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XXIII. On a controverted Point in Laplace's Theory of the 



Tides. 



To the Editors of the Philosophical Magazine and Journal. 



JCJ.S. Coast Survey Office, Washington, 

 GENTLEMEN, November 17, 1875. 



AS I have been introduced, and very properly, by Sir 

 William Thomson in his two recent tidal papers in the 

 September and October Numbers of the Philosophical Maga- 

 zine as an indorser and adopter of certain views and results 

 of Sir Gr. B. Airy contained in his ' Tides and Waves,' which 

 Sir William Thomson controverts, I beg leave to occupy a 

 few pages of your Magazine on the subject. 



The principal point in the first paper referred to is a defence 

 of Laplace's method of determining the constant A (2) in his 

 result, corresponding to K 4 in Airv's presentation of it, as 

 quoted by Sir William Thomson in § 6 — a method which has 

 been called in question by Airy, but which in this paper is 

 characterized as " Laplace's brilliant invention." 



With regard to Laplace's extension of a general formula, 

 which shows the relations between the constants, only follow- 

 ing K 4 back so as to make it give the relation between K 2 and 

 K 4 , where no such relation is found by equating the coeffi- 

 cients of like powers of <#, thus making it determine K 4 , it is 

 not necessary to add any thing to what has been stated by 

 Airy on page 278 of the Philosophical Magazine. The only 

 argument which Sir William Thomson advances in support of 

 this process of Laplace's is, that it is necessary to make 

 the meridional displacement of the water zero at the equa- 

 tor. His reasoning in § 7, from which he infers that Laplace's 

 constant satisfies this condition, is based upon the final ratio 

 between the constants, expressed by 



Kofc4-2= i 1 — — - 1 K 9 &. 



-(-a 



But this ratio is entirely independent of K 4 ; and whatever 

 value K 4 may have, equation (6) gives this same relation 

 between the constants, where k is so great that the term con- 

 taining k 2 in the denominator can be neglected. By Sir 

 William Thomson's reasoning, therefore, the condition that 

 the meridional component of motion must vanish at the equator 

 is satisfied with any value of K 4 whatever ; and hence this con- 

 dition does not determine it. 



A more direct and much more satisfactory method of pro- 

 ceeding in this case would be to substitute the expression of a*, 

 * Misprinted u in the equation. 



