DISCUSSION OF TIDES IN BOSTON HARBOR. OR 
From the last two of (42)-we get 
§K K; cos 3 4’ = $(5.0 
( K; sin 3 4’ =4$(5.2 
0 — 5.062) = .004 
\ 
es) 5.257 
7 
57 — ) =.000 
Hence 4’—0 and K;—.004 ft. This is the value of the constant or mean tertio-diurnal tide, and 
may be regarded as falling within the limits of the errors of observation, and consequently insensi- 
ble. 
With the preceding values of K;, which is the constant and principal part of Aj, and q,, the 
terms in the first of (42) are entirely insensible. We therefore obtain from (45), with the preceding 
mean values of H’; and H’,, 
(59) HH’, = 4(25.066+15.257) — 20.161 ft. 
for the mean height of the sea above the assumed zero of the tide-gauge. This, however, is not 
necessarily the same as the mean level obtained from observations made frequently at all times 
during the day, unless the tides follow the law of sines and cosines, or at least the parts above and 
below the mean level are symmetrical. It is simply the mean of the heights of high and low waters. 
We obtain from the second of (42), since the terms depending upon K; are insensible, 
(60) K, = $(25.066 — 15.257) = 4.904 ft. 
for the coefficient of the average or mean semi-diurnal tide. Consequently the mean range is 9.808 ft. 
With the preceding values of K;, 4,’, and Ky, (43) gives 
n=09, 2=37, 3 =— 0™.13, Qs = $7+0".13 
These values from the formule depending upon the eights are almost exactly the same as those 
obtained above from the observed times alone, (57). 
THE SEMI-MONTHLY INEQUALITY. 
28. If from each of the values of 2;, in the preceding table, we subtract By-+q,;—0" 42™.29, 
omitting the constant two days, and from each value of 2.’ we subtract Bo+q,.—k—6" 54™.84 
+0".4 cos 71, and call the differences 6 L, (46) and (47) will give, with these values of 5L and the 
corresponding values of the argument 7;—=2()—’), forty-eight equations of condition of the form 
6 L=M;, sin 71+ N;, Cos 41+ Myo Sin 19+ Nyo COS 710 
the angles 7; and 79 being the only ones included in the same argument. The tabular values of 
7, must be reduced to the time of low water for the twenty-four conditions obtained, from the low 
waters, by adding the mean change of 7, from high to low water, which in this case is 25™.24 
+0™.4 cos 7, the small term 0™.4 cos 7; being an inequality in the moon’s motion depending upon 
the argument of variation, which is the same as 7 or 2(~—y’). With these forty-eight equations 
we obtain, by the method of least squares, 
My, =—22™.25, N,=0".01, My=1™.96, Nj =—0™.37 
These values satisfy the forty- aieetht conditions, with an average residual of 0™.35 and a maxi- 
mum residual of 1.3. The residuals do not indicate any sensible term depending upon 37. With 
these values (48) gives a 
B, =— 220.5 = 09 
(60) : P i 
Bp= 2.0, £9) = 5° 
These comprise the constants belonging to two of the terms of the expression of Ls, (26). 
From (28) we get 
(61) T= Bp = 22.03 
for the age of the tide from the times. 
If, trom each value of H’, in the preceding table, we subtract H/,+ K,— 25.066 ft., (59) and (60), 
and also from each value of H’, we subtract H/)—K,—15.257 ft., aud call the residuals 6 H, (50) 
gives forty-eight equations of the form 
0 H= Ky (Mj cos 7) +-N; sin 7)+ Ky (M, cos 71+ N, sin 41+ Mig COS 419+ Nio Sin 7430) 
in which the minus sign belongs to the conditions obtained from low waters, and the tabular values 
of 7, for these conditions must be reduced to the time of low water as above. With these condi- 
tions we obtain, by the method of least squares, 
62 Ky My =—0.059 ft., K, M;= + 0.670 ft., K, Mj) =— 0.023 ft. 
) Ky Ni= 0.000 ft., Kk, Ni; =—0.124 ft., K, Nj —=+ 0.001 ft. 
