1890.] Harmonic Analysis of Tidal Observations. 307 



where, for example, 2(i + ii) denotes summation carried over the 

 half period made up of i acd ii. These totals are multiplied by 

 certain mean cosines and sines (whose values are given in F), and are 

 summed. The next process is multiplication by a factor <f> 

 (7r/4(n-f l)(m 1) of 4), of which the value depends on the number 

 of quarter-lnnar-anomalistic periods under treatment. The following 

 table gives the value of this factor : 



No. of i-lunar- 



anom. periods. ill. v. vii. ix. xi. xiii. 



<P 0-02950 0-01475 0-00738 0-00492 0-00369 0-00295 



The angle j is also required ; it depends on the time of the first tide 

 under reduction. If t be the time in hours since epoch to the first 

 tide, 



j = l-690-0-5444 t . 



For instance, in the example below the first tide is at 3 b 14 of 

 day |; this is S h 46 m , or 8 h 77, before epoch, so that t = 8 h '77 ; 

 then 



j = l-690+0 3 -5444x8-77 = +6-46. 



D. The Tides Kj, O, P. 



Summations are carried out over quarter-lunar periods numbered 

 I, II, III, &c., and totals are formed like those mentioned in C, and a 

 factor "ir (which differs slightly from $) is required in the formation ' 

 of S and S* ir . This factor depends on the number of quarter-lunar 

 periods under treatment, and the following table gives its value : 



No. of i-lunar 



periods. III. V. VII. IX. XI. XIII. 



* 0-02976 0-01488 0-00744 0*00496 0-00372 0-00298 



The angles i and Z are required ; they depend on the time of the 

 first tide under reduction. If t be the time in hours of the first tide 

 since epoch, 



t = l-705-0-549 t . 



Z=-O a -255+0-082* . 



For instance, in the example below we have, as shown in C, 

 t = -8 h '77, and 



t=+6-52, 

 Z=-0-97. 

 It is required to compute T and y- from 



