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From the right spherical triangles PBY and PAX we have the following 



fundamental relations: 



tan £' tan £' tan 77' tan n' 



(1) tan £ = = , tan 77 = = 



sinBY cos 77 sinAX cos £ 



2. Equation of the Spheric Line LM in Terms of its Intercepts. 



The arc of a great circle we will call a spheric straight line. Let the inter- 

 cepts be OL = a, OM = 0, and the angle OLM = <t>. Fig. 3. Then from the 

 right triangles MOL and PAL we have 



tan 



tan ti tan . V 

 , and tan <p = 



tan 77' 



sin a sinAL sin(o: — £) 



Equating these values of tan <p, and substituting the value of tan ??' from (1), 

 tan /3 tan 77 cos £ tan 77 



sma 



smacosl — eosasm£ sma — eosa tan £ 



Expressing each function in terms of tangents and reducing. Ave find the equa- 

 tion of the spheric line in the intercept form: 



(2) 



tan £ tan -q 



tan a 



tan p' 



= 1. 



