IN RELATION TO THE EARTH'S I X T K U N A L STRUCTUUK. 37 



force (the radius of the spin-mid being taken at unity, and tin- variation, assumed 

 slight, due to ellipticity, disregarded) icylii '/a\/l ic. The component of this 

 tending to tilt tin- spheroid about the axis in question is try dp (/nl/1 (U 2 cossr, 

 and its moment icy </u Ja nv \ ^u* cos or. 



Substituting the value of y (47), the above becomes 



n - ) ''' sin cos </ do fj*(l n*) cos* a 



-' J 



Integrating, first with reference to^from/.= 1 to ^=-(-1, then with reference 

 to a from (I to V? rr, we get, as the expression for the couple due to "the centrifugal 

 turee of the crowns of the tidal elongation," resisting the sun's action, 



fci M *f sin cos 



We have found (19) for the moment of the sun's force, producing precession, 



the expression 



-M(C A) sin cos 



/ 



a 

 and (46), (CA)=7te (b being taken at unity) and e, as already stated, is for 



1 > f- 2 



a homogeneous fluid spheroid = Making these substitutions, the above ex- 



*/ 



IIK -sjon becomes identical with (48). The processional force of the sun is, there- 

 fore, exactly neutralized by the centrifugal force of the tidal swelling. 



The theorem could, doubtless, be demonstrated for a revolving fluid spheroid 

 in equilibrium, of which the density of the strata varies. Without extending any 

 further the mathematical analysis, it will be sufficient to remark that the calculation 

 of tin- tidal elevations is, identically, that of equilibrium of form of the revolving 

 body subjected to a foreign attraction, and in the calculation the motion of rotation 

 is disregarded, and the centrifugal force, which expresses its entire effect upon the 

 form, alone considered. Under this point of view, equilibrium of form is, necessarily, 

 equilibrium (or stability) of position. For if any effective turning force exists, it 

 must, in order not to interfere with equilibrium of form, either be so distributed as 

 to give each individual particle of the spheroid its proper relative quantity of turn- 

 ing motion, or it must be a distorting force. The first alternative cannot be admit- 

 ted; the second is excluded by the hypothesis of equilibrium. Hence, there can 

 be no turning (or processional) force. 



The accuracy of the foregoing analysis is complete, 1 except that the consideration 

 of rdatirr. motion of the particles is excluded. But Laplace shows (p. 604, Vol. II, 

 Bowditch) that as the depth of the ocean increases, the expressions for the tidal 



1 There arc slight errors cf approximation : 1st, in the tidal expression (47) itself; 2d, in the above 

 integration which disregards the variation of the radius; and, 3d, ia the value of C A. They neutra- 

 lize each other in the final result. 



