TIDAL OBSEllVATlONS. 365 



(§ 24). Other averagings were performed according to the same rule for the 

 X, L, N, &c, reckonings respectively each averaging giving a group of 

 t^^"onty-four means. 



19. The next step was to find for each of these sets of averages the coeffi- 

 cients Aj, Aj, Bj, A^, Bj, &c. of the harmonic formulte, 



A„ + AjCOS ni+BjSin nt 

 + Aj cos 2>i^ + B, sin 2nt 



+ A^ cos 8n<+ B^ sin Snt, 



n denoting, as in § 2, the rate of increase of the hour-angle for each case ; 

 for instance y for the K tides, y—tr for the M tides, and so on. The condition 

 to be fulfilled is that the values of this formula calculated for <=0, < = 1 . . , 

 t=2'3 may agree as nearly as possible, on the whole, with the twenty-four 

 numbers of the group (the sum of the squares of the differences to be a 

 minimum*). The tabular forms and rules given by Mr. Archibald Smith, 

 and published by the Admiralty, for the harmonic reduction of the deviation of 

 ships' compasses, have been adopted mutatis mutandis, and have proved very 

 convenient. 



20. If, instead of including only seventeen coefficients, A^, A^, Bj, 



A^, B^, the calculation had been extended to A,j, B^, A^,,, so as to include 

 in all twenty-four coefficients, the calculated values would necessarily have 

 agreed with the twenty-four numbers given by observation. But thei'e was 

 no apparent probability that any thing more than accidental irregularities and 

 errors of observation could be represented by higher terms than A^, B^, and 

 therefore these were the highest included. The following Table exhibits the 

 results of this process for six series, the remaining series presenting similar 

 features. The columns headed " diifercnces " preserve the residues, however, 

 and may be referred to should further study of the subject indicate that use- 

 ful results are to be derived from them. The greatest of them is -055 of a 

 foot, and the maxima in cacli column are only from -J^ to -^-^ of a foot. 



Values of A^, B^, A^, &c., to first Approximation. 



* According to Laplace's method of " least squares," 



