44 U. S. COAST AND GEODETIC SURVEY. 



Then, for the functions of 1 in the coefficients of (100), we may 

 obtain from (132) to (135) the following: 



[cos* i/]o = i [1+2 cos Z+cos^* 7]o 



= \ [1 + 2 cos CO + cos^ oi + ^i^ (1 — 2 cos co — 3 cos^ (>i)'\ 



= i [(l+cosa))2 + ^^2 (1 + cos co) (1-3 cos co)] 



, 1 ••> ^1 1 — 3 cos 0) 



= cosHco + i^='cosHco y^--^ 



(143) 



cos' 



1 f-i , 1 -2 1 - 3 cos col 

 ^ L 1 + cos CO J 



[sin=^7]o = [l-cos2/]o 



= sin^ CO — ^ i^ (1 — 3 cos^ co) 



= sin^.[l + i.(^-^i^")] (144) 



[sin 1 cos^ hl]o = h [sin / + sin 7 cos 7]o 



, r . . , ., cos^ CO — ^ + i cos CO — 3 cos CO sin^ co~| 



= ■* sm CO + sm CO cos co + i ?' = 



' \_ sm CO J 



, r . . ■ ., 1 + 5 cos CO — 2 cos^ 00 — 6 cos^ co"! 



= i sin CO + sm CO cos co — t ^ ir 



^\_ sm CO J 



,r. . , ., (1 +COSC0) (1 +4COSC0 — 6cos^&)l 

 = i sm CO + sm CO cos co — t ^^ -j- — 



n 1-9 1 + 4 cos CO — 6 cos^ co") (145) 



= sm CO cos^ i CO 1^1 - 1 ^^ ^^ J 



[sin I sin^ i /]o = 2 [sin 7— sin 7 cos 7]o 



,r. . , ., cos^w — i — ^coscti + Scosctisin^oj"] 



= ^ smce> — smct»cosw + ^^^ -r- 



^\_ sm CO J 



,r. . , ., 1 — 5cosw — 2cos^co4-6cos^col 



= -i sm CO — sin CO cos co — t t^ 1- 



■* |_ - sin CO J 



,r, . , .„ (1 — cosco) (1 — 4cosco — 6 cos^co)"! 



= i sm CO — sm CO cos co — i ^'' ^ 



''L sin CO J 



. , , r, 1 .,1—4 cos 00-6 cos^ col (146) 



= smcosm='ico^l-i ^^ -^r^ J 



[sin 2 7]o = 2 [sin 7 cos 7]o = 



= 2 sin CO cos CO + i^ cot co[^ — 3 sin^ co] 



= sm2cori + i ? ^-^f " 1 (147) 



|_ * sin^ CO J 



