330 TIDAL WAVES. [CHAP. VIII 



that the determinantal equation (10) has a pair of zero roots. 

 In other words we have a possible free motion of infinitely 

 long period. The coefficients of Q 2 , Q 3 , ... on the right-hand side 

 of (18) then become indeterminate for er = 0, and the evaluated 

 results do not as a rule coincide with (19). This point is of some 

 importance, because in the hydrodynamical applications, as we 

 shall see, steady circulatory motions of the fluid, with a constant 

 deformation of the free surface, are possible when no extraneous 

 forces act; and as a consequence forced tidal oscillations of long 

 period do not necessarily approximate to the values given by the 

 equilibrium theory of the tides. Of. Arts. 207, 210. 



In order to elucidate the foregoing statements we may consider more in 

 detail the case of two degrees of freedom. The equations of motion are then 

 of the forms 



The equation determining the periods of the free oscillations is 



(a 1 \ 2 + c 1 )(a^ + c 2 ) + ^\^ = ........................ (ii), 



or a 1 a 2 \* + (a 1 c 2 + a 2 c l +p 2 )\ 2 + 0^ = ..................... (iii). 



For ' ordinary ' stability it is sufficient that the roots of this quadratic in X 2 

 should be real and negative. Since a lt a z are essentially positive, it is easily 

 seen that this condition is in any case fulfilled if c lt c 2 are both positive, and 

 that it will also be satisfied even when c 15 c 2 are both negative, provided /3 2 be 

 sufficiently great. It will be shewn later, however, that in the latter case the 

 equilibrium is rendered unstable by the introduction of dissipative forces. 



To find the forced oscillations when Q ly Q 2 vary as e t<r< , we have, omitting 



the time-factor, 



(G! - <rS) ft 



'} 



whence 



'"" ~^+<r~-^~u* (V) 



?2 = 



- o- 2 /3 2 



Let us now suppose that c 2 =0, or, in other words, that the displacement 

 q 2 does not affect the value of V T . We will also suppose that Q 2 = 0, i.e. 

 that the extraneous forces do no work during a displacement of the type q 2 . 

 The above formulae then give 



