886 REPORT— 1872. 



and neglecting the influcuec of land (§(, Jj), svc have [Thomson find Tait, 

 §§ 808, (18), (23)] 



c=j=;'cr'V(cos-Z and c' = j= w'-'a cos- 1 (a) 



Using these and (2), with the notation of (3) in (i), we find 



«+« = 4-eo3-^|-E"L( 2 ) cos[2(y-,,);'-2e-2P]+ -2-^co82y< 



+ (^-^=|^)'cos [2(y+^)^+26+2P]] 



+ T[C-nFT-^2[(,-.K-Q-a'-Q,]+^-L;li'cos2(,.-0) 



+ (lz_^^yeos2[(y + a)^-Q+6'+Q]]| (5) 



as the rigorous expression for the semidiurnal equilibrium tide-height, on the 

 supposition of no dry land, or of such a distribution as to make ^ = and 

 33 = 0. By taking the expressions given by physical astronomy for -^, P, ■^', 

 i, H, and Q, and expanding in series of simple harmonic functions of the time, 

 it is easy to obtain, in the form proper for the hai-monic analysis, a complete 

 expression for the whole astronomical semidiurnal tide-generating influence. 



The terms of (3) or (5), containing the factors /'^~^°"^Yand /ilZ^V, 



are, on account of these factors, necessarily very small. They show semi- 

 diurnal constituents with arguments 2(y + jj)< (solar) and 2 {y + (T)t (lunar), 

 which have not hitherto been investigated from observation, but Avhich, for 

 the case of the moon, and particularly in years when i is large, may be quite 

 sensible. 



51. The Table on the opposite page exhibits the comparative values of 

 the analyzed and equilibrium-theoretical semidiurnal tides referred to the 

 mean lunar semidiurnal tide as unity. The epochs of all of these tides 

 are expressed in hour-angles of mean solar time, and are referred to the 

 meridian of the place, except for Liverpool, Eamsgate, and Portland Break- 

 water, which are referred to the meridian of Greenwich. 



52. The following will illustrate the method at present employed in the com- 

 parison between the actual tide-heights as recorded and the heights as 

 furnished by the evaluated tide-constants. The residual differences (which 

 include instrumental errors of every description) show the amoimt of precision 

 arrivedat from the tide -components included in the analysis, and are useful 

 as a guide for the introduction of new arguments and the consequent evalua- 

 tion of new tide-components. The Tables are based on the analyzed values 

 of the tide-components of Kurrachee for the year 1868-69 alone, excepting 

 the R and T solar elliptic semidiurnal tides, which are the results of 1868- 

 1869 and folloM'ing year. 



In order to facilitate the computations of the heights. Tables showing the 

 value of the tide above or below mean level for each 15° of hour-angle for 

 the S tide should be formed, or for a less interval if it is contemplated com- 

 puting the tide-heights for more frequent intervals than each integral mean 

 solar hour, and for every degree of the M tide on account of the magni- 

 tude of 11, of this tide, and for every few degrees for the rest of the tide- 

 components. 



