320 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



and generally after Ke. 



[n (w— 2)— 8) K,—[{n—2){n—3)—2] K,_,+ftE„^=Qi. 

 From these equations we get the following expressions of 7v,. : 



lu=Iu \ (1) 



and generally, after Ka, 



^'■»=;-^-^— (T^zjy^- (3) 



This general expression is equivalent to Laplace's and Darwin's law 

 as given in my preceding paper, equation (2), but is more simple and 

 convenient in deducing any coefficient K^ from the last two preceding. 

 The one is reducible to the other by putting n=Ti+^. The general 

 law of (5) does not hold until alter iTe, but A'^ and Kg being obtained 

 from the direct equation of the coefficients of r^ and v^, then by 

 means of these, Kn is obtained, either directly from the equatiou of the 

 coefficients, or from the general expression of (o), and this law can be 

 extended forward, but not backward. For instance, Eq is not obtainable 

 from Ki and K2. As is usual in such cases, the general law is not ob- 

 tained until after several equations of the coefficients, and when the 

 values of K,^ are given directly in this way, and not by the general law, 

 the former must be taken, and the general law, which is a relation 

 found between the coefficients after Ee only, can not be extended back. 



Putting h for the amplitude of the real tide, we have, from what has 

 been stated above, 



h=Ev'-\-u=Ev~-\-EiV'-\-Eei'^ - +Zl„j'" . . (6.) 



Laplace extended the relation above, found to exist between the co- 

 efficients of r in (3), and after Eq only, back so as to make it, by means 

 of the continued fraction, determine the value of E^ and so the relation 

 between Ev'^ and u. This makes ^4 a determinate quantity, whereas 

 the equation of the coefficients of k^ gives iL4 = K4, an indeterminate 

 quantity. It is evident that any value of K4 satisfies the difiereutial 

 equation, and so, with the other coefficients depending upon it, is a so- 

 lution of the tidal equation. 



The extension of the general relation of (0) back so as to make it de- 

 termine K4, and the relation between Ev^ and u in (6), was regarded 

 by the writer in his previous paper as an extension of the law back where 



