DISCUSSION OF TIDES IN BOSTON HARBOR. 7 
13. Putting 
(21) i=2 D, (nt—y+/r) 
it is necessary, in the various tidal expressions, to know the principal constants in the following 
expressions: 2H 
2 
(22) += 5%; U; cos 7;, and == %, Q; sin 7; 
The principal of these are 
Up= 12.142, 
(23) Uj= 1742 —13.2 on, Qi; =52™.5 — 4034 6 p. 
: e—— 0500, 
U;= .0477— 0.56p, Q3—= 27.2 —148 op 
The other values of U; and Q; are small, and, their effects being generally insensible, they are 
omitted here. 
TIDAL EXPRESSIONS. 
14. If we put 
Y =the height of the tide at any time above mean level; 
L=the lunitidal interval in solar time; 
p=nt+o—7+ 8; 
7,==the time the maximum of any inequalities in the tides follows the maximum of the 
corresponding inequality of the disturbing force, 
Geary, gives the following tidal expressions corresponding with the general expression (7) of the po- 
tential of the disturbing force : 
(24) Y,=K, 4; Rj cos (7; —4;) cos (Sep —l) = A, cos (Sp—l) 
in which 
(25) (1+ F) Ri, =P;—E U;i 5 
=(e — By) D, Hi 
and 
(26) L,= 4; B; sin (4; — «) 
in which 
B= VM?—N? 
N; 
tan (¢;—4;) area 
(27) 1.035 0.164 
M;=— vist iE P; D; ap so Ry D; i 
Ni=E’ RB; 
The value of By is the mean establishment of the port belonging to the assumed transit. 
15. Of the constants in these expressions, E—Di K expresses the ratio between any change in 
i, the velocity with which the phase of the tide changes, and the corresponding change in the coefti- 
cient of the tide, and consequently the terms depending upon E show the effect of any change in 
the period of oscillation from the mean period. The constant F depends upon that part of friction 
in the theory which is supposed to affect the tides in a greater ratio than the first power of the _ 
velocity, and consequently affects the large tides more in proportion than the small ones, and, 
neglecting terms of a third order in the developments, affects the inequalities of the tides in a con- 
stant ratio. The terms depending upon F’ express the corresponding effect upon the lunitidal 
intervals. All the constants in the preceding expressions have different values for different values 
of s, and they should be written Ry, i), as,i), &¢., when it is necessary to distinguish them. 
16. From the second of (25) we get 
28 ™—=Bo+ 
\ 2 Di 7; 
The value of z given by this expression has been called the age of the tide from the heights. 
If we likewise put 
28! z',=Bo+= 
( = tp. Ni 
the value of zr’; in this expression has been called the age of the tide from the times. 
