=1/3 r7(^ 



Jo Jo 



24 U, S. COAST AXD GEODETIC SURVEY. 



common center at and the radius designated by r; then the vohime 

 of one of these infinitesimal solids will be 



dr . r d 6 . r sin 6 d (l) = r^ sin 6 d(j) dd dr (45) 



and the entire volume of the earth as included in the surface repre- 

 sented by equation (44) will be 



Volume = i \ r^ sin e d<f) dd dr (46) 



Jo Jo Jo 



la -'r v)^ sin 6 d4 dd (47) 



From (44) we may obtain 



r M a^ 1 ^ 



{a + u)' = a'\ l+i^|(3 cos^ ^-1) 



= a'[l+3/2^f, (Scos^ ^-l)+etc.1 ^^^^ 



the terms containing the powers of t above the third being neglected. 

 Substituting (48) in (47), 



Volume = 1/3 aM 1 +3/2 -^ ^(3 cos- 6-1) sin d4> dd 



= 2/3 a^ {d<t> = 4/3;r(j3 (49) 



As equation (49) represents the voliune of a sphere with radius a, 

 it is evident that a is the mean radius of the surface represented 

 by (44), and that u is the amount of the disturbance in the mean 

 surface due to the force under consideration. In other words, u is 

 the equilibrium height of the tide as referred to mean sea level. 



One of the conditions of an equilibrium surface is that the resultant 

 of all the forces at each point must be in the direction of the normal 

 to the surface at that point. Let us see if this condition is fulfilled 

 as to equation (44). In Figure 8 let P represent any point on the 

 surface defined by equation (44), and let i/' be the angle between 

 the radius vector and the normal at this point. If we imagine the 

 surface to be cut by a plane passing through the point P and the 

 centers of the earth and moon, it is evident that the trace of the sur- 

 face will intersect the arc of a concentric circle drawn through the 

 point P with radius r at an angle equal to the angle r/', and that 



uT 



tan^=-^ (50) 



From (41) and (44), 



^Ed 



a + yi^vi'^ cos-'d-Da (51) 



