44 REPORT — 1886. 



§ 3. The Reduction of the Results of Harmonic Analysis. 



We now suppose the harmonic analysis of the hourly means on the 

 'three sheets M, 0, S completed. 



The deduction of H^j, k^ and Ho, t:^ from the M and O sheets follows 

 exactly the same rules as in a long series of observations, and the reader 

 is referred to the Report of 1883 for an explanation. . 



With regard to the S sheet, the results of the harmonic analysis do not 

 separate the Sj tide from the Ko tide, nor the K, tide from the P tide, and 

 we have to employ theoretical considerations for effecting the separation. 



The semidiurnal tides will be taken first. 



The solar tide, as derived from a short series of observations, is of 

 course affected by the sun's parallax, and as the sun changes his parallax 

 slowly, the solar tide will follow the equilibrium law and vary as the cube 

 of the sun's parallax. Thus the height of the purely solar semidiurnal 

 tide as derived from our short sei'ies of observations will be p^H^ instead 

 of Hs, and this will be fused with the luni-solar tide K2. 



The schedules of the Report of 1883 thus show that we shall have as 

 the expression for this tide, compounded of S2 (with parallactic inequality) 

 and K2, 



A2 = P/H,cos(2<-».-,)+f"H"cos(2i+2/i-2.'"-K") . . . . (1) 



The theoretical ratio of H" to H3 is (see Schedule E, 1883) that of 

 •12662 to •45631, or 1 to 3'67 ; and the tides having nearly the same speed, 

 we may assume k"=k^. 



Hence : 



;io=H, \y, cos (2t-K,) +3^ cos (2^+ 2/i- 2 1/"— (.•,)[ 



=R, cos (2i -/:, + ;//) (2) 



■where 



f^ sin 2(7.-. /Q 3-67p, + f^cos2(/.— /Q 



■^^'' '^-3-67 p, + f cos 2(A-./')' ^~ 3-67 cos;// ^' ' ^"^^ 



If, therefore, the harmonic analysis of the S sheet for semidiurnal tides 

 has given the two components A2, B2 which are to define Rg, ^^ by the 

 equations 



A2=Rs cos Cs, B2=R3 sin Cs ; 



and if we put for p^ its value at the middle of the fortnight or month 

 as a mean value, and also put as a mean h^ @ , the value of the sun's mean 

 longitude at the middle of the fortnight or month, we get 



n 3-67 cos;// _ 



^~3-67p, + f" cos 2( ©-,/')' ''~^''^^' 



f" sin 2(0 )'") 1 



^^''"*^^'^=3-^67M^F^^^2^^'=70' a"'lH"=3^7H„ ."=., . (4) 



AVe now turn to the diurnal tides derived from the S sheet. 

 The schedules of the Report of 1883 show that we shall have as the 

 .expression for the tide which is compounded of Kj and P 



7ii=f'H'cos(i + 7i-,''-i7r-K-')+HpCos(i-7; + ^7r-K-p) ... (5) 

 The theoretical ratio of Hp to H' is (see Sched. E 1883) that of 19317 



