First Species " in Laplace's Theory of the Tides, 283 



and this whatever value, zero or other, we give to B (unless we 

 give it precisely the value found by Laplace's method below, 

 and then perforin each step of the calculation with infinite accu- 

 racy). Hence, whatever be the value of e, the series expressing 

 the solution converges for every value of p,< 1. Thus the solu- 

 tion is thoroughly satisfactory for the supposed case of two equal 

 polar islands of any finite magnitude. But the ultimate conver- 

 gence is shown by (13) to be the same as that of the series 



y + y+- ..— + ..., 



which is equal to 



1 



logy^ 



^ 



Hence, when fi= 1, the series for a' becomes infinitely great ; and 



da' 

 a fortiori it gives an infinitely great value for -j-, unless it has 



been calculated for precisely the particular value of B sought. 

 Hence equation (11) fails to determine this value. Thus the 

 solution fails for the very case for which it was sought, the case 

 proposed originally by Laplace, and taken by Airy and Ferrel as 

 the subject of their investigation — that is, the case of the whole 

 earth covered with water. Here Laplace's brilliant process, 

 referred to in an article in the preceding Number of the Philo- 

 sophical Magazine, comes to our aid marvellously. 

 Let 



5i±! = ^i (14.) 



We have, by (6), 



Hl _£{„ +1)(i+1)+ fi±g2±!>}. . „ 



From this equation applied to any moderately great even value 

 of * (greater or less great according to the degree of approxi- 

 mation required) , taking N !+2 = go , calculate N,-, and then, by 

 successive applications for smaller even values of i in order, cal- 

 culate Ni_ 2 , N;_ 4 , . . . N 6 , N 4 successively. Equations (7), with 



B 4 =-N 4 B 6 , (16) 



then give 



B « =e 3+2N 4 ' 



and 2H 



Be= - e 3TaN, 



