284 Prof. G. H. Dai-win. [June 10, 



where the summations 2 are carried out over the first semi-lunar- 

 anornalistic period, which may be designated as 1. 



In applying these equations to the next semi-period 2, the result 

 is got by writing a+(n + ~L~)e or a + "" for a, and b + ir for 6. 



Thus the equations are simply the same as (10), with the signs on 

 the left changed. 



The equations for semi-periods 3, 4, &c., will be all identical on the 

 right, with alternately + and signs on the left. 



Let the observations run over m semi-lunar-anomalistic periods ; 

 then double the equations appertaining to periods 2, 3, ... (TO!), 

 and add all the m equations together, and divide by 2(m 1), and we 

 have 



IT ~\ 



2/i cos V m = -B M sin (a -|e) Bjsin (b \e), - 



4(+l)(m-J) 



7T 



-r-, ^-, 7^2/4 sin F = -B cos fa ie) EI cos ( b -ie) . . 



4( + l)(m 1) J 



where 2 noAv denotes summation of the following kind : 



the numbers (1), (2), &c., indicating the number of the semi-lunar- 

 anomalistic-periods over which the partial sums are taken. 



Suppose the whole series of observations to be reduced covers 

 2m + 1 gwarier-lunar-anomalistic periods, which we denote by i, ii, 

 iii, &c. 



First suppose that the semi-period denoted previously by 1 consists 

 of i + ii, that 2 consists of iii + iv, and so on. 



Let t be the time of the first tide of the series, and since we 

 take noon of the first day as epoch, t cannot be more than a few 

 hours. 



Let j = \e(aaf)t := 1 0- 6903 (a w)t , a small angle. 

 Then a e=ff-art + n e= n 



. (12). 



Then denoting the operation - - =- - - 2 by S (the mark 



4(w + l)(m 1) 



indicating that the first tide included is nearly at epoch, when 

 (<r-T,r)t = 0), we have from (11) and (12) 



ft cos V m =-E n sin (&_/) +E, sin 



(13)- 

 S7i sin V m = E n cos ( f j ) EI cus ( fi 



