DISCUSSION OF TIDES IN BOSTON HARBOR. 39 
second, since the effects of the two terms depending upon E and F are in the same direction in the 
former conditions, and somewhat in the same proportion, while in the latter they are in contrary 
directions. 
When the terms depending upon F are sensible, there are three quantities to be determined, 
and consequently three conditions are necessary, and two of these conditions, if the heights alone 
are used, have to depend upon the small parallactic and declination inequalities belonging to the 
moon. Where circumstances are such as to make the terms depending upon EK and F small, as at 
Brest, the conditions are sufficient to determine them with adequate precision to get a pretty accu- 
rate determination of the moon’s mass; but in the case of the Boston tides these become terms of a 
first order, which, in a great measure, destroy the first and third inequalities, upon which the 
determination mainly depends, so that the problem becomes nearly indeterminate, and the condi- 
tions are not sufficient to give a reliable determination of the moon’s mass; for the small semi- 
monthly inequality observed is not wholly due to the solar tide, which is a kind of base in the 
determination, but a considerable part of it belongs to that term in the moon’s parallax depending 
upon the argument of variation, and which gives no advantage in forming the conditions. 
By the preceding method, in which conditions from the times are taken in, which our tidal 
expressions enable us to do, the great weight of the determination which rests upon the small 
declination inequality, which, in the case of the Boston tides, is reduced to 1.3 inches, is thrown upon 
the principal semi-monthly inequality of the lunitidal intervals, which is 22™.6, and which can be 
observed with much greater accuracy, proportionately, than 1.3 inches in the heights. The two other 
conditions from the times also give some little additional weight to the determination. The accu- 
racy of the determination, of course, depends very much upon the accuracy of the tidal expressions, 
and the probable error-may be greater than that which we have obtained from so few conditions, 
but still I think the determination is entitled to considerable weight. But it is very evident, from 
what has been stated, that the Boston tides are not at all favorable for an accurate determination 
of the moon’s mass, and that the same magnitude of errors in theory or observation makes the 
probable error of the determination several times greater than in the case of tides, as those of 
Brest, in which the magnitudes of the inequalities differ but little from what the equilibrium theory 
would give. 
COMPARISONS WITH THE DYNAMIC THEORY. 
41. In the oscillations of the first kind, which are oscillations of long period, the terms depend- 
ing upon BH, F, and G vanish, and the dynamic corresponds with the equilibrium theory, with which 
comparisons have already been made. The diurnal tides of Boston being very small, only the con- 
stants of the principal term depending upon the moon’s longitude have been determined from the 
observations; and, as it requires at least two conditions to determine the constants in the expres- 
sions (25) and (27) for each kind of oscillation, we have no means of making any comparisons of 
results obtained from the observations with those given by these expressions. The constants E and 
F having been determined for the semi-diurnal tide, (90), the first of (25) should give the value of 
R, for each of the inequalities of which we know the value of P;; and K, being known we then have 
K, R;, the coefficients of the inequalities. The third of (27) should likewise give M;—=5, cos(H,—4;). 
Three of the values of R; and M,, obtained from observation, belonging to the three principal ine- 
qualities, have been used in the six conditions by which the constants have been determined, and, 
from the smallness of the residuals, it is seen that, so far as these three inequalities are concerned, 
the observations are well represented by theory. 
The values of z, obtained from (28), with the angles of epoch a;, should all be equal, according 
to theory, taken in terms of a first order only. But, we see from (64), (68), and (71), these values 
differ considerably at Boston. Neglected terms of a second order, which are very large at Boston, 
are, no doubt, sensible in this case. It is difficult, also, to obtain the angles of epoch of small ine- 
qualities of long period with much accuracy from the observations, but in this case the differences 
seem to be rather great to be attributed to errors of observation. The other inequalities are too 
small to give a reliable value of c. 
The values of the angles of epoch E,, in the inequalities of the intervals, should be given by the 
second of (27), but no value for EF’ in the expression of N; can be obtained which will represent them 
