50 U. S, COAST AND GEODETIC SURVEY. 



The smii of the terms (187) and (188) may then be written 

 1/4 e sin / cos^ \I fcos (0-2P) +3 ^^\ ^j cos ^l 



I COS 1 I 



= 1/4 e sin / cos^ ^I\ cos 6 cos 2P + sin d sin 2P + 3 — ^-^r cos 6 

 L cos^ fi J 



; .^1/4 . sin / cosH/[9 ^ +6 ^cos 2P + l]i 



(190) 



sin 2P 

 -tan-'- 



X cos I _ cos / , ^ „ 



i e sin / cos^ ^7 



where 



cos {T+h-s + p-7rl2-u-QJ 



(191) 



7^ = ^[9 ^^.+ 6 ^^.cos 2P + 1> 



Qa, \ cos'' ii cos' il J 



= ["5/2-! tan' ^7+1 tan* ^7+3/2 (l-tan' ^7) cos 2p1* 



and 



/^ ^ 1 sin 2P . . sin 2P 



^" ^ o COS 7 ^ :;^^ ^''' 3(l-tan'i7)+cos2P ^l^^) 



3 ^^r-y + COS ii'^ 



cos' J 7 



Formula (190) represents the composite Mj tide, the mean speed and 

 period of which are determined by the V of the argument, which is 

 T+ h — s -hp — '^U- The u, which equals — v — Qu, may be shown to 

 vary between the limits of approximately ±70°, and will therefore 

 not affect the mean period as determined by the V of the argument. 



The period of this component is very nearly an exact multiple of the 

 period of the principal lunar component M2, and for this reason the 

 summations which are necessary in the analysis for the latter may be 

 conveniently adapted to the analysis for component Mj. 



Let 



e=T+h-s--l^-\-k-v (193) 



and P as before. 



The sum of terms (187) and (188) may then be written 



1/4 e sin 7 cos' i7 fcos (0 - P) + 3 ^4/ cos (^ + P)1 

 = 1/4 e sin 7 cos' ^7 Ul + 3 ^rrj) ^os cos P 



