26 DISCUSSION OF TIDES IN BOSTON HARBOR. 
the preceding, it is seen that K, and K, given by observation are less than the theoretical values 
while that of K,, the mean coefficient of the semi-diurnal tide, is nearly ten times greater. But in 
applying the equilibrium theory to the real case of nature, it has been usual to determine such 
constants as make the expressions best represent the observations instead of determining them from 
theory, and to depend upon the theory for the ratios of the inequalities to these constants. In the 
equilibrium theory, and also in the dynamic theory in the case of oscillations of mean level, we 
should have R,—P,. By comparing the preceding values of R, with those of P, (12) it is seen that 
while P, is wanting, R;=.504, and also that the value of R,. is greater than that of P,. The three 
values of (R,—P,) K, in these three cases give respectively —0.059 ft., —0.020 ft., and +0.003 ft. 
as the coefficients of inequalities in the mean level, belonging respectively to the arguments 71, 72, 
and 73, for which there are no corresponding inequalities in the disturbing forces. A semi-monthly 
inequality of mean level, corresponding with the first of the preceding, and with the same sign, 
though frequently much greater, has usually been found in all tidal discussions. 
The parts of the preceding inequalities without any corresponding disturbing forces are, no 
doubt, the effects of a quarter-day tide, which, with observations of high and low waters only, there 
are no means of separating from the semi-diurnal tide, and are not inequalities in the true mean 
level (§ 27); and the preceding inequalities are merely the inequalities in this tide, which varies 
with the semi-diurnal tide, and the effect of the constant part of this tide is contained in the value 
of H’,. Upon this hypothesis, if we assume the existence of a quarter-day tide with a coefficient of 
about three inches, it would account for these inequalities within the limits of the errors of observa- 
tion. We have already had indications of the existence of such a tide elsewhere (§ 34). The exact 
coefficient of such a tide can only be determined from observations made several times during the 
phase of the tide. 
If we compare the values of R, (83) with that of P, in (12), we see that the latter is a very 
inconsiderable part of the former, and that the difference corresponds to an annual inequality of 
mean level with a range of about four inches, for which there is no corresponding disturbing force. 
The whole of this inequality is given for the middle of each month in the last column of Table 
Vil. The maximum of this inequality occurs in October or November and the minimum in Feb- 
ruary. Such an inequality has been found at other ports. At Brest it is a little greater, with the 
maximum and minimum occurring a little later in the year. Dr. Bache found, from the discussion 
of the tides at Key West, an annual inequality with a range of about nine inches, and with the 
maximum in September and the minimum in February. 
These results should not be regarded as being at variance with the general tidal theory, but 
merely as being the effects of some circumstances or causes not taken into account in the theory ; 
and these effects are, no doubt, due to the annual changes in the currents of the ocean, produced by 
annual changes of temperature and of the winds. On account of the influence of the earth’s rotation 
there cannot be an annual change in the velocity or position of the currents of the ocean without 
a corresponding change generally in the mean level of the ocean at any port. The preceding results 
are very interesting in connection with this subject. I endeavored to give a full explanation of 
these inequalities, a few years ago, in the Proceedings of the American Academy of Arts and Sciences, 
Vol. VII, p. 31. 
By comparing the values of R, (85) with those of P, (18), it is seen that they differ very 
much, and consequently the equilibrium theory, applied to the Boston tides, gives very erroneous 
relations between the inequalities and the mean tide. While R;—.1388, we have P, —.4240, and 
hence the equilibrium theory would make the semi-monthly inequality in heights more than three 
times greater than it is. In the same way it is seen that it makes the inequality depending upon 
the moon’s parallax too small, while it makes that depending upon the moon’s longitude, or declina- 
tion, more than four times greater. 
37. In the equilibrium theory the lunitidal interval is expressed by 2 he (22), and the coeffi- 
a 
cients of the inequalities in the development by Q, (23). The mean establishment, referred to the 
nearest transit, should be 0, which does differ much from observation at Boston. But if we compare 
the preceding value of B, with that of Q, (23), we see that the equilibrium theory gives nearly two and 
a half times that of observation for the coefficient of the semi-monthly inequality. For the observed 
