8 DISCUSSION OF TIDES IN BOSTON HARBOR. 
The values of z and z/ cannot be the same unless a;—=<«;, which can only be the Ease when EF’ 
in (27) is equal 0. Hence, the difference depends on friction. 
17. In the eqailieiea theor y all terms depending upon U,q D; 7 and upon friction vanish, and 
(24) gives 
(29) Ne Ke 2 Peosin; 
in which, in the case of an ocean covering the whole earth with a fluid of insensible density, putting 
g for gravity, 
yl (OR 
e—— 
g 
In the case of nature the true values of K, differ a little from these, but these may be regarded 
as very near approximate values. With the values of C,, (12), (14), (18), (20), and with the value of 
(30) Gb S22 71.1 0p) ft., 
we get for the port of Boston, where 647° 40’, by neglecting the correction of the moon’s mass, 
(31) K,=—0.117 ft., K;=0.711 ft., K,—0.528 ft., K;—0.006 ft. 
This value of Ky is the mean or constant amount by which the mean level of the ocean is elevated 
by the moon and sun above the level which the water would assume in the case of no disturbing 
force. This value of Ky may be aiso used in the hydrodynamic theory, since, when s—9, the oscilla- 
tions depend upon the angles 7; in the expressions of Yo, that is, wpon the parallax and declination 
of the moon and sun, and hence are oscillations of long period compared with the diurnal and semi- 
diurnal oscillations. They are called by Laplace oscillations of the first kind. 
18. When s=1, (24) gives as the tidal expressions corresponding to (13) and (15) 
39 § Yi =K, sin (yg —a) cos (p —h) 
(22) 7 1¥4=K" sin (¢’—2’) cos (p/ —U,) 
in which 
p! — t+o0— yy! 
In this case we do not know the relation between C; and K,, and consequently K, can only be 
determined from observation. In this case the period of the oscillations is one day, and the oscil- 
lations are called by Laplace oscillations of the second kind. 
19. When s==2, (24) gives as the tidal expression corresponding with (17) 
(33) Y.—K, 4; Ry cos (7, —4;) cos (2 p—h)= A» cos (2 p—h) 
In this case the mean period of the oscillations is half of a mean lunar day, giving rise to the semi- 
diurnal tides. These Laplace calls oscillations of the third kind. The expression of Ly, in this case 
is derived from (26) and (27), putting s—2, and using the values of P;, D;7;, U;, and Q; in (18) 
and (23). : 
20. If we change the assumed transit from which Lis reckoned » transits forward, then the 
constant Bo is diminished n times 125 25™.24, and the whole of the corresponding lente kin the 
expression of L, is 
(34) k=—n (12" 25™.24-+-0™.4 Cos 71+3".0 COS 72 —2™.3 COS 73...) 
This expression is only approximate when a change of several transits is made. 
21. When s=3, the tidal expression corresponding with (19) is 
(35) Y;—K; cos (3 p—ly) 
in which K; must be determined from observation. In this case the period of the oscillations is 
one-third of a day, and, in accordance with Laplace’s method of designating them, they may be 
called oscillations of the fourth kind. 
There may be local circumstances, such as the shallowness of the harbor or river, producing 
quarter-day tides, but these do not depend upon any sensible term in the disturbing force. 
The tidal expression of the small term in the moon’s disturbing force depending upon the fourth 
power of the distance (3), since sin y=sin « sin g, neglecting the inequality of the node, is of the 
form 
(36) Y¥”=K” sin (g—a’) 
