HARMONIC ANALYSIS AND PREDICTION OF TIDES. 43 



From (114) 



cos l= (i + tan^^) ^^^~^'' ''''^' '' ^^' ^ ^^^^^ 

 sin ^ = tan ^ cos ^ = i cot co sin iV— J-i^ [cot^ ^ + h^ sin 2A^ (123) 



cos 2^ = 2 cos2 ^ - 1 = 1 - 2^2 cot^ co sin^ iV (124) 



sin 2^ = 2 sin | cos ^ = 2i cot co sin i\^- -i^ [cot^ w + ^] sin 2iV (125) 



From (118) to (125) 



cos 2^ cos 2v = \- 2^2 [cosec^ co + cot^ oj] sin^ N (126) 



sin 2^ sin 21/ = 41^ cosec co cot co sin^ N (127) 



cos 2^ cos v = \- ? [i cosec^ co + 2 cot^ co] sin^ N (128) 



sin 2f sin v = 2i^ cosec co cot co sin^ N (129) 



Since iV is an angle that changes uniformly throughout the entire 

 circumference, it may readily be shown that the mean value of sin N, 

 cos N, sin 2 N, and cos 2 iV is zero for each, since for each positive value 

 there will be a corresponding negative value in the same period. 

 Indicating the mean value of a variable by the subscript o, we may 

 now write 



[sin iV]o = [cos iV]o = [sin 2iV]o = [cos 2N]^ = {) (130) 



and 



[sin^ N]o = [h-h cos 2iV]o = i (131) 



since the mean value of the sum of several terms equals the sum of the 

 mean value of each term. 



Substituting (130) and (131) in formulas (112), (115) to (118), 

 (120), (124), and (126) to (129), and indicating the resulting mean 

 values by subscript o, the following may be obtained: 



' [cos /]o=(l-ii') cos CO (132) 



[cos^ /]o = cos^ CO + ?^2 (^ sin^ CO — cos^ co) 



== cos^ CO + i^ (^ — f cos^ co) (133) 



[sin 7]o = sin co + ^i^^^i^"^ (134) 



"• -'° ^ sm CO 



[sin I cos rio = sin co cos co + ^i"^ cot co [cos^ co — sin^ co — |^ — sin^ o}\ 



= sin CO cos CO + ^i^ cot co [^ — 3 sin^ co] (135 ) 



[cos v]o = 1 — i i^ cosec^ co (136) 



[cos 2z/]o = l — i^ cosec^ CO (137) 



[cos 2^]o= 1 -v" cot^ CO (138) 



[cos2^cos2j']o=1— "i^ [cosec^ co + cot^ co] (139) 



[sin 2^ sin 2j']o = 2i^ cosec co cot co (140) 



[cos 2^ cos i']o = 1 — i i^ [cosec^ a.' + 4 cot^ co] (141) 



[sin 2^ sin v\Q = i'^ cosec co cot co (142) 



72934— 24t 4 



