360 TIDAL WAVES. [CHAP. VIII 



The above investigation is taken substantially from the very remarkable 

 paper written by Lord Kelvin * in vindication of Laplace's treatment of the 

 problem, as given in the Mecanique Celeste. In the passage more especially 

 in question, Laplace determines the constant B by means of the continued 

 fraction for N^ without, it must be allowed, giving any adequate justifica- 

 tion of the step ; and the soundness of this procedure had been disputed by 

 Airyf, and after him by FerrelJ. 



Laplace, unfortunately, was not in the habit of giving specific references, 

 so that few of his readers appear to have become acquainted with the original 

 presentment of the kinetic theory, where the solution for the case in question 

 is put in a very convincing, though somewhat different, form. Aiming in the 

 first instance at an approximate solution by means of a, finite series, thus : 



Laplace remarks || that in order to satisfy the differential equation, the 

 coefficients would have to fulfil the conditions 



as is seen at onceby putting j& 2fc + 4 = 0, J 6 2fc + 6 =0,... in the general relation (13). 



We have here k+l equations between k constants. The method followed 

 is to determine the constants by means of the first k relations ; we thus 

 obtain an exact solution, not of the proposed differential equation (9), but of the 

 equation as modified by the addition of a term /3^? 2 j, + 2 i' 2A; + 6 to the right-hand 

 side. This is equivalent to an alteration of the disturbing force, and if we can 

 obtain a solution such that the required alteration is very small, we may 

 accept it as an approximate solution of the problem in its original form IF. 



Now, taking the first k relations of the system (ii) in reverse order, 

 we obtain B^ + z in terms of B&, thence B& in terms of 2 fr~i> and so on, until, 

 finally, B is expressed in terms of H'" ; and it is obvious that if k be large 

 enough the value of B. 2k + 2 , and the consequent adjustment of the disturbing 



* Sir W. Thomson, "On an Alleged Error in Laplace's Theory of the Tides," 

 Phil Mag., Sept. 1875. 



t " Tides and Waves," Art. 111. 



J "Tidal Researches," U.S. Coast Survey Rep., 1874, p. 154. 



"Recherches sur quelques points du systeme du monde," Mem. de VAcad. 

 roy. des Sciences, 1776 [1779] ; Oeuvres Completes, t. ix., pp. 187.... 



|| Oeuvres, t. ix., p. 218. The notation has been altered. 



H It is remarkable that this argument is of a kind constantly employed by Airy 

 himself in his researches on waves. 



