. The 

 angular length a can be found from the 

 relation sin a sin = sin ^ sin r. That 

 is, the equation of the tangent PT can 

 now be found since the two necessary 

 elements to fix the great circle, which is 

 the tangent, have been determined. 

 The inverse problem in which we are 

 asked to construct a curve where tan (p 

 = / (»", d) can also bë solved in all cases 



where the differential equation 



s m r 



dd 

 ~d7 



f{r,d) 



is solveable. 



XXXVI. A case discussed before in connection with the Car- 

 tesian system is better handled here. In the loxodrome where the 

 North Pole is taken at we have the simple relation 



dd 



tan (p = c = sin r 



dr 



/d r f d 



-. — = / — where id' r') are the co-ordinates of the 

 sin r •/ c 



