DISCUSSION OF TIDES IN BOSTON HARBOR. ilk 
Since the range of argument belonging to each group of observations is twice as great in this 
case as in the other cases, the coefficients of the inequalities, as obtained, are increased in the ratio 
of the sine to the are of the half range of the groups of observations. An explanation of this small 
correction has been given in (§ 23). This very small correction is insensible in the other cases. 
These constants satisfy the conditions, with an average residual of .016 ft. and a maximum ot 
.057 ft. As the preceding inequalities of the first degree are so small, those of the second degree 
depending upon 273; must be very small and may be neglected. 
From (28) we get 
113 
7 pS — == =—14,78 
(71) 3 ~ 460 78 
Since the maximum of the small tide depending upon the fourth power of the moon’s distance 
does not necessarily happen at the same time as that of the principal part of the semi-diurnal tide, 
the preceding value of K, R” represents the height of that tide at the time of the high water of the 
resultant tide. Hence, putting 
4’ = the lunitidal interval of the small tide ; 
q'’ =the time of the resultant high water after that of the mean semi-diurnal tide ; 
K, a= the coefficient of the tide ; 
we have 
1+ RY = V 12 a?222)aicos (ly — 2”) —1-+- a cos (Ly, —2”) nearly 
@ sin (Liy— 2”) 
1+<a cos (lL, —2”) 
tan q/ = =a sin (Ll, — 4”) nearly 
q 5 
From the preceding values of B” and <” and of R” and a’, we get 
q’ =— 2.5 sin (¢g+349) 
R’” = .0065 sin(¢+77°) 
These values of q’’ and R” cannot satisfy the preceding equations unless the angles are the 
same, whereas they differ 43°. But since the coefficient K, R’ of the small tide from which the 
value of a’ has been determined is only a small fraction of an inch, the discrepancy may be regarded 
as falling within the limits of the errors of observation. Assuming that the angles are equal, we 
then have, at the time of the maximum, q//—=— 2™.5, and R/’=.0065. 
With these values the preceding equations give 
.0065= a cos(L2—2”’) 
0218 =—a sin (Ly — 2”) 
from which we get 
(72) a= .023, L, — A’ =— 73° 
Hence K,a—0.115 ft. is the coefficient of the tide, and 73° expressed in solar time, which is about 
24 hours, is the time the high water of this small tide precedes the time of the high water of the 
principal tide. Hence the high water of this tide occurs at 11% 26™.5 —2> 30™= 8 56”.5, or about 9 
o’clock in lunar time. We also have in this case 
(73) Bi = 24 OF 49m — 2h 30™ = 14 238 12m 
for the mean establishment. 
INEQUALITIES DEPENDING UPON THE SUN’S ANOMALY AND LONGITUDE. 
31. From the means in the footings of Table IV we get the following table of average values, 
corresponding to the given values of v/ and 2¢’ belonging to the middle of each month: 
