HAKMO]!s^IC ANALYSIS AND PREDICTION OF TIDES. 45 



[1-3/2 sin=^ 7]o= 1-3/2 (144) 



= 1-3/2 sin^ oj - 3/4 ? (2 - 3 sin^ w) 



= (1 - 3/2 sin^ co) (1 - 3/2 i') ' (148) 



From (136) to (142) we may obtain the follovang: 



[cos(2^ — 2j^)]o = [cos 2^ cos 2j' + sin 2^ sin 2v]o 



= 1—1^ [cosec^ CO + cot^ CO — 2 cosec co cot ca] 

 cosec CO — cot co]^ 



= l-^2 



[cos 2v]o =\—i^ cosec^ oj = l—i 



-1-cosjoT ^ ^ _ ., rijZ^^^'l (149) 



sm CO J [_1 + cos coj 



(150) 



sm^ CO 



[cos (2^ — i')]o = [cos 2^ cos v + sin 2^ sin v]o 



= l — \i'^ [cosec^ CO + 4 cot^ co — 4 cosec co cot uj] 



= 1 — i i^ [cosec CO — 2 cot cop (151) 



1 



1 •f l-2cosco T 

 * |_ sin CO J 



[cos (2| + j')]o = [cos 2^ cos J/- sin 2^ sin i^jo 



= l — \ i^ [cosec^ CO + 4 cot^ co + 4 cosec co cot] 



* |_ Sin CO J 



(152) 



[cos i/Jo = 1 — i i^ cosec CO = 1 — i t^ rj-2~ (153) 



[cos 2^ = 1 -i^ cot^ co = 1 -^2 ~^ (154) 



By taking the products indicated on page 41 the mean values for 

 the variable factors of the coefficients of (100) may be obtained as 

 indicated below. Byj substituting cos* ^ -i as the equivalent of 

 1 — i i^, and 1 — 3/2 sin^ i as the equivalent of 1—3/2 i^, the results 

 are obtained in the forms adopted by Professor Darwin. The 

 numerical equivalents of the formulas are obtained by substituting 

 the values of co and i from Table 2. 



For terms {A) ^ to {A) ^ 



[cos* i /]o [cos (2^-2 v)]o = (143) X (149) = cos* i co [1 - ^ i^] ., „s 

 = cos" i CO cos* i i = 0.9154 ^^^^^ 



For terms {A)^. to {A)^^ 



[sin^ /]o [cos 2 j']o= (144) X (150) =sin2 co [1-3/2 i^] 

 = svo? CO [1-3/2 sin^ ^] = 0.1565 



(156) 



For terms (-4)26 to {A)^^ 



[sin /cos^ i /]o [cos (2 ^-i')]o= (145) X (151) .. „x 



= sin oi cos2 i CO [1 - 1 %"] = sin co cos^ ^ co cos* i i = 0.3800 ^^^ 



For terms {A)^^ 



[sin / sin2 ^ /j^ j-^qs (2 ^ + »')]o = (146) X (152) .. ^^. 



= sin CO sin^ i co [1 - ^ ^2] = sin co sin^ | co cos* i i = 0.0164 ^^^^^ 



