158 GEODESIC INVESTiaATIOIfS. 



It is evident the first side is a little greater than-^^; and we liave,witli rough 



approximate accuracy, for arcs of not more than 1°, the relation — 



2 



(34) 



' {(f)"--(^0} 



And .". also the rude unreliable approximates — ■ 

 ., = A . 2 = -1 ^, 



h (3o) 



= -P- 2 = — 



(See foot-note, Problem 5). 



We can express the arcs z^, z^.^, in the foUo'^'ing manner : — 



From the plane triangle S' S"Z^, we have — 



k . sin (arc ID^^) :=z R^ . sin z, 



■ / TT\ \ -K, . sin s, 



sin (arc Ix).,) =■ —^ '- 



7c 



B, cos ^2' ^ 



~ (F + R^^ — ^.k.E.sixv i 2') 



cos ^ 2^ ^ 



^ cos^ i 2^ . i 



r, = sin -M Z;2 ^.k . , -,) — I (^r —2') ; "^ 



VI 4--^- — . sin i 2/ I 



Similarly, ' ' \ (36) 



^ cos^ i 2" ^ i I 



«. = i (tt + 2") - sin - 1 (—-F 2.k . ^-J 



M + — ^ — _—— . sm ^ 2"/ J 



These expressions for z^ and z^, are rigorously accurate and Tcrj useful. 



/0\ From the triangle Sj,P D, we have 



\s_>/ cos Z" sin A!' = sin i, sin. D, 



cos I" sin A" 



sin L, =::- 



sin J), 



cos Z" sin ^" 



smi, = r~ ^ '. -)*••■ (^') 



< (sin A — tan \ 2" cos I" sin co)'^ + (cos I" sin w)^ > 



Similarly, 



. -.- ■ cos V sin ^' . „^ 



smL„= . ^ ; Ti ... (38) 



] (sin A" — tan i 2' cos Z' sin ct>)* + cos V sin co)^ > * 



Again, it is evident L, is the circular measure of the angle S'Z„C between 

 the fine S^^^ and the polar axis, and that 



