1892.] facilitating the Reduction of Tidal Observations. 375 



In this way we evaluate the luni-solar fortnightly and lunar fort- 

 nightly inequalities in the height of the water. 



The period of the moon is between 27 and 28 days, and if we erase 

 appropriately about one day in eight we are left with sets of 24 values 

 which may be taken as 24 equidistant values of the daily sums. 

 Accordingly we erase 46 daily sums out of 358, and write the 312 

 which remain consecutively into a schedule of 24 columns and 13 

 rows, of which each corresponds to a lunar anomalistic period. 



The 24 columns are summed and the sums analysed for the first com- 

 ponents. Finally, the components are to be divided by 24 times 13, or 

 by 312. In this way the lunar monthly tide is evaluated. 



But the result obtained in this way is, as far as concerns the tide 

 MSf, to some, and it may be to a large, extent fictitious. It repre- 

 sents, in fact, a residuum of the principal lunar tide M 2 . That this is 

 the case will now be proved. 



Suppose that t is an integral number of days since epoch, being 

 the time of noon on a certain day ; then the principal lunar tide M 2 on 

 that day may be written H m cos [2 (7 <T)( O +T) j, where T is less 

 than 24 hours. Then the daily sum for that day will be 



Now since t Q is an integral number of days 2 (7 ff) only differs 

 from 2(<r if) t by an exact multiple of 360 ; hence the argument 

 of the cosine may be written 2 (<r rj) t 23 (7 d) + m . 



But the true luni-solar fortnightly tide, which we may denote 

 lffcos[2(<r T?)( O + T) ], varies so slowly in the course of a day 

 that the daily sum is sensibly equal to 



24 Iff COS [2 (ff-rj) t + 23(<r iy) ]. 



It thus appears that the residual effect of M 2 is of exactly the same 

 form as that of MSf. It becomes, therefore, necessary to clear the 

 harmonic components, determined as described above, from the effects 

 ofM 2 . 



In order to determine the values of these clearances, I found the 

 values of cos 2 (<r rj)t and sin2(ff rf)t for every noon in a year of 

 369 days. I then erased the values selected for the treatment of 

 MSf and analysed the remaining values. In this way it was easy to 

 find the effect of the known M 2 tide. 



Suppose that AI, BX are the first harmonic components determined 

 by the treatment of a series of daily sums, and that A t , ^B x are the 

 corrections to be applied to them to eliminate the effects of M 2 , then 

 I find that if A m , B OT are the two components of M 2 as determined by 

 the previous method ( 6) of analysis, 



2 c 2 



