DISCUSSION OF TIDES IN BOSTON HARBOR. ; 9) 
22, The preceding are the tidal expressions belonging to the different kinds of oscillation taken 
separately ; but for a comparison of observations with theory, and also for the purpose of predic- 
tion formule, it is necessary to have expressions of the height of the tide and _of the lunitidal inter- 
vals belonging to the resultant of all the oscillations. If we put 
H,=the height of the mean leyel of the sea aboye any assumed zero plane, in the case 
of no disturbing force, 
then the expression of the height of high water at any time is H,+%.Y,;. If we put 
H,=the absolute height of the tide at the nth high or low water, n=1 belonging to the 
high water depending upon the upper transit ; 
1,=the corresponding lunitidal interval; 
and also put 
9 USE (fi 
(37) Al === Wp— Dy, 
k AMT =U 
we get, by combining the oscillations, 
(38) } Hy= H)+Ao+ A; cos 4 cos g, — A; Sin 4 Sin qy-++ Ay cos 2 Gy 
+A; 0S 3 4/ GOS 3 G,— Az Sin 5 4/ sin 3 qy 
in which q, must satisfy the conditions 
(39) 0—A, cos 4 sin q,+ Ay Sin 4 cos G,+4 A» SID qn COS Jn 
+3 A; cos 3 4’ sin 3 q+ A; Sin 3 4 COS 3 G 
In general, there are four values of ¢, which satisfy these conditions, two belonging to high waters 
and two to low waters. When A,, however, is very large, there are only two values which satisfy 
them, and then there is only one high and one low water in a day. i 
The value of A; is always small, and when A, is also small, as it is at all ports in the North 
Atlantic, the value of q at high waters is so small that we can put cos qg=1, and at low waters so 
nearly equal to 4x that we can put sin g=1. The preceding conditions then give 
H,=H)4+ Ay+ A; cos 4— A, Sin 4 sing, + Ay+ A; cos 3 4’— A; sin 3 d/ sin 3 qy 
(40) H,—H)+Ay+ A; Cos 4 cos @— Ay sin Jd— A, + A; cos 3 4’ cos 3 G+ A; sin 3 A’ 
H3;—H)+A,)— A, cos 4— A, Sin 4 sin g; + A, —A, cos 3 4‘’— A; sin 3 4’ sin 3 qs 
H,—Hy)+Ao+ Ai cos 4 cos qt A; sin d— A, + A; ¢08 3 4’ COS 3 G,— A Sin 3 4! 
~ They also give 
A, sin 4-3 A, sin 3 A’ 
Sing ——— 
t 4 A,+ A, cos 4+9 A; cos 3 4’ 
at high waters, and 
(41) 
A, cos 4+ 3 A; cos 3 A! 
~ HALF AG sin 449 A, sin 3 A’ 
When all the necessary constants are determined, the preceding equations (40) and (41) give 
H,, and q,, and then when L, is determined, (37) gives /,, L;, and Ls. 
All the preceding expressions are taken from the manuscript of a forthcoming paper on the 
theory of the tides, in which they are more fuily explained, but it would be impossible to give a 
complete and detailed explanation of them here. Few of them, however, depend upon any peculiar 
theory, and most of them can be verified by any one. 
COs q= at low waters. 
THE OBJECT AND PLAN OF THE DISCUSSION. 
23. The object of the following discussion is, first, to obtain directly from observation the con- 
stants of all the principal terms entering into the preceding expressions of H, and 2,, which, it will 
be seen by referring to (24) and (26), comprise all the constants Ry), 24,1), By, and ¢¢,, belonging 
to each one of the angles in the expressions; and, secondly, to obtain from these the general con- 
stants E, G, F, and F’ in the expressions (25) and (27), expressing the relations between all the pre- 
ceding constants, so that they may all be made to depend upon these few constants. It is not 
proposed to determine merely so many of the former as are necessary to determine the latter, thus 
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