HAEMONIC ANALYSIS AND PREDICTIOISr OF TIDES. 



25 



Then 



dr 



Td' 



3 Ma' 



2 Ed' 



sill 29 



(52) 



Substituting (51) and (52) in (50), 



, 3 Ma' sin 26 



tan i/' = 



2Ed' ^^, Ma'^ ,, ^. 

 1 + i YW^ ^^ 



3 Ma' 



sin 20 



[-* 



Ma' 



(3 cos= 0-1)+ etc 



] 



(53) 



2 Ed' -"-"[_- 2 £.^3 



Since jg is very small compared with unity, we may neglect the 

 higher powers in (53) and write 



(54) 



, , 3 Ma' . -_ 

 tan \(/ = ,y T^ 73 sm 26 



as the tangent of the angle between the radius vector and the normal 

 to the surface at the point P. 



T0W/1R05 MooN' 



Fig. 8. 



If we let 1^1 represent the angle between the radius vector and the 

 resultant of the forces under consideration at the same point P, we 

 have from (22), (23), and (30), 



3 fxMr 



tan ^1 



2 d' 



sm 26 



' ^E fiMr , . 



-^ 53- (3 cos 6-1) 



d' 



Ma^ . 

 = 3/2^sm20 



^ 



Mr^ 



1-^(3 008^0-1) 



(55) 



