DISCUSSION OF TIDES IN BOSTON HARBOR. -43 
interval depending upon /’, which is the principal part, the remainder being generally quite small, 
so that it need be applied as a correction only to a few of the principal terms. 
46. If we develop the preceding expressions, as in the case of the potential of the disturbing 
force, substituting for p, v, p’, and v’, their expressions depending upon the angles 7;, we should 
obtain expressions of the resultant of the lunar and solar tides similar to those in the preceding 
pages, (25) and (27), obtained from the resultant of lunar and solar disturbing forces, of which the 
constants should be equivalent, and the coefficients of the inequalities of the latter, divided by 
(1+F), should be equal to the corresponding ones in the former. 
In the comparison of the constants we get 
(100) V1+e” 956 K=K, 
The preceding must be used as a condition to determine K in (96). 
In the comparison of the coefficients of the semi-monthly inequalities in the development of 
(97) and (98), using the value of e/ (92) with those given by observation and by (25) and (27), it is 
found that the coefficient of the inequality in heights is too great by 0™.7, and that of the lunitidal 
intervals is too small by 1™.5. The theory with regard to this last expression and development 
seems to be in error by these amounts. But as our object here is merely to get the most convenient 
expression which will represent the observations with sufficient accuracy for practical purposes, 
this can be ‘obtained by changing a little the constants as obtained from the former conditions. If 
we take 
(101) f= 205, m=1.530, n=.410, F=.500 
with these values in (96), (97) will give the principal inequalities in the heights, except the effects 
of F, within 0™.2, and all the others with about the same accuracy as (25), and (97) will give such 
a value of (/ as, substituted in (99), will give the coefficients of the intervals within 0™.5, except 
the few discrepancies already mentioned in the comparisons with the expression of (27). The 
preceding constants, however, in this case do not quite have the relations to one another required 
by theory in the other developments. 
With the preceding value of e’, and the value of K, (60), (100) gives for the constant in (96), 
(102) : K=5.004 feet. 
47. Having obtained M and § from (96), the coefficient of the tide Q and the value of (7 in 
(97) and (98) are readily obtained by construction as follows: 
Take A B equal to M, and BC equal to S, making an angle with AB, bp 
produced to D, equal to 7,—«, which is twice y—y’ at a time x, previous to 
the time of high or low water, and then join AC. The latter is Q (97), or lp S< 
the coefficient of the tide, neglecting the effect of F. The angle B A C also BS 
is the value of 4’. Of course the same are readily obtained by a trig- \ 
onometrical calculation. One-half 5’, reduced to time and increased by 5th, 
is the part of the lunitidal interval in solar time depending upon /’. 
The values of M and § (96) contain only the parallaxes and declinations 
of the moon and sun as arguments, and very simple and convenient tables f 
may be constructed giving their values for any given arguments; and then 
by a very simple construction, or trigonometrical calculation, we get the 
coefficient of the tide and the principal part of the interval, and thus take 
in completely all the numerous terms arising from any developed form of 
expression. These terms do not consist only of the terms corresponding to - 
the terms given in the development of the disturbing forces in (12) and = |— 
(18), but a great many others of the same order as many given there. The 
coefficient of the tide thus obtained must be diminished by one-third of the 
inequality from the mean tide for the effect of friction depending upon 
F, which is the same as dividing the inequality by (1+-F). 
It now remains to determine the part of L depending upon / in (99), 4 
which is the lunitidal interval of the lunar tide. The constant of this is. B, determined by 
