PAPER BY WILLIAM FERREL. 321 



it is not applicable, and this is what was not clearly understood by his 

 correspondent. 



From (4) it is seen that the tidal expression consists of two parts, 

 one of which depends upon JT 4 , aud is independent of the tidal forces 

 contained in E, and the latter depends upon these forces. It is evident 

 that the former can exist without the latter. Also that being inde- 

 pendent of the forces, and dependent simply upon certain initial 

 motions which the sea may be supposed to have independent of the 

 forces, it must vanish when there is friction, and so K 4 must be put 

 equal to in the real case of nature. 



We come now to the second part of what we have proposed to con- 

 sider here, namely, the convergency of the series in the expression of 

 u in (3). Inasmuch as the vanishing ratio between consecutive values 

 of K n is unity, as is readily seen from an inspection of (5), it has been 

 said that the device of Laplace in the use of the continued fraction was 

 necessary to make the expression of u convergent at the equator where 

 v = 1, so as to give a finite value of u. It is true that the expression 

 at first is more convergent with a large value of K±, such as is given 

 by the continued fraction, but still the vanishing ratio in any case is 

 unity. But it can be shown that the expression gives a finite value of 

 u when we put 2f 4 = 0. 



We get by development, 



( i_v*)i=i-A v 2_Av 4 _iLv 6 .... -A nV «=i+2?A nVn :.(7). 



in which the relation between each coefficient A n and the preceding one> 



commencing with — -, is 



•j 



A»=~A n _ 2 • • • (8). 



Hence we have, when > = 1 



2»A n =-l (9). 



2:, +2 A »=-(l + XA n ) (10). 



in which n' is the exponent of any assumed term in the series. 

 The expression of (5) above may be put into the form, 



^=^^+^^-(^4^)^ ■ • (11) - 



From this, bv means of (8), we get for any coefficient for which the 

 characteristic is w', 



K n ,=^A n ,F n (12). 



-»n'-2 



80 A 21 



