HARMONIC ANALYSIS OF TIDAL OBSERVATIONS. 79 
first day. Strictly speaking, w should be taken as the same function of 
the longitude of the moon’s node, varying as the node moves; but as the 
variation is but small in the course of a year, uw may be treated as 
a constant and put equal to an average value for the year, which average 
value is taken as the true value of w at exactly mid-year. Together 
V+u constitutes that function which has been tabulated as ‘the argu- 
ment’ in the schedules B, C, F, H. 
Since V+w are together the whole argument according to the 
equilibrium theory of tides, with sea covering the whole earth, it follows 
that «/n is the lagging of ths tide which arises from kinetic ‘action, 
friction of the water, imperfect elasticity of the earth, and the distribution 
of land. 
It is supposed that H is the mean value in British feet of the semi- 
range of the particular tide in question. 
f is a numerical factor of augmentation or diminution, due to the 
variability of the obliquity of the lunar orbit. The value of f is the 
ratio of ‘the coefficient’ in the column of coefficients of the preceding 
schedules to the mean value of the same term. Tor example, for all the 
solar tides f is unity, and for the principal lunar tide M,, f is equal to 
cos‘ 41 /cos* 5w cost di; for as we shall see below, the mean value of this 
term has a coefficient cos* }w cost 47. 
It is obvious, then, that, if the tidal observations are consistent from 
year to year, H and « should come out the same from each year’s reduc- 
tions. It is only when the results are presented in such a form as this 
that it will be possible to judge whether the harmonic analysis is pre- 
senting us with satisfactory results. This mode of giving the tidal 
results is also essential for the use of the tide-predicting machine. 
We must now show how to determine H and « from R and ¢. 
It is clear that H=R/f, and the mode of determination of f from the 
schedules bas been explained above, although the proof has been deferred. 
If V, be the value of V at 0» of the first day, then clearly 
—f=V,tu—r. 
So that 
k=C€4+V,4+4. 
Thus the rule for the determination of « is: Adil to the value of ¢ the 
value of the argument at 0 of the first day. 
It is suggested that it will henceforth be advisable to tabulate Rand (, 
80 as to give the results of harmonic analysis in the form R cos (nt—¢) ; 
and also H and x, so as to give it in the form fH cos (V+u—«c), when 
the results will be comparable from year to year. 
A third method of presenting tidal results will be very valuable for 
the discussion of the theory of tidal oscillations, although it is doubtful 
whether it will at present be worth while to tabulate the results in this 
proposed form. This method is to substitute for the H of the second 
method FK, where F is the mean value of the coefficient as tabulated in 
the column of coefficients in the schedules—for example, in the case of My 
we should have F=} (1 —4e?) cos! }w cos‘ $7, and in the case of S, we 
should have F=4 .5 cost}. When this process is carried out it will 
enable us to compare together the several K’s corresponding to each of 
the three classes of tides, but not the several classes inter se. 
