DISCUSSION OF TIDES IN BOSTON HARBOR. ; 29 
two tables. If from each value of 2, in the preceding table we subtract By+q,—=0" 42™.24, as 
obtained from the preceding table, and from each value of 2’, we take By+q,—k—6" 54™.84 
+1™.5 cos 72, putting 0 L for the residuals, (46) and (47) give forty-eight equations of the form 
OM, sin jo+No cos qo+ My sin qutNu COS Ai 
the angles 7, and 7, being included in the same argument. In this case the value of k (34) includes 
the constant and the inequality depending upon 7», the value of » being 4 as before. In this case, 
to obtain the values of 7 belonging to low water, we must add 3°9.3+0°.5 cos 72 to the tabular 
values of 7 given for high water. These conditions give, by the method of least squares, 
M, ==} 43) Nz = om2, My —— 0™.6, Nu — 02.8 
These values satisfy the conditions, with an average residual of 0.3 and a maximum residual ot 
0™.9. With these values (48) gives 
( By —6™.2, Gy ==> 27 
(By=1™0, 41 123° 
The preceding are the values of ¢ and < when 7 is given for a time two lunar days after the 
transit C (§ 26). In order to reduce them to the case in which 7» is given for the time of transit C, 
we must subtract the mean changes of 7) and 7, in two lunar days, which is 27°.1 in the former 
and twice that, or 54°.2, in the latter. Hence we get, in this case, 
(66) eg— 45°.6, ey—— 119 
(65) 
The preceding are the constants belonging to two more of the terms in the expression of L; (26). 
Tf, now, as in the preceding case, we subtract H’,+ K, from each value of H’; in the preceding 
table, and H’/,—K, from each value of H’,, with these forty eight residuals and the corresponding 
values of 7, (50) gives forty-eight equations of the form 
O EKG (M, cos no+N, sin q2)+ K, (M, cos jo+N2 sin jotMi cos jutNn sin 711) 
in which the minus sign belongs to low waters. From these forty-eight conditions we obtain 
Ky M,=— 0.033 ft., K, M,=+ 0.771 ft., K, My —-+ 0.051 ft. 
Ky N,=—0.014 ft., K, No=-4+ 0.204 ft., K, Ny; =-+ 0.014 ft. 
With these values (49) gives 
K, R, =— 0.036 ft., Ro, 2) =—0.308, a0,2) = 38° 
(67) K, R, =+ 0.797 ft., Re, 2) =0.1624, (2, 2) —==29° 51! 
K, ti i= + 0.053 ft., Re 11) = 0.0107, (2, 11) — 450 
These values satisfy the conditions, with an average residual of 0.020 ft. and a maximum resid- 
ual of 0.049 ft. These are a part of the constants belonging to the terms in the expressions of Ao 
and A, (24). 
In order to reduce the preceding values of the angle of epoch to transit C, we must subtract 
27°.1 from the first two and 54°.2 from the last one. 
From (28) we get, with the reduced value of a.=299.9 — 27°.1, 
(68) T)—= 24.034 a et 
INEQUALITY DEPENDING UPON THE MOON’S LONGITUDE. 
30. By taking the average of all the values of 2 and H of each transit in Tables I and II, and 
then taking the half sum and the half difference of the values of the two transits, we get the follow- 
ing table of average values belonging to the given longitude, from which the inequality depending 
upon (}—y’) is eliminated, and consequently they contain only the inequality depending upon the 
moon’s longitude. 
