282 Prof. G. H. Darwin. [June 19, 



Then taking the mean of these equations over a period begin-. 

 ning with t = and ending -when t ir\, we have (writing 

 A p = E, p cos p, Up = l&p sin p ) 



jS/t cos V p '= A p + \E q cos (a 

 V = B \Esin a 



where X and are certain constants, depending on the sum of a 

 trigonometrical series. 



Again, if we take means from t = TTJV to t = 2Tr/v, the second terms 

 have their signs changed. 



Hence the difference between these two successive snms will give 

 \E q cos (a ? ) and \E q sin ( ? ). There will be usually two terms 

 such as those typified by q, and we shall then have to take two other 

 means, vi/,., one beginning at ?r/2v and ending at 37r/2i>, and the other 

 beginning at 3ir/2v and ending at 5ir[2v. From the difference of 

 these sums we get \E q sin (* f ? ) and \E f cos ( ? ). From 

 these four equations the two E q 's and the two * ? 's are found. The 

 solution is a little complicated in reality by the fact that it is not 

 possible to take t = exactly at the beginning of the series, because 

 the first tide does not occur exactly at noon, but this is a detail which 

 will become clear below. 



When all the A's and I?'s or _R's and 's have been found, the posi- 

 tion of the sun and moon at the epoch, found from the Nautical 

 Almanac, and certain constants found from the Auxiliary Tables in 

 Baird's ' Manual of Tidal Observations,'* are required to complete 

 the evaluation of the H's and /c's. 



The details of the processes will become clear when we consider 

 the various tides. 



It may be worth mentioning that I have almost completely 

 evaluated the F's and Gr's, which give the perturbation of one tide 

 on another, in the case considered in the Appendix. Without giving 

 any of the details of the laborious arithmetic involved, it may suffice 

 to say that the conclusion fully justifies (he omission of all those 

 terms, which are neglected in the computation as presented 

 below. 



4. The tides N and L. 



These are the two lunar elliptic tides. 



For the sake of brevity all the tides excepting M 2 , N, L are 

 omitted from the analytical expressions. 



Since V a = F OT -(> )*, Fi= 7* +(-)/, 



* Taylor and Francis, Fleet Street, 1886. 



