DISCUSSION OF TIDES IN BOSTON HARBOR. 37 
inequality also of 6™.2 depending upon 7, the equilibrium theory gives none; and throughout the 
smaller coefficients there are large proportionate differences between the theory and observation. 
38. The values of the angles of epoch in the equilibrium theory should be 0 when referred to the 
nearest transit. The values of a and 22 (83) both nearly vanish for the transit C occurring two lunar 
days earlier, and the value of a; would vanish for a transit nearly a day and a half earlier. This 
also applies to the dynamic theory. But it has been shown that these inequalities, except a small 
part of the second and third, do not depend upon any corresponding term in the disturbing force, 
but are probably the effects of quarter-day tides, resulting from the circumstance of a shallow sea, 
and depending upon the magnitude and epoch of the semi-diurnal tide. Hence the values of the 
angles of the epoch should correspond somewhat with those of the semi-diurnal tide in (85), which 
they do as nearly as could be expected, since it is impossible to determine the values of the angles 
accurately for so small inequalities. The preceding comparisons are sufficient to show how inad- 
equate the equilibrium theory is to represent the observed inequalities of the Boston tides. 
DETERMINATION OF THE GENERAL CONSTANTS. 
39. It is now proposed to determine the constants in the general tidal expressions (25) and (27) 
These being known, these expressions then give the special constants belonging to each inequality. 
Among the constants to be determined is the correction of the moon’s mass, dy, contained in the 
expressions of P;, U;, and Q;, (18) and (27). The diurnal tide in the port of Boston being very small, 
only the coefficient of the principal inequality has been determined from observation, and conse- 
quently we have no means of forming conditions enough to determine these general constants 
belonging to this tide; and they are of no consequence, since the effects depending upon them in 
this case must be insensible. We can, therefore, only determine these constants for the semi-diurnal 
tide. 
With the values of R; (85) belonging to the first three inequalities, and the corresponding values 
of P; and U;, the first of (25) gives the following conditions : 
1388 (1+ F) =.4305 — 24.0 0 4 —(.1742 — 18.2 0 ») B 
(88) 1624 (14+ F)=.15214 3.604 .0500 i 
0225 (14+F)=.0985 4+ 1.00 —(.0477—0.5 0 »)E 
Also with the values of M,=B; cos (¢;—«;), belonging to the same inequalities, the third of (27) 
gives 
— 22™,9 — 52". 51.4034 6 w+ (21".75 —1210 0 »)E—0™.6 
(89) 4 — +( 4°.10-+ 187 6 4)E—0.3 
5™.3—— 2",24+ 148 d0p4+( 57.164 5004) K—O0.1 
By giving proper relative weights to these two sets of conditions from the heights and from 
the times, we can combine them in the determination of the constants, and thus obtain the most 
probable values which all the conditions give, and get some idea of their probable errors. These 
six are the only conditions which can be formed having much weight in the determination. 
A solution of the first three conditions, (88), depending upon the heights of the tides, gives 
(90) 6 p=—.000283, E=1.408, and F=.365 
With this value of 6 1, (9) gives for the moon’s mass, 
p=.013— .000283—.012717 = aa 
In order to determine the most probable value of the three preceding constants belonging to 
all of the preceding conditions, we shall multiply the first three by 300 and then substitute the pre- 
ceding values plus the correction belonging to the new conditions. We thus get 
—1650 4 »—51.60 4E—41.644F= 0 00 
+1080 4 2415.00 4 B—48.72 4F= 0 B\ + .02 
MO Apis ees 0 © je ae 
4.9333 4421.75 4E = 08 9 s\2o 
4+ 2624 p+ 4104E BED 208 f' 221.03 
+ 28 4p4 5.1645 = 03 + 0.29 
