320 REPORT— 1882. 



+ cos [_n{t^ + 23) - £] + i cos [«(/„ + 24) - £] | 



= _i_ 5 { ^^^^ cos [nit, + 12) - 

 L sin -1^ Ji 



- icos \nt^ - f] - i cos [7i(/„ + 24) - e] | 



, -r, sin 12 51 r /i , To\ n 



[The formula derived from the rule in the Report has sin \n in place of 

 tan ^)?., and <„ + IH in place of t^ + 12.] 



If 0** is noon, then t^ + 12 is midnight of the day in question. The 

 speed n is generally given in degrees per m.s. hour ; but if we write v for 

 24ji, and r for ^'j {t^ + 12), so that v is the speed in degrees per m.s. day, 

 and r the time in days from the epoch up to the first midnight in the 

 year, then n{t^ + 12) = v -. The corresponding term for successive days 

 is to be found by writing r + 1, r + 2, . . . for -. If, then, we wish to 

 clear the daily mean from the influence arising from the tide of speed n, 

 it is obvious that we must subtract .}^ B sin 12 n cot ^n cos [)(r 4- i) — ej 

 from the diurnal mean corresponding to the (i + l)th day. The same 

 should be done for each short-period tide, which experience shows may 

 exercise a sensible effect on the I'esult. But in the present instance we 

 have to trace the results of non-clearance. 



The next step is to find the mean height of water for the whole year. 

 There are altogether 365 X 24 -f- 1 heights given, being those at each 

 hour from 0^ of day 1 to 24*^ of day 365. The proper mean is — ^^ — — 

 of the sum of the 365 x 24 -f- 1 heights less the mean of the heights 

 at 0^ of day 1 and 24*^ of day 365. From the manner in which the 

 diurnal means were formed this is clearly -3^ 5 of the sum of the diurnal 

 means. 



Now when there is no clearance, it was shown above that the diurnal 

 mean for the (i + l)th day contains the undue term: — 



, T) sin 12 J2. I- , , .^ -1 

 ^^^lanl^''°^^"(^ + ^)-^^ 



And therefore the annual mean contains the undue term : — 



1, sin 12 72- sin 182 f r , , ,oo\ n 

 ^ x„„ 1-' ■ 1 cos [.(r + 182) - £] 



365X24 tani?i sin ^ 



Since 12 n is identical with ^v, sin 12?i divides out with sin |)', and 

 the result is exactly what we should have found if the mean had been ob- 

 tained by considering the series of hourly values throughout the year, with 

 attention to the rule of only taking a half of the first and last terms. 

 This is obviously correct. 



It will be shown below how much this improper term in the expression 

 for mean water will affect the result. 



The next step is to find how much the diurnal naeans depart from 

 mean water ; these differences are then to be the subject of harmonic 

 analysis for the purpose of extracting the long-period tides. 



Each diurnal mean is subtracted from the annual mean, leaving resi- 

 duals oh. For the (i + ly^ midnight the undue term in ch is clearly 



