Theory of the Fortnightly Tide. 139 



supposing that <? 2 = and also that Q 2 = 0, while Q 1 remain- 

 ing finite is proportional to e i<rt , as usual. We find 



so that in the case of a disturbance of very long period when 

 a approaches zero, 



Since a 2 is positive, q x is less than its equilibrium value ; and 

 it is accompanied by a motion of type q 2 , although there is 

 no extraneous force of the latter type. 



It is clear then that in cases where a steady motion of 

 disturbance is possible the outcome of an extraneous force 

 of long period may differ greatly from what the equilibrium 

 theory would suggest. It may, however, be remarked that 

 the particular problem above investigated is rather special in 

 character. In illustration of this let us suppose that there 

 are three degrees of freedom, and that c 2 , c 3 , Q 2 , Q 3 are 

 evanescent. The equations then become 



(cj — o" 2 ^)^ + ia{3 12 q 2 -j- iaff u q 9 = Q. 1? 

 — aa 2 q 2 +i^ 21 q! +i/3 23 q s = 0, 

 *-aa % q z +ip&qi + i/3 32 q 2 =0; 



whence, regard being paid to (2), 



* 2 (a 3/ V + a 2 /V) 



yi 





"When <7 = 0, the value of q x reduces to Qi/c 1? unless /3 23 = 0, 

 so that in general the equilibrium value applies. But this 

 is only so far as regards q lm The corresponding values of 

 q 2 , q z are 



q 2 =—q 1 /3 31 //3 32 , 7s = - Qi Ai/As ; . . (8) 



and thus the equilibrium solution, considered as a whole, is 

 finitely departed from. And a consideration of the general 

 equations (1) shows that it is only in very special cases that 

 there can be any other outcome when the possibility of steady 

 motion of disturbance is admitted. 



It thus becomes of great importance in tidal theory to 

 ascertain what steady motions are possible, and this question 

 also has been treated by Lamb (§207). It may be con- 

 venient to repeat his statement. In terms of the usual 



