IN RELATION TO THE EARTH'S INTERNAL STRUCTURE. 37 



force (the radius of the spheroid being taken at unity, and the variation, assumed 



slight, due to eUipticity, disregarded) n'-i/d^i JvrV l—fi'. The component of this 



tending to tilt the spheroid about the axis in question is u"// (/i.i '/cjV 1 — /i" cos cr, 



and its moment n^y d(i dxs /<y 1 — ,«" ^^* ^• 



Substituting the value of >j (47), the above becomes 



11^ sin cos (If^i dxTi f-C (1 — u~) cos" ct 



2/-V 



Integrating, first with reference to ,(/ from ^:^ — 1 to |(/=:-|-l, then with reference 

 to cT from to 2 7t, we get, as the expression for the couple due to "the centrifugal 

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



(48) 27t""-f sin cos Q 



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

 the expression 



M (C—A) sin cos 



Q 



and (46), (C — J)^ - rre [h being taken at unitv) and e, as already stated, is for 

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



prcssion becomes identical with (48). The precessional 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 Avhich the density of the strata varies. Without extending any 

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

 of the 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 precessional) force. 



The accuracy of the foregoing analysis is complete,Vxcept that the consideration 

 of rehitice motion of the particles is excluded. But Laplace shows (p. 604, Vol. IT, 

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



' Tlicro are slight errors of approximation : 1st, in the tidal expression (47) itself; 2d, in the above 

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

 lize each other iu the final result. 



