42 U. S. COAST AND GEODETIC SURVEY. 



Since 



tan (^^-0=i + taniVtan^ ^^^^^ 



we have from (107) and (108) 



^ sin •^ cot CO tan JV+ (cos ^ — 1) sin iV" 

 ~" sin i cot w + cos i cos iV+ sin iV tan N 



_ sin i cot (o sin JV— sin^ ^-i sin 2 iV 

 "sin i cot io cos iV— 2 sin^ ^i cos^ iV+1 



(109) 



tan V 



For the computations of tables of the values of /, p, and $, with N 

 as the argument, formulas (103), (104), and (105) will be found 

 especially convenient. Formulas (106) and (109) provide for the 

 direct computations of v and ^ independent of each other. For the 

 computations of the mean values now sought it will be found desirable 

 to modify formulas (103), (106), and (109) and represent the values of 

 /, V, and $ in approximate forms that are more easily developed. 

 By Table 2 it will be noted that i is small, being equal to 5.145°, or 

 0.090 of a radian; therefore, using the radian as the unit angle, the 

 sine of i, or of any fraction thereof, may be taken as approximately 

 equal to the angle itself. 



Then 



sin i = i (110) 



cosi=l-2 sinH'i = l-i? (Ill) 



Substituting (110) and (111) in (103), (106), and (109), and develop- 

 ing to the second power of i, we may obtain the following: 



Cos /=cos a> — -isin a> cos N—^i^ cos oj (112) 



i sin N 



(1 — ^i^} siaco + i cos co cos N 



= i cosec CO sin N—^i"^ cos w cosec^ co sin 2 N (113) 



i cot 0) sin iV— j? sin 2 N 

 ^^"^^-i cot CO cos iV-i? cos^ N-\- 1 



= i^cota;siniV-i? [cot^ oj + ij sin 2iV (114) 



From (112) 



cos^ 7=cos^ CO — i sin 2co cos N+i^ (sin^ ca cos^ iV— cos^ co) (115) 



COS^ Oi COS^ N 



sin 7=(1 — cos^ 7)*= sin co + t cos oo cos N+^i"^ ^r- (116) 



sin I cos I 



=i sin 2co + ^ cos 2co cos N+ ^ i^ cot co [cos 2co - cos^iV (1+2 sinM] (117) 



From (113) 



cos J' = rrr^ — rri = 1 - i^^ cosec^ co sin^ 'N (118) 



(1+tan^v)'' 



sin i' = tan j'cosi' = icoseccosin iV— ^i^coscocosec^cosin2iV (119) 



cos 2v = 2 cos^ V - 1 = 1 - 2^2 cosec^ co sin^ N (120) 



sin 2^ = 2 sin v cos v = 2i cosec co sin N— i^ cos co cosec^ co sin 2N (121) 



