1890.] Harmonic Analysis of Tidal Observations. 319 



(f.) Table of Sums for N and L. (See p. 320.) 



Maintaining the divisions i, ii, iii, &c., it is now necessary to sum 

 each of the four times 18 vertical columns in each of the xiii divi- 

 sions, to subtract the 18 columns of the 3 rd quadrant from the 

 18 columns of the l rt , and to subtract the 18 columns of the 4 th 

 quadrant from the 18 of the 2 nd . The 2 nd 4 th columns have then 

 to be reversed. 



Since in this case nearly all the H.W. fall in the 2 nd quadrant, 

 and nearly all the L.W. fall in the 3 rd quadrant, it is easy to write- 

 down at once 2 nd 4 th , and 1 st 3 rd , as shown on the next page. 



In this the signs of the L.W. entries have to be reintroduced, 

 but as the L.W. lie mostly in 3 rd quadrant, which enters with nega- 

 tive sign, they become positive again. It thus happens that nearly 

 all the columns come out + ; there are, however, a few in xii. 



(g.) Table of Sums for M 2 . (See p. 321.) 



We now disregard the sub-divisions i, ii, iii, &c., and sum the 

 4 times 18 columns into grand totals, stopping the summations, 

 however, at the end of 6 semi-lunations (i.e., at 171 H.W. and 

 171 L.W.). 



It would hardly be wise to attempt in this case the subtractions 

 lt_3rd } 2 nd 4 th , without the intermediate steps. 



The following table (p. 321) gives the results. 



(h.) General Rule for Cosine and Sine Summations. 



For ' cosines ' the 18 numbers required are derived from 

 (lt_3rf^_(2i-4 th , reversed). 



For ' sines ' the 18 numbers required are derived from (1 st 3 rd ) 

 + (2 nd 4 th , reversed). 



(i.) Evaluations q/'N and L (continued). (See p. 322.) 



Cosines. From Table (f) of Sums enter the 18 ' cosine ' numbers, 

 in accordance with (h) in xiii vertical columns, and perform the 

 operations indicated in the example on page 322. 



The column of ' cosine factors ' are those given in F, and 4> is 

 given in C for xiii ^-luuar-anomalistic periods. 



