214] 



LAPLACE'S SOLUTION. 



361 



force which is required to make the solution exact, will be very small. This 

 will be illustrated presently, after Laplace, by a numerical example. 



The process just given is plainly equivalent to the use of the continued 

 fraction (17) in the manner already explained, starting with j+l=k, and 

 jy k =(3/2k (2k + 3). The continued fraction, as such, does not, however, make 

 its appearance in the memoir here referred to, but was introduced in the 

 Mecanique Celeste, probably as an after-thought, as a condensed expression of 

 the method of computation originally employed. 



The following table gives the numerical values of the coeffi- 

 cients of the several powers of v in the formula (19) for %/H'", in 

 the cases ft = 40, 20, 10, 5, 1, which correspond to depths of 7260, 

 14520, 29040, 58080, 290400, feet, respectively*. The last line 

 gives the value of QH" f for v = 1, i.e. the ratio of the amplitude 

 at the equator to its equilibrium-value. At the poles (z/ = 0), 

 the tide has in all cases the equilibrium-value zero. 



We may use the above results to estimate the closeness of the approxima- 

 tion in each case. For example, when /3 = 40, Laplace finds B^ = ~ '000004/jT'" ; 

 the addition to the disturbing force which is necessary to make the solution 

 exact would then be -00002/T' V 30 , and would therefore bear to the actual 

 force the ratio - -00002 j/ 28 . 



It appears from (19) that near the poles, where v is small, the 

 tides are in all cases direct. For sufficiently great depths, @ will 



* The first three cases were calculated by Laplace, I.e. ante p. 360 ; the last by 

 Lord Kelvin. The results have been roughly verified by the present writer. 



