RECORD AND REDUCTION OF THE TIDES. 67 



The preceding tables (No. 2) contain the individual and mean values for interval 

 and height, for high and lovp vv^ater, and the moon's upper and lower transit. The 

 mean, in some cases, was improved by the application of Peirce's criterion for the 

 rejection of doubtful observations ; a few other rejections were made, as stated, in 

 order to obtain a well-balanced mean; of 982 observations of the interval, but 17 

 were thus rejected. 



Half -mo7ithly Inequality . — For the comparison of the observed with the theoretical 

 values, it is customary to use the forms of the equilibrium theory or of the wave 

 theory,' certain modifications being necessary to produce an agreement between 

 these theories with observation. According to the equilibrium theory the formula 

 for the position of the pole of the tidal spheroid is : 



tan.2e' = - J''^ ''-^^ , 



h' + h COS. 2 <p 



where h and h' are the elevations of the spheroid due to the sun and moon respect- 

 ively, the angular distance of the moon from the sun and 6' the angular distance 

 of the pole of the spheroid (or of high water) from the moon's place. In reality, 

 however, the pole of this spheroid follows the moon at a certain distance, the mean 

 value X' of which is known as the " mean establishment" (also fundamental hour, 

 corrected establishment), and which corresponds to a distance of the sun and moon 

 of 4) — a instead of ^. This retroposition of the theoretical tide has been called 

 the age of the tide. For the comparison of the observed and computed values 

 for the half-monthly inequality in time, we have the formula : '^ 



tan. 2 (6' - X') = - ^^^ipr.^) 



^ h' + hcos.2{^ — a) 



This inequality goes through its period twice in each mouth. Proper values have 



to be found for the ratio — and the angle a. 



h' " 



The observations of 480 high waters furnish us with the following values, derived 



from the preceding tabulation on form No. 2 : — 



' An account of the Equilibrium, Laplace's and the Wave Theories, will be found in the Eucyclopcedia 

 of Astronomy, forming a portion of the Bncyclopisdia Metropolitana, London, 1848 ; article " On Tides 

 and Waves," by G. B. Airy, Esq., Astronomer Royal. 



' Phil. Trans. Royal Society, 1834, Part I. On the Empirical Laws of the Tides in the Port of 

 London, with some Reflections on the Theory ; by the Rev. W. Whewell. 



See also Phil. Trans. Royal Society, 1 836, Part I. Researches on the Tides, fourth series : On the 

 Empirical Laws of the Tides in the Port of Liverpool. By the Rev. W. Whewell. 



