198-199] TWO DEGREES OF FREEDOM. 331 



In the case of a disturbance of long period we have o- = 0, approximately, and 

 therefore 



1 



The displacement ^ t is therefore Zm than its equilibrium-value, in the 

 ratio 1 : l+^ 2 /a. 2 c 1 ; and it is accompanied by a motion of the type q 2 

 although there is no extraneous force of the latter type (cf. Art. 210). 

 We pass, of course, to the case of absolute equilibrium, considered in 

 Art. 165, by putting /3 = 0. 



199. Proceeding to the hydrodynamical examples, we begin 

 with the case of a plane horizontal sheet of water having in the 

 undisturbed state a motion of uniform rotation about a vertical 

 axis*. The results will apply without serious qualification to 

 the case of a polar or other basin, of not too great dimensions, on 

 a rotating globe. 



Let the axis of rotation be taken as axis of z. The axes of x 

 and y being now supposed to rotate in their own plane with the 

 prescribed angular velocity n, let us denote by u, v, w the 

 velocities at time t, relative to these axes, of the particle which 

 then occupies the position (x, y, z). 



The actual velocities of the same particle, parallel to the in- 

 stantaneous positions of the axes, will be u ny, v + nx, w. 



After a time Bt, the particle in question will occupy, relatively 

 to the axes, the position (x + uSt, y 4- v8t, z + w$t), and therefore 

 the values of its actual component velocities parallel to the new 

 positions of the axes will be 



Du 



u+j-t-n(y + v&t), 

 JJt 



v +- ~ St + n (x + uSt), 



* Sir W. Thomson, "On Gravitational Oscillations of Rotating Water," Phil* 

 Mag., Aug. 1880. 



