ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 161 



Thus we find from the formulae (51), with \/ca = 2-4400, 



m = 6-104, Ml =9753, W e = - 23'25, MJ = - 144-97, M! O = - 426'9, 

 Nj = 1-180, NJ = 11-180, N> = 29-180, N} = 55'180, NJ = 89'18. 



It will be convenient for us now to introduce the following abridged notation : 



M* +I X* +2 ... ad inf. 



E; 



F' 



_<_ 

 Ml 



- NJU - M' n _ 2 - 



N- - M' n _, - N' n _ 2 - 



(64), 



the last two continued fractions terminating with the partial quotient which involves 

 a* in the numerator, and either M* or N^ in the denominator. 

 From these definitions of the quantities e, f, E, F, we have : 



r 



c n 



; -f., 



"n 1 



>ji r~" 

 " <" +1 < 



(G5); 



"NT" -- F, 

 " r> -i 





while the period- equation may be written in the forms : 



M^ - F'_, -/.'+! = "1 

 N?, - EJ_, - e' n+l = I 



(GG). 



Suppose that we neglect /{ ; making use of the numerical values obtained above 

 for the quantities M, N, a, by successive applications of the formulae (65) we obtain 



loge! = nO-7556, log/ 9 ] = 1-2229, log el = n07829, 

 log/ 1 = 0-9666, log 4 = 5iO-9649, log/ 1 = 0-5606. 



In like manner, if we neglect e\ , we find 



log/ 1 = 1-2498, logcj = wO'7801, log/ 1 = 0'96G8, 

 log e l 6 = nO'9649, log/ 1 = 0'5606. 



Now the two values of / ! obtained by these methods are respectively the 6th and 

 VOL. cxci. A. y 



