ON HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 333 



approximately ; similarly the effect of (21) on K„jsln«m reduces to 



-^^^°l2V"(7;-l)'^-*[ (26) 



approximately. 



On combining (7) with (25) and (26) we obtain as the corrections to 



E„j cos f,n and R„ sin e^ 

 required by (3) 



- F R cos (k - 6), F R sin (« - e), (27) 



respectively, where 



sinjNw(<r,/(ro — l)ir} 



F = 



N«(<rr/(ro — l)ir 



If these corrections are small compared with R^, they give 



8 R„i = - F R cos (k + e„i - e), 

 8 e,„ = F R sin (k + €„ - e), 



which we notice involve only the relative phase em — 



. «=«(n-^)(^;-i)-. (28) 



(29) 



Danoin's Method for Solar Constikients. 



9. This method consists in applying the S series isolation process and the least 

 square rule to different sets of 30 consecutive days' record and then analysing the 

 resulting sets of values for yearly and half-yearly harmonic variations. 



When a year's record is available, 12 sets of 30 days are used, and from the 

 results values of 



^o, Kj, T,, K,, P,, Ssa, Sa, A„, S4, Sj, S, 



are immediately taken. Residues from M„ are allowed for. When less than a 

 year's but as much as half a year's record is available, Ssa is neglected. 



Analysis of Hourly Heights for Long Period Constituents. 



10. There are two methods in general use, and we shall refer to them as the B.A. 

 method and Darwin's short method. The B.A. method is used by the Survey 

 of India. 



The principle of the B.A. method is the least square rule applied to daily mean 

 heights, using one decimal place for the multiplying sines and cosines. The residues 

 from all primary astronomical constituents are allowed for. 



Darwin's short method uses the principle of isolation and proceeds on a plan 

 similar to the assignments, the daily means taking the place of the given hourly 

 heights. Residues from M2 are allowed for, but no great accuracy is claimed for the 

 method ; Darwin gave it as a much less laborious process than the B.A. method. 



Darwin's Method for Harmonic Analysis of High and Low 

 Water Observations. 



11. If C denote the height of the water at time t, it will be given by an equation 

 of the type 



f = R„ cos ((r„<-e(,) + 2 Rr cos (M-eJ. • . . (30) 



r 



where <r„ is no longer the speed of S,, but that of any constituent conveniently chosen 

 to play a special part in the analysis. 



At the time of high or low water we have 



= R„ sin (<T„t - f J + 2 I'-R^ sin (cr,t - t ), . . . (31) 



and if we let 



^1' ^2 t,, . . . . , \ ^32"» 



Sl> fjl • • • • fs. • • • • f j ' ' N ' 



