38 DISCUSSION OF TIDES IN BOSTON HARBOR. 
The solution of these by the method of least squares gives 
4 p= .000120 + .00021, 4 H——.005, and 4 F=.001 
with the residuals given above. The first three residuals divided by 300 and multiplied by 60 
inches, the coefficient nearly of the mean tide, give the real residuals in inches, the greatest of which 
is only .024 inch. The last three are the residuals belonging to the times in minutes. The multi- 
plication of the conditions from the heights by 300 makes the relation between the two kinds of 
residuals in general about the same as that of the probable errors of the two kinds of observations. 
With the preceding corrections we get for the constants belonging to the new set of conditions, 
n—=0.012717 + (.000120-4.00021) — + 
~ T7941.8 
(aa eenGeiee 
F—0.365 + .001 —0.366 
(91) 
The preceding residuals show the accuracy with which theory represents the three principal 
inequalities of the heights and the times. This may be somewhat accidental in this case, and it 
remains yet to be determined by an application of the theory to other ports, whether it will repre- 
sent the inequalities of the times with such accuracy that conditions deduced from them should 
have any weight in the determination of the moon’s mass. The preceding probable error, obtained 
from so few conditions, cannot be relied upon as showing with much certainty the real probable 
error of such determinations in general. 
The preceding value of E for the port of Boston is four or five times as great as it is in most 
Kuropean ports. This extraordinary value, in the first of (25), diminishes the values of R, and R;, 
and consequently the coefficients of the first and third inequalities, to less than one-third of what 
they would be by the equilibrium theory, but increases the second inequality, since U, is negative, 
and makes it greater than what the equilibrium theory would require, and even greater than the 
semi-monthly inequality. In like manner this large value of E, in the third of (27), diminishes the 
coefficient, and consequently the range, of the first or semi-monthly inequality in the lunitidal 
intervals, so that at Boston it is only about half as much as in European ports generally, and less 
than half of what the equilibrium theory requires. 
The value of F above, in the first of (25), tends to decrease all the inequalities, and in the first 
and third is in the same direction with the effect of the term depending upon E; but in the second 
inequality the effects of the two terms are in contrary directions, but that depending upon E is the 
greater, and consequently the second inequality is greater than the conditions of a static equilibrium _ 
would require. Whether the effect of the term depending upon F is due to friction, as we have 
supposed, (§14), or to some other cause, it is evident that the preceding conditions cannot be satis- 
fied without such a term. 
40. In the equilibrium theory both E and F vanish, and in this case the first of either (88) or 
(89) furnishes a condition for determining the correction of the moon’s mass, the former from the 
heights, and the Jatter from the times. It is well known that this theory, even in European ports, 
where the deviation from the relations of the equilibrium theory is comparatively small, gives very 
unreliable determinations of the moon’s mass; but in the port of Boston it would give a mass more 
than double the true mass, as may be seen from a mere inspection of the conditions in this case. 
The preceding conditions are based upon the hypothesis of a small correction only, and consequently 
fail in this case, except that they indicate that it must be very large. 
In the equilibrium theory only one condition is necessary for determining the moon’s mass, 
which can be based upon the first and principal inequality. Where the term depending upon E 
has a sensible effect, a second condition is necessary, which may depend either upon the second or 
third inequalities. Laplace, in his last tidal investigations in the fifth volume of the Mécanique 
Celeste, used the first and third inequalities. Airy, in the discussion of the tides of Ireland, formed 
conditions from the first and second inequalities, but obtained a mass very much too great. When 
the terms depending upon F in the preceding conditions have a sensible value, it is readily seen 
that these two sets of conditions must give very different results, and that the conditions based 
upon the first and third inequalities must be much better than those based upon the first and 
