10 DISCUSSION OF TIDES IN BOSTON HARBOR. 
making all the rest depend upon theory, but to determine all that are of much importance practi- 
cally, so that they may be used in testing the accuracy of the general theoretical expressions, and 
for constructing approximate formule of prediction independently of any theoretical relations 
between the constants, or used by any investigator for verifying and improying any tidal theory. 
All the constants and relations being determined which are of theoretical importance, the most con- 
venient practical formulz will be constructed from the results for the prediction of the times and 
heights of high and low water, together with tables for facilitating the computations. 
The plan adopted for determining the constants from the observations is to apply Lubbock’s 
method of averages to circular arguments throughout, instead of to arguments of parallax and 
declination, and then to use the constants thus obtained to determine the constants belonging to 
any other forms of expression into which it may be thought advisable to put the results. As the 
quantities H,, and /, are the only ones which are directly observed, corresponding to any given 
time or values of the arguments, these must be determined from observations for all parts of the 
arguments separately by so grouping the observations that the effects belonging to all the other 
arguments are eliminated, and then, by means of the conditions (33), (37), (40), and (41), the con- 
stants belonging to each argument, as well as the general constants independent of any arguments, 
can be determined. 
To obtain the values of H,, and 4, belonging to the different parts of any argument 7, alone, all 
the observations within certain limits of the argument as from 7/; to 7//; have been grouped together, 
and the averages of all these have been taken as the normal values of H,, and 4,, belonging to the 
averages of all the corresponding values of the argument, which, when the number of the observa- 
tions is considerable, does not differ much from 43(7/;-+7/’,), the mean of the two limits. If these 
limits should be somewhat wide, a slight correction to the averages of the observations may be 
necessary, which is easily applied. When the observations extend over a long period, and have 
been regularly made, the effects of all the inequalities belonging to other arguments are completely 
eliminated, since the periods of all the other arguments being different, the observations falling 
within certain limits of any one argument, are equally distributed through all parts of the other 
arguments, and all the plus and minus effects cancel one another; and this is especially the case in 
a series of nineteen years, which is very nearly a multiple of the periods of all the principal argu- 
ments in the tidal expressions. In a long series, also, the inequalities due to the winds and baro- 
metric changes, and whatever other abnormal disturbances there may be, are very nearly eliminated. 
24, If the normal values of 2, and H, have been obtained from all the observations without 
regard to the arguments of the inequalities, then all the inequalities in 2, and H, depending upon _ 
these arguments disappear, as also A; and the inequalities of Aj, A,, and A; in (40) and (41). 
Putting 
—s (AAs) } HH, = 
My—4$ (Ao +1) 
we in this case obtain from (40), 
$(H’, + H’,) =H), + Ko + K; cos 5 4/ cos 38 @—K; sin 3 4’ sin 3 
4(H,— H’,) = K,—K; cos 3 4’ cos 3 q—K, cos 3 4! cos 3 
4(H, —H; )=K; cos 3 4’ 
ae — H,)=K; sin 3 4’ 
and, from (41), omitting 9 A, in comparison with 4 A, in these very small quantities, 
3 K, sin 3 4! 
4 Ky 
3K, cos 3 4! 
4 Ky 
Since sin q;=sin q;, and sin @=sin q, we have », ¢,.—=%7, and hence from the first of (37) we 
get Lp. —=+(5,4,—27) when 4, are all reckoned from the upper transit; but if two of them, as is 
iui 2 
(42) 
sin q@=— 
\ 
(43) < 
( 
usual, are reckoned from the lower transit, putting = —=12" 25™.24 in solar time, we get, since 
L.= By in this case, f 
(44) Bo=4(2y Ap —12" 25.24) —$ (0, 42/7 — 6» 12.62) 
which is the mean establishment of the port. 
