20 U. S. COAST AISTD GEODETIC SURVEY. 



As the moon's parallax varies inversely as its distance d, the terms 

 containing the reciprocal of d^ are said to depend upon the cube of the 

 moon's parallax and those containing the reciprocal of d^ upon the 

 fourth power of the moon's parallax. Assuming the approximate 



numerical value of -? as before, it is evident that the above terms in- 

 volving the fourth power of the parallax will generally be only about 

 2 per cent of the entire tide-producing force and are therefore of little 

 relative importance. They will, however, be given further attention. 



For convenience, the force depending upon the fourth power of the 

 moon's parallax will be treated separately from the principal tide- 

 producing force, which depends upon the cube of the parallax. They 

 may be expressed separately, as follows : 



Tide-producing force depending upon the cube of moon's parallax 



iJiM:r 

 Vertical component = ^3 (3 cos^ d—l) (22) 



Li Mr 

 Horizontal component = 3/2 —^^ sin 2 d (23) 



Tide-producing force depending upon the fourth power of moon's 



parallax 



aMr"^ 

 Vertical component = 3/2 ^^ (5 cos^ — 3 cos d) (24) 



Horizontal component = 3/2 .4 (5 cos^0 — 1) sin0 (25) 



Similar expressions for the tide-producing force of the sun may be 

 obtained by substituting the mass of the sun for M and the distance 

 of the sun for d. Because of the greater distance of the sun the terms 

 depending upon the fourth power of its parallax will be negligible. 



The relation of the tide-producing force of the sun to that of the 

 moon will be approximately 



Mass of sun ( mean distance of moon) ^ _ ^ . ^ (c,n\ 



Mass of moon (mean distance of sun) ^ 



Examining formulas (22) and (23) for the prijicipal tide-producing 

 force it will be noted that the vertical component becomes zero when 

 cos 6= ± Vi, and the horizontal component becomes zero when 6 = Q, 

 90, or 180°. The vertical component has a maximum positive value 

 when = or 180° and a maximmn negative value when = 90°, the 

 latter force being only one-half as great as the maximum positive 

 value. The horizontal component is at a maximum in the positive 

 direction when = 45° and a maximum in the negative direction 

 when = 135°. These forces all become zero at the center of the earth 

 where r is zero. 



To express these forces in terms of gravity, 



let g = mean force of gravity on earth's surface 

 a = mean radius of earth 

 £'=mass of earth 



then g = ^, and ij. = -=^ (27) 



