HARMOE-IC ANALYSIS AND PREDICTION" OF TIDES. 21 



The substitution of this value of n in equations (22) to (25) will give 

 the forces in terms of gravity. If we assume r to be equal to the mean 

 radius of the earth and d to be the mean distance of the moon, we may 

 obtain numerical values from Table 2, Vv-^hich, when substituted in (22) 

 and (23), will give the following expressions for the approximate tide- 

 producing force of the moon : 



Vertical component = 0.000,000,056 (3 cos^ 6-1) g (28) 



Horizontal component = 0.000,000,084 sin 2 ^ (29) 



The tide-producing force of the sun will be 0.46 times as large. 



7. TIDE-PRODUCING POTENTIAL. 



The potential at any point due to a force is the amount of work that 

 would be required to move a unit of matter from that point, against 

 the action of the force, to a position where the force is zero. This 

 amount of work will be independent of the path along which the unit 

 of matter is moved. If the force being considered is the gravity of 

 the earth, the potential at any point will be the amount of work 

 required to move a unit mass, against the force of gravity, from that 

 point to an infinite distance from the earth's center where the force 

 of gravity becomes zero. With the symbols as in the preceding 

 section, we have according to the law of attraction 



Force of gravity on or above earth's surface = -^ (30) 



The amount of work required to move a unit mass against this force 

 through an infinitesmial distance dr 



= ^^dr (31) 



The total amount of work necessary to move this particle from a 

 point r distance from the earth's center to infinity is the gravitational 

 potential at that point, and will be here designated by Vg. Then, 



n 



00 



^Cj^dr^^-^ (32) 



JrT-r 



The tide-producing potential at any point in the earth is the amount 

 of work required to move a unit mass, against the tide-producing 

 force, from that point to the center of the earth where the tide-produc- 

 ing force becomes zero. If we assume the particle to be moved along 

 the radius of the earth directly to the center, we will be concerned with 

 only the vertical component of the tide-producing force, since the 

 horizontal component would not affect the amount of work required 

 along this path. 



Considering, first, the force dependmg upon the cube of the moon's 

 parallax, the amount of work necessary to move a unit mass against 

 the vertical component, formula (22), through an infinitesimal dis- 

 tance — dr toward the center of the earth equals 



^ (3 cos' 6-1) rdr (33) 



