204-205] FORCED OSCILLATIONS. 341 



(1), we notice that when o- = 2^ they are satisfied by = f, Q=iE. To deter- 

 mine R as a function of r, we substitute in the equation of continuity (2), 

 which gives 



d(hR) 8-1,- , , ,. , 



~^r -~ hR =-x(r) ........................... (iv). 



The arbitrary constant which appears on integration of this equation is to be 

 determined by the boundary- condition. 



In the present case we have x(r) = O 8 /a*. Integrating, and making R=0 

 for r = a, we find, 



^ = ^V-rV (2 ^ +s ' +e) ........................ (v). 



The relation = iR shews that the amplitudes of R and are equal, while 

 their phases differ always by 90 ; the relative orbits of the fluid particles are 

 in fact circles of radii 



described each about its centre with angular velocity 2n in the negative 

 direction. We may easily deduce that the path of any particle in space is an 

 ellipse of semi-axes rT described about the origin with harmonic motion in 

 the positive direction, the period being %7r/n. This accounts for the peculiar 

 features of the case. For if have always the equilibrium-value, the hori- 

 zontal forces due to the elevation exactly balance the disturbing force, and 

 there remain only the forces due to the undisturbed form of the free surface 

 (Art. 200 (3)). These give an acceleration gdz^dr, or n?r, to the centre, 

 where r is the radius-vector of the particle in its actual position. Hence all 

 the conditions of the problem are satisfied by elliptic-harmonic motion of 

 the individual particles, provided the positions, the dimensions, and the 

 ' epochs ' of the orbits can be adjusted so as to satisfy the condition of con- 

 tinuity, with the assumed value of The investigation just given resolves 

 this point. 



205. We may also briefly notice the case of a circular basin 

 of variable depth, the law of depth being the same as in Art. 189, 

 viz. 



Assuming that R, 0, all vary as e ^ +s + , and that h is a function of 

 r only, we find, from Art. 202 (2), (3), 



Introducing the value of h from(l), we have, for the free oscillations 



