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



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

 obtained by writing a+(w + l)e for a, 6 + (w + l)e for b, and 

 c+(w + l)z for c; that is to say, a + 7r for a, & + TT for I, and 

 c + 153-0706 or C + TT- 26'9294 for c. 



If, therefore, we put e = 26 0> 9294, we obtain the result from (25) 

 by changing the signs on the left and writing 7 e for 7. 



The equations for semi-periods 3, 4, 5, &c., will be alternately + 

 and on the left, and identical as regards the terms in and /3, but 

 with 7 2e, 7 3e, 7 4e, &c., successively in place of 7. 



Let the observations run over m semi-lunar periods ; then double 

 the equations appertaining to periods 2, 3 ... (m 1), add all the m 

 equations together, and divide by 2(m 1). 



The terms in Tip will involve the series 



sin s i n / \ I o s i n / 9^-4- 4- s * n ( ( "H ") 



cos^ cos ^ ' ' cos^' ' cos ^ ^ 



sin|(m l)esin , _- L( 

 ' cos^ 7 2 ^ m 



This is equal to 

 S 

 Then if we put 



- - -ir~i, 

 (m 1) tanfe 



our equations (25) become 



X = 11436, 



\ 

 1) 



= .R'sin* J2 sin/3 fiRp sin (7 ^(m 



osj3 fiRpCOS (7 ^(m l)e), 

 where 2 now denotes summation of the followin kind : 



... (26), 



Suppose the whole series of observations to be reduced covers 

 exactly 2i + 1 qua/rter-hm&r 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 denote the time of the first tide of the series, and since noon 

 of the first day is epoch, t cannot be more than a few hours. 



Tpf ilp af 1.704 at ~\ 



I Jl U v ^C ^^ " VQ ~ -I- / V/TKJ U l/Q) 



> (27), 



and k=z-(<t-Zrit = l -4497-(ff-2ry)f ;J 



i and k are clearly small angles. 



