ON HARMONIC ANALYSIS OF TlDAL OBSERVATIONS. 881 



In the B.A. assignment all the given heights are used and the maximum 

 difference between the time of a place in the summations and that of the height 

 assigned to it is ir/24(r or half a special hour. When the times of two of the given 

 heights are nearer to that of a place than this interval, both are assigned to it, while 

 when no height has a time as near as this interval, none is assigned to the place. 

 The former case occurs regularly when <r < o-„, the latter when o- > a„. In both cases 

 the mean is formed by dividing by the total number of heights used in the 

 summation. . 



In Darwin's tidal abacus the height at each mean solar noon is assigned to that 

 place in the summation whose time differs from this noon by less than half a special 

 hour, while any two consecutive heights belonging to the same mean solar day are 

 assigned to consecutive places in the summation. As in the B.A. assignment all the 

 given heights are used so that two heights are assigned to certain single places 

 . . . when (T<(T„ and no heights are assigned to certain places . . . 

 when cr >(r„. . 



Let us now examine the effects of these averaging processes on the various 



constituents. 



Taking firstly the term 



R cos {mart - e) (1*V 



the summation for the special hour h will give 



R Vcos f ^ + m<rT, - .) .... (11) 

 N ^ V 12 / 



8=0 



where t, denotes the excess of the time of the (s + l)th height in the summation 

 over that of the place to which it is assigned, and N the number of terms in the 

 summation. 



Now, for given values of o-, m, and N, the sums 



N-l N-1 



i 2 c°s imcTs), — 2 sin (wo-T,) . . . (12) 

 j=0 »=o 



may be determined accurately by direct addition. This would be laborious and the 

 sums are replaced by integrals. For the B.A. assignment they are replaced by 



t^[ cos Onar^dr, _ sin imaT)dT, 



J — T J — T 



(13) 



respectively, where T is half a special hour. The eflfect is to replace (11) by 



sin maT 



mtrT 

 For the abacus assignment (II) is replaced by 



RCOS ('^-«) 0*) 



sin "^W2'l sin{K<r/(ro-l)^} r cos / ^f ^ ("(1 - - V ^1 '^ " * i • O^) 

 jnir/24 7re((r/(ro- l)ir [ 12L " V <^o/ J J 



Taking next the term 



B.COS Qlffrt-t), (16) 



the summation for the special hour h will give 



N-l 



l2-|»4(-,4)V'-]-r ■ ■ • <"> 



which we shall refer to as the residue from (16). 



Again, for given values of a, Vr, n and N, the series 



N-l / , \ N-l 



i-2cos|«.,^2jr^^^'jj_ 1 2«i°|«'^^(v + ^')}' • ('^) 



