164.] FREE OSCILLATIONS IN AN OCEAN OF UNIFORM DEPTH. 199 



We have still to express the condition that (47) should be the 

 equation of a bounding surface. This condition [(10) Art. 10,] 

 becomes in the present case 



* dt ~ dr 

 whence 



2-*^ {&quot;-&amp;gt; ............... &amp;lt;&amp;gt; 



Eliminating 8 between (48) and (49) we find 



provided 



g 



............ (50). 



The motion consists therefore in general of a series of superposed 

 oscillations, the periods r of which are obtained by putting n = l, 

 2, 3, &c., in (50). 



The longest period is that for which n = 1, in which case 



2 _27r 2 62 3 + a 3 . f ?V \ 



r - ~ ~Y ~V^ \ * + P( b * - a * -&quot; 

 If (7 = p, we have T X = oo , as we should expect. In fact 



(Tj infinitely small) is the equation of a sphere of radius b whose 

 centre is near the origin. The fluid and solid are then equivalent 

 to a single spherical mass of uniform density, so that there is 

 always equilibrium when the surface is of the form (52). 



If p &amp;gt; a, TI is imaginary, the value of r, 2 given by (51) being- 

 then negative. This indicates that the equilibrium of the ocean, 

 when in the form of a sphere concentric with the earth, is un 

 stable. The ocean would in fact, if disturbed, tend to heap itself 

 up on one side*. 



- * See Thomson and Tait, Nat. Phil. Art. 816. 



