ON THE METHOD OF DEDDCINa NUMEEICAL VALUES OP TIDES. 319 



On the Method of Harmonic Analysis used in deducing the Nu~ 

 inerical Values of the Tides of long 'period, and on a Mis-print 

 in the Tidal Report for 1872. By Q. H. Darwin, M.A., F.R.S. 



[A communication ordered by the General Committee to be printed in cxtenso among 



the Keports.] 



I HAVE recently been carrying out a laborious reduction by the method of 

 Least Squares of the observations of the tides of long period at a number 

 of stations. The results, which seem to have an important bearing on 

 the question of the rigidity of the earth's mass, will appear as § 848 in the 

 new edition of Thomson and Tait's ' Natural Philosophy,' now in the 

 press. 



Subsequently to the completion of the calculations Professor J. C. 

 Adams discovered a misprint in the Tidal Report of 1872, which forms 

 the basis for the method of harmonic analysis which has been applied to 

 the tidal observations. 



On inquiring of Mr. Roberts, who has superintended the original 

 computations, Professor Adams learnt that the erroneous formula has 

 been used in all the reductions of the long-period tides. 



It now therefore becomes necessary to investigate the amount of 

 error which may have been incui'red. 



The erroneous formula occurs near the middle of page 371 of the 

 ' Report of the British Association ' for 1872, in the instructions for 

 clearing the diurnal means from the influence of the shoi't-period tides ; 

 in the first of the two formulee for that jjurpose, the factor, sin 12;i/sin \n, 

 should obviously be replaced by sin 24<}i/sinii. 



I shall now therefore go over the process of clearing, shall evaluate 

 the eSect of non-clearance, and shall indicate what I conceive to be im- 

 provements in the method of the Tidal Report. 



On any day, the height of water above the datum line is measured at 

 0^, l"^, 21* . . . . 24'* ; then the mean height of water for that day may 

 be taken as 174th of the sum of the 25 heights less the mean of the 

 heights at O^* and 2^^. In the Report the instruction is to take yr^th of 

 the sum of the heights at O*", li» . . . . 23^^ At the time that I wrote 

 this paper I conceived this to be less accurate than the method here 

 suggested. But after conversation with Sir William Thomson, I per- 

 ceive that the rule of the Report is just a shade more accurate than that 

 here suggested. If, however, it were practically more convenient to find 

 the heights of the tides of short periods at each midnight (see below) 

 instead of at 11.30, then the rule here given would be practically prefer- 

 able. As the computations in this paper have been made from the rule 

 as here suggested, I leave the analysis in the form in which it stood 

 originally. 



We wish to find how much this diurnal mean is influenced by a short- 

 period tide, for which the expression is H cos (nt — e). 



Now let t^ be the time at O"* of the day in question, measured from 

 the epoch when ^ = 0. 



The uncleared daily mean has obviously involved 



2i ■ 



'< icos [n^o— t] + cos [?</<, + 1)— £] + 



