TO ASSIST WORK ON THE TIDES. 933 
that the average of the results of the formula over twenty-hours will give, to a 
very close degree of approximation, the value of L at the mid-hour. L was found at 
intervals of twelve hours, and intermediate values obtained by linear interpolation. 
It should be noted that L has to be added to the residue. 
(2) The second method, used for Liverpool, was applied to (¢)*,—(¢)%, instead 
of ¢,”, with an appropriate formula. The residual value of L is very small. 
(8) The third method has been used for reducing the second half of the 1918 
records for Newlyn. This method avoids the introduction of long-period con- 
stituents, and involves less labour than the first method. 
Referring to § 12, we have 
¢,= C, cos 30°¢—S, sin 30°¢ = R cos (30¢ + a)° 
where R cos a=C, R sin a=S, and R is slowly-varying. 
Then ¢,= cR? cos (60¢ + 2a + x)° 
=C, cos 60°f—§, sin 60°%, 
where C,=(e cos x)(C,?—S,”) - (¢ sin x)(2C,Ss) 
and S,=(e cos'x)(2C,8,) + (¢ sin x)(C,?—S,2). 
Now ce cos x and ¢ sin x are constants, being, for Newlyn, —.0012 and .0120 
respectively. It is sufficient, therefore, to calculate C,?—S,? and 2C,S, at intervals 
of six (or even twelve) hours, and thence to evaluate C, and S, by linear inter- 
polation, ‘The hourly values of ¢, are then given by 
C, cos 60°¢ —S, sin 60°, 
and since cos 60°¢ is either +1 or +3, and sin 60° is either +.866 or zero, it is 
desirable to write down 340, and .866 §,; the calculation of ¢; is then a very simple 
matter. 
This method is superior in every way to either of the first two methods. 
§ 16. Analysis of observations.—Before proceeding to discuss the analysis of 
residues it is desirable to consider the principles governing the methods in vogue 
for the harmonic analysis of tidal observations. A full discussion of these is given 
by Professor Proudman in the Report for 1920, and a brief statement only is here 
required; the matter may be considered under three headings :—(1) the isolation 
and analysis of the principal solar series; (2) the ‘assignment’; (3) the length of 
record to be included. 
(1) If the constituent sought be of the principal solar series then it repeats itself - 
at intervals of twenty-four mean solar hours, or of some sub-multiple thereof. Hence 
the mean of the heights at intervals of twenty-four hours will tend to isolate the 
_ height due to the solar series of constituents at the given hour of the day. Other 
_ constituents will, in the long run, eliminate themselves. The twenty-four means so 
_ obtained may be submitted to harmonic analysis by the least square rule, and the 
_ separate constituents obtained. If ¢is the tidal height at time ¢, measured in mean 
solar hours, N the number of mean solar days included in the record submitted to 
analysis, 15° the speed in degrees per mean solar hour, then the whole of the 
_ processes give 
AAS 
| ° 24N ¢ 
; 
An = S C cos 15°nt Ws, 20°04) 
1 - 2 
Bn= ion = ¢ sin 15°nt (S15 2) 
(2) The ‘assignment’ is a process whereby heights at successive mean solar 
hours may be utilised in ‘special time.’ When the constituent sought is a principal 
solar constituent the method of analysis is that stated above: a similar process for 
any other constituent would involve a knowledge of heights at ‘special hours’ in 
1921 3 
