HARMONIC ANALYSIS AND PEEDICTION OF TIDES. Ill 



and 



-, — ^ . — 7-7 ■ — ^ sm {12x364(as + a) + ll.oas + o;s} 



has [_ sm 12(as + a) 



n=365 



S [ys]n sin 24(rt-l)a = 



J_ - sinj^asfsin 12x365(as + ft 

 48 " sin ^< 



sin 12x365(as — a) . ,-,r,..na4r ^ , n c , i~l /'.i/)i\ 



^ — ^TTT — --^T — ^ sm {12x364(as-a) + 11.5as + o;s} (441) 



sm 12(0-8 — a) ' ^ 'J 



Now let 



A's =^s COS as 

 and (442) 



A"s= —A^ sin as 



then (440) and (441) may be reduced as follows: 



n=365 

 S [Vsln COS 24(72-1) a 

 n=l 



1 sin 12as fsin 12x365(a« + a) riov.oc^/ , ^ , n - i 



= 7^ -^ — I — - '• — T157 , — X — ^ COS {12x364(as + a) + ll.oas} 



48 sm -las |_ sm 12 (0.5 + a) ' 



i 12x365(as- 

 sin 12(a.s — «) 



1 sin 12as fsin 12x365(0-3 + a) . movxo^^^ , ^ , n r i 



• 7^ —■ — -. — -■ — TTT7 i — \ — sm {12x364(as + a) +11.5as} 



48 sm ^as L sm l2(as + a) ^ ' 



sir^22g6Mc^:a)^.^^^2x364(as-a) + 11.5as}1^''s (443) 

 sm 12(as — a) J 



and 



n=365 



S iVsh sin 24(71-1) a 



1 sin 12as Fsin 12 x36o(as + a) . i-,r,..oaAr , \ , n c 1 



= — -^ — J — ^ -. — -TTT , — ^ sm {12x364(as-ra) + 11.5as} 



48 sm -^-o-s [_ sm 12(as + a) ^ "t 



, sin 12x365(as-a) iio^.oaAr \ < -n n \~\ a' 



-\ ^^^ .^^^ ::^; — cos {12 x364(as — a) +11.5as} M- s 



1 sin 12rA„ Fsin 12y 3P.5('o,„-l-o,) . . . 



sin 12x36o(as — «) • nowoc-i/- \ , n k 



sm {12x364(as — «) + ll-5a 



Sn2(as-a) ""' U^AOU.VC.s-«.-rxx..UsiJ^'s 



1 sin 12as fsin 12 x 365 (as + a) /ion/q«-i/ , ^ , ^-, k \ 



T^ —■ — i -■ — To? — ; — \ cos {12x364(as + a) + 11.5as} 



48 sm ^o-s L sml2(as + a) ^ 



_sinl^x365(a^ cos {12 x364(a3-a) + 11.5as} 1 ^"s (444) 



sm 12(as — aj J 



Formulas (443) and (444) represent the clearances for any 

 long-period component A due to any short-period component ^s- 

 The first must be subtracted from terms corresponding to 

 S^/n cos 24 (n — l)a and the latter from terms corresponding: to 

 S?/n sin 24 {n — l)a of formula (436) before solving the latter. 



In (443) and (444) the coefficients of A'^ and A"s, which 

 for brevity we may designate as C, C" , S', and S", respectively, 

 contain only values that are constant for all series and may therefore 



