On a controverted Point in Laplace's Theory of the Tides. 277 



The results which I have now given, and indeed all the results 

 of this pape^ have been deduced not only from the observations 

 which I publish, but from very many others ; so that my Tables 

 may be considered to represent the average of a very extended 

 series of researches, though they are not really so. 



[To be continued.] 



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

 By Sir George B. Airy, Astronomer Royal. 



To the Editors of the Philosophical Magazine and Journal. 



Gentlemen,, 



IN a paper published in the last Number of the Philosophical 

 Magazine, Sir William Thomson has, with great personal 

 courtesy towards myself, objected to my imputation (in the 

 article " Tides and Waves " of the Encyclopaedia Metropolitana) 

 to Laplace, of a serious error in a portion of his investigation of 

 the theory of the Tides. I had at first proposed to myself to 

 enter into a discussion of Sir William Thomson's methods; but 

 I soon found my thoughts taking the same course as in the 

 original article ; and I finally judged it best to do little more than 

 to refer to that article, with perhaps a slight expansion of the 

 verbal remarks. 



The part of the Mecanique Celeste in which the investigation 

 in question occurs is Livre IV., Chapitre Premier, Article 10; and 

 the part of my own Essay, in which I have endeavoured to put 

 Laplace's investigation into a clearer form, is Section III. Art. 110. 

 And the following is the course of the process. Supposing the 

 depth of the sea to be uniform, and making (for the present) no 

 assumption regarding its boundaries, there is to be satisfied a 

 certain differential equation of which every term can be expressed 

 as a multiple of an even power of sin 6. The quantity to be de- 

 termined is called ga. To obtain its value by the method of 

 indeterminate coefficients, the form is assumed, 



ga = \i 2 sin 2 6 + K 4 sin 4 6 + &c. -r K 2fc sm 2 *<9 + K 2ifc+2 sin 2 * +2 + &c, 



or, if^ = Gsin 2 + a"', 



a'" = (K 2 - G) sin 2 6 + K 4 sin 4 6 + &c. + K 2/f sin 2 *0 



+ K 2/f+2 sin 2 *+ 2 0-f &c. 



Substituting this in the differential equation above mentioned, 

 and comparing the coefficients of the different powers of sin 0, 

 the following subordinate equations are obtained : — 



