320 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



and generally after E 6 . 



{n («_2)— 8) K n —[(n—2){n—3)— 2] E n _,+pE n ^=0. 

 From these equations we get the following expressions of K n : 



K A =K, I (4) 



and generally, after 7T,;, 



This general expression is equivalant 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 E n from the last two preceding. 

 The one is reducible to the other by putting n=.2i-\-i. The general 

 law of (5) does not hold until after E 6 , but E 4 and K 6 being obtained 

 from the direct equation of the coefficients of y 4 and r G , then by 

 means of these, E s is obtained, either directly from the equatiou of the 

 coefficients, or from the general expression of (5), and this law can be 

 extended forward, but not backward. For instance, E G is not obtainable 

 from Ki and E 2 . 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 E n 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 E 6 only, can not be extended back. 



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

 been stated above, 



h=zEv' z +u=Ev 2 +E i v i +E 6 v 6 - +E n v n . . (6.) 



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

 efficients of v in (.3), and after E 6 only, back so as to make it, by means 

 of the continued fraction, determine the value of 2f 4 and so the relation 

 between Ev 2 and u. This makes K 4 a determinate quantity, whereas 

 the equation of the coefficients of v* gives ^L4=K4, an indeterminate 

 quantity. It is evident that any value of K 4 satisfies the differential 

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

 lution of the tidal equation. 



The extension of the general relation of (5) back so as to make it de- 

 termine K 4 , and the relation between Ev 1 and u in (6), was regarded 

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



