ON THE METHOD OF DEDUCING NUMERICAL VALUES OF TIDES. 323 



to 365. By aid of ttis rule the formula for undue influence becomes 

 universally applicable. 



The manner in which this inconvenient result is brought about is 

 curious. 



The finite integrals S, which arise in the harmonic analysis, are of 

 course the representatives of the integrals which arise in the analysis of 

 continuous functions. In general, the distinction between 2 and /would 

 be quite insensible. 



Now f cos Xr cos(Xr -.)dr = , ^-^ ( ^^^fh(" + ^^ 7 ^U 



+ 



1 r sin [3641 (" _ X) _ ,] 1 



2(v - A) I + sin[|(i' - A) + e] j 



If ii'' ± ^) ^6 860° the correspondinof term within { } vanishes. It 

 would seem therefore at first sight, that the result of the finite integral, in 

 which the term becomes large instead of vanishing, must be erroneous. 



The discrepancy enters in the assumption practically made in the 

 treatment through diurnal means, that because the long-period tide 

 varies slowly it is sufficiently accurate to estimate its height only once 

 in the twenty-four hours. ^ 



The expression il iJi^ cos A(t + ?') is of course intended to be, and in 







J = 365x21 



general is, a sufficiently close approximation to Si ch'j cos ^:g X(t +j), 



j = ' 



where j is the number of m.s. hours since 0"^ of day 1, and tJi'j is the 

 departure from mean water at the_y'tli hour, and where we take a half of 

 the first and last terms. The term due to the tide n in dh'j is of course 

 J2cos [n(t + j) — £]. Then if we effect the summations involved in this 

 expression, we shall find that one of the two terms vanishes when 

 ■|(A + >') = 360°, just as it does in the continuous integral. 



It follows therefore that the method by diumial means leads to a 

 considerably different result from that by hourly values, or by con- 

 tinuous integration. This does not constitute a reason for discarding 

 the method of diurnal means in these special cases, but it is necessary to 

 pay careful attention to the terms introduced in that process. 



The five long-period tides for which the harmonic analysis has been 

 practically carried out are those of speeds 2(t, (t — 'ut, 2a — 2>/, 2(7, rj. 

 We have to consider what values of n added to or subtracted from these 

 speeds will give resulting speeds of y — ij or 2y — 2;;, that is to say of 

 15° or 30° per m.s. hour. 



The accompauyiDg schedule gives the values of (n) which give these 

 results. 



Amongst the resulting speeds there are only a few Avhich correspond 

 with actually existing sensible short-period tides. These are marked in 



' The point of view from which I here regard the problem of clearing the daily 

 means is, as I learn from Sir William Thomson, somewhat different from that sup- 

 posed to be adopted in the Tidal Report. It is there supposed that the results of 

 the previous harmonic analysis for the tides of short period are utilised to subtract 

 the tide of short period from eveiy hourly value of the height of water, but instead 

 of making twenty-four subtractions for each day, a twenty-fourth part of the sum of 

 the quantities to be subtracted is taken from tlie diurnal mean. From this point of 

 view a result practically identical with that in the te.Kt may be obtained. 



r 2 



