DISCUSSION OF TIDES IN BOSTON HARBOR. 41 
sion, the corresponding changes when small may be regarded as exactly proportional. This is 
equivalent to supposing that in the development of the tidal coefficient K, which is a function of 0s 
by Taylor’s theorem, only the first term depending upon 47; of which the coefficient is D; K or E, is 
sensible, and that all the others, depending upon the square and higher powers of 4%, may be 
neglected. ‘This is, no doubt, true of Brest, and of most Huropean ports, but we are hardly safe in 
assuming that it is strictly true in the case of the Boston tides; for we have seen that the effect 
of the first term, that depending upon BH, is a quantity of the first order of the inequalities, and in 
the ease of the semi-monthly inequality amounts to more than a fourth part of the whole tidal 
coefficient, and hence it can hardly be supposed that the second term depending upon (di)? is 
entirely insensible. If such terms are sensible they must cause slight deviations of observation 
from theory, and also affeet the determination of the moon’s mass. The smallness of the residuals, 
however, do not indicate that the effect of such terms, if at all sensible, can be of much consequence. 
PREDICTION FORMULA AND TABLES. 
43. With the preceding tidal expressions the heights and times of the tides may be computed 
for any given time; but although these expressions are in a form most suitable for the discussion 
of the tide observations and comparisons with theory, and for the study and investigation of the 
tidal theory, yet on account of the great number of arguments which would have to be used, and 
number of terms taken in, resulting from the developments, the whole result can be put into a 
more suitable form for prediction, containing but few arguments. For this purpose we shall 
determine from the constants belonging to the resultant tides of the moon and sun, which have been 
obtained directly from comparisons of observation with the tidal expressions, certain constants 
belonging to the lunar and solar tides considered separately, and then combine these separate tides 
into a form of expression similar to that of the resultant expression of the potentials of the 
disturbing forces of the moon and sun, (5) and (6). 
By neglecting the effect of friction depending upon F, which decreases the larger tides a little 
more in proportion than the smaller ones, the first of (25) may be used, with the value of E which 
we have obtained, in determining the relative magnitudes of the lanar and solar tides. In this 
case the quantity corresponding to U; is D; 7,=.426, twice the difference of the velocities of the 
moon and sun in right ascension. » If we therefore put e/ = the ratio between the solar and lunar 
tide, since ¢ (8) is the ratio between the solar and lunar disturbing forces, we shall have 
(92) ¢ =e (1—.426 E)—0.180 
using the values of e, ov, and E in (10) and (90). Hence the-solar tide is decreased in the ratio of 1 
to 1—.426.E, or of 1 to .401, in consequence of the sun’s having a slower motion in right ascension 
than the moon. 
44, If we now consider the lunar tide alone, omitting that part of the effect of friction which 
depends upon F, since in this case the inequalities in the disturbing force depend upon the moon’s 
parallax and declination only, we shall have in (17) 
(93) 3 P; cos n=O(14"? a — sin’ vy) 
in which C is a constant and p is the mean parallax, 0p is the excess of the parallax above the 
mean parallax, and v is the moon’s declination, as heretofore defined. In this case the values of 
P; in the first member, and the corresponding values of Uj, are different from those already given 
in the case of the resultants of the potentials, and are found to be 
P,=.0250, C—O Os} 
(94) P,==.1638, U,—=— .0500 
P;=.0860, Cj 0397 
omitting the terms depending upon the moon’s node and other small secondary terms. With these 
values the first of (25) gives, omitting F, 
R,=.1638 + .0503 x 1.408 —=.2342 
(99) R3—=.0860—.0397 x 1.408—=.0302 
