DISCUSSION OF TIDES IN BOSTON HARBOR. fell 
In the first two equations of (40), the terms depending upon A; being the products of two fac- 
tors, which are both generally small, are, for the most part, entirely insensible, and may be omitted. 
If we then put 
H’, =the mean height of the sea above the assumed zero plane, 
we Shall have 
(45) A =H)+ Ky) =4 (4-H) 
25, If we obtain the values of 2, for any values of 7; by grouping the observations in such a 
manner that all the inequalities belonging to the other arguments are eliminated, we have from (26) 
for high waters, 
(46) 4, —Bo— 1 = 5; B; sin (4, —=;) =; (My sin 7;+N; cos 7) 
and for low waters, 
(47) 4,— Bo — p+k= §; B sin (4; —2;) = 5; (M, sin 7,N; cos 7;) 
in which : 
(48) BV MP=EN} and tan ¢,—— Mh 
M; 
In these expressions +; includes only the terms the angles ef which may be included in the 
same argument; that is, the angles which are multiples of the first. The value of & in (34) must be 
used, putting »—4 and taking only the inequality belonging to 7;. 
Haying obtained from observations m values of 4% or 2/,, or of both, corresponding to m values 
of the argument, the preceding equations give m conditions for determining, by the method of least 
squares, the value of the constants B; and ¢. 
If we in like manner obtain the values of H,, for any values of 7;, we obtain from the first and 
third of (40), omitting the small terms just referred to (§ 24), 
H,— H)=Ao+ A> for high waters, 
and from the second and fourth of (40), 
H’,— H)= A,— A, for low waters. 
But in this case we have (24), 
Ay=Ky+ Ky 5; Ri cos (7; —a,) —=Ky+ Ky 3; (M; cos 7,4 N; sin 7;) 
A ,—=K2+K, 3 Rj cos (7; — 21) =K,+ Ky 5; (M, cos 7;+N; sin 7;) 
in which 4; is limited as above, and in which 
bs 
(49) Ri = V M?+N, and tan a,= = 
The values of R;, M;, We., differ in their different connections with K, and K,. From the pre- 
ceding we get, for high waters, 
5 H’,— H,— K,=K, 5; (Mj cos 4;-+N; sin y;)+ Ky ¥, (Mj cos 7;4+N; sin 7;) 
( lat —)a + KO IKey x (MM cos nitN; sin 7i)—Ke a (M; cos nitN;, sin 7i) 
With m values of H’, and H’,, corresponding to m values of the argument, we have, from the 
preceding, m equations of condition for determining the constants Rg, i) and ag i). 
From the first of (87) and from (41) we obtain, when 2; and A, are reckoned from the lower 
transit, 
¥ § 4 (4y—/53)—= sin hh 
(51) TL pp weed 
U5 (21) =— cos w 
i 
(50) 
— 
Ss 
when A, is small in comparison with Ay. 
We also obtain from (40), in the case in which A, is not eliminated in the grouping of the 
observations, 
oe ; $(Hi—H;)= + Aj,cos44+A,;cos3 4 
Se 4 (H,—H,) =— A; sin 4+ A, sin 3-4 
In these equations A;is always so small that the values of K, and 4’, obtained by the last two 
conditions of (42) for the constant and principal part of Aj, can be used without any sensible error. 
With m values of 2, for both high and low waters of both transits, (51) and (41) give m equa- 
