n 



ON THE METHOD OF DEDUCING NUMEEICAL VALUES OF TIDES. 321 

 , 8inl2n r , sin 182)' r- ^ , -iqon -\ r / , ■\ -\^ 



Then unless ^y differs by an exceedingly small amount from 180° or 360°, 

 the first of these terms within ( } is small comj^ared with the second, 

 except for those terms (if any) in which )'(r + i) — £ is very nearly equal 

 to 90° or 270°, whilst r^r + 182) - e is nearly equal to zero or 180°. 



Even in the case of the K tides, in which the diurnal speeds are 

 respectively 360° 59' 8", and 721° 58' 16", the coefficient of the first term 

 is small, being respectively -00339 and •00340 for the two K tides. 

 Hence the improper term in the residual ^/t, is to a close degree of ap- 

 proximation. 



, T-, sin 1 2 w p ^ , .-, -, 



— ^r B ; — cosl I (r + i) — t\ 



tan -ku 



The ch's have now to be submitted to harmonic analysis for extracting 

 the long-period tides, but as the values of ch are given discontinuously 

 for each midnight, the continvious integrals, which arise in such analysis, 

 are replaced by finite integrals. 

 It is now assumed that 



ch = I,[A cos Xt + B sin Xr] 



Where S denotes summation for all the values of A, which the theoretical 

 development of the tide-generating potential, or other considerations 

 indicate as likely to give sensible results. In the tidal report there are 

 five values of X, namely 2it, cr — tn-, 2(a — rj), 2>;, rj. I here suppose \ to 

 be expressed in degrees per m.s. day. 



Now let \i be the speed of the special long-period tide for which we 

 are searching ; and Ai, B^ the amplitudes of the components which it is 

 required to evaluate. Then 



c/icos A,7 = ^.-lj(I + cos 2\ir) + ^Bi sin 2/\,r 



+ 2[^ cos Xt cos XiT + i? sin \r cos X,r] 



?/i sin Xj- = 1 Z)i(l — cos 2X,r) + ^ ^i sin 2\,- 



-f 2[^ cos Xt- sin X,r + B sin Xr sin Xjt] 



Where il denotes summation for all values of X except Xj. By means 

 of the known numerical value of X,, cos X/r -f i) and sin X^(r + i) may 

 be computed for every midnigiit in the year, and the results multiplied by 

 the corresponding ch;, whether cleared or uncleared of undue influence. 



Now if we form 2 ch; cos [X,(r + /)], it is obvious that A^ will be 



multiplied by a coefficient differing but little from 183, and that the 



coefficient of B^, and all the other coefficients of the ^'s and L"s, will be 



' = .11-, 1 

 small. Similarlj^, the coefficient of 7?, in 2 cV;,- sin [X,(r -f i')] will be 



1=0 



nearly 183, and all the other coefficients small. Proceeding in the same 

 way with each long-period tide we shall obtain twice as many equations 

 as there are long-period tidts, in each of which there is oiie unknown 

 A or B with a coefficient nearly equal to 183. These equations may be 

 solved by successive approximation, and the two components of each tide 

 evaluated. 



We must now trace the undue term through this process. 



1882. T 



