280 Sir William Thomson on the " Oscillations of the 



lunar fortnightly (declinational) ; and for it a is about ^4 of #>. 

 which rnakes/=- 5 -V- Even for this, and more decidedly for the lunar 

 monthly (elliptic) and solar semiannual (declinational) and annual 

 (elliptic), a good approximation to the result might be obtained 

 by taking a = 0. Laplace does not enter on the integration of 

 the equation, but contents himself by pointing out that an 

 infinitesimal degree of friction will, when c- = 0, cause the actual 

 tide-height to be the same as the equilibrium tide-height, and 

 that even for the lunar fortnightly the actual height must be 

 sensibly the same as the equilibrium height if there is enough 

 of friction to reduce in a fortnight a free oscillation to a small 

 fraction of its original amount. The result of any tide-genera- 

 ting influence of sufficiently long period would obviously be 

 more and more nearly in exact agreement with the equilibrium 

 theory the longer the period, were it not for the earth's rotation. 

 But, because of the earth's rotation, a long-period tide does not 

 approximate to agreement with the equilibrium tide if the water 

 be perfectly frictionless ; and the solution of the beautiful "vor- 

 tex problem " thus presented is what is aimed at by Airy* and 

 Ferrelt in their integration of the preceding equation for the case 

 cr = 0, in which it is reduced to the comparatively simple form J 



d /1 — LL 2 da'\ . . _ 



^(-/-^)- 4eM=4e0 • • • • (2) 



* (e 



Tides and Waves " (Encyclopedia Metropolitana), art. (97). 

 t "Tidal Researches" (Appendix to United-States Coast-Survey Re- 

 port, 1874), § 151. 



X [Note added, Bristol, September 2, 1875.]— Without this simplifica- 

 tion, the equation (1) is susceptible of nearly as simple a solution as with 

 it. Assume 



1 da' vir . 



This gives 

 and 



/* 2 -/ 2 dfx 



« = 2^(K l -_3-/ 2 K i -i) 



so that, to determine the coefficients Kj, we have the equation of condi- 

 tion 



if 



9=20,7**. 



This is a particular case of an almost equally simple solution of Laplace's 

 general equation of Tides, which has been communicated to the British 

 Association at its meeting now concluded, and will be published also in the 

 November Number of the Philosophical Magazine.] 





