HAEMONIC ANALYSIS AND PREDICTION OF TIDES. 



19 



relationship. The denominators of (15) and (16) are to be con- 

 sidered as positive. 



Substituting (14), (15), and (16) in (12) and (13), and designating 

 the forces in the direction OP and perpendicular to the same as the 

 vertical and horizontal components, respectively, we have 



Vertical component of tide-producing force at P 





cos —-J 





cos 6 



(17) 



Horizontal component of tide-producing force at P 



sin 6 



,xM 



T j.»2\ 3/2 



2 -n COS ^ + 3^ ) 



— sm 



d^^^'' ' d 

 By Maclaurin's theorem the fraction 



1 



(18) 



l-2^cos^ + - 



^V 



may be developed into a series arranged according to the ascending 



powers of -n, which has a value of approximately 0.017 when r is 



taken as the mean radius of the earth and d as the mean distance of 

 the moon from the earth. When d is taken as the mean distance 



of the sun from the earth, the value of -^ is considerably smaller. In 



equation (19), which follows, the terms involving the higher powers of 



-n are relatively unimportant and may be neglected. 



1 



1-2 ^cos0 + ^ 



^^2 = 1+3 cos ^^ + 3/2 (5 cos^ ^-1) ^ 



+ 1 (7 cos^ 0-3 cos e) ^ + etc. 



(19) 



Substituting (19) in (17) and (18) and neglecting all terms contain- 

 ing powers of -^ above the fourth we obtain 



Vertical component 



I Mr 



Mr'' 



^ (3 cos^ 0- 1) +3/2 ^^ (5 cos^ 0-3 cos d) 



Horizontal component 



}xMr 



Mr-" 



3/2 ^ sin 2 + 3/2 '^^ (5 cos^ 0-1) sin 



(20) 



(21) 



