4 DISCUSSION OF TIDES IN BOSTON HARBOR. 
the harbor for ten miles ranges mostly from five to ten fathoms, and beyond that in the open bay 
for many miles there is only a very gradual increase of the depth of the sea. The vast expanse, also, 
of comparatively shallow ocean over the Banks of Newfoundland, over which the greater part of the 
tide originating in the deeper ocean has to travel, no doubt affects the character of the tides. In 
any casual considerations of the character of the tides and of the results of this discussion, these 
general circumstances of the tide-station should be considered, but in any critical study of them, 
of course, the charts of the Coast Survey should be consulted, in which all the minute circumstances 
are accurately laid down. 
EXPRESSIONS OF THE DISTURBING FORCES. 
3. In the discussion of tide observations it is necessary to have some form of expression of the 
disturbing forces, and to know something of the tidal expressions corresponding with them. Theory 
must furnish the arguments to be used in any discussion, but different forms of expression give dif: 
ferent arguments. In the complete solution of the tidal problem it is necessary to have a develop- 
ment of the disturbing forces into a series of terms containing angles, which increase in proportion 
to the time, and the corresponding tidal expression has then a similar form; but very accurate 
approximate tidal expressions may be obtained in which the expressions contain circular argu- 
ments, but in which the angles do not increase exactly in proportion to the time. Such expressions 
may be obtained which contain a much smaller number of sensible terms than in the case in which 
the development is required to contain only terms with angles increasing exactly in proportion to 
the time; and the arguments in the expression, still being circular arguments, are much preferable 
to parallax and declination arguments, since all the observations, within certain equal limits of the 
argunent, have nearly the same number of observations; and the results obtained from a discus- 
sion of the observations in this way are of much more importance in any theoretical study and 
investigation of the tides, since the constant coefficients and the angles of epoch show the relations 
between each term in the tidal expression and the corresponding term in the disturbing force. 
4. Since the forms of expression of the disturbing forces used in this discussion, and likewise 
the notation, are for the most part entirely different from those contained in any treatise on the 
tides to which reference could be made, it is necessary to give them here. 
If we put 
2 =the potential of the moon’s disturbing force ; 
pits mass; 
ry =its distance from the center of the earth; 
7 =its right ascension ; 
v =its declination ; 
wo —the terrestrial longitude of any place; 
é=its polar distance ; 
n t==the earth’s rotatory motion ; 
we have, from the development of 2, 
(1) 2=%,N, cos s (nt-+o—y) 
in which 
= U. , Gy ao. ~ 
N=> ,(1—3 cos? 6)(1—3 sin’ v) 
7 
N, =~ sin 26 sin 2v 
PB 47° 
(2) Byam .. 5 
N,—=—— sin? 0 cos? v 
4. rs 
3 [ . 
N;=— , sin’ 0 cos? v 
\ Colt fe 
There is also a small term depending upon the fourth power of the moon’s distance, which pro- 
duces a sensible effect, of the form 
es 1p Al 2 . 
GN — iz COS 0 sin? 0 cos? V sin ¥ 
F- 
