22 U. S. COAST AND GEODETIC SURVEY. 



Then, designating the tide-producing potential due to this force by 

 Vt, we have as the total amount of work necessary to move the particle 

 to the center of the earth — 







Ft=- f^ (3 cos 2^-1) rdr = i-^'^ (3 cos ^ 0-1) (34) 



The same result will be obtained by assuming the particle to be 

 moved again against the horizontal component of the tide-producing 



force until it reaches a position where cos 9 = o ^^'^ the vertical com- 

 ponent becomes zero. From this point it can be moved directly to 

 the center of the earth without additional work. 



For that part of the tide-producing force depending upon the fourth 

 power of the moon's parallax let the potential be designated by 

 Vf. Then, assuming a unit mass to be moved against the vertical 

 component, formula (24) , directly to the center of the earth, we have 



Vt=- 3/2^^^(ocos3 0-3cos0)(Zr = i^^~^(5cos3 0-3cos0) (35) 



This potential may also be obtained by assuming that the particle 

 is first moved against the horizontal component, formula (25), to a 

 position where = 7r/2 and the vertical component becomes zero. 



Similar expressions for the tide-producing potential of the sun 

 may be obtained by substituting in the above formulae the mass and 

 distance of the sun for M and d, respectively. The tide-producing 

 potential of the sun which involves the fourth power of its parallax 

 is negligible. 



8. SURFACE OF EQUILIBRIUM. 



A surface of equilibrium is a surface at every point of which the 

 sum of the potentials of all the forces is a constant. On such a 

 surface the resultant of all the forces at each point must be in the 

 direction of the normal to the surface at that point. If the earth 

 were a homogeneous mass with gravity as the only force acting, the 

 surface of equilibrium would be that of a sphere. Each additional 

 force will tend to disturb this spherical surface, and the total deforma- 

 tion will be represented by the sum of the disturbances of each of the 

 forces acting separately. In the following investigation we need not 

 be especially concerned with the more or less permanent deformation 

 due to the centrifugal force of the earth's rotation, since we may 

 assume that the disturbances of this spheroidal surface due to the 

 tidal forces will not differ materially from the disturbances in a true 

 spherical surface due to the same cause. 



Let us first consider the surface of equilibrium due to gravity and 

 the principal tide-producing force of the moon. Designating the 

 potential due to these two forces by V, we have as the condition of a 

 surface of equilibrium, 



F= Vg + Ft = a constant (36) 



Substituting the values of Vg and Ft from (32) and (34), 



F= ^ -h i ^^^ (3 cos^ 6-1) =& constant (37) 



