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COAST SURVEY REPORT FOR 1868.
APPENDIX No.
PREPARED
By WILLIAM FERREL.

We Nees) Sle ACn aa) Se © OPAC STIS Wives

DISCUSSION OF TIDES IN BOSTON HARBOR.

PREPARED

By WILLIAM FERERREL.

FROM THE COAST SURVEY REPORT FOR 1868.

DISCUSSION OF TIDES IN BOSTON HARBOR.

CAMBRIDGE, MASSACHUSETTS, May 2, 1870.

DEAR Str: I have the honor to submit the following report of a discussion of the Boston dry-
dock tide observations. On account of the completeness of the set of observations and the
importance of the tide-station, as representing a peculiar, and, in relation to the tidal theory, a very
interesting, type of tides, extending along the whole coast of New England, it has been thought
advisable to give all the labor to the discussion necessary to obtain the most accurate results, and
also to give a somewhat full report of them. Some applications of the results have also been made
to theory, and practical formule and tables constructed for the prediction of the tides.

THE OBSERVATIONS AND LOCALITY.

1. The series of observations used in this discussion embrace a period of nineteen years, com-
mencing with the 1st of July, 1847, and extending to the 1st of July, 1866. The series is very
nearly complete, the observations of the times and heights of both high and low water having been
made with great regularity both day and night, and for every day of the week; so that it rarely
happened that even a single observation was lost. Only on two occasions were there any interrup-
tions of any consequence in the whole series. During the latter part of May, and the first part of
June, 1854, one week’s observations were lost, owing to the observer’s being on duty to check the
mob at the time of the Burns trial. The observations of ten days, also, of the latter part of Jan-
uary, 1865, were lost on account of the illness of the observer.

The first part of the series of observations was made with a tide-gauge consisting of marks des-
ignated by copper figures set in the stone wall at the entrance to the dry-dock. The observer
commenced usually about a half hour before each high and low water and noted the readings every
five minutes until the stand, then the duration of the stand, and then again he noted the readings
every five minutes until he felt sure the tide had changed. In 1860 a box-gauge was substituted,
which was used until the series was closed, on the 30th of September, 1866, the full series having
been commenced in June and continued about nineteen years and four months.

The observations seem to have been accurately made, so far as can be judged from the group-
ing of them in the discussion; but this is rather an uncertain criterion, since numerous astronomi-
cal inequalities, and also meteorological irregularities, necessarily cause a considerable range in the
observations of each group. The most accurate test of the faithfulness of the observer is found in
the slight irregularities caused by the diurnal tide. These irregularities in both the heights and
times of the tide at the Boston dry-dock are quite small, and only sensible to ordinary inspection of
the observations at certain periods; yet these irregularities can generally be readily distinguished
in the recorded observations, amid all the other numerous irregularities, at such times as theory
indicates that they should be most easily seen. It is not probable that the observer knew either
the periods of these irregularities, or their character, and hence their occurring in the record must
have resulted from a careful observation of the tides.

This discussion has not been made from the original records of the observer, but from what are
called the first reductions of the tide observations, in which are given the apparent times of the
moon’s meridian transit, aud the apparent times and absolute heights of high and low water, and
the lunitidal intervals, as obtained from the observer’s record. These reductions, so far as I had
any means of testing them, seem to have been carefully and accurately made.

2. On account of the shallowness of the greater part of the harbor, and even of the bay for
many miles beyond, and the interruptions of numerous islands, and the narrowness of the channel
leading to the Boston dry-dock, the tide-station must be regarded as a somewhat inland one, and
some of the characteristics of river-stations should be found in the observed tide. It is seen, from
consulting the Coast Survey chart of the bay and harbor, that the average depth of the channel in

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-

Or

DISCUSSION OF TIDES IN BOSTON HARBOR.

5. If we likewise put
; 2’— the potential of the sun’s disturbing force,
and let »’, rv’, »’, and v’ denote the same with regard to the sun, which the same letters without an
accent do with regard to the moon, we shall have

(4) 0/ = ¥,N’, cos s (nt+o0— y’)
in which the expressions of N’, are the same as those of N, with v, 7, , and v accented.

In the preceding expressions the origin of ¢ must be such as to make the angle nt-+a—y or
nt+o—y’ equal to iz, that is, some multiple of z, when the moon or sun is on the meridian of the
place with regard to which the force is considered. The values of s may be 0, 1, 2, &c., but there
are no terms producing sensible effects in which s is greater than 3.

6. In the comparison of tides with the forces producing them, it is necessary to either analyze
the result and tide of the moon and sun in each port into its component parts, or to have the result-
ant of the component forces of the moon and sun with which to compare them. The latter is pref-
erable in tidal discussions and investigations, since the developed expressions of the resultant of
the forces of the moon and sun being obtained, and all the constants accurately determined, these
expressions, depending mostly upon celestial circumstances, serve, with a very few convenient mod-
ifications, for every port; whereas if the former method is adopted, a troublesome analysis, and a
determination of the constants belonging to each component of the tide, must be made for each port.

By combining the preceding components, we get

(5) Q+0'=3,VNZ+N2+2N, N% cos s () — 7) coss (nt +a—w-+ f)
in which -
N’, sin Ss (b — yp’)
N, +N’, cos s (p—v’)

(6) tan S ,=

7. If we put
Q,—the part of the preceding expression belonging to s,
its development may be expressed in the following form:
(7) 2.— CO, 3; P; cos 7; cos s (nt+G— yp + fz)
in which the angles 7; and n¢-+-o—7-+/, do not increase exactly in proportion to the time on
account of the variable motions of the moon and sun in their orbits, and the obliquity of the eclip-
tic. The latter angle also varies with the changing value of @, which expresses the angle in right
ascension between the moon and the position of a disturbing body which would represent the result-
ant of that part of the forces of the moon and sun belonging to the characteristic s. In the pre-
ceding expression ©, is the constant or average value of the coefficient of cos s(nt4+o—y-+ fs),
and is independent of any of the inequalities. Its value depends upon terrestrial circumstances,
and consequently is different in different ports. The constants P; and the angles 7; depend upon
celestial circumstances only, and consequently are the same for every port. The constants P; are
different for different values of s, and should be denoted by Piz, i) when it is necessary to distinguish
them. LP, is the constant of integration and is equal to unity. ‘
8. We shall now give the angles and the numerical values of the constants, and also the mean
values of the first derivatives of the angles in terms of the radius, belonging to the principal terms
in the preceding expression of 2, for each value of s. In the expressions of the angles the following
notation is used:
» =the moows mean anomaly ;
v= that of the sun;
yg =the moon’s longitude;
g'= that of the sun;
w == the longitude of the moon’s ascending node.
From the notation (§§ 4 and 5) we also have ;
» — 1’ =the difference in right ascension between the moon and sun, usually expressed by the
apparent time of moon’s transit.

6 DISCUSSION OF TIDES IN BOSTON HARBOR.

We shall also put, for the mean values of r and 7’,

Z, _ 3H
4 ys
3p!
(8) Caer:
UH ]
e=—
Z
If we put, in terms of the earth’s mass,
(9) p=.0138+0 4
in which 6p is the correction of the assumed mass of the moon, we shall have
(10) €= 4380 — 33.8 0 p
9. With the preceding constants and notation, when s—0, (7) gives :
(11) 2Qo—= Cy 5; P; cos 7;

in which, omitting the correction of the moon’s mass,
Co= .254 (1—3 cos? 0)(1+¢) Z

P, 114, 42=”
(12) Pee ie 73==2 9
7 OG 74—!
Pe 09o% Is—=2 ¢!
Pp=— .025, 16—= 0

The term belonging to 11, in this case, is wanting.

10. When s=1, the development of the resultant of the moon and sun in the general form of
(7) is not sufficiently convergent for practical purposes, and therefore expressions must be obtained
for the moon and sun separately in this case. The only terms, in the case of the moon, which we
shall have occasion to use in this discussion, may be most conveniently expressed in the following
form, # vanishing in this case :

(13) 2;—=C;, sin ¢g cos (nt-+a— 1)
in which
(14) 1=.181 sin 2 6Z
In the case of the sun we shall likewise have
(15) 2,—=C%, sin ¢! cos (nt-+-a— »’) :
in which
(16) C’, =.731 e sin 2 0Z
11. When s=2, (7) gives
(17) 2,—=C, ¥, P; cos 7; cos 2 (n t-+a — y+ fr)
in which
C,= .9564V1+2@ sin? 0 Z
Pi, = 4305 — 24.0 6 p, My =2 (vp — yp’), Dim —.426
P,= 15214 3606p, 12 =, Di 42 = .229
P; = 0985+ 1.06p, 13 =29, D; 73 = .460
1, = .0093— 1.0 D pty N4 — ie
(18) P; = .0053— 1.00p, 75 =2 9",
Ps =—.0375, 76 =,
Pa Oalo; 77 =29—v, Di 47 =-462
Pe 008d; 73 = + 7, Di 73 = .655
Py = _ .0085, 79 =11— Ney Di 79 —.197
Py=— 0470 4+ 4.70, 710 == 2 1, Di G10 852

The unit of time in the preceding derivatives is one solar day.

12. When s=—3, (7) gives only one term producing any sensible effect upon the tides, which may
be expressed by

(19) 2;—=C; Cos 3 (nt-+-o—/)
in which

(20) C;=.0146 sin’ 0Z

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.

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’)

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
O%
tod

10 DISCUSSION OF TIDES IN BOSTON HARBOR.

making all the rest depend upon theory, but to determine all that are of much importance practi-
cally, so that they may be used in testing the accuracy of the general theoretical expressions, and
for constructing approximate formule of prediction independently of any theoretical relations
between the constants, or used by any investigator for verifying and improying any tidal theory.
All the constants and relations being determined which are of theoretical importance, the most con-
venient practical formulz will be constructed from the results for the prediction of the times and
heights of high and low water, together with tables for facilitating the computations.

The plan adopted for determining the constants from the observations is to apply Lubbock’s
method of averages to circular arguments throughout, instead of to arguments of parallax and
declination, and then to use the constants thus obtained to determine the constants belonging to
any other forms of expression into which it may be thought advisable to put the results. As the
quantities H,, and /, are the only ones which are directly observed, corresponding to any given
time or values of the arguments, these must be determined from observations for all parts of the
arguments separately by so grouping the observations that the effects belonging to all the other
arguments are eliminated, and then, by means of the conditions (33), (37), (40), and (41), the con-
stants belonging to each argument, as well as the general constants independent of any arguments,
can be determined.

To obtain the values of H,, and 4, belonging to the different parts of any argument 7, alone, all
the observations within certain limits of the argument as from 7/; to 7//; have been grouped together,
and the averages of all these have been taken as the normal values of H,, and 4,, belonging to the
averages of all the corresponding values of the argument, which, when the number of the observa-
tions is considerable, does not differ much from 43(7/;-+7/’,), the mean of the two limits. If these
limits should be somewhat wide, a slight correction to the averages of the observations may be
necessary, which is easily applied. When the observations extend over a long period, and have
been regularly made, the effects of all the inequalities belonging to other arguments are completely
eliminated, since the periods of all the other arguments being different, the observations falling
within certain limits of any one argument, are equally distributed through all parts of the other
arguments, and all the plus and minus effects cancel one another; and this is especially the case in
a series of nineteen years, which is very nearly a multiple of the periods of all the principal argu-
ments in the tidal expressions. In a long series, also, the inequalities due to the winds and baro-
metric changes, and whatever other abnormal disturbances there may be, are very nearly eliminated.

24, If the normal values of 2, and H, have been obtained from all the observations without
regard to the arguments of the inequalities, then all the inequalities in 2, and H, depending upon _
these arguments disappear, as also A; and the inequalities of Aj, A,, and A; in (40) and (41).
Putting

—s (AAs) } HH, =
My—4$ (Ao +1)
we in this case obtain from (40),
$(H’, + H’,) =H), + Ko + K; cos 5 4/ cos 38 @—K; sin 3 4’ sin 3
4(H,— H’,) = K,—K; cos 3 4’ cos 3 q—K, cos 3 4! cos 3
4(H, —H; )=K; cos 3 4’
ae — H,)=K; sin 3 4’
and, from (41), omitting 9 A, in comparison with 4 A, in these very small quantities,
3 K, sin 3 4!

4 Ky
3K, cos 3 4!

4 Ky

Since sin q;=sin q;, and sin @=sin q, we have », ¢,.—=%7, and hence from the first of (37) we
get Lp. —=+(5,4,—27) when 4, are all reckoned from the upper transit; but if two of them, as is

iui 2

(42)

sin q@=—

\
(43) <
(

usual, are reckoned from the lower transit, putting = —=12" 25™.24 in solar time, we get, since
L.= By in this case, f

(44) Bo=4(2y Ap —12" 25.24) —$ (0, 42/7 — 6» 12.62)
which is the mean establishment of the port.

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-

12 DISCUSSION OF TIDES IN BOSTON HARBOR.

tions of condition for determining A, and 4 for m values of ¢ =47;, When A, and A; and also 4’ have -
been determined from preceding conditions.

With m values of H,, also, for both high and low waters of both transits, (52) gives, when A;
and 4’ are known or insensible, m equations of condition for determining A, and 4 fo m values of
g from observations of the heights only.

From (32) we have

(53) A,=K, sin (g — z)=M sin 9+ N cose
in which
(54) K,=V M+? and tan ——
7 J

With m values of A;, determined by the preceding equations, (53) gives m equations of condi-
tion for determining R and «. ,

From the first of (41) we obtain, approximately, when A, is small in comparison with- A2, and
consequently sin q is small,

and from the second of (41),
__ Arcos 4 cos 4

AAS en
When @ and @—3= are very small we can use for A, its mean value K,, and then with the
preceding value of A, we get

(2

Ky sin 4
ae ( i 2K, sin (g—a)
i)
; ) jy = SOS I (g—2)
4K

The values of q; and q, ave the complements of the preceding expressions respectively.
TABLES OF AVERAGE OR NORMAL VALUES.

26. The following tables contain the averages of groups of observations taken within certain
limits of two arguments, and arranged, with reference to the averages of the arguments to which
the observations correspond, in the form of tables of double entry. By summing these average
results in two ways, the averages of all the observations contained within the limits of each group
of either argument are obtained. The arguments have been for the most part divided into twenty-
four equal parts, and the mean of the two limits of each division has been taken as the average of
the values of the arguments, except in the case of (y—vw’), in which the true average has been
obtained. From these tabular results all the constants in this discussion have been obtained, and’
they might be treated in various other ways and many important results obtained which have not
been brought out here. ;

An explanation of the notation contained in the headings of these tables may be found in sece-
tions (4), (8), and (11). The values of all the arguments except 7 are given for the time of the
moon’s apparent transit over the Washington meridian, happening a little more than two days
before the time of high water, and which is the transit C according to Lubbock’s notation. For the
sake of convenience in grouping the observations, the values of 7 were used for a time two lunar
days later, for which a reduction must be made when it is necessary to have the values of the argu-
ments all referred to the same time.

To these tabular values a constant of 2 days must be added to the lunitidal intervals, and also 20 -
feet to the heights of high water, and 10 feet to those of low water, for the absolute heights above
the assumed zero of the tide-gauge.

DISCUSSION OF TIDES IN BOSTON HARBOR. 13.

TABLE I—Containing average values belonging to the Arguments (x) —W')=4 and d=4 75

UPPER TRANSITS.

(py —W/)=0h.0m. . . . . Oh. 30m. (Ww—w') =0h. 30m. . . . . 1h. 0m. (Ww —wW') =1h.0m . . . . 1h. 30m.
: : :
(w—w)} Ay XN H, Obs. |(w—w)} Ay de H, | By [Obs |@—v)| a, | As H, | He
h. m.| h. m.| h. m.| Ft. h. m.| h. m.| h. m.| Ft. | Fe. h. m.| h. m. | h. m.| Ft. | Fe.
0 12; 0 40 6 47 | 5.60 | 4.58 14) 0 43|] 0 33 6 43 | 5.62 | 4.48 15 1 16|} 0 26) 6 38) 5.40} 4.94
0 12} 0 43) 6 46 | 5.16 | 4.87 13) || 0 45) 0 39 6 45 | 5.38 | 4.92 12 1 22) 0 31) 6 40 | 5.14 | 5.03
0 16) 0 45] 6 49)/ 5.41 | 5.15 14 0) 53)]) 0 38 6 44) 4.88 | 5.19 10 1 14 0 34) 6 42) 5.33 | 4.80
0 17) 0 46 6 51 | 5.41 | 5.03 10 0 42.) 0 38 6 42) 5.04 | 5.47 12 ik 118} 0 35} 6 42 | 5.32 | 5.22
® TG OF Ze5)) BH Ay) Beale) ah ales 10 0 42) 0 41 6 44 | 5.19 5. 06 10| 1 16] 0 33] 6 40) 5.01 | 491
0 il 0 45) 6 48] 5.27] 4.97 11 0 41 0 39 6 46 4.89 | 5.06 10 1 14 0 32) 6 43) 5.15 |) 5.16
6 0 17} 0 40} 6 46 | 5.10 | 5.21 11 0 47 0 34 6 43 | 5.04 | 5.17 10 ib 5 |) @ By) 6 34 | 4.69 | 5.47
oO) 0 13 0 40) 6 45 | 4.98 | 5.00 LON OS Ae 0 280) Gis 5. 07 | 4.94 ait 1 16 | 0 29 6 (37 | 5.19 |S. 21
BG) 0 16) 0°33] 6 43] 5.20) 4.63 11 0 41 0 34/ 6 42/ 5.12} 4.98 |) a abl |) Oy Su 6 39 | 5.08 | 5.07
Ba) 0 11 0 35 6 46 | 5.44 | 4.52 12) 0 47) 0 32 6 38 | 4.96 | 4.79 mi) a 7G 0 32 6 42 | 5.39 | 4.59
Buy 0 18) 0 39) 6 51 | 5.53) 4.70 11 0 49 0 30 6 38 | 5.41 | 4.62 10 ib als) 0 3st 6 42 | 5.30 | 4.37
By 0 17) 0 40 6 52 | 5.62 | 4.54 12} 0 47 0 36 6 46 | 5.72 | 4.59 10 CSO O29) 6 37 | 5.62 | 4.64
AD) 0 17) O 41 6 53 | 5.91 | 4.71 13 0 49) 0 37 | 6 47 | 5.94 | 4.63 12 | 1 7) 0 35 6 46 | 5.75 | 4.32
Ai) 0 19} 0 40 6 53 | 5.82 | 4. 12) 0 46} 0 36 6 47 | 6.07 | 4.36 11 | 1 14 0 33) 6 48'||5.55 | 4.51
6G) 0 17) 0 41 6 56 | 6.26 | 4. 12>) 0 42 0 38 6 93 | 5.79 | 4.55 13 lon Omso 6 50 | 5.72 | 4.54
Be) 0 13) 0 47 7 02) 6.07 | 4 13} 0 45 0 38 | 6 53} 6.01 | 4.51 12 | ik al || (0) 383 6 46 | 5.89 | 4.87
a 0 12] 0 49] 7 00) 6.23 | 4. 10| 0 44] 0 38] 6 52|5.63/-4.80] 11] 1 11] 0 34] 6 50/610] 4.52
At) 0 16) 0 42) 6 56) 5.74 | 4. 12} 0 45) 0 34 6 49 | 6.05 | 4. 40 9 1 14) 0) 3f 6 44 | 5.67 | 4.63
A) 0 15 0 37 (Gy |) BEE)! 2b 11 0 47] 0 36 | 6 52 | 5.82.] 4. 62 10 iL at) 0 38 6 50 | 5.67 | 4.70
AD) 0 11 0 34) 6 50] 5.95} 4. 12 0 42 | 0 30) 6 43 | 5.66 4. 65 lil) al ae | 0) 26 6 41 | 5.89 | 4.28
6x3) 0 12) 0 38) 6 54) 6.01 | 4 9 0 43 0 34 6 47 | 5.91 | 4.28 9 | i a) @ Pf) 6 40) 5.67 | 4:22
Bi) 0 17) 0 34) 6 45) 6.12) 4. 11 0 41 0 29) 6 43 5. 88 | 4.45 i0|| 2 8 0) 22 6 33 | 5.46 | 4.39
pt) 0 20) 0 37] 6 46] 5.60} 4 10 0 45) 0 28 6 38 | 5.73 | 4.31 is} || ab aly 0 22 6 34 | 5.74 |) 4.66
Bi) 0 19 0 38} 6 45) 5.59 | 4 11 0 42°) 0 29 G 39 | 5.53 | 4.14 re} |) iy | 0 28 | 6 37) 5.39 | 4.55
0 15.2) 0 40.3) 6 50.3) 5.63 | 4. 275| 0 44.8] 0 34.6] 6.44.5] 5.51 | 4.71 264 114.4) 0 30.9) 6 41.4) 5.46 | 4.73
(w —wW’) = 1h. 30m. 2h. Om (w—wW') =2h.0m. . . . . 2h. 30m. (Ww —wW') =2h.30m. . . . . 3h. 0m.
(w—W)| Ay re H, | Hy | Obs. ((w—w')| Ay Xo H, | Hy f Obs. (W—w)| Ay dy H, | WH
h. m.| h. m.| h. m.| Ft. | Ft h. m.| h. m.| he m | Ft. | Ft. h. m. | h. m.| h. m.| Ft. | Ft.
1 ot 0 26 6 35 | 5.10 | 4.84 10) |) $2 it 0 22 6 31} 5.00 | 5.24 11 2) 4 0) 20 6 31) on84 |) SyoL
1 46) 0 25 6 34 | 5.08 | 4.65 12))) 2) 16) |) 0 425) }) 6, (34 || 4°93) 5.26 13 2 45] 0 25| 6 35 | 4.97 | 5.24
1 40} 0 33 6 41 | 5.42 | 4.94 13 2 13') 0 26) 6 33 | 4.84 | 5. 76 14 2 47 | 0 23 6 33 | 4.73 | 5.56
1 44 0 29) 6 40] 4.71 | 5.31 14) 2 20) 0 28 6 37) 4.7 5. 7 12) 2 48/ 0 23 6 30 | 4.52 | 5.62 j
1 45 || 0 26 6 35 | 4.79 | 5.51 12) 2 16 0 26 6 34 | 4.80 | 5.90 12) |, 2) 41 0 24 6 33 | 4.44 | 5.65
1 43) 0 31 6 35 | 5.13 | 5.68 12] 2 17 0 26 6 33) 4.74) 5.69% 12] 2 49 0 19 6 27) 4.44 | 5.97
1 43) 0 32] 6 37 | 4.56 | 5.24 10'} 2 15 0 25 G 34 | 5.34 | 5:50 LON 2435S O23 6 30) 4.87 | 5.94
1 46 02% 6 38 | 4.98 | 5.38 9 2 16 0 20 6 28) 4.7 5.58 | 10) 2 41 0 16) 6 25 | 4.59 | 5.38
1 46 0 20 6 28 | 5.04 | 5.04 11 2 16 0 16 6 25) 4:72) 5.21 8 10 | 2 43) 0 19 6 29 | 4.95 | 5.38
1 42) 0) 20) 6 28) 5.29 | 4°88 12) 2), 13 0) 421 6 26] 4.87] 5.02% 10] 2 46 0 18] 6 30] 4.76 | 5.23
1 43) 0 31] 6 38) 5.10 | 4.91 PY) batt 0 25 6 34) 4,95 | 4.91 | 9] 2 40) 0 23 | 6 32) 5.16 | 5.79
1 42) 0 28} 6 38) 5.29 | 4.82 BNI) Bale) 0 23 @ 6B G05 | 505]! Ti) @ oh | 0 25] 6 33) 5.06 | 5.21
43) |e O26 6 39 | 5.52 | 4.56 T3255) | 0 27 6 40) 5.18) 5.03% 12) 2 44 0 25 6 37 | 5.31 |) 5.238
1 45 0 30) 6 42) 5.54 | 4.90 14] 2 19 0. 28 6 40)5.16] 4.98] 12 Q AQ) 02211) (Gi) \|on teh oso:
1 48) 0 27 6 41 | 5.50 | 4.70 di PPO O Pe @ SE) GB |) Gy abt } 11 2 42) 0 23] 6 40 | 5.47 | 5.09
1 43) 0 30 6 49 | 5.58 | 5.00 14 ZL 030) 6 41) 5.48]) 5.09) 12) 2 43 0 25 6 42 | 2. 32 | 4.93
1 44/] 0 31 6 44 | 5.67 Zh |B -alb} 0 28 6) 7-39 || 5534 | 5.29) 9) 13) |) 2), 45 0) 123 6 37 | 5.44 | 5.33
1 44] 0 27) 6 43) 5.61 UP) BG 0 21 Gio ||omdle ono) 10} 2 40 0 23 6 41 | 5.59 | 5.24
1 44) 0 25] 6 38) 5.55 10) 2 14 0) 21 @ Si | aye 4, 9. 11 2 44 0 21 6 36 | 5.19 | 5.38
1 44] 0 17 6 33 | 5.64 12) 2) 14) 0) 24) 6 30) 5.77 | 4055 11 2 52) 0 18] 6 34 | 5:19} 4°80
1 45] 0 21 6 35 | 5.63 11 2°19 | 0) 15 6 38 | 5.63 | 5.03 | 11} 2 44 Omer 6 30 | 5:17 | 5.02
1 42) 0 24) 6 38/573 12) 2 14 0 26 6 38|5.46| 4.86} 11 PA TZBY 1 0) iG) 6 27 | 5.19 | 5.08
145) | OM 2251 Gerda) Parole 10) |) 2 20 0 23 6 36 | 5.71 | 4.85 LOS 28 47 0) 13°) 6) 27) | 5:35") 5.22
1 49] 0 25) 6 36) 5.42 TOS | aa) 0 20) 6 33 | 5.44 | 492 10 2 42) 0 13] 6 25] 5.08 | 5.32
1 44.4) 026.4) 637.4! 5.31 283 | 215.6| 0.23.9] 6 34.4) 5.20| 5.19] 269] 2 44.6) 020.7) 6 32.4] 5.05 | 5.32

14 DISCUSSION OF TIDES IN BOSTON HARBOR. =

TABLE I—Containing average values belonging to the Arguments (4) — w’) = 4m and 6 = 473—Continued.

UPPER TRANSITS.

(W— wW')=3h.0m. . . . . 3h. 30m. (YW — yy’) = 3h. 30m. . - . . 4h. Om. (W—wW') =4h.0m. . . . . 4h. 30m.

@ - : :
Obs. |(Y—W)| ds | A | H | H fons. y—y)| ry At Weel ees ely! baru) pe
° h. m.| h.m.| h. m.| Ft. | Ft. h. m.| h. m.| h. m.| Ft. | Ft. h.m.| h m.| h. m. -
7.95 2) 3) 13 0 19 6 30 4.92 | 5.44 10 | 3 44 0 26 6 32) 4.54 | 5. 64 11 4 12 0 25) 6 35) 4.23
22:0) 12 3 18 0) 22 6 32 | 4.39 | 5.46 11 3°45; 0 19 6 32 | 4.66 | 5.78 12 4 12 0 28 | 6 38 | 4.18
37.5 12 3 19 0 25 6 34 | 4.46 | 5.67 11} 3 45) 0 22] 6 32) 438] 5.66 11 A 12 0 26 | 6 41 | 4.40
92.5 10 3 17 0 21 6 34} 4.68] 5.78 12 3 42) 0 23 6 33 | 3.72 | 5.86 14 4 17 0 31] 6 38] 3.97
7.5 14) 3 16 0 25 6 32 | 453 | 6.12 13} 3 44 0-23) 6 33| 441! 6.08 12 4 2 0 26 6 34) 3.68
§2.5 12) 3 16 0 18 6 31 4.93 | 5. 64 11 3 44 0 19 6 27 | 4.28 / 6.08 12 4 13 0 21 6 2 4.10
7.5 11 3 16 0 20 6 30| 4.78 | 5.83 14 3 45 0 16 6 26 | 4.10 | 6.18 13 4 21 OF LG 6 32 | 4.01
112.5 11 35 0 17 6 27 | 4.73} 5.87 10 | 3 45) 0 22 6 31) 4.83) 5.95 12 4 14 0 22 6 31) 4.59
210 11 Seelam tO melt 6 24 | 4.62} 5. 64 10 3 42 0 14 6 21) 4.18 | 5.74 10 4 17 0 23 6 36 4.66
142.5 12 3 13 0 15 6 23} 4.78 | 5.68 11 3 47 0 10 6 27 | 4.83 | 5.57 12 4 16 0 19 6 31) 4.45
157.5 11 3 13 0 20 6 29 | 4.84 | 5.46 11 3 44) 0 20 6 23) 4.73 | 6. 04 12 4 1£| 0 27 | 6 40
172.5 10) 3 23 0 23 6 36 | 4.83 | 5.54 11| 3 45 0 26 | 6 35 | 4.85 | 5.45 11 4 15 0 26 6 41
187.5 il 32 0 22 6 31 | 4.91 | 5.39 11 3 40 0 26; 6 39) 4.89 | 5.73 11 4 15 0 27} 6 45] 5.03
202.5 12| 3 14 0 22 6 39 | 5.14] 5.46 1343047 0 29; 6 43) 4.99 | 5.86 13 4 16 0 31 6 44
217.5 13 3 14 0 25 6 40 | 9.07 | 9.37 13 3 42 0 24 6 41) 5.01 | 5.77 12 4 16 0 33) 6 50
232.5 14 oy slic 0 26 6 41) 5.07 | 5.73 13 3 44 Omi 6 42) 4.90 | 5.45 11 4 15| 0 30 6 45
275] 12) 3 22] 0 26] 6 47) 5.42)5.25] 13) 3 43) 0 22| 6 36|4.86/590} 15] 4 14] 0 25] 6 47
262.5 12 3.9 0 24} 6 37 9.15 | 5.43 14 3 45; 0 20 6 37] 5.12) 5.64 12 4 2 0 30 6 44
2171.5 13 3 16 (iby 6 33 | 6.30 | 5.48 NTs) Abel On 17 6 33 | 5.17 D: to) 12 4 16 0 19; 6 36
292)5) 9 3 15 0 15 6 28) 5.63 | 5.42 10 | 3 42 0 13 6 31] 5.13 | 5.96 12 4 16 0 15 6 26
307.5 12 3 18 0 16 6 28) 5.14 | 5.36 2 3°45 0 16 6 28) 5.19 | 5.22 11 4 15 0 11 6 31
322. 5 12 3 17 0 10 6 21 | 5. 08 | 5. 22 12 3 44 0 17 6 32) 4.80 | 9. 12 10 4 19 0 13 6 30
337. 5 11 3 13 0 18 6 27 | 495 | 5.33 11} 3 45] 0 20 6 32) 5.19 | 5.40 11 4 17 0 15) 6 30
362.5 aD) (3) 2% 0 17 6 2 | 4.66 | 5.61 12 3 47 0 17 6 32) 4.92 | 95. 35 11 4 17 0 16) 6 28
280 | 3 15.8} 019.8) 6 31.8 4.92} 5.55] 280] 3 44.2) 0 20.3) 6 32.6) 4.74 | 9.7L 283 415.9) 0 23.2) 6 36.7
| | | |
(WY — ww’) = 4h. 30m. . 2 . . 5h. Om. (— wv’) = 5h. Om. . . . . 5h. 30m. | (wy — W’) = 5h. 30m.
o l 7 7 7 7 =
Obs. |(W—W)| Ar » | | Hy | Obs. \—w)]| A | d2° | H.| He | Obs. (Y—)| A | Ae
= =| Se }
° h. m. | h. m.| h. m.| Ft. | Ft. h. m.| he m.| ho m.| Ft. | Ft. hem.) h. m.) h. m.
1.9 12 4 45 0 31 6 43 | 4.43 | 5. 68 11 3 fd 0 33 6 45 | 4-21 | 5. 72 10 5 44] 0 42] 6 53] 4.27
22.5 11 4 43 0 35) 6 46 | 4.60) 5.66 12 5 16 0 36 6 49 | 4.10 | 6.20 10 5 42 0 38 6 952
Sto 16 4 46 0 31 6 42 4.03 | 6.05 11 | o> 18 0 41 6 53 | 4.21 6.28 10 5 46 0 44) 6 53 | 4.34
52.5 12 4 47 | 0 32 | 6 42/ 3.89 | 6.02 11) 5 14 0 40) 6 51 | 4.03 | 5.95 12 5 39 0 46 6 54
67.5 12 4 46 | 0 30 | 6 42) 3.9. 3. 88 12). > 16 0 39 6 47 | 3.68 | 6. 37 13 5 43 0 46 6 38
82.5 14 4 43) 0 27 | 6 37] 4.27] 6.43 ASie oO WT 0 35 6 45 } 3.92 | 6.49 13 5 46 0 33 6 50
97.5 11 4 47| 0 20 6 28] 4.00 | 6.33 12°)|\ 3 a6 0 27 6 42 4.43 | 6.43 14 5 43 0 36 6 46
112.5 12 4 42) 0 20 6 30} 4.61 | 5.91 12 5 16 0 26 6 40 | 4.05 | 6,55 13 5 46; 0 29 6 44
127.5 12 | exe) iy) 6 28) 4.32 6.53 11] 5 18 0 23) 6 37 4.24 | 6. 22 11 59 42, 0 29) 6 41 4.38
- 142.5 li} 4 45 0 20 6 31) 4.43 |) 6.15 11 5 17 0 2 6 42) 4.72 | 5.89 12 5 41) 0 32] 6 46 | 4.40
157.5 12) 4 46| 0 20 6 35 4.31 | 6.06 10 5 13 0 29 6 44) 4.74 | 5.62 11 5 43 0 34 6 49 | 473
W2a0) 9 4 45] 0 31] 6 45 | 4,48] 5.49 11 5 16 0 29 | 6 46 | 4.70 | 5.94 i 5 45) 0 40 6 57} 4.89
87.5 11 4 44 0 32] 6 43) 4.47 | 5.59 11 5 15 0 32] 6 51 | 4.61 | 5.57 11 5 44 0 40 6 56 | 5.03
202.5 11 4 AT 0 40} 6 52] 4.85 | 5.41 10 5 14 0 41 | 6 58 | 4.67 | 5.80 11 5 46 0 48 6 59 | 4.83
2179 13 4 42 0 36 6 50 | 4.69 | 5.76 L2iko, Bhd 0 41 7 Oj} 4.56 |} 5.40 12 5 44 0 51 7 8] 5.06
232.5 12 4 47 0 30| 6 51 | 4.84 | 5.85 14 5 16 0 48 7 35| 4.7% | 6.01 12 5 47 0 47 7 7| 4.86
247.5 13 4 44 0 33 | 6 49} 4.58 | 6.09 12 5 18 0 35 6 58 | 4.97 | 5.90 14 5 45| 0 47] 7 12] 4.70
262.5 12) 4 39) 0 22 | 6 41} 4.87} 6.19 13 2 12 0 37 6 55 | 4.71 | 5.91 15 3 40 | 0 35 6 55 | 4.95
2.9 151) 4: E45: 0 24 6 44 | 4.87 | 5.84 14 5 19 0 32) 6 51 | 4.58} 6.07 12 5 47) 0 36| 6 56} 5.05
292.5 11 4 45 © 16 | 6 34) 4.79] 6. 2 oO) phe 0 21 6 41 | 4.80 | 5.99 13 5 44] 0 33 6 52] 5.43
307.5 10 | 4 44 | 0 17| 6 32) 4.84] 5. 12'| 5 (26 0 25 6 42 4.69 | 5.83 13 5 45] 0 27 6 44) 4.51
322.5 10); 4 45 0 19). 6 35 4.79 | 5. 12)| > —16 0 19 6 36 | 4.52 | 5.82 11 5 50 0 28 6 43 | 4.29
337.5 i 4 47 O22 6 39 | 4.72) 5. 125) 75 a3: 0 29 6 45 | 4.69 | 5.78 11 5 44 | 0 29 6 43 | 4.35
352.5 10 4 46 0 28 | 6 41) 4.45 | 5. 12))\ 7: AIS 0 30 6 44 | 5.05 | 5.57 13 5 47 0 36) 6 49 | 4.34
283 | 4 44.8 0 26.4 6 40.0| 4.50 | 5.91] 283 | 5 15.6) 0323) 6 47.8) 4,48 5.97 | 288 5443 0 37.9) 6 528 4.46

DISCUSSION OF TIDES IN BOSTON HARBOR. 1 by;
TABLE I—Containing average values belonging to. the Arguments (1) —xp')=4m and ¢=473;—Continued.
UPPER TRANSITS.
(W—w/)=6h. 0m. . . « . Gh. 30m. (w—wW) 6h. 30m... . Th. Om. (W—w!) =Th.0m. . . . . th. 30m,
Obs. |(@—w!)| Ay as H, | Hp | Obs. |(w—w)} Ay As H, | He | Obs. (ew) ds Re H, | He
hem.) he m.| hem.) Ft. | Ft. hem.) hem. || hom.) Bte | Pe. ho om.) home| hom. | Be | Ft.

12) 6 14) O 46 6 59 | 4.47 | 5.93 11 6 45 0° 47 7 1) 4.48) 5.67 11 uf sale al. Pal 7 G6 | 4.07 | 5.75
11 6 10) 0 48} 6 57} 4.32 | 6.00 12 6 42 O54 7) 6H aa) ons ist uf ait 0°57 | % 9 | 4.45 || 5.52
11 Gieln | Ole O20 hen een den 4s Oe 6.43 10 6 44 0 58) 7 8) 3.84) 5.93 1L te aly 123 V7 14} 4.25 | 5.80
12) 6 14 0 56) 7 4+) 3.84 | 6.17 13 6 46 iW) 7 9 | 4.30) || 5.98 11 7 16 | ie RA) 2b a} |) Go)
13 6 19 OF 535) VS 393252) 16252 12 6 44 0 56 7 11 | 3.79 | 5.67 9 7 195 ES OD ie 14 OG monso)
13 6) 15) 07 48')) '% 1) 3549)) 6.45 13 6 46 0 57 7 10 | 3.92} 5.84 11 i alg) 1 8} 7 12) 3.60} 6.10
12} 6 17] O 44) 6 57] 4.05 | 6.43 13 6 43 0 54 7 17) 3.89} 5.89 11 7 14 il eat 7 10 | 4.08 | 6.03
13 6 18 0 45] 6 59) 4.11) 6.56 14 6 42 0 45 6 58 | 4.33 | 6.14 13 viet) 0 57 7 10} 4.2: 6. 28
ip) || @ Jalsl 0 30} 6 43 + 4.00 | 6.43 14 6 46) 0 47] 6 59 | 4.23] 6.13 11 i BR Woes 7 5 | 4.45 | 6.05
11 6 18 0 40 6 53 | 4.64 | 6.02 11 6 45 0 41 6 55 | 4.48 | 6.43 11 ti aly 0 48 7 4) 4.65 | 5.84
60) 13 | © Zl 6 55 | 4.69 | 5.93 12) 6 43 0 49 WS 13) 4585) 19: 97 apt | ie 12 OR or 7 #4) 4.75 | 5.50
no. 13 6 16 0 45 7 0 | 4.92 | 5.98 11 6 49 0 50 7 7 | 4.84 | 5.56 10| 7 14 0 55 7 12) 5.25 | 5.51
Bb) 9 6 14) 0 47). 7 31] 4.90 } 5.80 10 G48 LO She Ohh oO 5246 10 7 13 0 54 7 14 | 5.27 | 5.58
iB) 12 6 14 0 53 Mf tla |) PAO) |) Gy, (7a 11 6 40 0 53 7 11 | 5.00 |} 6.00 11 7 13 wb) By % 24 | 5.38 | 5.53
A) 11 @ ‘ir DON GH 47859 5579) 11 GREAG RSL sel: 7 20 | 4.92 | 5.60 12 7 14 i il U 24 | 5.20) | 5.22
A) 11 6 14 0 59 Me alee |) 2S! | bs, 763 11 6 41] 0 58]°7 18 | 5.04 | 5.59 11 eel 9) 110) YT 28 | 5.11 } 5.55
Ai) 12 6 15 0 51 % 14) 5.12) 5.81 12} 6 46 | bt 2) 7 24] 498 | 5.34 12 7 15 Uf Ye |) We rt | ZSEE) |] Byte
2.5 12 6 18 0 50) 7 12] 4.69 | 6.08 12 6 43 | 0 50 7 12] 4.96 | 5.90 iil |) asl iO) 7 26} 4.84 | 5.35
0 11 6) 19 0 48) 7 10) 4.68 | 5.94 13 6 43 | 0 46 w 6)) 9: 19) 55.80 13 allt al al UT 21) 4.72 | 5.54
AD) 11 6 15) 0 42) 6 59) 4.44 | 6.04 12) 6 46) 0 46 Y 4) 4.65 |) 5. 72 10 7 18 0 56 7 12) 4.85 | 5.69
60) 12 Gal |) WO By 6 56 | 4.94 | 5.93 11 6 38) 0 44) 7% 51) 4.61 |} 5.61 12 Ww U2) WO 2H] ty |) ZEEE) a Ger
mo) 10 6 16 0 37 6 53 | 4.68 | 6.00 11 6 47] 0 37} G6 55) 4,7 | 5.58 iB} |} ye alee 0 54] 7 7) 4.65} 5.67
Bb) 11 6 16 0 35 6 49 | 4.41 | 5.57 13 6 42] 0 42 | 6 59 | 4.43 | 5.72 14 7 16 0 49) 7 3] 4.61 | 5.63
2.5 9 6 18 0 34 6 50 | 4.31 | 5.51 10 6 47 0 48 7 0} 4.20 | 5.64 11 eel) 0 51 U5 | 40370 |) 5567
276 6 15.5) 0 45.9} 7 0.9) 4.46 | 6.03} 283 6 44.2) 050.8) 7 6.3! 4.54 | 5.79 271 715.4) 0 58.4) 7 13.1) 4.62 | 5.72

(W—wW') = 7h. 30m. . . . . 8h. Om. (~—w') =8h. 0m. . . . . 8h. 30m. (ve —w!) = 8h. 30m. . . ... Oh. Om.
Obs. \(y—y')| Ay re H, | H, fF Obs. |((—w’)) AY re H, | HH. fObs. |(w—w)) AY rz H, | H
hem.) h. m.| he m.| Fe. | Te. he m.| he m.| hem.) Ft Tt. h. m.| he mm.) he am.) Tt | Fe
To tv 49 0) 59 q 11 | 4.32 | 5.60 11 8 19 Th |) ee BE CECE) Galt) 13) 8 495 9) 7 14 | 5.00} 5.13
12) 7v 47 1G 1 7 13) 4.65 | 5.54 11 Gi Tey |) al | 7 Y 15 | 4.53) 5254 11 8 48 1 4 7 14 | 5.44 | 5.24
10 tT 43 ay £3} 7-19 | 4.27 | 5.64 10 8 13 il, sit % 195) 4012) )'5254 11 8 44 1 5 7 15 | 4.71 | 5.36
11 tT 47 16 7 16) 4.44 | 5.71 9) 8 16 il “al 7 19 | 4.47 | 5.85 10 8 43 DL) 185)) 159) 4578) os20!
11 VU 43 iB} % 12°) 4.35 | 5.85 all &} as) |) ab iY 7 U7 | 461 | 5.53 11 8 42 i “all 7 15) 447 | 5.33
11 7 44) 1 4 T U7 | 4.24 | 5. 60 13 } akey |} al al % 22) 4.08 | 5.87 13 GB} W¥Y |) at ak @ 11 || 4.62 | 5.41
6 12) 7% 43 il ab GW G1} |) Ley. || 6 BB} 13 8 16 il lit MW 18>) 3599 );5254 12 8 47 1 7} 7 14) 4.50 | 5.17
BO) 14 7 43 0 59 % 10 | 4.28 | 5.72 12 8 20 i, 8 7 14 | 4.34 | 5.66 13} 8 44 1G 7 14) 4.67 } 5.13
BO) 12) 7 45 0 59 7 9) 4.60 | 5. 66 13 i ale) al) % 11 | 4.64 | 5.5L 14 8 43 i Pat Heel | 42 OOM ost0
2.5 14 7 40 0 54 % 5 || 4.82 | 5:58 14 (S\ wale 1 4 7 15 | 4.73 | 5.38 14 8 46 Lt 3) 7 12) 5.04) 5.08
-3 11 7 43 0 58)].% 12] 5.14 | 5.94 1B} }| fe} alge 2) 24 15.09) }5220. 11 8 46 OPPS Sa en ON fon 29) Pon
NO! 12) 7 45) 0 59 Y 15) 4.73 | 5.56 10 8) 20) i 7 17 | 5.43 | 5.06 10 8 44) 1 9 7 26 | 5.64 | 5.32
Bu) 13} |) we) 28) al @ % 21/5231) 5)19) 12 8 16 eo) 7 21 | 9.26) |, 5.06 9 8 47 BI) Ue BL |} BED || & 02
&) 12} 7 41 aby 33 © 21 \5.34 | 5. 26 12)) 8 16 i m8} 1238 |Fono25|/4a80 iL 8 43 SPS) ee Be |) GEO || cL Gt
) 10 7 47 Yoo) 7. 27) 5.45 5. 48 11 8 14 1 10 % 4265|"5.59))|/ 74597 10 8 43 1 2) @% 19) 6:08 | 4.43:
Ab) 1G 47 1G G9) 7 28 | 5.63 | 5.08 11 8 19 al 5 318} @ 3/5235) 4.95 11 8 47 113) U Sil }) GEES} || G0)
60) 10 G42) at! & 7 31) 5.13 ) 5.01 12))/ 8 13 1.13) 7% 27|/5.28 | 5:5: 11 8 45 LP LON 278 )165219) 4590)
nO) 14) 7 41 D9 > 265|05. 09) o0S 1} || hy aly 9 We GIL ]| SERES | Geary 10 8 46 Py Pal |) Gs aS) |) Ge ales
60) 13) 7 41 LY 1) 7% 20) 493) 5.44 11 acy ab 7 23) 5.15 | 4.76 13 8 43 iy W) 25 15:40 | 4052
BOY 13) 7 46) 0 58} 7% 18) 5.26} 5.30 LS Sh Lz, i) 8 % 18 |) 5.12 |) 5225 12 8 46 1 49) 7 24 | 5.51 | 4.66
Ap) 14 48 ONS OH anol on los Oralmonc 14} 8 18 1 3) % 19 | 4.93 | 5:15 13 8 45] 0 56) 7 12) 5.65 | 4.78
2.5 11 % 46) 0 58] 7 12) 4.95 | 5.48 11 8 16 0 57 % 9) 5.05 || 5573 12 8 41 0 56 7 9 | 5.68 | 5.07
0) 13 & 465), (0) vol! % 8) 4.77) 5.29 127) 8 12°) 0) 59 % 11 | 5.06 || 5530 13 8 44) 0 58] 7 13) 5.25) 5.36
5S) 9 7 46) 0 57 7 9 | 4.91 | 5.44 11 8 12 150) Y 11) 4.97 || 5:36 12) 8 43 1 2) 7 12) 4.94 | 5:26
284 | 7 44.2) 1 1.0) 716.1) 4.84 | 5.49 281 815.8) 1 5.9} 7 18.6) 4.87 | 5.33 7 280 8.44.5] 1 4.8) 7 17.1) 5.22 | 5.06

DISCUSSION OF TIDES IN BOSTON HARBOR.

TABLE 1—Containing average values belonging to the Argqunents ( —)') =4m and ¢ =473;—Continued.

UPPER TRANSITS.

(w—w')=9h. Om. . « « « Dh. 30M. _ (b—W)=9h. 30m. . . « « 10K. Om. (v—wW')=10h.0m. . . « « 10K. 30m.

l(~—wW)) ry Ag Hy, | Hz | Obs. |(w—w’)| Ay 2 Hy (w—w')) AY Ag H, | He

h. m.| h. m.| hom. | Ft. | Ft. h.m.| h.m.| hom. | Fe. h. m.| h. m. | h. m Ft. | Ft.

An) 9 18) 0 58 % 5 | 5.25) 4.56 11 9) FASe i ell eal 7 11) 5.65 10 ii 2D) 70 So) anes tors
.o OSLO ih el shir4. 92015505 LOD PROV 445 AT! 2s 7, S19 a aL8) 10 13| 0 59) 7 9) 5.10} 5.06
at) 9 18 iby alt 7 14! 4.78 | 5.07 10 9 40 1 5) 7 13) 4.58 10 11 1 3} 7 10) 5,39 | 5.24
A) ye i ths aki} 7 21) 4.63 | 5. 41 10 9 44) 1 8 7 14 | 5.01 10 16) 1 2) 7 8| 4.84) 5.42
5) 9) Ho. ASE) 2 SNA S68a ros A6 12) 9 46); 1 6F 7 10) 483 10, Ly a ese ed LL SSO oan
4b) 9 16 1 12 |- 7 15) 4.85 || 5.36 10; 9 46 DSH 7G 554168) LO 5 el SO) he 7S) | ASes7 or 9)
Ay) 9 13] 1 6) 7 14] 4.54] 5.34 9) P42H 20! POS 7) 75) 15:108 LO) Se Ls Sele a eo WOR OON eone0)
Aly ORLA | LIS OW PARR aan 5 04 12 | 9 47| 0 59) 7 7) 4.90 10 1%) 0 58} 7 3) 4.74 5.08
BG) 9 19) 1 6) % 14) 4.61) 5.46 12| 9 46 abt al 7 6 | 5.26 10 12) 0 56 7 5 | 5.04] 4.78
A) Gy aly 0) PSS) Gnas 28n 5: 08 12/49) $42/)) ot 9 10) M19) 5532) 10) 9125), 40 5S) 4 1G |P5525) rosa
Bb) 9 14| 1 2) 7 14| 5.36 | 4.88 13 | 9 43)) 10 (570% LE 5.152 2 10 20) 0 56} 7 5) 5.47 || 4.92
As) 9 nit 0 59) 7 16) 5.55 | 4.65 14) 9 47 A) PLO SISSIES oss pS: 10 17) 0 58) 7 9) 5.65 |) 423
pe Oe eels eons LOD Ds dale Co 11 9 45 uit fs NOM Ty ebsHon |e LO) Le |) eal 7 16 | 6.03 | 4.19
Bi) OLA a abe es BLT S278) 502 11 | 9 50 1 ONG 7 24| 5.94) 4. 10 16} 1 O| 7 14} 6.04 | 4.57
Bb) 9 Az) f AS |) % 3271) 15236) 4. 96 12} 9 46] 1- 8| 7 23] 5.93] 4. DO Ga 1 SS Say er SS GNM aso:
If 9 16) 2 6) % 122)116:42)) 4.36 125/69) PAS Sen 7 VRE 539% vA 10 17/1 5S} 7 18) 6.02 | 4:61
a0) 9) fs 1 12) 7 32) 5.29 | 4.97 1 9 43 LPNS 7, S228 6.230 4= LO Ron el eee 7 16 | 5.90 | 4.45
iO wy all a) Bi 7 25.| 5.83 | 4. 64 11 | 9 44 il! ae 7 23 | 5.72) 4. TN aM at he SEE Gp Gala Zs zal
ae 9 S16 A ee BOS ES a RG 11} 9 43] 0 58} 7 13) 5.91] 4 10 14| 0 56] 7 12] 6.09 | 5.09
A) 9. VA OQ A ASS 74s |) 454 11) 9 41 LON ie: 4 S163 | ae LOLS FIMO Sony 5.63 | 4.39
m5) ID. PLAN) 70) ot |: SUG ba53n | Aid 12| 9 41] 0 59] 7 14/ 5.98) 4. LO) 9165) 108555 7, 5.63 | 4, 37
at) 9 AS a al 7 12) 5.16 | 4.65 15| 9 46| 0 57 7 10) 5.76 | 4. 10 16) 0 53 7 10 | 5.95 | 4.57
ne) 9) 19, ft os 7 12) 5.27 | 5.01 12 9 52°\) 0) 525) 7%) 451 5.43) a 10 19] 0 57] 7 10) 5.73 | 4.43
mo 9 18 aly a 7 12) 5.08 | 4.89 10 9 44 0 58 7 715.60) 4 10 12 0 55) 7 5) 5.63 | 4.97
915.2) 1 4.8] 7 16.2) 5.21 | 4.94 276 9 44.9) 1 1.5) 7 12.9) 5.49 | 4, 10 15.0) 059.1) 7 9.8| 5.50| 4.80

— ~

(/— WW’) = 10h, 30m. . . «© « 11h. Om. (W—wW')=11h,0m. . . . . 11h. 30m. (b—wW/) =11h. 30m... . . 12h.0m.
Obs. |(ws—w')| A, Ae Hy | Hy | Obs. |(w—w)) Ay re H, | He | Obs. |\(W—w’)| Ay Ae 18h, |) ashy
|

h.m.| hom. | he m.| Fe. | Ft. hom. | hom. | him | Ft. | Ft. h. m. | h. m.| h. m. | Ft. | Ft.

Bt) 14]}10 45) 0 54 7 4] 5.57 | 5.30 13} 11 19 0 50) 6 59 | 5.66 | 4.66 10 | 11 46 0 47] 6 55 |"5.84 | 4.25
Le) 13} 10 45] 0 56 7 1) 5.38 | 5.04 11} 41 19) 0 52) 6 581 5.32 | 4.87 10) 11 43] 0 41 6 58 | 5.58 | 4.78
15 10/10 48} 0 53 Re) oss 76) 10} 11 13 0 50 6 57] 5.49 | 4.56 12}11 40 0 52) 6 54) 5.32) 5/21
eo) 8/10 47} 0 58 % ) | 5.36 | 4°75 if ak} 0 53 6 58] 5.53 | 4.62 11)11 43] 0 50 6 55 | 5.28 | 4.91
E14) 11 | 10 43/ 0 59 1 S19) ont 92 Py iL 120 Hod! 6 59 | 4.94 | 4.95 10) 11 46 0 49 6 53 | 4.83 | 5.25
PA) 10} 10 44) 0 56 6 58 | 5.06 | 5.17 Gy) Tok ae Os Gt! 6 58 | 5.05 | 5.04 9} 11 42] 0 50 6 55 | 5.03 } 5.22
A) 10} 10 46 0 54 7 O| 4.92 | 4.89 10; 11 14) 0 48 6 51 | 4.85] 5.11 10 | 11 44 0 44 6 48 | 5.15 | 5.02
2.5 11 }10 43) 0 49 6 57 | 5.44 | 4.83 SL aL LG 0 46 6 54 | 5.35 | 5.04 11 | 11 47 0 44) 6 45) 5.15 | 477
Ab) 15|10 44 0 50 7 0] 5.28 | 4.63 13 | 11 18 0 45 6 55 | 5.08 | 4.62 10} 11 49} 0 45) 6 50) 5.50 | 470
Av) 13 | 10 44; 0 50 6 57 | 5.42 | 4.46 10}11 18] 0 40 6 51 | 5.42 | 4.45 9/15 40 0 42) 6 50} 5.62} 4.47
i) 12|10 47 OmoL 7 #1/) 5.86 | 4.41 1S) a Pay, 0 47 6 58 | 5.47 | 4.49 13 | 11 44 0 42) 6 50) 5.79 | 4.14
54) 11 | 10 46 0 53] 7 47) 5.98 | 4.66 12 | 8) 13 0 49) 7 ’ 0 | 5.74 | 4.24 13 | 11 44 0 47 6 59 | 5.82} 4.60
AG) 13 | 10 43 0 53 7 7) 6.06 | 4.17 14) 11 13 0 55 7 6 | 6.29 | 4.14 13 |11 46) 0 44] 6 55] 5.81 | 4.51
ie) 13 ||10 42) 0 58 7) 13) 6.25 | 40917 13 || 5) 0 54 7 9 | 6.43 | 4.06 13 | 11 48 0 49 Y 1) 5.88 } 4,30
5 9/10 40) 1 1 1A 9))\\to: 9% |S 1D || a1 Ps 0 54 7 10 | 6.02 | 4.55 12]11 45) 0: 48) 7 6. 28 | 4,20
<3) 11|10 44 1 4 7 «17 | 6.25 | 4.54 12 | 11 14 0 56 7 10 | 6.17 | 4.39 10|11 46) 0 44) 7 1) 6.14) 4.33
5 13) 10 43} 0 58] 7 16 | 6.27 | 4.13 10} 11 18] O 54 7 10 | 5.82 11 |11 47 0 52) 6 59 | 6.21 | 4.20
5 10/10 48) 0 54 @ 0) | 6.50 }\74562 12 | 41. 17 0 43 7 0] 6.23 11|11 46) 0 46) 7 3/631] 414
5 11 | 10 42) 0 53 7 10 | 5.84 | 4.39 12) 41 14 0.50 87> 7 5.72 10} 11 45} 0 43 6 58 | 5.82} 4.36
5 11 | 10 47 0 48 7 #%1/{ 5.94] 4.98 11}11 13} 0 45 7 Oj} 6.19 10 | 11 45} 0 48) 7 41] 6.08} 4,22
5 12)10 43) 0 41 7 5 | 5.66 | 4.07 10;}11 13] 0 44 6 53 | 6.10 12/11. 43) 0 38) 6 51-|'5.62 | 4.64
5) 12} 10 46 0 47 7 3] 6.08 | 3.79 12} DLy 15 | 0n43 6) 57 | 5.92 11) 11 44) 0 33] @ 47 / 5.73) 3.86
Ab) 12|10 39] 0 49 7 Oj 5.97 | 4.24 13} 11 14 0 44 6 59 | 5.78 14/11 48) 0 41) 6 49 | 5.61 | 4.04
2. 5 15} 10 45) 0 49 WT 1 5:63") 4531 15} 11 19 0 47 6 54 | 5.65 13) 11 51 0 45) 6 53 | 5.78 | 4.41
280 | 1044.3] 053.7] 7 4.3] 5.72| 458] 278/11 15,3] 0 49.0| 6 59. 7| 5.68 | 268 | 11 45.1) 0145.4) 6.55.3) 5.67 | 4.58

DISCUSSION OF TIDES IN BOSTON HARBOR. 17

TABLE Il—Containing average values belonging to the Arguments (—7')=4m and 6=4 73.
LOWER TRANSITS.

(p—w')=0h. 0m. . . « « Oh. 30m. (wW—wW') =0h. 30m... . . Ih. Om. (wW—W)=1h.0m. . . « . 1h. 30in.
Obs. |(W—w’)) Ag Ay Hy | Hy, | Obs. |(~—y’) a3 Ay H, | Hy } Obs. \(y~—w’) A3 Ay Hg | Hy
| |
6 h.m.| h.m.| hom.| Ft. | Ft. hem. | hom.| him. | Ft | Ft | him. | him.) him. | Fe | Fe.
7.5 14 0 17) 0 40) 6 50} 5.67 | 4.60 13 0 48) 0 33] 6 46] 5.87] 4.05 LON) 1h 9155) (00285) 16) 139) | 53904532)
me) 11 0 22) 0 38) 6 51) 6.11} 4.09 11 0) 47>)" (0) 33: 6 47 | 6.00 | 4.28 13 1 ail 0 .32| 6 44] 5.90 | 4.31
At) 8] 0 17] 0 40} 6 52) 5.95 | 4.31 12 0 41) 0 34 6 48 | 6.01 | 4.51 15 1 15] 0 30 6 44) 5.86] 4.5
AMY 13) @ il}; @ Be) 6 54) 6.11 | 4.05 13 0 46) 0 35] 6 47] 6.08} 4.45 8} 1 17] 0 30) 6 45) 6.02) 4.8
60) 10} 0 20 0 38| 6 54] 6.23 | 4.43 11 0 S1 0 39 6 50 | 5.88 | 4.37 10 1 14) 0 24) 6 42) 5.72) 4°30
2.5 9 0 14) 0 42) 6 58] 5.95 | 4.49 10 0 43] 0 33) 6 50] 6.19 | 4.18 10 4 0 26 6 48] 5.82 | 4.52
A) 11 0 17) 0 38] 6 51} 6.31} 4.39 10 0 44) 0 34] 6 50] 6.18 | 4.40 11 1 14 0 30) 6 50) 5.62) 4°55
Ab) 12} 0 30| 0 34] 6 48] 6.10} 4.10 11 0 44] 0 27) 6 43] 6.07 | 4.50 11 1 16 OMT G6 38] 5.59 | 4.61
ab) 10] 0 20] 0 33} 6 47} 5.57} 4.30 10 0 48] 0 28] 6 42] 6.13} 4.25 10} 1 15) 0 19| 6 31) 6.01 | 4.36
2 11 0 18) 0 38] 6 52) 5.93 | 4.53 10 0 44 0 33 6 47 | 5.49 | 4.64 10 1 12] 0 29] 6 42] 5.58 | 4.20
Ab) 12} 0 16) 0 34 6 47 | 5.92 | 4.19 12 0 44} 0 30 6 43 | 5.86 | 4.31 13 1 14 0 33) 6 46) 4.99 | 4.66
2 13 0 12) 0 43 6 53 | 5.88 | 4.62 13 0) 42)) 0 3 6 46 | 5.60 | 4.54 1P}) ah ay |) Pa 6 39 | 5.57 | 4.82
Aby 13°'} 0 12| 0 44) 6 S51} 5.56 | 4.76 14 0 43 0 38 6 48 | 5.21 | 4.85 14 1 16 0 33| 6 43 | 5.40 | 4.78
Be) 12'))" 0) 135) (0) 46 6 53 | 5.51 | 4.98 12) 0 47] 0 38) 6 48] 5.32 | 4.70 120 8h 0) 3h 6 44 | 5.04 | 5.12
Ab) 15} 0 14] 0 48] 6 56] 5.11 | 5.08 13 0 47| 0 42] 6 50] 5.43 | 5.27 12 1 14 0 37| 6 43 | 5.02 | 5.25
Ab) 12} 0 £2) 0 48) 6 56} 5.38} 5.14 10 0 49 0 42| 6 51 } 5.26 | 5.33 9 1 10 0 39) 6 48 | 5.29 | 4.92
BO) it} 0 18] 0 43] 6 50] 5.38 | 5.07 10 0 41 0 49 6 52 | 5.22 | 5. 60 TPs |} ah a3) @, ¢8l 6 49 | 4.87 | 5.57
Bi) 11 0 18) 0 49} 6 51} 5.20 | 5.25 11 0 48 0 44 6 49 | 5.03 | 5.54 11 if ab) 0 35} 6 43] 5.20 | 5.74
3 10) 0 13) O 427) 6 52) 4.69 | 5.25 10 0 40 0 41 6 44 | 4.93 | 5.35 11 1 16) 0 32) 6 41 | 5.07 | 5.67
BY) 11} 0 15] O 38) 6 53) 5.03 | 5.20 11 Q 44 0 30 6 43 | 5.03 | 5.01 11 il atl 0 30} 6 36) 5.21) | 4°99
Ab) 11 0) 27) 0) 45 6 44 | 5.13 | 5.33 11 0 45) 0 36 6 38 | 5.24 | 4.55 11 1 16 0 28] 6 37) 4.95 | 5.09
BD} 12) 0 13) 0 36) 6 45) 5.66 | 4.20 11 0 45) 0 23 6 35 | 4.95 | 4.84 10 1 16} 0 34] 6 40} 5.16 | 4.93
Ab} 14 0 10; O 36 6 46 | 5.27 | 4.39 13 0 46) 0 34 6 43 | 5.67 | 4.60 9 i i) 0 30| 6 39) 5.02 | 4.94
5) 14 0 16) 0 37 6 46 | 5.85 | 4.07 13} 0 49); 0 31 6 41 | 5.39 | 4.45 21871) 00529, 6 44 | 6.13 | 4.64
277 0 15.7/ 0 40.4| 6 50.8] 5.64 | 4.63} 278 | 0 45.2! 0 35.1] 6 45.9/ 5.59 | 4.69] 263) 114.7) 0 30.6) 6 42.3) 5.46 4. 84
(W—wW!) Th. 30m. - . . . 2h.0m. (p—wW!)=2h.0m. . . . « 2h. 30m. (W—W')=2h. 30m. . . . . 3h.0m.
Obs. |(W—wW')| 3 M4 Hz; | Hy j Obs. |(P—w’)| Ag QW Hz | Hy | Obs. |(W—wW!)| 3 M4 H; | Hy
— |
® h.m. | h. m. him Ft. Ft. h.m.| hem. | hom. | Fe. Ft. hem. | hom.| him. | Ft. Ft.
7.5 11 1 40 0 25) 6 37 | 5.37) 475 TSH) Qe L5H Ol 235) 16) 6325185533)),5509 11 2 44 0 24) 6 33 | 5.34 | 5.19
2259 15 1 44 0 27) 6 39) 5.76 | 4.89 Wl |) P} ale l| WPS} 6 37 | 5.68 | 4.88 9 2 48) 0 2 6 30 | 5.03 | 4.82
37.5 13 1 49| 0 28) 6.41) 5.77) 4.97 12 2 18) 0 24 6 35 | 5.42} 4.98 12| 2 42] 0 25| 6 41) 5.48 | 5.00
52.5 11 1 44) 0 24 6 43 | 5.72 | 4.49 153 || Fl) WO pal 6 38 | 5.66 | 5.06 15| 2 45) 0 24 6 38 | 5.28 | 5.35
->f 10] 1 42) 0 30 6 40 | 5.89 | 4.63 11 2 15) 0 23] 6 36] 5.46] 5.24 11 2 43) 0 18| 6 32'| 5.52) 5.41
-oH 14 1 45 0 26) 6 38) 5.66 | 4.71 12)|/ 2° 15 0 19| 6 34 | 5.47 | 5.00 10| 2 45] 0 18] 6 32) 5.74) 5.48
.54 10 AS OF 19 6 35 | 5.88 | 4.79 @) |} 783 |) Oa} 6 33 | 5.56 | 4.53 14} 2 44) 0 15] 6 33] 5.55 | 5.40
5 11 1 44) 0 22) 6 36) 5.48) 4.94 9) 2 16 Oman 6 33 | 5.43 | 5.10 9|} 2 46} 0 13) 6 30 | 5.78} 5.09
.5 | 9 1 46] 0 20| 6 35] 5.76 | 4.76 11 Qa ONL) 6) 32>) S71 |) 4591 11 2 46) 0 8] 6 23) 5.41 | 5.21
5] 13] 1 44] 0 20) 6 31 | 5.48] 4.39 11 2 15 0 16) 6 3L| 5.40} 4.79 1 OB Zl O mH] G gs] 09 | be
ong) LO 1 47] 0 28] 6 39 | 5.23 | 4.96 MH) Oo ms} @ Wel) GO 62 | 5. 22 | 4. 84 10| 2 46) 0 15] 6 28) 5.17) 4.83
5} 11 1 45 0 28 6 38 | 5.43 | 4.67 13)|| 2 13 0 29] 6 40] 4.94 | 5.02 ash |) py Aut 0 22) 6 33 | 4.92 | 5.25
By) ale} |) ale} 0 30 6 43 | 5.25 | 4.82 11} 2 17] 0 24) 6 32) 4.93)) 4.90 12] 2 45) 0 24) 6 35) 4.81 | 5.37
Sq 11 1 45| 0 33) 6 40] 4.91 | 4.76 18} || a4) @ eh) 6 38 | 4.83 | 5.03 13} 2 41 0 27) 6 37] 4.63 | 5.41
5] 13 1 41 0 34) 6 42) 4.77 | 5.13 13) 2 10) 0 25.) 6 36) 4.77) 5.47 12| 2 45°] 0 27) 6 39 | 4.37 | 5.28
5 13 1 47 0 35| 6 41 | 5.01 | 5.60 13] 2 18] 0 29) 6 37} 4.60 | 5.38 12) 2 44) 0 26 6 37 | 4.61 | 5.9L
5 12) 1 45) 0 34 6 42] 4.88 | 5.58 12}; 2 18 0 29} 6 36 | 5.03} 6.03 u 2 48/ 0 33] 6 40 | 4.83 | 560
5 4 10 1 43 0 33) 6 40) 5.25) 5.62 1} 9} abl 0 30 6 37 | 4.64) 5.35 12] 2 43] 0 25.| 6 36) 4.54) 5.85
{| il 1 43) 0 30) 6 37] 4.93 | 5.27 11 by ies || O 2H) G sh 4.7 6.00 11 2 47 | 0 21 6 31 | 4.98 | 5. 82
5 9) 1 40 0 34) 6 39] 5.16 | 5.45 rol |) 95 ales 0 23) 6 33 | 4.74} 5.39 10} 2 43| 0 23| 6 30°) 4.77 | 5.58
.51 12 1 45 | 0 23 6 33) 5.16 | 5.20 TD || 2 aE Oa) 6 32) 4.51 | 5.10 10)) 2) 41) 0) 21 6 32 | 4.76 | 5.65
5 1 1 46) 0 23) 6 35) 4.96 | 4.78 10} 2 14 0 16) 6 28) 4.96} 5.43 LOD ese a | 0 18] 6 29] 5.01 | 5.19
Oa))) it 1 44 0 20] 6 34) 5.43 | 4.85 12} 2 15} 0 22] 6 30] 5.06 | 5.25 11 2 46) 0 24) 6 36 | 5.08] 5.27
5 11 1 41 0 25 6 33 | 5.47 | 4.57 12) |) @ wey) O Bp 6 30 | 5.27 | 5.09 1l 2 46) 0 24 6 36 | 5.35 | 5.09
274 1744.4) 0 27.1) 6 37.9| 5.36 | 4.94 | 279 214.6] 0 22.8 6 34.0) 5.14] 5.17] 268) 2 44. § 0 21.3) 6 33.4! 5.09 | 5.33

Se)

18 DISCUSSION OF TIDES IN BOSTON HARBOR.

TABLE Il—Containing average values belonging to the Arguments (1) — 1’) =4m and ¢=473—Continued.

LOWER TRANSITS.

(W—W) =3h.0m. . . . . 3h. 30m. (W—wW') =3h. 30m. . . . . 4h.0m. (W—W) =4h.0m. . . . . 4h.30m.

¢ L
. (Y—p)| ag vy Hz | Hy }Obs. |((w—wy’)| Ag vy H’ Obs. |(~W—wW')| Ag vg H;
° | h. m.| h. m.| h. m.| Fe | Ft. h. m.| ho m}) he m.| Fe. h. m.| he m.| he m.| Ft.

7.54 ar ite ae) 6 32] 5.30] 5.40 12) 3° 43) 0 20 6 32 | 4.74 aby |) ek aly 0 28 6 42) 4.84
22.54 3 9| 0 2] 6 36} 5.41 | 5.42 13} 3 44] 0 22] 6 37] 4.98 12} 4 16) 0 25) 6 40) 471
37.5 f 3 13] 0 16] 6 33} 5.14 | 5.38 12} 3 45) 0 19 6 38] 4.96 12))| 4 15)5)) (0) 28 6 43 | 4.90
52.5 3 15] 0 18] 6 24} 5.23 | 5.45 1h) Sy 25 |) A) BES) @ SS}\) GER} 12); 4 15] 0 24 6 43 | 4.82
67.5 3 11 0 18) 6 35) 5.30} 5.09 14 3 45] 0 20) 6 40) 5.02 15) 4 11 6 22) 6 39] 5.04
82.5 3) 254) 0) 12))) 16) 28) 5537 || 5520 135) 3) 465), 0) 15) 16) 2915169 13) e eon ere 20 6 34) 4.91
97.5 3°16] C 16) 6 30] 5.07 | 5.45 9} 3 46) 0 10) 6 29) 5.24 13} 4 12} O 14] 6 30] 5.07

112.5 | 3 13) 0 14 6 31] 5.24 | 5.04 11 3 41 0 13 6 30 | 5.02 13 | 4 17 0 12/ 6 31] 4.87
127.5 3 15] 0 13 6 27) 4.92 | 5.37 12;| 3 47) 0 15) 6 327) (5.00 13) 4 16 0 13 6 28 | 4.60
142. 5 § 3 145) (0° 25) 6 26))/'5.09) 5.38 10 3.45! 0 14 6 28 | 4.93 10} 4 14 0 12) 6 30} 4.80
157.5 | 3°14] 0 17 6 28 | 4.55 | 5.40 11 3 44 0 21 6 34 | 4.92 13} 4 14] 9 13} 6 30] 4.72
172.5 i 3 14) O 22) 6 33) 4.85 | 5.37 11 3 45] 0 22 6 33 | 4.56 10} 4 18) 0 25/ 6 40) 4.52
187. 5 | 3 15] 0 28 6 37 | 4.79 | 5.49 11 3 50] 0 26 6 38 | 4.39 10} 4 16] 0 30 6 39) 4.11
202.5 3 17] 0 2 6 35 | 4.37 | 5.62 12) 3 44) 0 27 6 38 | 4.28 11 4 15} 0 30 6 44 | 4.38
217.5 | 3 14 0 24 6 36 | 4.79 | 5.50 13 3 44) 0 25 6 39 | 4.06 bby er ae by 0 31 6 41 | 4.03
232.5 } 3 16)|) 0) (27 6 38 | 4.52 | 5.7 13} 3 45) 0 29 6 39 | 4.34 13} 4 18) 0 33 6 43 | 3.93
247.5 | 3 13 0 2 6 37 | 4.43 | 5.89 13 3 50 0 28 6 38 | 4.16 11 4 15} 0 30 6 39 | 4.03
262. 5 By ay) (On Pat 6 31) 4. 6. 3 45] 0 28 6 38 | 4.30 12) 4 10 0 22) 6 35] 4.16
77.5 3) 135) 10/523 6 33) 4 6. 3 44 0 22 6 35 | 4.08 12; 4 15] 0 24) 6 34| 4.67
292. 5 | 3 15) 0 22) 6 32) 4 6. 3 48] 0 19 6 24) 4.84 10} 4 13 0 21 6 33 | 4.34
307. 5 § 3 16; 0 16 6 27) 4. 5. 72 3 49} 0 15| 6 32] 4.91 11 4 15 Waly 6 30) 4.47
322.5 | 3 18] 0 15 6 28} 4. 5. 3 48) 0 16 6 30) 473 9| 4 14] 0 24 6 34) 451
337.5 § 3 16) 0 18 6 31) 4 5. 3 47) O 14 6 30 | 4.76 9| 4 16} 0 20 6 34 | 4.88
352. 5 3 13} 0 16 6 225] 5. 5. 3 42) 0 25 6 37 | 5.37 12; 4 14) 0 24) 6 36) 4.62
314.5] 019.4) 6 31.4] 4.89 | 5. 3 45.6) 0 20.3) 6 33.9) 4.74 283 414.8] 0 22.6 6 36.3\ 4.58

(W— wy’) = 4h. 30m. . . . « Sh. Om. | (P— wy’) =d5hk.0m. . . . . 5h. 30m. (Wh — W)=5h. 30m. 6h. Om.
6 z =
Obs. |(P—wW)| Ag Ry | (ei8bsclleasth (W—wW)| dg a, | Hz | Hy | Obs. \(p—w)] ag ay Hy
|

° h m.| ho m.) ho m. | Ft. | Ft. h. m.| h m.| h. m.| Ft | Ft. hm.) h. m.| he m.| Ft.

7.5 11 4 44 0 27 6 44) 4.83 | 5. 5 12] 0 35] 6 48] 4.78 | 5.69 10} 5 44 0 39 6 53 | 478
22.5 9 7 4 48] 0 °33| 6 5.00 | 5. oy alz 0 39 6 50 | 4.85 |] 5.8 11 5 46 0) 455) 7 at 4582
37.5] 13] 4 43] 0 32] 6 52] 4.641] 6. o 135) 0, 34 6 52 | 4.93} 5.70 12) 5 44 0 44 7 5| 4.79
52.5 13 4 44) 9 33 6 52) 4.81 | 5. 5 14] 0 36} 6 52] 4.49) 5. 10 5 46 0 42) 7 1) 5.06
67.51 13] 4 45| 0 25] 6 53| 461] 6. 5 15] 0 34| 6 56| 4.91] 5. 13] 5 46 0 42) 7 1) 4.64] 5.
82.5 12} 4 44) 0 22) 6 40} 4.77] 5. 5 14 0 30 6 51 | 4.87] 5. 9| 5 46 07535) 6 54 | 4.64] 6
97.5 12) 4 49; 0 21 6 38 | 4.87 | 5.95 5 16; 0 16 | 6 37 | 4.91 | 6. 10| 5 44 0 S28)), (6) SIN AS Telos

112.5 11 4 50; 0 17 6 32) 4.42) 6. 5 15] 0 18 6 39 | 4.72] 5. 11 5 44 0 26 6 43 | 4.82] 6.
127.5 10 4 48) 0 20 6 35 | 4.90 | 5. ° 13! 0 18) 6 39] 4.56) 5. 11 5 45 | 0 28 6 46 | 4.44] 6.
142.5 11 4 44/) 0 22) 6 37] 4.46 | 5. 35 14} 0 20 6 35 | 4.72] 5. 12| 5 47 0 32 6 49 | 4.58 | 5.
157.59 11 4 46) 0 19} 6 36) 4.84] 5. 3 16] 0 27 6 49 | 4.60) 6. 12) 5 49 0 32 6 46 | 4.49 | 5.
125) 125) 4° 47) 0) 321] 16) 42) 4:93 1/5, 5 16] 0 33] 6 48] 4.36] 6. 11 5 44 0 34 6 48 | 4.15 | 5.
187.54 10 4 46| 0 32} 6 42} 4.19] 5. 5 13] 0 41] 6 51} 3.90} 5. 12| 5 42; 0 41 6 56 | 4.07 | 5.
202.5 11 4 44 0 32) 7 44/ 3.84] 5. > 17) 0 48 6 57 | 4.06 | 5. 10}; 5 49 0 49) 7 0} 3.96) 6.
217.5 i 11 4 49 0 40 6 55 | 3.89 | 6 5 17/ 0 44 6 59 | 4.20 | 6. 9) 5 44) 0 50 6 50 | 3.90 | 5.
232.58 12) 4 47 0 40 6 50 | 3.76 | 6 DL 19 | 0 40 6 52 | 3.99 | 6. 134) 3) 41 | 0 56 7 #4) 3.61) 6.
247.5 f 11 4 45) 0 37 6 45 | 4.05 | 6 D> 12 | 0 45) 6 55/3.61] 6 14 3 47| 0 48 7 33.76) 6.
262.54 12] 4 51 0 31 6 45 | 4.30 | 6. 5 12| 0 36 6 46 | 4.05 | 6. 13) 5 43 0 49 6 58 | 3.97 | 6.
277.5 | 12))| 4 45) 0 30)) 6 43 | 4.10 | 6. 3 18] 0 28 6 40 | 4.18 | 6.5 11 5 48/ 0 46) 6 53} 3.56) 6.
292.5) 13| 4 43) 0 22] 6 34 | 4.05 | 6.2 > 144) 0 28 6 43 | 430) 6 11 5 43 0 33 | 6 50) 4.35 | 6.
307.5 f 2 4 44] 0 20] 6 36] 4.53] 6 5 12) O 28) 6 43] 4.30 | 6. 11 5 44 0 34 6 50 | 4.12] 6.
322.5 10 4 42) 0 20 6 34 | 4.69 | 5. 3 13) 0 2 | 6 39 | 4.40 | 5. 13} 5 45] 0 32] 6 47] 4.42) 3.
337.5 H 12 4 46 0 27} 6 42 4.59 | 5. 5 17| 0 33| 6 46| 5.05] 6. 11 oO) 454) 0) 31 6 48 | 4.62 | 5.
352.5 10] 4 44 0 27 6 43 | 5. 08 | 5. . 45 0 32 | 6 48 | 4.67] 5. 11 5 43 0 38/| 6 53 | 5.00 | De
f 4 45.8) 0 27.6) 6 42.9) 4.48 | 5.99 5 14.3) 0 32. 0 6 49. 2) 4.48 | 5. 271 5 45.0) 0 39.1) 6 53.8) 4.39 | 6.
| | |

DISCUSSION OF TIDES IN BOSTON HARBOR. 19

TABLE II1—Containing average values belonging to the Arguments (4 —w')—=4 m and ¢=n3—Continued.

LOWER TRANSITS.

(W—wW')=6h. Om. . . . . Gh. 30m, (wW—wW') =6h. 30m. . . - . Th. Om. (W—wW!) =Th. Om. . . . . Th. 30m.
Obs. |(~—w’)| Ag ay | Hy | Hy 7 Obs. \(Y—W)| As A H; | Hy } Obs. |((Y—w’)| Ag va Tsk. |} 33h
}
h. m.| h. m.| h. m.| Ft: | Fe. h, m.| he m.| ho m.| Ft. | Ft. h. m.| he m.| ho m.| Fe. | Fe.
13} 6 13] 0 46) 7% 2 5.07] 5.69 12) 6 45} 0 52) % 8) 4.99 | 5.81 10} 7 14} 0 51 @ 12) )/5.19 | 5: 13 |
11 6 16) 0 48) 7 4] 4.91 | 5.85 DQ) |) 6) 445 0; 4 te 65.19) 5x48) RI TON ie 1) 0) ba.) 5) 5535) 55151)
6 10) 6 17) O 52) 7% 10) 4.86 | 5.53 11 6 47} 0 50 Ue 12S ALO Oat LON te 16 0 56) 7 18) 5.15 | 5.17
2.5 10} 6 13| 0 44) 7% 91] 5.12 | 5.92 12| 6 42| 0 57] % 14] 4.90/5.73] 12) 7 15] 0 O Me Q1 fon 5536
.5 14] 6 10} 0 45) 7 4,72 | 5.80 13 6 43 0 57} 7 20) 4.70 | 5.90 10 COLTS ON 5257 12h ESS SOR 5362
Be) 14} 6 li 0 42) 7 1/4) 4.66 | 5.86 16 6 45) 0 55] 7 15] 4.75 | 5.46 f 11 7 15} 0 58) 7% 19} 5.01 | 4:96 |
b 5) 16} 6 12) 0 39 i 4,82 | 5.83 14) 6 49 O@ 47| 7 7) 4.94|)5.868 14] 7 16) O 54] 7 14} 5.00/ 5.68
Aas) 12} 6 12} 0 29) 6 52) 4 89 | 5.97 12; 6 43) 0 46] 7 3] 4.54 | 6.26 11 a OFA ieronln4eriiall sone, |
An) 13} 6 15} 0 34} 6 54/| 4.64) 5.80 13| 6 49 0 43| 6 59 | 4.66 | 6.09 14 7 16} 0 47) 7 51] 4.90 | 5.55 |
By 11 6 15} 0 27] 6 43) 4.35 | 5.99 11 6 44) 0 40 7 O| 4.68 | 5.54 D2 te D2 0 147 UT 35 | 4.54 | 5.7
Bb) 9| 6 18] 0 46) 6 56) 4.75 | 6.90 12) 6 44] 0 45) 7% 1) 4.61 | 5.79 120 16 0 48) 7 3} 4.51 | 5.64 |
No) 11 6 14) 0 42) 6 56] 4.34 | 5.75 11 6 43| 0 51 to 4) 4560 1 on78 12) % av) 0) 56 7 14.749) 15.90
5 13 6 15| 0 47] 6 Sa | 4.41 | 5.96 11 6 41 0 52) 6 56) 4.26) Sg 12) 7 17 ual % 12 | 4.26 | 5.62
. 5 10/} 6 18] O 51 7 6 4.05 | 5.69 11 6 49 m Oi WT) Coie Ge) aI af Te) i al 7 12) 4.53 | 5.44
Ar) 12} 6 12 1 2] 7% 11) 4.14 | 6.42 12) 6 42) 0 59 7 14 | 4.00} 5.8L 125), 16 1 10 % 20) 4.13 | 5.99
Boy 12| 6 12] 0 54] 7 5) 3.80) 6.08 11 G47 Shi 1994503) 16505 10 7 14 ZL 2] 7% 14) 3.85 ) 5.22
Ab) 11 6 18) 1 2) 7% 12) 3.93') 6.40 11} 6 45) 1 S| 1 13) 4.04) 6.09 1) |) all 1 9j| 7 18) 3.80} 5.64
Ar) 14 6 Be) 0) 52)\) 7 11369) || 6:30 13 6 45 1 4) 7 12) 3.73 | 6.05 11 7 14 1 12) 7 19 | 4.05 | 5.64
Bb) 15| 6 13] 0 46 6 58 | 4.16 | 6.39 14} 6 48 0 58) 7 7) 3.78) 6.18 | 12))| % 16 1 U4 7 19 | 4.06 | 5.86
. 3 12} 6 15] 0 46) 7 O|} 4.14] 6.16 IB CO OP Gee SERS GO) Fakes ef IG OL a VY 10) 4.25 | 5.91
5 14 6 12] 0 40 6 52 | 4.18 | 6.24 13} 6 48) 0 50} 7 5} 4.25] 6.23 | IPSN aby) WO BSN 7 |) CBR) Ge!
A) 11 6 15] 0 36} 6 53) 4.34 | 5.87 10} 6 49) 0 50} 7 41] 4.50} 5.86 10 7 12) O 47}! 6 59) 4.75 | 5.59
A) 11 6 13 0 38 6 55 | 4.57} 5.91 10} 6 46)| 0 42) 6 56) 4:79 | S72) 12) 7 13) O 52) % 6) 4°83'| 5.62
5 12)) 6 19 0 43) 7 #O|} 4.74] 5.60 11 Gia Ta On4 7s eNO eOOniPoLOn ool ime LON lt) LON Olmo2e mete elon locOSi|tostol
291 6 14.0) 0 44.6] 7 0.2) 4.47] 5.96} 289 6 46.0) 0 52.6) 7 7. 0| 4.51 | 5.85 | 278 | 714.8) 0 56.4) 7 11. 6| 4.64 | 5. 59
— 4
(P—y') =Th. 30m. . . . . Bh. Om. (P—w') =8h. Om. 8h. 30m (wW—w’)=8h. 30m. . . - . 9h. Om.
Obs. |(W—')} dg Ay | He | Hy | Obs. |(Y—)] > ag Ay | He | Hy ] Obs. |(Y—W] dg A; | Hs | Hy
h. m.| h. m.| h. m.| Ft. Ft h. m.| h. m.| h. m.| Ft. | Ft. h. m.| h. m.| h. m.| Ft. | Ft.
25 10 7 41 0 57 % 13)) 5:37} 5.36 11 8 17 ft 2) |) % 19} 5:26) 5:08 10} 8 44) 0 56 7 12) 5.82 | 4.53
Bb) 11 7 47 4 e210 |F52319)|\o180 11 8 17 1 3] 7 19) 5.64) 4.88 11 ZB) ak 7 18} 5.31] 4.83
At) 10 1 45 i al UT 24) 5.69 | 5.16 12 8 14 tO} 7% 22 | 5.69 | 5.37 SIRABE pe bes4 7 21) 6.25 | 4.46
By) 9 Tt 43 1 2) 7 27) 4.93) 5.40 || fey aly 1 2) 7 24) 5.71) 4.47 12 8 46 1 7] 7 30) 5.46 | 5.23
6b) 9 GW AN al By Ge OB) BGI 1] GBB 11 ff) 76} )) ab at % 22\| 9.57 || 4.93 8 45 1 2) 7 21) 5.23 | 5.05
2.5 11 t 42) 1 5 7 30} 5.18] 5.57 11 8 14 1 3 % 24 | 5.32 | 5.26 11 8B 44 uw 7 25 | 5.65 | 5.10
Bis) 13 (Ain | Les Oba cel Oma O 2493 13 8 14 52) YE PHL eK |) GREP UW) |) Gy ee a} 7 19 | 5.18} 4.80
ms) 13 7 44) 0 55) 7 15} 4.79 | 5.45 TSR 81250 a) PONS Sel 45977 12} 8 49 if al 7 20 | 5.17} 4.80
Ab) 12) 7 49] 0 54] 7 9 | 4.83] 5.90 13} 8 17) O 56 7 12) 5.14) 4.97 13] 8 48} O 58) 7 18] 5.15] 5.01
Bi) 12 7 46 0 52) 7 10) 4.94 | 5.50 137) (8 15) Oat) 1 2h 5. 41 12} 8 44 1 2) 7% 17] 5.07) 5.43
A) 12) 7 45] 0 59} 7 11] 4.87 | 5.63 11 8 12) 0 58) 7 12)) 4.61) 5.66 13} 8 48] 0 56} 7 8} 5.13) 5.18
£0) 11 7 47) 0 58} 7 9} 4.95} 5.28 IPP) {3} ay at oy’ 7 12°) 5.00) 5.61 13 8 43) 1 3 7 15 | 5.08 | 5.28
A) 11 7 44 1 0} 7 13] 4.73] 5.53 11 G aks) al ©) 7 16 | 4.78 | 5.62 13 | 8 46 1 12) 7% 18) 4.82) 5.34
A) 9 T 47 1 13) 7 21 | 4.45 | 5.50 12 8 12 ip) 7 18) 4.55 | 5.60 12| 8 44 i ght 1 16 | 4.73 | 5.32
er) 11 7 43) 1 12) 7 QL) 4.46) 5.74 11 8 18} 1 12] 7 20} 4.49] 5.40 12} 8 49 1 iv) 7 24) 4.62) 5.44
Bey 11 7 46 1 17) % 24) 4.17) 5.72 11 8 14 ite 1% 6235)174536))) 5.79 11 8 46 1 16) 7 24) 4.26 | 5.52
BR) 12 7 44) 1°16 7 22) 4.24 | 6.00 10 8 16} 1 10 VY 16 | 4.31 | 5.48 12} 8 46 1 12) 7 21) 4.51 | 5.46
BBY 11 7 48 D9) 228) 45351) 5.167, 12} 8 13) 2 14 T 21) 4.09 | 5.40 10 8 44) 1 13) 7 22) 4.64 | 5.51
Bb) 11 Ay 4a DE % 2013567 || 5.74 11 Cleon | weet 7 19 | 4.14 | 5.52 13 8 49) 1 15) 7 19] 4.14) 5.40
no) 13) 7 46 1 5| 7 12) 4.48) 5.84 13 BLS Ete SN 4S SBie Pons 13} 8 43} 1 5] 7 13] 4.44 | 5.29
BY 11 7 45 i 7 13 | 4.63 | 5.49 gy |] Go. yf) 7 16 | 4.65 | 5.53 13} 8 46] 1 6] 7 14) 4.99 | 5.20
Bb) 15 7 45) 0 57 7 (W| 4.88) 5.77 14} 8 15 1 0 7 13'| 4.86 | 5.42 11 ets} ah al 7 13) 4.79 | 4.87
A) 13 7 43 0 S1 7 35] 5.10) 5.36 163) fe} all |) ae 7 13 | 5.00 | 5.24 10 8 46) 0 55)) % 12)) 5.37 || 4.64
5 12 7 46 0 57) 7% 12) 4.97% | 5.21 11 SieiZa| ete Lelinedpeslonl ons wallos00 10|} 8 46) 0 56} 7 15) 5.59 | 4.81
273 7 45.2) 1 2.5) 717.0) 4.81 | 5.52 9 287 8 14.6) 1 4.6) 717.5) 4.95 | 5.31 275) & 45.9) 1 5. 4) 718.2 5.06 | 5.10

20 DISCUSSION OF TIDES IN BOSTON HARBOR.

TaBLE II.—Containing average values belonging to the Arguments (7 —1')=4 m and ¢=+4 n3—Continued.
LOWER TRANSITS.

(W—wW') -9h.Om. . . « . 9h. 30m. (W—wW') = 9h. 30m. . . . . 10hk.0m. (W—wW’) =10h.0m. . . «. . 10h. 30m.
g
Obs. |(W—wW')| Ag Ay Hy Obs. |(Y—w’)| Ag Ay H; | Hy (W—wW)] Ag vg Hz | Hy,
| ° h.m.| ho m.| h. m.| Ft. h.m.| h.m.| hi m.| Ft. | Ft. h. m.| h. m.| h. m.| Ft. | Ft.
| 7.5 UBY || ME BY) ab ee ih ep Pdi) | Gate 13| 9 48] O 57} 7 13) 5.80} 4.53 10 20} 0 56) 7 7/} 5.92] 4.30
| 225 LON OP on LON eve OR os onin4. 1235)59)49 PL 2) 4) 16) S560) 4088 10 16} 0 54) 7 8] 6.28) 3.91
37.5 11 yay |) a ah | ee GAUL GEG} Ch 10} 9° 46) 0 59) 7% 14) 5.90} 4.30 10) 1331) 92)! 19) | Ss93N rae
52.5 11 Gy kK) al ech) ef |) GRY | 2h 8| 9 49 ZL 3%) 7 17) 5.80) 4057 10 14] 0 58] 7 18] 6.53} 4.06
67.5 14))/ 9) D4) a 35% = 23>) 5.182 |/"4 5b eee c 1 0} 7 18] 6.03 | 4.46 10 14) 0 57] 7 18} 6.07 | 4.83
82.5 11 9 13] 0 51 7 13) 5.87) 4 11} 9 49/ 0 56] 7 48] 6.06} 4.7 10 16] 0 53) 7 13] 6.10} 4.10
97,5 LOT ONL GH MOR OBR Nin ton ro.don| 04 9| 9 44) 0 58] 7 17] 5.94 | 4.87 10 12] 0 52] 7 15] 6.00 | 4.20
112.5 11 SPSLTA | Ooo ere ton i5. 904: 9) 9 44) 0 59] 7 12] 5.86] 4.41 10 13} 0 44] 7 61} 5.80} 4.49
127.5 Ty A ae) ela Pie 1) GEE) Hic 13] 9 44] 0 58/ 7 13} 5.36) 4.91 10 17} O 51 tf \9..95°| 4,27
142.5 14S SOF ISRO Oe) cella F5.260 4s: 12| 9 47] O 58) 7 11] 5.58} 4.80 10 12) 0 S11) 7 6) 5.97) 4.52
157.5 UPy || By alee ah Bh ce a8 GEBSS Ge 12} 9 47 | 1 2) 7 11) 5.42) 4°86 10 10] 0 52) 7 6| 5.63] 4.57
172.5 TPA ate) ab eee | ih ok bya | as 12} 9 46) 0 58; 7 9) 5.47) 4.7 10 14] 0 S57} 7 6| 5.66} 4.96
187.5 il SO eta east ros 44 I RO; 10} 9 42 La 1G Pon S8uFon0n, 10 13) 1 2] 7% 10) 5.48) 4°97
202.5 ily) || ale aE aki) |) es IY) GEES |) DD ON 425 ON a6 HPSS oly it) SES ak Gy) 7p ae) Ge283 || ae sil
217.5 9) 9 Ov) 2 14) % 20) 5.03 ]-5. il GB) EEE ah fats) Galt |) GSTs} LOR ELO tS Shae cal RoxsGaleombt
252. 5 UKD)|) -G) aks ab GES ae DAR) Zee |) Gy 10} 9 43 1 6| 7 13) 4.74 | 5.18 10 13} 2 9] 7 14) 5.10) 5.38
247.5 9) 9 16) 1 14) 7% 21) 4.69 | 5. 109) (95 46 5() ee TB 20) 4275))| 5558) MQ ate ah BD rf ON ea kh
262.5 THY | GQ ae) ab abe 7 17 | 4.46 | 5. 11 9 45) 1 9} 7 13] 4.85 | 5.12 10 14) 1 10} 7% 15) 5.03} 5.29
277.5 11 O15. ee eae ii eta eae alan os 10} 9 49} 1 8] 7 11] 4.64) 5.19 10 19} 0 56} 7 3) 4.74 | 5.18
292. 5 TRY |) OG) aE) a ich ete aE eR) |] 11 9 44 Lo 7) © 135) 42539) 4.95, 10 14 1 O} 7 6) 5.14 | 4.64
307. 5 120) TOF UG SRN reeds aS (22085; 12} 9 45) 0 58) 7% 9) 5.18) 4.87 10 15} 0 54) 7 3) 5.08 | 4.78
322.5 9) 9 15] 0 58] 7 10] 5.40) 4 13 | 9 44 1 4) @ 12°) 5,33) 4:87 10 13] 0 54] 7 5} 5.46} 4.48
337.5 135) (9) 135) 2 0%) (7 135) 5.455) -4° 14} 9 41 0 S7| 7 11) 6.25] 4.80 10 15} O 57| 7 8] 5.47 | 4.68
351.5 13 CD Pr S |) GRGES pct 13} 9 43) 0 59) 7 13} 5.51 | 4.89 10 18; 0 53) 7 5)5.77| 4.39
272 | 915.5) 1 3.6) 7 16.5) 5.34 268 | 9 55.4) 1 2.6) 713.8) 5.42 | 4.88 276 | 10 14.7) 0 57.4) 7 9.7] 5.61 | 4.67
(W—wW') =10h. 30m. . . . . 11h. Om. (W—wW')=11h.0m. . . . . 11h. 30m. (Y—w') =11h. 30m. . . . . 12h.0m.
o T
Obs. |(Y—w')| Ag Ay Hz | Hy Ay | Hs | Hy j Obs. (~—wy)| ag v4 H; | Hy
° h. m.| h. m.| h. m. | Ft. Ft. m.| h.m.| Ft. | Ft. hm h. m.| hi m.| Kt. | Ft.
¥ 6x4) 9/10 45} 0 49) 7 4) 6.13 | 4.42 ol 7 3) 6.04 | 4.49 13) 11 44) 0 42) 6 55 | 6.01 | 4.04
22.5 11} 10 42} 0 52] 7% 4) 6.26 | 4.06 02] 7 4/ 6.08) 425 13} 11 50} 0 42) 6 57 | 6.18 | 4.70
37.5 13} 10 Sl 0 55) 7 9} 5.94 | 4.15 49} 7 5 | 6.09 | 4.32 14)11 50) 0 45) 6 59) 6.22) 4.33
52.5 10/10 44] 0 50) 7 10) 5.94 | 3.98 49 7 31 5.93 | 4.24 9/11 48) 0 41 6 57 | 6.04 | 4.01
7.5 8/10 46) 0 53] 7 13 | 6.17 | 4.51 DON Zin 16 6.04 | 4.45 11}11 42) 0 44) 7 3] 6.26 | 3.96
82.5 11}10 46) 0 54| 7 12 6.18 | 4.32 46) 7 0} 6.15 | 4.24 9 | ll 48/ 0 49} 7 3) 6.09 | 4.31
97.5 12}10 45] 0 45] 7 3) 5.84 | 4.60 46 | 6 57} 6.06 | 4.18 12)}11 45] 0 38] 6 55} 6.31 | 4.25
112.5 10} 10 46] 0 48) 7 6/ 6.02] 4.51 40 | 6 56 | 5.95 | 4.26 9] 11 42) O 32] 6 51) 614) 4.12
127.5 9/10 47) 0 45] 7 3) 5.85} 4.09 44 6 58 | 6.23} 4.05 12) 11 43] 0 38] 6 54] 5.92] 4.32
| 142.5 11}/10 40] 0 50) 7 3/ 5.75] 4.51 43 | 6 57] 5.95 | 419 14|11 46] 0 40} 6 52) 5.79 | 4.30
157.5 15|10 45} 0 SL} 7% 1) 5.99} 4.59 42) 6 57 | 5.87 | 4.41 12; 11 49/ 0 42) 6 56] 5.66 | 4.40
| 172.5 12/10 44) 0 59] 7 31} 5.21) 4.65 52 | 6 59} 5.60 | 4.73 11} 11 45| 0 45}. 6 52) 5.84} 4.37
| 187.5 14/10 45) 0 59) 7 7) 5.35 | 4.74 54] 7 1'| 5.48] 4.76 13} 11 44] 0 47| 6 54] 5.47) 4.31
| BG} TW) AH) a Ip ee el) | Bh rk) 58} 7% 4 | 5.48 | 5.25. 9) 11 44) 0 SL} 6 59) 5.67) 4:55
| Ab) 10 50} 1 S| 7 10) 5.08) 5.17 58) 7 41] 5.28] 5.40 12/11 41) 0 53] 7 1) 5.31 | 4.76
2.5 LO) ASH oS ON S8uos81) ito. 14 1 7 5 | 5.28} 5.10 13}11 43] 0 54] 7 0] 4.99] 5.40
Ab) IEC A abe ON gt S | 4.80 | 5.09 56) 7 1) 5.33] 5.31 11 {11 47|°0 52} 6 59 | 5.18} 5.38
2.5 10 46] 0 58) 7% 4} 5.26} 5.02 56] 7 1 4.89 | 4.80 9/11 48] 0 48] 6 54] 5.02 | 5.12
| -5 10 48} 0 58] 7 3) 4.92) 5.57 51 6 59 | 4,96 | 5.22 12/11 45) 0 46} 6 51) 5.15 | 4.68
5 10 44] 0 54] 6 58] 4.98 | 5.00 44 6 53 | 5.10} 4.81 9/11 46] 0 43) 6 46] 4.88 | 5.15
| By WW 43] 0 50) 6 59/| 4.99 | 4.73 50 | 6 56 | 5.56 | 4.41 8/11 48] 0 45| 6 45 | 5.30 | 4.68
i) 10 45) 0 52) 6 57) 5.48 | 4.71 46 | 6 53 | 5.21 | 4.33 10 | 11 47| 0 39} 6 50} 5.46 | 4.49
} Be) 10 47} 0 52) 7 1) 5.49 | 4.28 46 | 6 56 | 5.73 | 4.08 12/11 39] 0 41] 6 51 | 5.85] 4.31
Bo) 10 49 0 50 4% 2) 5.91 | 4.33 50 6 58 | 6.00 | 3.88 12 | 11 42] 0 43 6 53 | 5.49 | 4.14
| 10 45.5} 0 53.8) 7 4.8) 5.59 | 4 64 47.2) 6 59.8) 5.68 | 4.55 7 269 | 11 45.8) ji 0 44.2) 6 54.8) 5.68 | 4.50

DISCUSSION OF TIDES IN BOSTON HARBOR. 21

TaBLE I1I—Containing average values belonging to the Arguments (x) —') =4 7 and 72

COMBINED TRANSITS.

n= 15° 7, = 30° n,= 45° “m= 60°
(w-wh) |
Obs.) 4 An HH; H’, | Obs Ny | HY, | Ble |Obs. | Ay | AY | HY | He | Obs.) Ay NEY ||| 28l ast
|
h. m h. m.| h.m.| Ft. Ft. h.m.| h.m.| Ft. | Ft. hom.| h.m.| Ft. | Ft. h.m.| him.| Ft. Ft. |
0 15 21) 0 40 6 51 6.51 || 3:52 21 0 34 | 6 45 | 6.38 | 3.57 20 | 0 37 | 6 49 | 6.67 | 3.65 23 | 0 39 | 6 52 | 6.43 | 3.79
0 45 23} 032) 6 46 6.33 | 3.49 22) 035 | 6 46 | 6.54 | 3.45 20 | 0 35 | 6 47 | 6.50 | 3.59 20 | 0 33 | 6 45 | 6.43 | 3.77 |
1 15 23} 0 30 6 42] 6.32] 3.73 21 0 33 | 6 46 | 6.22 | 3.56 23) | 0 29 | 6 42 | 6.27 | 3.76 22 | 0 32 | 6 46 | 6.28 | 3.93
1 45 22:} 027) 638] 615) 4.09 22} 0 31 | 6 44 | 6.10 | 3.83 23 | 0 28 | 6 42 | 6.04 | 3.87 20 | 0 27 | 6 41 | 6.15 | 4.20
mel) 21 024) 637) 6:15) 415 23 0 23 | 6 37 | 5.93 | 4.30 26 | 0 27 | 6 41 | 6.01 | 4.18 21 | 0 26 | 6 38] 5.79 | 4.31
2 45 23} 020) 6 34) 5.59 4.46 23.| 0 21 | 6 34 | 5/85 | 4.45 20 | 0 24 | 6 37 | 5.69 | 4.50 22) 1/0 25 | 6 38 | 5.70 | 4.56
3 15 24 020] 635] 5.62 4. 88 22 | 0 23 | 6 38 | 5.64 | 4.56 24 | 0 20 | 6 36 | 5.67 | 4.67 24 | 0 21 | 6 36 | 5.52 | 4.74 |
3 45 24) 019 635 | 5.43] 5.03 23) 0 21 | 6 37 | 5.45 | 4.89 23 | 0 22 | 6 38 | 5.49 | 4.96 23 | 0 23 | 6 36 | 5.32 | 5.01
4 15 23 0 23 6 40} 5.38] 5.07 95 | 0 21 | 6 35 | 5.39 | 5.25 25 | 0 25} 6 43 | 5.32 | 4.95 24 1 0 26 | 6 42) 5.22 | 5.11
4 45 25] 024) 640) 5.15} 5.34 23 0 29 | 6 44 | 5.21 | 5.23 23 | 0 27 | 6 43 | 5.03 | 5.22 24 | 0 29 | 6 46 | 5.03 | 5.31
5 15 20 | (0°28 643) 4.99 | 5.33 24) 0 28] 6 47 | 5.35 | 5.23 24 | 0 33 | 6 51 | 5.19 | 5.39 24 | 0 33 | 6 52 | 5.06 | 5.47
5 45 26 035} 651] 5.15] 3.51 22] 0 36 | 6 53 | 5.04 | 5.33 23 | 0 39 | 6 55 | 5.17 | 5.36 24 | 0 40 | 6 57 | 4.97 | 5.37
6 15 26) 038) 658) 5.31 5. 58 25 0 41 | 6 56 | 5.15 | 5.16 25 | 0 40 | 6 57 | 5.14 | 5.38 271043] 7 11) 5.06 | 5.40
6 45 24 045] 658] 5.24] 5.08 23 | 044) 7 1) 4.99 | 5.11 23] 046) 7 5 | 5 28 | 5.24 28/0 49/7 6 | 5.13 | 5.23
t is 26 0 49 ¢ Wi) ese i) ebarr 25| 049) 7 7) 5.27) 4.97 26 | 0 52 | 7 14 | 5.16 | 4.84 25 | 0 53 | 7 14 | 5.07 | 4.99
7 45 22°) 053°) 7 11 5.78 | 5.03 22} 055 | 710) 5.44 | 4.64 241057] 711 | 5.12 | 4.83 25] 0 58 | 713 | 5.29 | 4.95
8 15 24) 056) 711 5.84 | 4.73 24) 057 | 712) 5.70 | 4.30 23 | 0 56 | 7 11 | 5.62 | 4.30 22 | 0 58 | 7 14 | 5.40 | 4.80
8 45 PR} 1) (0 Gi? 710} 6.13} 4.18 23 | 0 58 | 7 14 | 6.08 | 4. 41 26 | 0 59 | 7 11 | 5.78 | 4.44 23 | 0 58 | 7 10 | 5.60 | 4.58
9 15 23) 0 56 7 8] 6.21 4,12 22] 058) 711 | 6.22) 4.18 22 | 0 57 | 7-12 | 6.14 | 4.11 22 | 0 59 | 7 14 | 6.00 | 4.07
9 45 22) 0/54 || 7 6 6. 64 3. 88 23] 054) 7 g | 6.51 | 3.87 22 | 0 58 7 11 | 6.33 | 4.08 21)058|)7 9 | 6.07 | 4.22
10 15 24) 052) 7 4) 6.46 3. 87 22) 052) 7 5) 6.45 | 3.79 22) 0 54/7 6) 6.40 | 3.90 22 | 0 52) 7 7 | .6.53 | 3.98
10 45 25} 047) 7 O| 6.59 3. 46 21 0 47; 7 1/] 6.79 | 3.82 24) 050] 7 3/ 6.85} 3.78 19} 052|7 7] 6.47 | 3.84
11 15 21 0 44) 656) 6.61 3.47 23} 0 45 | 6 59 | 6.70 | 3.39 23 | 0 46 | 6 57 | 6. 74 | 3.58 22) 0 42) 7 2) 6.62 / 3.89
11 45 21 0 37 650 | 6.57] 3.48 22] 0 42] 6 56 | 6.78 | 3.32 21 | O 41 | 6 54 | 6.80 | 3.56 25 | 0 43 | 6 57 | 6.37 | 3.76
561 | 0 37.9) 6 52.3) 5 89.4) 4 42.61 546 | 0 39. 0/6 53.6) 5. 882) 4.372] 555 |0 40. 1'6 54.8) 5.850) 4.427) 552 0 40.8 6 56.0) 5. v4. 904

n= 75 12 = 90° 72 = 105° ny = 120°
(w—y’) j
Obs.| A4 Ao HY Obs.| 4 | HY, | He | Obs.| 4 Wo | Hy || Ee | Obs. | Ae | Ao || HG | He
| |

h. m. h.m.| hem. Ft. | hm.| h.m.| Ft. | Ft. hom. | h. m. Ft. | Ft. h.m.|h.m.| Ft. | Ft
0 15 21 0 42 | 652) 6.31) 4. 23} 0 38) 6 50 | 6.05 | 4.10 21 | 0 37 | 6 48 | 5.74 | 4.21 22 | 0 40 | 6 52 | 5.53 | 4.82
0 45 23| 035] 6 48) 6. 21 0 38 | 6 44 | 5.92 | 4.49 20 | 0 33 | 6 45 | 5.43 | 4.30 23 | 0 33 | 6 44 | 5.55 | 4.64
ih 65 22) 0 31 6 43 | 6. 20 | 0 32/| 6 46 | 5.88 | 4.42 23 | 0 30 | 6 42 | 5.64 | 4.59 18 | 0 32 | 6 43 | 5.25 | 4.59
1 45 24 0 28) 6 41 5. 22] 0 28) 6 39 | 5.78 | 4.45 22 | 0 26 | 6 36 | 5.44 | 4.89 22] 0 28] 6 41 | 5.33 | 5.06
9 15) 22) 025] 6 38) 5. 22 0 26 | 6 49 | 5.66 | 4.85 24 | 0 28 | 6 39 | 5.40 | 4.99 24 | 0 27 | 6 39 | 5.34 | 5.35
2 45 25) 0 21 6 36) 5. 22) 0 24 | 6 34} 5.50 | 4.83 22 | 0 23 | 6 36 | 5.48 | 5.25 17 | 0 24 | 6 35 | 5.16 | 5. 42
Silo) 20 0 22) 6 36] 5. 23) 0 25] 6 37 | 5.25 | 5.01 93 | 0 22 | 6 38 | 5.31 | 5.26 22 | 0 21 | 6 34 | 5.09 | 5.69
3 45 23 | 022) 6 40) 35. 22] 0 24 | 6 42 | 5.09 | 5.39 93 | 0 24 | 6 37 | 4.98 | 5.51 24 | 0 23 | 6 37 | 4.97 | 5.84
4 15 22} 026) 6 40) 5. 20 | 0 28) 6 41 | 5.00 | 5.45 24 | 0 28 | 6 44 | 4.79 | 5.31 23 | 0 27 | 6 40 | 4.74 | 5.93
4 45 25 029) 6 44 4. 23 | 0 32] 6 48 | 4.88 | 5.77 22 | 0 33 | 6 48 | 4.69 | 5.84 21 | 0 32 | 6 44 | 4.56 | 6.04
& aby 22) 035 6 51 4. 27 0 34 | 6 50 | 4.75 | 5.68 25 | 0 36 | 6 Sl | 4.71 | 5.83 24 | 0 36 | 6 52 | 4.47 | 5.97
5 45 24 0 41 7 0 4. 23) 0 44] 6 59 | 4.83 | 5.67 21 | 0 42 | 6 58 | 4.45 | 5.96 2210 46/7 2] 4.49 | 6.11
6 15 22) 048) 7 3 4. 23) 048) 7 5] 4.71 | 6.00 26/048) 7 4| 4.67} 5.96 24)052)7 6} 4.41] 6.15
6 45 24 OO i Ol) ws 25) 053) 7 8) 4:73 |°5. 54 92) 0 57 | 712 | 4.81 | 5.84 23 | 0 55 | 7 13 | 4.63 | 5.99
7 15 26) 057] 714) 4 24) 058] 715 | 4.75 | 5.38 26 | 0 58 | 7 14 | 4.64 | 5.52 25] 1 6 | 719} 4.62} 5.88
7 45 25 ) Bel |) (7 1 || Bh 22) 1 2)| 7207) 5.13 | 5.17 25)1 4) 719 | 4.68 | 5.54 24)1 5] 7 21 | 4.83 | 5.65
8) 15 24) 11 wv 18!) 5: 24 1 4/7 21 | 5.04 | 4.90 95 |1 7) 721 | 4.93 | 5.17 22) 1 9| 7 22] 4.61 | 5.33
8 45 12h Onlin: 22} 1 6) 718] 5.30 | 4.90 26) 1 5] 718 | 4.98 | 5.04 25] 110 | 7 22 | 4.87 | 5.23
9 15 DE) ee aS ire) 24) 1 5) 717) 5.43 | 4.84 OBS |) ah We |i i) |) GG || Zoe) O30 7 1884895) | (5.210
9 45 056) 710)| 5.73 23) 1 0/712) 5.77 | 4.67 22)1 3) 715] 5.48 | 4.62 24|/1 3) 718) 5.21 | 4.92
10 15 Of] T Bl Ges 25) 054/7 8] 5.80 | 4.38 24/058) 7 6 | 5.58 | 4.68 22)1 1/713] 5.04 | 4.88
10 45 050) 7 2) 614 23) 0 51)7 3] 5.86 | 3.99 22:)052) 7 4 | 5.66 | 4.53 24/0 56| 7% 7 | 5.59 | 4.90
11 15 047) 658] 6.30 22 | 0 47 | 6 57 | 6.01 | 4.11 25} 0 51 | 7 2) 5.83 | 4.56 24) 051) 7 2] 5.44 | 4.65
ll 45 045] 657] 6.33 23 | 0 41 | 6 54 | 5.90 | 4.03 23°) 0 44 | 6 54 | 5.90 | 4.36 23 | 0 46 | 6 58 | 5.69 | 4.69
0 42.0) 6 56.2) 5 56. 6| 548 | 0 43.416 57.0) 5. 376| 4.917] 559 (0 44. 0/6 57.1) 5.182) 5.112) 565 |0 45. 4/6 58.4) 5.015) 5. 372

|

DISCUSSION OF TIDES IN BOSTON HARBOR.

TABLE I1I—Containing average values belonging to the Arguments (x) — p') =m and y:—Continued.
COMBINED TRANSITS.

N2—= 135° N2=150° N= 165° n2= 180°
(Y—w') T 7
Obs. | A4 Ao H, HH’, f{ Obs.| Ay Aly | Hy | He f Obs. | A4 A’, | Hy | WH’, | Obs.) A | Avo | HG | He
h.m. h.m.| hem Ft. Ft. hm.|h.m.| Ft Ft. hm.|hom.| Ft. | Ft. h.m.|h.m.| Ft. Ft.
0:15 24) 045} 654) 5.40 4,96 23 | 0 42) 6 55 | 5.05 | 5.30 22] 0 44 | 6 53 | 5.07 | 5.45 23 | 0 44 | 6 52 | 4.95 | 5.39
0 45 24] 0 38 6 49 5. 22 4,98 21 0 38 | 6 46 | 5.16 | 5.29 24 | 0 39 | 6 43 | 5.03 | 5.35 22 | 0 37 | 6 435 | 4.82 | 5.31
i 215) 24) 0 35 6 46 5. 21 5. 03 24 | 0 33 | 6 45 | 5.03 | 5.40 20 | 0 36 | 6 43 | 4.99 | 5.46 21 | 0 32 | 6 41 | 4.65 | 5.43
1 45 22] 029) 6 40 5. 30 5.12 22] 0 30 | 6 42 | 5.06 | 5. 22 25 | 0 31 | 639 | 4.73 | 5.65 24 | 0 33 | 6 41 | 4.54 | 5.66
215 21 0 26 6 37 5. 04 5. 41 23 0 29 | 6 36 | 4.71 | 5.47 20 | 0 24] 6 31 | 4.44 | 5.73 21 | 0 23 | 6 36 | 4.68 | 5.96
2 45 24) 023] 6 35 4, 87 5. 44 20 | 0 22} 6 33 | 4.85 | 5.64 23 | 0 24 | 6 34 | 4.74 | 6.16 23 | 0 24 | 6 33 | 4.65 | 6.30
3.15 20) 026) 6 39 4.76 5, 89 23 | 0 24) 6 33} 4.54 | 5.77 20 | 0 23 | 6 32 | 4.54 | 6.01 20 | 0 20 | 6 33 | 4.35 | 6.25
3 45 22} 926] 6 38 4. 66 5. 85 22] 0 21 | 6 36 | 4.55 | 6.04 22 | 0 25 | 6 36 | 4.36 | 6.22 21 | 0 18 | 6 28 | 4.13 | 6.32
415 22) 0 26 6 41 4. 62 9. 92 24] 027) 6 39 | 4.29 | 6.16 23 | 0 28} 6 40 | 3.88 | 6.15 24 | 0 30 | 6 41 | 4.11 | 6.50
445] 24 032) 6 47 4,41 6.09 21 0 36 | 6 44 | 4.29 | 6.29 20 | 0 34 | 6 46 | 3.88 | 6.34 23 | 0 33 | 6 44 | 4.08 | 6.71
51 22 | 0 36 6 53 4.32 6. 21 23 | 0 42 | 6 45] 4.15 | 6.45 20 | 0 34 | 6 49 | 4.16 | 6.47 23 | 0 37 | 6 SI | 3.98 | 6.59
5 45 21 0 44 6 56 4. 23 6.18 23 | 0 43 | 6 59 | 4.09 | 6.39 20/0 46/7 0} 3.99} 6.71 21 | 0 47 | 6 59 | 3.87 | 6.50
61 23 | 0 53 %) 4,27 6. 07 24) 050) 7 5} 4.20} 6.26 23] 053] 7 5 | 4.07 | 6.61 22] 052] 7 8 | 3.90 | 6.69
6457 23 P68) 005: 4.37 5. 88 24 054)7 9) 401 | 6.15 25} 1 0] 715 | 4.07 | 6.32 23 |1 0] 714 | 4.00 | 6.49
TAS 4 21 fe perdae ea ede 4G} 6,11 24] 0 8} 7 23 | 4.09 | 6.05 21)1 1) 7 16 | 4.04 | 6.35 24) 110) 7 25 | 4.14 | 6.23
ASH 825;\)) ol! 198) e724! 4.74 5. 92 22} 113] 7 26] 4.35 | 6.00 25] 1 11 | 7 26 | 4.29 | 6.12 23) 114) 7 27 | 4.11 | 6.24
8 15 26 114) 7 26 4.74 5. 81 25 113 | 7 26 | 4.53 | 5.84 26 | 117] 7 30 | 4.32} 5.79 22) 117) 7 28 | 4.32} 6.11
B45 4 25 110 72 4. 86 5. 48 22 1 14 | 7 25 |+4.52 | 5.41 25 | 117 | 7 23 | 4.23 | 5.67 26:+1 17 | 7 28 | 4.36 | 5.90
915 24 114 12% 4. 94 5. 25 110/723 4.93 | 5.38 22) 114 | 7 25 | 4.62 | 5.50 23 | 116 | 7 26 | 4.66 | 5.60
9 4 24 1 8; 718 5.18 3. 24 ababt | 7 20 | 4.98 | 5.38 25 | 110 | 7 20 | 4.84 | 5.46 23 | 112 | 7 23 | 4.77 | 5.70
10 15 25 ayaa! t12 5:20 | 5. 24 £6 | 716 | 4.88 | 5.28 23|1 4)714 | 4.89 | 5.36 25) 1 7) 717 | 4.94 | 5.49
10 4 23) 058] 7 8 5. 09 4. 24 ata | t 9 | 5.08} 5.15 24/1 0) 710 | 4.90 | 5.04 24 LON S02) a a7
11 15 22 | 053 7 2) 5.38 5. 24 055/7 5] 5.09 | 5.00 24) 054) 7 4) 5.10} 5.13 20 | 055 | 7 3} 4.87 | 5.22
1145} 23) 045] 6 56 0. 23 4. 22] 0 50 | 6 58 | 5.22 | 5.28 24/051) 7 O| 5.18 | 5.12 24 | 0 50 | 6 56 | 4.85 | 5.23
554 | 0 47. 5| 6 59.9) 4.854) 5. 553 | 0 47.86 59.5) 4. 652) 5.692] 546 |0 48. 3/6 58.9) 4.515) 5.840) 550 |0 48. 7/6 59.5) 4. 448] 5. 966
N2= 195° y= 210° Nz = 220° N2= 240°
(wp)
Obs nv; Ao HH, HH’, | Obs.| Ay WV, | Hy | Hg } Obs.| Ay Ny | HY, | H’e J Obs.} Ay A, | HY, | Hs
h. m. h.m.| hem. Ft. Ft h.m. | h.m. Ft. Eitan | meni hom.| hom.) Ft. | Ft.
015 24 047) 6 56 4. 80 5. 37 25 0 42 | 6 52] 4.71 | 5.50 5.05 | 5. 60 22 | 0 41 | 6 51 | 5.03 | 5.45
0 45 23) 0 39 6 47 4.75 | 5.55 24 0 39 | 6 47 | 4.84 | 5.48 4.79 | 5.58 23 | 0 40 | 6 49 | 4.86 | 5.54
115 2 032) 6 41 4. 68 5. 58 23 | 0 37] 6 45 | 4.76 | 5. | 4.86 | 5.64 24 | 0 30 | 6 38 | 4.89 | 5.78
1 45 21 0 27 6 36 4. 63 9. 72 25 | 0 27} 6 38} 4.70} 5. 4.49 | 5.75 24 | 0 25 | 6 33 | 4.52 | 5.58
2s) 25 | 025) 6 34 4.32 | 5.99 22] 025} 6 35 | 4.37 | 5. 4.60 | 5.99 23 | 0 23 | 6 32 | 4.48 | 5.97
2 45 19 ON23 6 33 4.47 6. 08 24 0 22) 6 3L | 4.36 | 6.15 4.30 | 6.03 26 | 0 20 | 6 31 | 4.28 | 5.92
3:15 22))| 0/25 6 32 | 3.99 6. 31 22) O 20} 6 32) 4.30] 6.39 4,22 | 6.21 25 | 0 20 | 6 29 | 4.13 | 6.18
3 45 22) 0 21 6 33 4, 44 6. 56 22} O 20 | 6 29 | 3.99 | 6.38 3.99 | 6.54 23 | 0 23 | 6 34 | 4.27 | 6.38
415 22) 023) 6 36] 3,96 6.45 25} 0 22} 6 33 | 3.87 | 6.63 4.02 | 6.50 25 | 0 18 | 6 32} 3.95 | 6.36
4 45 20 0 30 638] 3.91 6. 62 24 0 27 | 6 41 | 3.90 | 6.50 3.57 | 6.78 25 | 0 28 | 6 38 | 3.81 | 6.59
5.15 21 037) 653] 3.79 6. 89 19 0 30/6 43 | 3.86 | 6.74 3.81 | 6.57 20 | 0 32 | 6 46 | 3 75 | 6.65
5 45 19 | 045] 658) 4.04 6. 88 24 0 43 | 6 56 | 3.74 | 6. 3.81 | 6.66 19 | 0 40 | 6 53 | 3.68 | 6.49
6 15 22 052) 7 8] 3.96 6.72 20 052)7 41} 3.90) 6. 3.77 | 6.71 22) 0 44) 7 O| 3.74 | 6.52
6 45 24 D1 2) % 15) 3.88 | 6.71 23 0 56 | 7 12 | 3.95 | 6. 3.79 | 6.61 20) 056) 7 9 | 3.97 | 6.66
715 22 ie 5 Teak) 3.88 | 6.08 221 Te el Sai42055 1162 3.91 | 6.44 22/1 0/711 | 4.02} 6.31
45 22 111 723 4.02 | 6.24 23 113 | 7 28 | 4.39] 6. 4.21 | 6.10 23]1 8] 7 20) 4.16 | 6.17
15 25 116] 730 4.27 5. 89 20) 116} 7 27 | 4.23 | 6. 4.11 | 5.77 23 | 111 | 7 20 | 4.33 | 5.86
8 45 25 119] 7 29 4.45 | 5.97 116 | 7 28 | 4.26 | 5. 4.44 | 5.72 24/1 8) 719 | 4.49 | 5.76
9 15 227) LSB eeraen |) 64557, 5. 81 Da Pye pay)| Z Essa | ay 4.56 | 5.66 22 | 113] 7 22 | 4.48 | 5.67
9 45 25 113) 7 24)- 4.72) 5.67 1 13 | 7 22 | 4.80 | 5. 4.91 | 5.53 22)}1 9] 719 | 4.92 | 5.48
15 24); 110] 718] 4.89 5.45 1 8/718) 4.75] 5. 4.91 | 5.68 22)1 4/711) 4.74 | 5.46
10 45 24) Si) WW Si} SAR) 10 i554! D4 | Wiel) |PARSSalinos 4.84 | 5.51 25 | 059 | 7% 9) | 5.18) 5.27
15 24 O'S50|) 7 4 4, 83 5. 27 059) 7 8| 5.07] 5. 4.98 | 5.42] 26 | 052) 7 1/ 4.90 | 5.46
11 45 25 | 050] 6 57 4, 82 5. 39 0 49 | 6 59 | 4.96 | 5. 4.96 | 5. 60 22 | 0 48 | 6 57 | 5.16 | 5.32
545 | 0 48.7) 6 59.3) 4.374) 6,014 0 47. 5/6 58. 4) 4.379) 6.047 547 |0 45.86 56.7) 4.398) 6. 025} 552 |0 44.7/6 55.2) 4. 406) 5. 953

DISCUSSION OF TIDES IN BOSTON HARBOR.

COMBINED TRANSITS.

Taste [l—Containing average values belonging to the Arguments x) —y' =} m and y:—Continued.

---—
N2z= 300°
y—y'
AY HY, HH H’y | Obs. | Ay | Hy

h. m. h. m. Ft. | A Ft. Ft. h.m.| h.m.| Ft.

0 15 0 43 5.14 5. 6 3: 25 4 76 21 | 0 38 | 6 49 | 5.43

0 45 0 36 4.79) 5. 6 5.18 ) . 5. 12 24 | 0 31 | 6 42 | 5.47 | 4.70

1 15 0 32 5.00] 5. 6 5.04 | 5. 23 6 5. 5.09 Q1 | 0 27 | 6 37 | 5.52] 4

1 45 0 26 4,85 5. 6 4.97 | 5.6 6 De 5. 20 24 | 0 23 | 6 34 | 5.39 | 5.

2 15] 0 23 4.64] 5. 6 4,72 | 5. 22 6 5. 5. 46 25 | 0 21 | 6 30 | 5.07 | 5.

2 45 0 18 49 | 95. 6 4. 64 | 5. 25. | 0 6 5. 5. 86 21 | 0 14 | 6 29} 5:18 | 5.

3 15 0 18 4.12] 5. 6 4.51 | 5. 25 | 0 6 4. 9.92 23 | 0 15 | 6 26 | 4.86 | 5.

3 45 0 16 4.10 6 6 4.15 | 6. 26 | 0 6 4. 5. 94 26 | 016) 6 31 | 4.65 | 5.

4 15 0 20 3. 96 6. 6 4.17 | 6. 23 | 0 6 4, 6. 09 25 | 0 18} 6 29 | 4.33 | 5.

4 45 0 24 4.00] 6. 6 4, 6. 24/0 6 4. 6. 23 22) 019 | 6 34 | 4.34 | 5.

5 15 0 31 3. 83 6. 6 4, 6. 25.| 0 6 4, 6.15 25 | 0 25 | 6 42 | 4.41 | 6.

5 45 0 37 3. 82 6. 16 | 4. 6. 24) 0 6 4, 6. 34 24 | 0 31 | 6 45 | 4.32 | 6.

6 15 0 40 3.73 | 6. 6 4, 6. 48 21 | 0 6 4, 6. 23 23 | 0 40 | 6 54 | 4.29 | 5.

6 45 0 52 3.94] 6. 7 4, 6. 23 | 0 7 4. 6. 08 231047] 7 1} 4.72] 5.

7 15 0 59 4.31 6. 7 4. 6. 23 | 0 7 4. 5. 70 26)048) 7 5 | 4.54] 5.

1 45 i 4 4.30 | 6. ng) | 4. 5. 22] 0 7 4. 5. 64 221053] 7 6) 4.74] 5.

8 15 1 9 4. 63 5. 7 4. 3. 22) 1 7 4. 5. 50 221 0 59 | 7 10 | 4.98 | 5.

B 45 9) 4.67 5. 7 | 4. 5. 20 | 1 7 5. 5. 41 19 | 0 56 | 7 15 | 5.09 | 5.

@) ils} 1 ¢4 4. 64 5. 7 | 5. 20) 1 ri 5. 5. 06 221 PL ON 12) Konami.

9 45 al ak 4.76 | 5. 7 oO: 22) 1.0159} 7 5. 5. 02 23: | 059) 7 9 | 5.56 | 4.
10 15 12 5.07 | 35. rat 5. 21 | 095 1-7 by 4. 82 23:|055 ) 7 6) 5.62 | 4.
10 45 0 55 4.94] 5. ‘a 5. 22) OSL | 7 3. 4.97 24105017 0/574] 4.
11 15 0 51 5. 08 5. 6 58 4. 23 6 3. 4, 86 21 | 0 46] 658 | 5.88 | 4.
11 45 0 47 5. 24 5. 6 50 4, 23 | 0 41 | 6 5. 4 80 21 | 0 40 | 6 51 | 5.70 | 4.

0 42.3 4.502] 95. 6 51.6 0 38.3 5.510! 550 |0 37.16 49.8) 5.045] 5.
N2 = 360°
y—w'
ON HY Ea Hy My H’, | H’, | Obs.| A4 Pye fee = Ce s Cy

h. m. h. m. Ft. Ft. h. Ft. h. h. Ft. | Ft. h.m.| hom.| Ft. Ft.
OS 0 35 5. 73 4.54 20 0: 3 4. 0 6 6.15 | 3.82 22 | 0 37 | 6 50 | 6.32 | 3.56

0 45 0 30 5. 17 4. 20 0 5. 4.2 0 6 6.36 | 4.02 21 | 0 35 | 6 47 | 6.31 | 3.79

i 16} 0 26 5. 46 4% 20 0 3. 4, 0 6 5.99 | 4.21 20 | 0 31 | 6 43 | 6.12 | 3.79

1 45 0 24 5, 37 4. 23 0 3. 4. 0 6 5.86 | 4.51 23 | 0 26 | 6 37 | 6.16 | 4.

2 15 018 5.40 | 5. 25 0 6 32 | 3. 4. 0 6 5.80 | 4.68 23 | 0 23 | 6 36 | 6.02 | 4

2 45 0 18 5. 10 5. 23 0 6 30 | 5. 5. 221 0 6 5.07 | 4.45 24 | 0 22] 6 37 | 5.81 | 4
3) 15) 0 15 4.89] 5. 24 0 6 29 | 5.2 5. 21/0 6 5.49 | 5.05 24] 0 21 | 6 34 | 5.58 | 4

3 45 0 14 5.03) | 5: 0 6 28 | 4. 9. OF 22) 0 6 9732) || 0.03: 25 | 018 | 6 31 | 5.16 | 5.

4 15 017 4.64 | 5. 0 6 30 | 4. 5. 24 | 0 6 5.01 | 5.33 25 | 0 20 | 6 33 | 5.08 | 5.

4 45 0 21 6° 4.38] 5. 0 2 6 35 | 4. 5. 25 | 0 6 4,83 | 5 58 22 | 0 25 | 6 38 | 5.00 | 5.
5 15 0 24 6 4.54] 5. 0 6 41] 4. 5. 27 | 0 6 4.84 | 5.43 24 | 0 29 | 6 44} 4.91 | 5.

5 45 0 31 6 4.49 5. 0 6 50 | 4. 5. 25 | 0 6 4.88 | 5.52 26 | 0 32 | 6 48 | 4.97 | 5.

6 15 0 37 6 4.47 3. 0 6 52] 4. 3. 22 | 0 6 4.99 | 5.40 23 | 0 38) 6 55 | 5.24 | 5.

6 45 0 46 7 4.62] 5.5 0 6 58 | 4. 5. 24 | 0 6 4.94 | 5.28 21 | 0 43 | 6 58 | 5.28 | 5.

ab) 0 47 ri 4. 81 o. 0 ion | os oy 3 | 0 Re2) on Ba one 23) 0 44) 7 2) 5.37 | 5.

7 45 0 53 q 0. 12] 3: 23 0 US) GER |) eb 0 7 8) 5.51 | 4.92 25) 053)7 8} 5.46] 4.

8 15 0 56 7 9.23 | 4. 22/1) (0) nO os 4. 0 7 6) 5.51 | 4.45 24/055)7 9) 5.71) 4

8 45 0) GB) 5. 30 4. 24 0 7 12) ) 5. 4, 21 | 0 7 7 | 5.67} 4.40 22) (056) | 7% 8) an 908 \r4:

() ab) 0 55 7 5. 55 4. 237) 0 UT || & 4, 22 | 0 7 7] 5.86 | 3.98 22 | 053} 7 6] 6.00 | 4.

9 45 0 54 7 5. 80 4. 23) 0 gb & 4. 23] 0 7 3 | 6.21 | 3.88 221/053) | 7) S0t6529) 103!
10 15 054] 7 5. 56 4. 0 Tt 3. 231 0 V2) 6.11) 3:89) 22) 047)7 3} 6.58) 3.
10 45 22) 0 46 7 6. 06 4. 0 a0) 4, 20) 0 7 0} 6.40 | 3.56 22 | 045) 7 0 | 6.62 | 3.
whl 71} 22) 045) 6 6.04 | 4. 0 6 53 3. 20) 0 6 55 | 6.24 | 3.60 23 | 0 46 | 6 58} 6.41 | 3.
11 45 24) 0 41 65 5. 81 4, 0 6 53 3. 0 6 49 | 6. 3. 61 23 | 0 41 | 6 54 | 6.61 | 3.

554 | 0 35.9 215, 35. 4 8. 0 35.7 9 4.5937 551 0 37. 2/6 51.0) 5.772) 4.

24

DISCUSSION

OF

TIDES IN

BOSTON

HARBOR.

TABLE 1V—Containing average values belonging to the Arguments ¢'—=+ 75 and Q, or to the month and year.

COMBINED TRANSITS.

JANUARY. FEBRUARY. MARCH.
Year. =a] tas ; ]
Ay Xo Hy H'; “YY Ne HY | H, | H2
ho m.| Fe | FE] hk. m.| hm ; he m.| hm. | 2 | FR
6 52 4. 87 5. 03 0 41 Gao) 5.25 0 38 6 53 5.05 5. 04
6 46 4. 89° 4, 83 0 41 6 55 5.17 0 40 6 54 5.14 | 4.86
6 54 5. 37 Ont 0 39 6 54 5.10 0 41 6 55 5.16 | 4.99
6 54 5. 01 4.92] 0 40 6 53 4. 81 0 43 6 56 5. 18 5. 05
6 54 5. 00 5. 44 0 36 6 50 4, 83 0 40 6 55 4, 93 4.01
6 46 5. 57 5. 67 0 38 6 53 0.19, 0 32) 6 45 5. 07 4, 88
6 47 4,89 4.97 0 37 Gaol! 4.70 0 35) 6 49 4.78 4.78
6 48 4.97 5.28 0 39] 6 58 5. 29 0 44) 6 56 4.96 5.01
6 48 5.14 5. 63 0 31 6 44 4,81 0 36 6 50 4.77 5. OL
6 46 5.21 5. 43 0 4 6 54 4. 69 0 40 6 53 4. 88 4.88
@ 633 4.74 5. 33 0 48 6 52 4.36 0 41 6 54 4.75 4.92
6 50 4.69 5. 24 0 35 6 51 4.89 0 40 | 6 54 4.90 5. 44
6 50 4.53 5. 21 0 40 6 53 4. 59 0 39 6 54 4.74 5.05
6 53 5. 01 5. 30 0 41 6 55 4.61 0 38 6 52 4.98 5. 04
6 54 4.88 o. 12 0 37 6 48 5. 02 0 41| 6 53 5. 37 5.47
6 52 4. 64 5.14 0 40 6 49 4.54 0 45 6 55 4. 66 5. 11
6 56 4,23 4.90 0 47 6 57 4. 48 0 45) 6 57 5.17 5. 50
6 42 5. 21 4, 96 0 53 6 47 5. 21 0 44 6 54 5. 12 4.95
Th es3 4.98 4,95 0 59 6 55 | 4. 44 0 43 6 55 4. 66 9. 21
651.0 4.933 3. 128) 0 40.7) 6 52. 3) 4. 841 0 40.8) 6 53.4 4, 962 5. 058
APRIL. MAY. JUNE.

No A, HH’, ry r'o HH, HH’, ry A’e HH, H’,
m.| h. m. Ft. Ft. h. m.} h. m. Ft. Ft. h. m.| h. m. Ft. Ft.
41 6 54 5. 05 4.90] 0 43 6 55 5. 39 5. 08 0 45 Giron 5. 45 5. 17
44 6 56 4.92 4.70 0 46 6 58 5. 10 5. 07 0 50 iO) 5. 39 5. 41
40 6 53 5.12 4,88 0 42 6 54 5. 06 5. 21 0 48 6 58 5. 69 5. 12
46 6 57 5.31 5. 33 0 43 6 54 4.98 5. 09 0 45 6 54 5.19 5. 43
41 6 54 5. 63 5.39% 0 42 6 54 5.13 4,99 0 45 6 58 4.99 0.17
39 6 352 5. 00 4.97} 0 42 6 54 5.05 5.16 0 45 6 56 5. 02 5. 07
41 6 54 4,99 5.03 0 43 6 55 4.76 5.16 0 46 6 59 5. 01 5.47
42 6 55 4.81 4.95] 0 42 6 54 5.15 5. 46 0 42 6 56 4.93 5. 34
40 6 54 4.79 5.19 0 41 6 53 5. 00 5.439 0 44 6 56 4.90 5. 43
39 6 51 5. 08 5. 36 0 43 6 57 4. 89 5. 24 0 49 7 5.10 5.45
34 6 56 4.72 5. 34 0 46 6 57 4.72 5. 36 0 46 6 56 4. 82 5. 34
39° 6 «=O 4.89 6.31 0 40 6 52 4 88 5.11 0 41 Gol 4.83 4.93
38 6 53 4.75 5. 06 0 45 6 56 4,97 5. 06 0 47 6 58 5. 19 4.35
44 6 56 5.3 5. 42 0 42 6 54 5.18 5. 34 0 45 6 56 4.98 5, 28
43 6 56 4,85 5.31 0 46 6 59 4. 92 5. 30 0 46 6 59 5.18 5. 42
37 6 48 4.96 5. 36 0 45 @ Ae 5. 21 5. 46 0 49 6 58 5. 20 5. 34
45 6 57 5. 79 5. 69 0 42 6 56 5. 37 5. 35 0 43 6 538 5.13 9. 12
46 6 59 5.14 5. 12 0 47 e W 5. 46 5.58 f 0 51 ts o. 19 . 25
38 6 32 4.85 O20) 0 44 6 57 5. 21 5. 35 0 47 uk 5. 12 5.18

41.4 6 54.1 5. 000 5.177} 0 43.4) 6 55.5 5. UTS) 5. 2534 0 46.0; 657.8) 5.090] 5. 225

DISCUSSION

OF TIDES IN

BOSTON HARBOR.

25

TABLE 1V—Containing average values belonging to the Arguments ¢'=4 73 and Q, or to the month and year—Continned,

COMBINED TRANSITS,

JULY. AUGUST, SEPTEMBER.
Year.
My No HH, H's, ONT Ney |j “aswA HH’, NA 2 Hi | He
|
h. m.| hom. Ft. Ft. |} h. m.| h. m Ft. Ft. | h. m.| h. m Ft. Ft.
0 43 6 57 5. 287, 5. 06 0 44 6 56 5. 35 4.93} 0 46] 6 58 5. 40 5. 35
0 49 U at 5. 32 5.19 0 48 @ Bt) 0: 29/4) 10, 14 0 46) 6 59 5.28 5. 50
0 54 ad 8 5. 09 5. 24 0 54 if) | 5. 39 5.43 § 0 48) 6 59 5.27 5. 37
0 45 6 55 5. 20 5.22 9 0 48 6 56 | 5. 38 5.35] 0 46 | 6 57 5. 37 5. 35
0 45 6 57 5. 37 5.367 0 45 6 55) 5.33 5.18], 0 46) 6 57 5.23 9. 22
0 47 6 59 4.95 | 5. 11 0 48 = 0 5. 09 5.23 0 48 | 6 58 5. 35 5.49
0 47 6 58 5.04 | 5.43 0 48 | 6 57 5. 16 5. 86 0) 42) || 6) (55 5. 07 5, 63
0 44) 6 55 4. 87 5.237 0 43) 6 54 5. 00 5.39 0: 43) 6 55 B03} || 5.2)
0 45) 6 57 5. 09 5. 28 0 42 6 54 | 5,15 0.421 0 42) 6 5d Sle eon gO
0 46 6 54 5.05 5. 56 Oa e IS GS Tee cos29) 5. 91 0 42) 6 53 5.47 5.73
0 53 6 59 5. 07 5.56 | 0 40 6 51 | 4.76 5.90] 0 43 | 6 53 4.90} 5.74
0 46 6 56 5. 02 5.45 7 0 46 6 54 5. 03 5.33] 0 43 6 53 4. 87 5. 33
0 42 6 54 5. 04 5. OL 0 42 6 51 5. 12 4.99} 0 41 6 52 4.98 5.16
0 49 t- 2 4.94 5.24] 0 44 6 56) 4.96 5.17} 0 45 6 57 4.75 5.05
0 49 tou 4, 92 5. 23 0 48 al | 4. 88 5.29} 0 46 65% 4. 84 5.16
0 49 Hires 5. 20 5. 44 0 47 6 58) 5.00 5.16% 0 43 6 55 4.91 5. 06
0 47) 6 57 5.19 5. 14 0 48) 7 OF} © 5.25 5.05] 0 53 gerd 4.99 5. 24
0 49 if 8 5. OL 5.19] O 54 7 10) 4.88 4.90} 0 47 ¢. 8 4. 94 5. 13
0) 52 | U | 4.91 5. 06 050; 7 5 4. 83 5. 06 0 50) 7 6 4.91) 5.01
047.6) 6587/5. 02) 5.2 ; 0 46.5) 6.57.8] 5.113 fe 3L0} 0 45.3) 6 57.2) 5.091) 5.360
OCTOBER. NOVEMBER. DECEMBER,
Year. ] Tr =a =
My Ny HH, HH’, vy Ng H; HH, DNaT dp HY H’,
i}
h. m. Ft. Ft. | h. m.| he m. Ft. Ft. h. m.| he m. Ft. Ft.
6 56 5.22 0.23 f| 0) 41’ G A |) B20) 5.38] 0 41 (bb) 5.17 5. 18
6 oF 0.30 |) 9. 72/9) 0 40 6 54 5. 07 5.39 7 0 37 6) 52 0. 23 5. 48
6 99 5. 76 5. 64 0 38 6 51 5. 76 | 5.46} 0 41 6 55 5. 36 5. 16
6 52 5.21} 5.25) 0 40 6 53 5.16 5.189 0 41 6 54 5. 30 5.21
6 55 5, 31 5. 30 0 37 6 51 0. 27 5. 40 0 35 6 49 4.93 5. 20
6 53 5. 31 5.85} 0 42] 6 55 5. 34 5. 68 0 32 6 47 5. 30 5.45
6 49 4, 86 5. 37 0 33 6 48 5. 21 5. 23 0 36 6 51 5. 34 5. 39
6 43 5. 09 5.707 0 38 GO Gil} e119 5.52 7 0 40 6 54 5. 03 5. 26
6 49 5.16 5.777 0 35) 6 48 4.97 5. 64 0 30 6 45 4. 93 5. 34
6 53 5.17 5. 80 Q 45 Genin | moet 5. TL 0 41 6 42 0. 21 5. 25
6 SL 5. 34 Onli 0 38 6 51 5.15 5.37 | 0 39 6 54 4.77 5. 07
6 50 5,02 5.627 0 37] 6 49 5. 38 5.37 0 38 6 36 4,90 5, 20
6 52 4.90 5.42 0 40 6 53) 4. 83 5.427 0 39 6 50 4.79 5. 58
6 56 4. 83 5.18} 0 43 6 57 5. 08 5. 63 0 40 6 54 4. 93 5. 37
6 59 4.99 5.23} 0 47 6 59 5.31 5.43 1 0 43 6 55 4. 84 4, 94
6 53 5. 00 5.28] 0 44) 6 52 4.97 5.167 0 42] 6 53 4.72 5. 00
% 3 5.15 5.529 0 41 6 52 5. 02 5.54] 0 42 6 53 4. 84 5. 36
eae, 5.05 5.26] 0 41 6 57 5.14 5.12] 0 36 6 49 5.10 5.10
6 58 5. 04 5. 04 OQ GB) ¢ w 5. 41 5.24] 0 43 6 57 5.19 4.96
6 54.5) 5.143) 5.4697 0 40.6) 6 53.3) 5. 202| 5. 4251 0 38.7] 6 50.8) 5.046 5. 240.

26 DISCUSSION OF TIDES IN BOSTON HARBOR.

THE CONSTANT OR MEAN TIDE.

27. From the footings of Tables I and II we get the following table of average values of all the
observations contained within certain limits of the argument (— wv’), and corresponding to the given
average of the argument. These values, consequently, are independent of the effects depending
upon any of the other arguments, and their inequalities depend only upon the argument (~—v’),

TABLE Y.
i
UPPER TRANSITS. LOWER TRANSITS. COMBINED TRANSITS.
Obs.| (W—w’)| A » | Obs.) (YW) | As As Hz, | Hy | Obs.| (W—wW)) Ay Ne) ee
| f | |
h. ™. ln. m. \h. m. | re | hom. |h. m. Vey | me) me hom. |hom. |hom | Fe | Ft
273 | 015.2 | 0 40.3 | 6 50.3 | 5.63 277 | 015.7 | 0 40.4 | 6 50.8 | 5.64 | 4.62} 550) 015.4] 0 40.3} 650.5] 5.63] 4.60
275 | 0 44:8 | 0 34.6 | 6 44.5 | 5.51 275 | 0 45.2} 035.1) 6 45.9) 5.59) 4.69] 550| 0 45.0) 0348)6 45.2) 5.55] 4.70
964 | 114.4 | 0 30.9 | 6 41.4 | 5.46 263 | 114.7] 030.6 | 6 42.3) 5.46] 4.84] 527] 114.6/030.8|/6 41.8) 5.46] 4.78
985 | 1 44.4 | 0 26.4 | 6 37.4 | 5.31 274 | 144.8 | 027.1) 637.9] 5.36 | 4.94] 559] 1446/|026.7/637.6] 5.33] 4.95
283 | 215.6 | 0 23.9 | 6 34.4 | 5.20 279| 214.6|0228)6340/5.14| 5.17] 562} 2151/]0234/6342] 5.17] 5.18
269 | 2 44.6 | 0 20.7) 6 32.4 | 5.05 268 | 2 44.6 | 0 21.3 | 6 33.4 | 5.09] 5.33] 537] 2 44.6] 021.0] 6329] 5.07] 5.32
a0 | 315.8) 019.8] 6 31.8 | 4.92 295 | 314.5] 019.4] 6.31.4] 4.891555] 565) 315.2/019.6|6 31.6] 4.90] 5.55
280 | 3 44.2 | 0 20.3) 6 326) 4.74 281 | 3 45.6 0 20.3}6339/4.74|5.73] 561| 3449/020.3]6 33.2] 4.74] 5.72
283 | 415.9 | 0 23.2 | 6 36.7 | 4.58 283} 414.8/0226|6 36.3) 4.58] 5.81] 566] 415.3)0229|]6365| 4.58] 5.85
983 | 444.8 026.4 | 6 40.0 | 4.50 272) 445.8] 0 27.6 | 6 42.2] 4.48] 5.99] 555) 445.3) 027.0) 641.1] 4.49] 5.95
983 | 515.6 | 0 32.3 | 6 47.8 | 4.48 273 | 514.3] 032.0] 6 47.2) 4.48) 5.99] 556] 515.0} 0321)647.5| 4.48] 5.98
988 | 5 44.3 | 037.9 | 6 52.8 | 4.46 am | 5 45.0] 039.1} 653.8) 4.39| 6.00] 559) 5 44.7|/0385|653.3| 4.43] 6.03
976 | 615.5 | 045.9) 7 0.9 | 4.46 291} 614.0] 0 44.6|7 0.2) 4.47] 5.96] 567| 614.7) 045.3|7 0.6] 4.46] 6.00
283 | 6 44.2) 050.8|7 6.3 | 4.54 289) 6 46.0)0526)7 7.0) 4.51])5.85] 572) 6 45.1)051.7|7 66| 4.52) 5.82
271 715.4 | 058.4 713.1 | 4. 62. eg | 714.8|056.4|711.6/ 4.64] 5.55] 549| 715.1] 0574/7124] 463] 5.63
984] 7442/1 1.0) 7161) 4.84 273 | 745.2)1 25/7170) 4.81)5.52) 557] 7447)1 17/7165) 4.82] 5.50
281] 8 15.8/1 5.9] 718.6 | 4.87 287) 814.6) 1 4.6) 717.5) 4.95) 5.31] 568) 815.2)1 5.3) 7180) 491) 5.32
980] 8 44.5)1 4.8) 717.1 | 5.22 275 | 845.9)1 5.4| 718.2) 5.06]5.10] 555] 845.2]/1 51/7176) 5.14] 5.08
282] 9152/1 4.8) 7 16.2 | 5.21 272) 915.5/1 3.6) 7165] 5.34] 4.90} 554] 915.3/1 42/7163) 5.27] 4.92
276) 9449/1 1.5) 7129 | 5.49 268} 945.4]/1 26/713.8| 5.42] 4.88] 544] 945.1]/1 20/7133] 5.46] 4.83
$75 | 1015.0 | 059.1|7 9.8 | 5.50 276 | 1014.7] 057.4) 7 9.7) 5.61) 4.671 551] 10149|0582]/7 9.8] 5.55) 4.73
980} 10 44.3/053.7|7 4.3 | 5.72 264| 10 45.5) 0538) 7 4.8) 5.59| 464] 544| 1044.9] 0537/7 4.5] 5.66] 4.61
78 | 11 15.3 | 0 49.0 | 6 59.7 | 5.68 280 | 1115.1 | 0 47.2| 6 59.8 | 5.68] 4.55} 558) 1115.2) 048.1] 6 59.7] 5.68] 4.53
268 | 11 45.1 | 0 45.4 | 6 55.3 | 5.67 269 | 11 45.7 0 44.2| 6 54.8 | 5.68 | 4.50 537 | 11 44.7 | 0 44.8} 655.0] 5.67] 4.54
Means...) 0 42.37] 6 54 67 5.070 | Means..| 0 42. 2 6 55.00] 5.062) 5. 257] Means..| 0 42.29| 6 54.24/ 5.066| 5.257
|

In the footings of this table the inequalities having the argument (~—¥v’) are also eliminated,
and we have results belonging to the mean tide. Since the diurnal tide depends upon g, it is also
eliminated, and we have left only the constant part of the other oscillations.

With the preceding mean values of 4’; and 4’, we get from (44), supplying the omitted constant
of 2 days,

(56) Bo = 4 (2% 0" 42™,294 24 6b 54™.84 — Gh 12™,62) — 24 Ob 42™,25

which is the mean establishment of the port belonging to the assumed transit. In order to reduce
this to the transit immediately preceding high water, we must add the constant part of k (34), put-
ting n=3, and we thus get
Bo= 2? 08 42™.25 — 1¢ 132 13™,.14 = 02 115 267.53
From the first of (37) we get, since L.—=B, in this case,

9
Q2=6 54.67 —0 42.25 —142— 07.20
Sq@=0 42.22—0 42.2
Gs—=6 55.00 — 0 42.25 = $2407.13
in which z=12 lunar hours or 12" 25™.24 in solar time. Hence all the intervals between high and low
and low and high waters in the mean tide are almost exactly one-fourth of a lunar day.

(57)

DISCUSSION OF TIDES IN BOSTON HARBOR. OR

From the last two of (42)-we get
§K K; cos 3 4’ = $(5.0
( K; sin 3 4’ =4$(5.2

0 — 5.062) = .004
\
es) 5.257

7

57 — ) =.000

Hence 4’—0 and K;—.004 ft. This is the value of the constant or mean tertio-diurnal tide, and
may be regarded as falling within the limits of the errors of observation, and consequently insensi-
ble.

With the preceding values of K;, which is the constant and principal part of Aj, and q,, the
terms in the first of (42) are entirely insensible. We therefore obtain from (45), with the preceding
mean values of H’; and H’,,

(59) HH’, = 4(25.066+15.257) — 20.161 ft.
for the mean height of the sea above the assumed zero of the tide-gauge. This, however, is not
necessarily the same as the mean level obtained from observations made frequently at all times
during the day, unless the tides follow the law of sines and cosines, or at least the parts above and
below the mean level are symmetrical. It is simply the mean of the heights of high and low waters.

We obtain from the second of (42), since the terms depending upon K; are insensible,

(60) K, = $(25.066 — 15.257) = 4.904 ft.
for the coefficient of the average or mean semi-diurnal tide. Consequently the mean range is 9.808 ft.

With the preceding values of K;, 4,’, and Ky, (43) gives

n=09, 2=37, 3 =— 0™.13, Qs = $7+0".13

These values from the formule depending upon the eights are almost exactly the same as those

obtained above from the observed times alone, (57).

THE SEMI-MONTHLY INEQUALITY.

28. If from each of the values of 2;, in the preceding table, we subtract By-+q,;—0" 42™.29,
omitting the constant two days, and from each value of 2.’ we subtract Bo+q,.—k—6" 54™.84
+0".4 cos 71, and call the differences 6 L, (46) and (47) will give, with these values of 5L and the
corresponding values of the argument 7;—=2()—’), forty-eight equations of condition of the form

6 L=M;, sin 71+ N;, Cos 41+ Myo Sin 19+ Nyo COS 710
the angles 7; and 79 being the only ones included in the same argument. The tabular values of
7, must be reduced to the time of low water for the twenty-four conditions obtained, from the low
waters, by adding the mean change of 7, from high to low water, which in this case is 25™.24
+0™.4 cos 7, the small term 0™.4 cos 7; being an inequality in the moon’s motion depending upon
the argument of variation, which is the same as 7 or 2(~—y’). With these forty-eight equations
we obtain, by the method of least squares,
My, =—22™.25, N,=0".01, My=1™.96, Nj =—0™.37

These values satisfy the forty- aieetht conditions, with an average residual of 0™.35 and a maxi-
mum residual of 1.3. The residuals do not indicate any sensible term depending upon 37. With
these values (48) gives a

B, =— 220.5 = 09
(60) : P i
Bp= 2.0, £9) = 5°
These comprise the constants belonging to two of the terms of the expression of Ls, (26).

From (28) we get

(61) T= Bp = 22.03
for the age of the tide from the times.

If, trom each value of H’, in the preceding table, we subtract H/,+ K,— 25.066 ft., (59) and (60),
and also from each value of H’, we subtract H/)—K,—15.257 ft., aud call the residuals 6 H, (50)
gives forty-eight equations of the form

0 H= Ky (Mj cos 7) +-N; sin 7)+ Ky (M, cos 71+ N, sin 41+ Mig COS 419+ Nio Sin 7430)

in which the minus sign belongs to the conditions obtained from low waters, and the tabular values
of 7, for these conditions must be reduced to the time of low water as above. With these condi-
tions we obtain, by the method of least squares,

62 Ky My =—0.059 ft., K, M;= + 0.670 ft., K, Mj) =— 0.023 ft.
) Ky Ni= 0.000 ft., Kk, Ni; =—0.124 ft., K, Nj —=+ 0.001 ft.

28 DISCUSSION OF TIDES IN BOSTON HARBOR.

These values satisfy the conditions with an average residual of .025 foot and a maximum of
06 foot. The residuals do not indicate any sensible term depending upon the angle 37 in the
oscillations of the third kind, as may be also inferred from theory.
With the preceding values (49) gives
Ky Ry —— (059 ft., Ro, i) = 0.504, 40,1) = 0
(63) K,R, =+ 0.681 ft, Rey = 0.1888, ae1) ——10°29’
Ky Fp ——— 01023; ft.. Re, 10) —=— 0.0047, 42, 10) =— 20°
These are the constants of two of the terms in the expression of A, and of one of the terms of
Aoy (24).
In obtaining the values of R from K R the values of Ky and Ky, (31) and (60), have been used.
It will be remembered that R expresses the ratio of the inequality to the mean tide in each kind of
oscillation. As the inequality Rei) of the second degree is scarcely sensible, it is not probable
that there are sensible inequalities of the third degree.
From (28) we get, expressing the value of a; in terms of the radius,

183 14.60

°6

(64) 7 —= 21,03 —
for the age of the tide from the heights.

INEQUALITY DEPENDING UPON THE MOON’S MEAN ANOMALY.

29. From the footings of Table III we get the following table of average values of all the
observations contained within the limits of each of the twenty-four equal divisions of 72,in which the
middle of the division is taken as the value of 7 belonging to the averages. In these footings the
inequality depending upon (—y’) is eliminated, and they consequently contain only the inequality
depending upon 7p:

TABLE VI.

Obs. Ne Ny Ne H, H’,
h. m. h. m Feet. Feet.
561 15 0 37.9 6 52.3 5. 894 4. 426
546 30 0 39.0 6 53.6 5. 882 4. 378
509 45 0 40.1 6 54.8 5. 850 4. 422
552 60 0 40.8 6 56.0 5. 771 4. 554
501 75 0 42.0 6 956.2 5. 566 4. 692 ji
548 90 0 43.4 6 57.0 5. 376 4.917
599 105 0 44.0 6 o7.1 5. 122 5. 112
365 120 0 45.0 6 58.4 5. O15 9. 372
ood 135 0 47.5 6 59.9 4. 854 5. 552
553 150 0 47.8 6 59.5 4. 652 5, 692
O46 165 0 48.3 6 53.9 4.515 5. 840
530 180 0 48.7 6 59.5 4. 448 5. 966
545 195 0 48.7 6 59.3 4. 374 6. 014
559 210 0 47.5 6 58.4 4.379 6. 047
547 225 0 45.8 6 56.7 4. 398 6. 025
552 240 0 44.7 6 55.2 4. 406 5 953
549 255 0 42.3 6 53.5 4. 502 5. 797
556 270 0 40.3 6 51.6 4. 676 5. 630
553 285 0 38.3 6 50.1 4. 840 5.510
590 300 0 37.1 6 49.8 5. 045 5. 281
5o4 315 0 35.9 6 49.0 95. 215 4. 992
546 330 0 35.4 6 48.9 5. 408 4. 802
556 345 0 35.7 6 49.4 5. 605 4. 593
551 360 0 37.2 6 310 5. T12 4. 480
13, 254 Means. 0 42.24 6 54.84 5. 068 5. 252

From the footings of this table the constants of the mean tide might also be obtained, as in the
preceding case, but they would not differ sensibly, as may be seen by comparing the footings of the —

an

DISCUSSION OF TIDES IN BOSTON HARBOR. ; 29

two tables. If from each value of 2, in the preceding table we subtract By+q,—=0" 42™.24, as
obtained from the preceding table, and from each value of 2’, we take By+q,—k—6" 54™.84
+1™.5 cos 72, putting 0 L for the residuals, (46) and (47) give forty-eight equations of the form
OM, sin jo+No cos qo+ My sin qutNu COS Ai
the angles 7, and 7, being included in the same argument. In this case the value of k (34) includes
the constant and the inequality depending upon 7», the value of » being 4 as before. In this case,
to obtain the values of 7 belonging to low water, we must add 3°9.3+0°.5 cos 72 to the tabular
values of 7 given for high water. These conditions give, by the method of least squares,
M, ==} 43) Nz = om2, My —— 0™.6, Nu — 02.8

These values satisfy the conditions, with an average residual of 0.3 and a maximum residual ot
0™.9. With these values (48) gives
( By —6™.2, Gy ==> 27
(By=1™0, 41 123°

The preceding are the values of ¢ and < when 7 is given for a time two lunar days after the
transit C (§ 26). In order to reduce them to the case in which 7» is given for the time of transit C,
we must subtract the mean changes of 7) and 7, in two lunar days, which is 27°.1 in the former
and twice that, or 54°.2, in the latter. Hence we get, in this case,

(66) eg— 45°.6, ey—— 119

(65)

The preceding are the constants belonging to two more of the terms in the expression of L; (26).
Tf, now, as in the preceding case, we subtract H’,+ K, from each value of H’; in the preceding
table, and H’/,—K, from each value of H’,, with these forty eight residuals and the corresponding
values of 7, (50) gives forty-eight equations of the form
O EKG (M, cos no+N, sin q2)+ K, (M, cos jo+N2 sin jotMi cos jutNn sin 711)
in which the minus sign belongs to low waters. From these forty-eight conditions we obtain

Ky M,=— 0.033 ft., K, M,=+ 0.771 ft., K, My —-+ 0.051 ft.
Ky N,=—0.014 ft., K, No=-4+ 0.204 ft., K, Ny; =-+ 0.014 ft.

With these values (49) gives
K, R, =— 0.036 ft., Ro, 2) =—0.308, a0,2) = 38°
(67) K, R, =+ 0.797 ft., Re, 2) =0.1624, (2, 2) —==29° 51!
K, ti i= + 0.053 ft., Re 11) = 0.0107, (2, 11) — 450
These values satisfy the conditions, with an average residual of 0.020 ft. and a maximum resid-
ual of 0.049 ft. These are a part of the constants belonging to the terms in the expressions of Ao
and A, (24).
In order to reduce the preceding values of the angle of epoch to transit C, we must subtract
27°.1 from the first two and 54°.2 from the last one.
From (28) we get, with the reduced value of a.=299.9 — 27°.1,

(68) T)—= 24.034 a et

INEQUALITY DEPENDING UPON THE MOON’S LONGITUDE.

30. By taking the average of all the values of 2 and H of each transit in Tables I and II, and
then taking the half sum and the half difference of the values of the two transits, we get the follow-
ing table of average values belonging to the given longitude, from which the inequality depending
upon (}—y’) is eliminated, and consequently they contain only the inequality depending upon the
moon’s longitude.

30 DISCUSSION OF TIDES IN BOSTON HARBOR.

-TaBLe VII.
Combined transits. The differences of transits.
Obs.
% Ay Ng Hy H’, (Ay re As) Qe— Na) (8, — Hg) (H,— Hy)
® h.m. h.m. Ft. Ft. m. m™. Ft. Ft.
555 7.5 0 41.7 6 53.4 5. 182 5. 123 +2.0 —2.8 |—0.486 | + 0.234
553 22.5, 0 43.5 6 55.1 5. 148 5. 145 BD) 4.0 0. 719 0. 391
555 37.5 0 44.8 6 57.95 5. 106 5. 233 3.4 4.7 0. 822 0. 481
556 52.5 0 45.4 6 57.8 5. 043 5. 282 4.8 4.8 0. 886 0. 589
347 67.5 0 44.2 6 57.0 4.991 5. 334 5.1 3.0 0. 992 0. 523
548 82.5 0 42.5 6 55.2 4,998 5. 355 yal 5.4 0. 963 0. 636
550 97.5 0 39.4 6 52.4 4.999 5. 365 5.0 4.3 0. 888 0. 540
545 112.5 0 36.8 6 50.3 5. 029 5. 334 5.6 IEG) 0, 667 0. 482
552 127.5 0 33.9 6 48.7 5. 024 5. 234 3.5 2.5 0. 567 0. 383
558 142.5 0 36.5 6 49.1 5. 075 5.199 peal —2.2 |—0. 235 0. 160
563 157.5 0 39.1 6 51.5 5. 122 5.178 24 +0.1 |+0.001 | + 0.042
553 172.5 0 43.0 6 54.4 5.131 5. 168 +0.3 2.7 0.227 |—0. 068
566 187.5 0 45.2 6 56.9 5.131 5.157 —2.0 2.8 0. 555 0. 230
558 202.5 0 47.9 6 99.9 5. 086 5. 215 2.5 3.1 0. 739 0. 352
563 217.5 0 49.1 6 61.9 5. 056 5. 291 3.8 4.1 0. 830 0. 501
562 232.5 0 50.4 6 63.0 5. 015 5. 390 3.1 4.5 0. 942 0. 578
563 247.5 0 49.6 6 62.0 5. 007 5. 434 4.0 5.5 0. 945 0. 576
557 262.5 0 46.4 6 59.0 4. 997 5. 427 6.4 3.7 0. 872 0. 511
552 277.5 0 43.5 6 56.5 4. 924 5. 403 4.2 | 5.1 0. 852 0.519
541 292. 5 | 0 40.4 § 53.1 4.998 | 5. 304 3.9 2p} 0. 769 0. 479
558 307.5 0 38.0 6 50.7 5. 056 5. 220 4.7 1% 0. 5382 0. 400
545 322.5 0 36.5 6 49.4 5.101 5. 136 2.3 +01 0. 328 0. 206
569 337. 5 0 37.4 6 50.3 5. 168 5. 097 1.6 —1.6 |+4 0.042 |—0,115
546 352. 5 0 39.2 6 51.0 5. 203 5. O71 —1.5 |e 3.9 | —0.291 | + 0.081
Means. 0 42.35 6 54.83 5. 067 9. 295 |

The inequalities in the values in the preceding table are affected by the tide produced by the
small term in the moon’s disturbing force depending upon the fourth power of the moon’s distance.
The expression of this tide, and also of the lunitidal interval, contains the angle ¢, (33); and hence
the expressions representing the inequalities in the preceding tabular values must contain such an
angle. By proceeding in the same manner as in the preceding cases, we obtain from the preceding
table forty-eight equations of the form,

6L—M" sin g+N” cos ¢+M; sin 43+N; Cos 75
These equations give
M”=—2™.1, N’——1™.4, Neo N;==(0".5
From these we get by (48),
Oe, e// —__ 3.40

(62) B, =5".3, ass we

These constants satisfy the conditions, with an average residual of 0™.4 and a maximum of 1™.4.

From the values of H’; and H’, inthe preceding table we obtain, as in the preceding cases, from
(50) forty-eight equations of the form

6 H= Ky (Mz; cos 734 N3 sin 73) + Ky (Ms; cos 73+N; sin 73+ M” cos g+N sin ¢

From these forty-eight conditions we obtain

K,M;—=—.021 ft., K, Ms = + .107 ft., M” = + .030
Ky No>=+ .005 ft., K, N;=—.012 ft., NY — + .007
With these values (49) gives
Ky R; === 022 ft., Ro, 3)—.190 Zo, 3) =—149
(70) K, R; = 109 ft., Re. 3) .0225 2233) ———— 6° 30!

K, R’= 032 ft., R” =.0065 al! = ==-++ 13°

DISCUSSION OF TIDES IN BOSTON HARBOR. ilk

Since the range of argument belonging to each group of observations is twice as great in this
case as in the other cases, the coefficients of the inequalities, as obtained, are increased in the ratio
of the sine to the are of the half range of the groups of observations. An explanation of this small
correction has been given in (§ 23). This very small correction is insensible in the other cases.

These constants satisfy the conditions, with an average residual of .016 ft. and a maximum ot
.057 ft. As the preceding inequalities of the first degree are so small, those of the second degree
depending upon 273; must be very small and may be neglected.

From (28) we get

113

7 pS — == =—14,78
(71) 3 ~ 460 78

Since the maximum of the small tide depending upon the fourth power of the moon’s distance
does not necessarily happen at the same time as that of the principal part of the semi-diurnal tide,
the preceding value of K, R” represents the height of that tide at the time of the high water of the
resultant tide. Hence, putting

4’ = the lunitidal interval of the small tide ;
q'’ =the time of the resultant high water after that of the mean semi-diurnal tide ;
K, a= the coefficient of the tide ;
we have

1+ RY = V 12 a?222)aicos (ly — 2”) —1-+- a cos (Ly, —2”) nearly
@ sin (Liy— 2”)
1+<a cos (lL, —2”)

tan q/ = =a sin (Ll, — 4”) nearly
q 5

From the preceding values of B” and <” and of R” and a’, we get
q’ =— 2.5 sin (¢g+349)
R’” = .0065 sin(¢+77°)

These values of q’’ and R” cannot satisfy the preceding equations unless the angles are the
same, whereas they differ 43°. But since the coefficient K, R’ of the small tide from which the
value of a’ has been determined is only a small fraction of an inch, the discrepancy may be regarded
as falling within the limits of the errors of observation. Assuming that the angles are equal, we
then have, at the time of the maximum, q//—=— 2™.5, and R/’=.0065.

With these values the preceding equations give

.0065= a cos(L2—2”’)
0218 =—a sin (Ly — 2”)
from which we get

(72) a= .023, L, — A’ =— 73°
Hence K,a—0.115 ft. is the coefficient of the tide, and 73° expressed in solar time, which is about
24 hours, is the time the high water of this small tide precedes the time of the high water of the
principal tide. Hence the high water of this tide occurs at 11% 26™.5 —2> 30™= 8 56”.5, or about 9
o’clock in lunar time. We also have in this case

(73) Bi = 24 OF 49m — 2h 30™ = 14 238 12m

for the mean establishment.

INEQUALITIES DEPENDING UPON THE SUN’S ANOMALY AND LONGITUDE.

31. From the means in the footings of Table IV we get the following table of average values,
corresponding to the given values of v/ and 2¢’ belonging to the middle of each month:

32 DISCUSSION OF TIDES IN BOSTON HARBOR.

TaBLE VII.
|
Month. u 2 g! My Ne HH H’, |:(H,+ H’.)/t(H';—H’.)| d Hy
h. m. Feet. Feet. Feet. Feet. Feet.
DEENA cote soocsuososbosaspsincesoscbosedsgac5 6 51.0 4, 933 5.190 20. 061 4. 861 —. 039
February 6 52.2 4, 841 5. 060 19. 950 4. 890 —. 200
March ..-. 6 53.4 4, 962 5. 058 20. 010 4, 952 —. 140
April ..- 6 54.1 5. 000 5.177 20. 092 4.915 —. 058
May ..- 6 55.5 5. 075 5. 253 20. 164 4.911 +. 014
June. 6 57.8 5. 090 5. 225 20. 157 4. 932 +. 007
July - 6 58.7 5. 082 5. 266 20.174 4. 908 +. 024
August ....-.-...--- 6 57.8 5.113 5. 300 20. 206 4. 906 +. 056
September 6 57.2 5. 091 5. 360 20. 225 4.865 | +.075
October==-e-se——s— 6 54.5 5. 143 5. 469 20. 306 4. 837 +. 156
November 6 53.3 5. 202 5. 425 20. 313 4, 888 +.163-
December 6 50.8 5. 046 5. 240 20. 143 4.903 —.007
6 54.9 8, 058 5. 202 20. 150 biel Waepocosiae~

From this table we obtain, as in preceding cases, twelve values of 6L, which, together with the
corresponding values of v/ = 7, and 2 ¢g/ =7;, in (46) and (47), give twelve equations of the form

OL= My, sin y4+ N, cos 44+ M; sin 75+ Ns; cos 75
From these we get, in the same manner as in preceding cases,
{ Bs =— 3.9, 24 == — 73°
1B; —=— 0.6, &5—=— 10°
In the sanie way with the values of $(H’; — H’2), (50) gives twelve equations of the form
0H = Ky (My cos 44+ Ny sin 474+ Ms cos 75+ N; sin 75)

(74)

From these conditions we get

(75) { K, Ry = 0.0378 ft., Ry =— .0077, a4 —=— 65°
UK, R; = 0.0093 ft., R; = — .0019, a5 = — 58°
In the same manner we obtain from the values of $(H,+ H’.),
(76) § Ky Ry=0.126 ft., R,= 1.08, a4 = 2549
( Ky Rs = 0.073 ft., R; = 0.62, ds;—= 98°

Since the arguments in the preceding table change so little from high to the following low
water, $(H’,;+H’,) may be taken as a normal yalue of the mean height, and 4 (H’,— H’,) as the
coefficient or semi-range of the tide.

In the preceding table 6H’, is the difference between any value of H’, and its mean value, and
the column expresses the annual variation of mean level.

INEQUALITY DEPENDING UPON THE MOON’S NODE.

32. By summing the values of 24), 2/2, H;, and H's, and taking the averages so as to eliminate the
annual inequalities, we obtain the following table of averages for each year, in which the value of
w belonging to the middle of the year is given:

DISCUSSION OF TIDES IN BOSTON HARBOR. oo

TABLE VIII.

Year. o 4 re HY, H, |4(H’, + H’,)|4(H4—H’2))

h. m. h. m. Feet. Feet. Feet.

0 42.5 6 55.7 5. 21 5. 24 4.99
0 44.3 6 56.4 5.27 5.17 5.05
0 42.4 654.6 | 5. 21 5.17 5.02 |
0 42.1 6 54.3 | 5.16 5.19 4.99
0 41.8 6 54.8 5.15 5. 30 4.93
0 39.2 6 52.0 5.13 5.82 | 4.90
0 40.0 6 53.1 4.98 5. 28 4. 83
0 39.5 6 53.0 5.05 5. 37 4, 84
0 40.2 6 51.0 5. 04 5. 49 4,73
0 41.6 6 53.4 4,99 5.42 4.79
0 41.8 sIG RON 4. 86 5. 33 4.77
0 39.4 6 52.2 4.90 5.24 4.83
0 42.5 675555 4. 86 5.11 4.88
0 44.0 6 56.4 4.99 5.22 4.89
0 43.6 6 55.1 5. 00 5. 25 4.88
0 45.2 6 55.5 4.97 5. 26 4.85
0 45.1 6 58.7 5. 02 5.16 4.93
0 46.7 6 58.9 5.14 5.11 5. 01
0 42.6 6 54.9 5. 048 5. 249 20. 150 4. 897

In this case, as in the preceding one, the argument changes so slowly that we can take $ (H’,
4+ H’,) as the mean level, and $ (H/;—H’,) as the mean range, corresponding to the given value of w.
From the preceding values of 2’; and i’, we obtain, in the same way as heretofore, eighteen
equations of the form
oL=M, sin 76+ Ng COS 7
which gives by (48),
(77) Be=—2™.5, &6 == — 509
From the last column we obtain eighteen equations of tle form
5H=M, cos 7¢-+Ne Sin 7
which, by (49), gives
G8) K, Rg =— 0.112 ft., Rg =— .0235, ag —=—11°

INEQUALITIES DEPENDING UPON 73 AND 7p.

33. If in Table III we combine all the values of 241, 2/2, H’;, and H’2, in which 7;+72—7°.5, and
then all those in which 7,-+-7.—22°.5, and so on; and likewise combine all those in which 7,—72
=7°.5, and then all those in which 7,—7,—22°.5, and so on, we get the following table of averages,
corresponding to twenty-four values of the argument 7;+72—7s, and also to twenty-four values of
the argument 7,—72—7, in which the inequalities of all the other arguments are eliminated:

5*

34 DISCUSSION OF TIDES IN BOSTON HARBOR.

TABLE IX.
Ng | Ay | A'2 H, H's No AY r’2 HH’, HH,
|

° h m. h. m. Feet. Feet. ° h. m. h. m Feet. Feet.
1.5 0 41.7 6 54.3 5, 104 9. 232 7.5 0 43.4 6 55.9 5. 195 5. 123
22.5 0 43.0 6 55.0 5. 099 5. 242 22.5 0 45.0 6 57. 7 5.119 5. 147
37.5 0 42.2 6) go a82) 5. 073 5. 247 37.5 0 45.1 6 57.8 7 5. 095 5. 206
o2. 01 0 43.4 6 55.9 5. 036 5, 207 52.5 0 45.3 6 58.0 5. 082 5. 254
67.5 0 43.4 6 55.8 5. 015 5, 247 67.5 0 45.8 6 58.2 5.017 5. 253
82.5 0 43.8 6 56.9 5. 060 5, 252 82.5 0 45.5 6 58.7 5. 034 5. 295
97.5 0 43.9 6 56.6 5. 030 5. 270 97.5 0 45.4 6 57.8 4. 998 5. 372
112.5 0 44,2 6 56.1 5. 029 5. 283 112.5 0 44.7 6 57.2 5. 016 5. 337
127.5 0 43.4 6 56.0 4. 978 5. 259 127.5 0 43.8 6 56.5 4.999 5. 386
142.5 0 43.5 6 55.9 5. 082 0. 202 142.5 0 43.2 6 55.3 4. 985 5. 364
157.5 0 43.2 6 55.8 5. 025 5. 290 157.5 0 41.5 6 54.3 4.961 5. 390
172.5 0 42.8 6 55.0 5.030 5. 278 172.5 0 41.2 6 53.8 4, 943 5. 343
187.5 0 42.5 6 55.2 5. 089 bass 187.5 0 40.4 6 52.6 4, 926 5. 352
202.5 0 42.2 (iS GEE 5. 107 5.2 202. 5 0) 73925) Givolend 4, 997 5. 332
217.5 0 41.4 Giosns 5. 039 0. 28 217.5 0 38.7 6 51.8 4. 989 5. 267
232.5 0 41.3 6 54.4 5. 067 5. 232.5 0 39.9 6 52.2 5. 060 5. 290
247.5 0 40.8 6 53.8 5.111 5. 2 247.5 0 39.1 6 51.9 5. 108 5. 260
262.5 0 41.0 6 53.5 5. 090 5.2 262.5 0 39.1 6 51.0 5.111 5. 196
277.5 0 40.6 6 53.6 5. 077 5. 277.5 0 39.3 6 51.9 5. 130 5. 227
292. 5 0 40.4 6 53.0 5. 067 5. 292.5 0 395 G25 1 5. 144 5. 143
307.5 0 41.2 6 53.7 5. 102 5. 4 A) 0 40.3 6 52.8 5. 167 5. 147
322.5 0 40.3 6 53.8 5. 108 3. Pas) || (251653 6 54.8 5. 183 5. 130
37. 5 0 41.8 6 93.7 5. O80 5. 5 0 42.4 6 55.0 5.183 5. 103
352. 5 0 41.5 6 54.5 5. 095 i) 352.5 0 44.2 6 57.1 5. 139 5. 128
Means 0 42.2 6 54.38 5.066 | Means. 0 42.3 6 54.8 5. 066 5. 252

By the methods heretofore used, we obtain from these tabular results,
By =1".5, €g =— 259, Ky R; = .0265 ft., Ry = .0054, ag —= — 45°
Bj —=3™, ég== 240, Ky Ry = .1196 ft., Ry = .0240, ag —= — 21°

As has been stated, the values of 72 in Table III belong to a time two lunar days after transit C,
and hence the values of the argument 7, are too great for the assumed transit ©, by the change of
72 in that time. For the same the values of argument 7 are too small by that amount. The pre-
ceding values of the epochs have been reduced to transit C by subtracting 279.1 in the former
case, and adding the same amount in the latter case.

In the same manner the constants might be found for the terms depending upon the arguments
m+73 and y;—73, OF 72+73 and 72—73, but the effects depending upon these arguments must be still
smaller than those depending upon 7 and 7. This one case will serve to show that these effects
are of very little importance.

(79)

DIURNAL TIDE.

34. It has been found that the terms depending upon A; are insensible, and hence with the
twenty-four values of (H,—H;) and (H,—H,) in Table V, (52) gives twenty-four sets of equations
for determining the twenty-four values of A; and 4, belonging to the twenty-four values of g. With
these values of A, and the corresponding values of ¢ in the table, (53) then gives twenty-four equa-
tions of condition for determining, by the method of least squares, the values of M and N, with
which we then obtain, by (54),

(80) K, =— 0.58 ft., a==—19°.7
for the coefficient of the diurnal tide and the value of the angle at the epoch.

From (28) we get, using the value «f By in (56),

fe)
(81) t= 22,03 — —___—04,504

for the age of the diurnal tide, 13°.18 being the daily motion of the moon in longitude.

DISCUSSION OF TIDES IN BOSTON HARBOR. 35

The preceding twenty-four equations of conditions also give 4—31°. Expressing this value
in solar time, we get from (37) L.—L,—2" 8™ for the time the high water of the diurnal tide precedes
that of the semi-diurnal tide.

We consequently have

(82) Bo = 2? 08 49m — 2h Gm — 14 29h 34m
for the mean establishment of the diurnal tide belonging to transit C.

It is evident from (41) that $(2; — 23) =q@ and 4(4.—/,) =42—q. With the preceding values
of K; and 4, and Ky (60), we get from (55), expressing ares in time,

a=+ 1.8 sin (g—a)
$z— @—=— 3.0 sin (g — a)

in which a must have the value above. Hence these expressions should represent 4(2;—A;) and
$(42—2,) in Table VII. The angle of the epoch is about right, but the coefficient of the former is
nearly one minute too small and that of the latter a little too great. These slight discrepancies are,
no doubt, caused by the existence of a small quarter-day tide, which has not been taken into account
in (41), from which the preceding expressions have been deduced.

RECAPITULATION OF RESULTS.

35. For the general tidal expressions, (24) and (26), the following constants have been obtained,
in which the values of the epochs are for transit C:
In oscillations of mean level, in which s—0,

pS OA Rie,

Ri= 0.504, q== . @
R= 0.308, et
(3) R;= 0.190, a;—=— 14
R.= 1.08, y= 4
Rs —! 0:62) ds = — 82
In diurnal oscillations, in which s—1, (32),
(84) KG OF Sitiies a—=— 199.7, Bo= 14 220 34™
In semi-diurnal oscillations, in which s—=2,
K,= 4.904 ft., Bo = 22 02 497.25,
R, = (0.1888, a, —=—10°.5, B, =— 22".6, ec 0°
Ro = 0.1624, d =—+ 22.7, By =. O82 & == 459.6
R3; = 0.0225, C= OSs) 5 = TAs S=
R, =— 0.0077, a, =— 65°, B, =— 3™9) &4 =— 713°
(35) <R; =—0.0019, a; =—58°, B; =— 0"6, «© =—10°
\Re =— 0.0235, ag =—119, Bs =— 2.5, « =—50°
R, = 0.0054, ag =—459, = Is £3 —=— 250
Ry = 0.0240, a —=— 219, By = =H QS We
Rypy=— 0.0045, 239 = — 29, Bio = BD ; €10 = 5°
Ry= 0.0107, a = — 99; i OS ey — 77°
In tertio-diurnal oscillations, in which s—3,
(86) K,= 0.004 ft. :
In the semi-diurnal tide, depending upon the fourth power of the moon’s distance, (36),
(87) K” =.0065, al! — 77°, ye BE IS

The absolute values of the coefficients of the tide are K, R;.
COMPARISONS WITH THE EQUILIBRIUM THEORY.

36. The values of the constants K, in the equilibrium theory, and also in the dynamic theory in
the oscillations of mean level, are given approximately in (31). By comparing these values with

26 DISCUSSION OF TIDES IN BOSTON HARBOR.

the preceding, it is seen that K, and K, given by observation are less than the theoretical values
while that of K,, the mean coefficient of the semi-diurnal tide, is nearly ten times greater. But in
applying the equilibrium theory to the real case of nature, it has been usual to determine such
constants as make the expressions best represent the observations instead of determining them from
theory, and to depend upon the theory for the ratios of the inequalities to these constants. In the
equilibrium theory, and also in the dynamic theory in the case of oscillations of mean level, we
should have R,—P,. By comparing the preceding values of R, with those of P, (12) it is seen that
while P, is wanting, R;=.504, and also that the value of R,. is greater than that of P,. The three
values of (R,—P,) K, in these three cases give respectively —0.059 ft., —0.020 ft., and +0.003 ft.
as the coefficients of inequalities in the mean level, belonging respectively to the arguments 71, 72,
and 73, for which there are no corresponding inequalities in the disturbing forces. A semi-monthly
inequality of mean level, corresponding with the first of the preceding, and with the same sign,
though frequently much greater, has usually been found in all tidal discussions.

The parts of the preceding inequalities without any corresponding disturbing forces are, no
doubt, the effects of a quarter-day tide, which, with observations of high and low waters only, there
are no means of separating from the semi-diurnal tide, and are not inequalities in the true mean
level (§ 27); and the preceding inequalities are merely the inequalities in this tide, which varies
with the semi-diurnal tide, and the effect of the constant part of this tide is contained in the value
of H’,. Upon this hypothesis, if we assume the existence of a quarter-day tide with a coefficient of
about three inches, it would account for these inequalities within the limits of the errors of observa-
tion. We have already had indications of the existence of such a tide elsewhere (§ 34). The exact
coefficient of such a tide can only be determined from observations made several times during the
phase of the tide.

If we compare the values of R, (83) with that of P, in (12), we see that the latter is a very
inconsiderable part of the former, and that the difference corresponds to an annual inequality of
mean level with a range of about four inches, for which there is no corresponding disturbing force.
The whole of this inequality is given for the middle of each month in the last column of Table
Vil. The maximum of this inequality occurs in October or November and the minimum in Feb-
ruary. Such an inequality has been found at other ports. At Brest it is a little greater, with the
maximum and minimum occurring a little later in the year. Dr. Bache found, from the discussion
of the tides at Key West, an annual inequality with a range of about nine inches, and with the
maximum in September and the minimum in February.

These results should not be regarded as being at variance with the general tidal theory, but
merely as being the effects of some circumstances or causes not taken into account in the theory ;
and these effects are, no doubt, due to the annual changes in the currents of the ocean, produced by
annual changes of temperature and of the winds. On account of the influence of the earth’s rotation
there cannot be an annual change in the velocity or position of the currents of the ocean without
a corresponding change generally in the mean level of the ocean at any port. The preceding results
are very interesting in connection with this subject. I endeavored to give a full explanation of
these inequalities, a few years ago, in the Proceedings of the American Academy of Arts and Sciences,
Vol. VII, p. 31.

By comparing the values of R, (85) with those of P, (18), it is seen that they differ very
much, and consequently the equilibrium theory, applied to the Boston tides, gives very erroneous
relations between the inequalities and the mean tide. While R;—.1388, we have P, —.4240, and
hence the equilibrium theory would make the semi-monthly inequality in heights more than three
times greater than it is. In the same way it is seen that it makes the inequality depending upon
the moon’s parallax too small, while it makes that depending upon the moon’s longitude, or declina-
tion, more than four times greater.

37. In the equilibrium theory the lunitidal interval is expressed by 2 he (22), and the coeffi-

a
cients of the inequalities in the development by Q, (23). The mean establishment, referred to the
nearest transit, should be 0, which does differ much from observation at Boston. But if we compare
the preceding value of B, with that of Q, (23), we see that the equilibrium theory gives nearly two and
a half times that of observation for the coefficient of the semi-monthly inequality. For the observed

DISCUSSION OF TIDES IN BOSTON HARBOR. 37

inequality also of 6™.2 depending upon 7, the equilibrium theory gives none; and throughout the
smaller coefficients there are large proportionate differences between the theory and observation.

38. The values of the angles of epoch in the equilibrium theory should be 0 when referred to the
nearest transit. The values of a and 22 (83) both nearly vanish for the transit C occurring two lunar
days earlier, and the value of a; would vanish for a transit nearly a day and a half earlier. This
also applies to the dynamic theory. But it has been shown that these inequalities, except a small
part of the second and third, do not depend upon any corresponding term in the disturbing force,
but are probably the effects of quarter-day tides, resulting from the circumstance of a shallow sea,
and depending upon the magnitude and epoch of the semi-diurnal tide. Hence the values of the
angles of the epoch should correspond somewhat with those of the semi-diurnal tide in (85), which
they do as nearly as could be expected, since it is impossible to determine the values of the angles
accurately for so small inequalities. The preceding comparisons are sufficient to show how inad-
equate the equilibrium theory is to represent the observed inequalities of the Boston tides.

DETERMINATION OF THE GENERAL CONSTANTS.

39. It is now proposed to determine the constants in the general tidal expressions (25) and (27)
These being known, these expressions then give the special constants belonging to each inequality.
Among the constants to be determined is the correction of the moon’s mass, dy, contained in the
expressions of P;, U;, and Q;, (18) and (27). The diurnal tide in the port of Boston being very small,
only the coefficient of the principal inequality has been determined from observation, and conse-
quently we have no means of forming conditions enough to determine these general constants
belonging to this tide; and they are of no consequence, since the effects depending upon them in
this case must be insensible. We can, therefore, only determine these constants for the semi-diurnal
tide.

With the values of R; (85) belonging to the first three inequalities, and the corresponding values
of P; and U;, the first of (25) gives the following conditions :

1388 (1+ F) =.4305 — 24.0 0 4 —(.1742 — 18.2 0 ») B
(88) 1624 (14+ F)=.15214 3.604 .0500 i
0225 (14+F)=.0985 4+ 1.00 —(.0477—0.5 0 »)E
Also with the values of M,=B; cos (¢;—«;), belonging to the same inequalities, the third of (27)
gives

— 22™,9 — 52". 51.4034 6 w+ (21".75 —1210 0 »)E—0™.6
(89) 4 — +( 4°.10-+ 187 6 4)E—0.3
5™.3—— 2",24+ 148 d0p4+( 57.164 5004) K—O0.1

By giving proper relative weights to these two sets of conditions from the heights and from
the times, we can combine them in the determination of the constants, and thus obtain the most
probable values which all the conditions give, and get some idea of their probable errors. These
six are the only conditions which can be formed having much weight in the determination.

A solution of the first three conditions, (88), depending upon the heights of the tides, gives

(90) 6 p=—.000283, E=1.408, and F=.365
With this value of 6 1, (9) gives for the moon’s mass,

p=.013— .000283—.012717 = aa

In order to determine the most probable value of the three preceding constants belonging to
all of the preceding conditions, we shall multiply the first three by 300 and then substitute the pre-
ceding values plus the correction belonging to the new conditions. We thus get

—1650 4 »—51.60 4E—41.644F= 0 00
+1080 4 2415.00 4 B—48.72 4F= 0 B\ + .02
MO Apis ees 0 © je ae
4.9333 4421.75 4E = 08 9 s\2o
4+ 2624 p+ 4104E BED 208 f' 221.03

+ 28 4p4 5.1645 = 03 + 0.29

38 DISCUSSION OF TIDES IN BOSTON HARBOR.

The solution of these by the method of least squares gives
4 p= .000120 + .00021, 4 H——.005, and 4 F=.001

with the residuals given above. The first three residuals divided by 300 and multiplied by 60
inches, the coefficient nearly of the mean tide, give the real residuals in inches, the greatest of which
is only .024 inch. The last three are the residuals belonging to the times in minutes. The multi-
plication of the conditions from the heights by 300 makes the relation between the two kinds of
residuals in general about the same as that of the probable errors of the two kinds of observations.

With the preceding corrections we get for the constants belonging to the new set of conditions,

n—=0.012717 + (.000120-4.00021) — +

~ T7941.8
(aa eenGeiee
F—0.365 + .001 —0.366

(91)

The preceding residuals show the accuracy with which theory represents the three principal
inequalities of the heights and the times. This may be somewhat accidental in this case, and it
remains yet to be determined by an application of the theory to other ports, whether it will repre-
sent the inequalities of the times with such accuracy that conditions deduced from them should
have any weight in the determination of the moon’s mass. The preceding probable error, obtained
from so few conditions, cannot be relied upon as showing with much certainty the real probable
error of such determinations in general.

The preceding value of E for the port of Boston is four or five times as great as it is in most
Kuropean ports. This extraordinary value, in the first of (25), diminishes the values of R, and R;,
and consequently the coefficients of the first and third inequalities, to less than one-third of what
they would be by the equilibrium theory, but increases the second inequality, since U, is negative,
and makes it greater than what the equilibrium theory would require, and even greater than the
semi-monthly inequality. In like manner this large value of E, in the third of (27), diminishes the
coefficient, and consequently the range, of the first or semi-monthly inequality in the lunitidal
intervals, so that at Boston it is only about half as much as in European ports generally, and less
than half of what the equilibrium theory requires.

The value of F above, in the first of (25), tends to decrease all the inequalities, and in the first
and third is in the same direction with the effect of the term depending upon E; but in the second
inequality the effects of the two terms are in contrary directions, but that depending upon E is the
greater, and consequently the second inequality is greater than the conditions of a static equilibrium _
would require. Whether the effect of the term depending upon F is due to friction, as we have
supposed, (§14), or to some other cause, it is evident that the preceding conditions cannot be satis-
fied without such a term.

40. In the equilibrium theory both E and F vanish, and in this case the first of either (88) or
(89) furnishes a condition for determining the correction of the moon’s mass, the former from the
heights, and the Jatter from the times. It is well known that this theory, even in European ports,
where the deviation from the relations of the equilibrium theory is comparatively small, gives very
unreliable determinations of the moon’s mass; but in the port of Boston it would give a mass more
than double the true mass, as may be seen from a mere inspection of the conditions in this case.
The preceding conditions are based upon the hypothesis of a small correction only, and consequently
fail in this case, except that they indicate that it must be very large.

In the equilibrium theory only one condition is necessary for determining the moon’s mass,
which can be based upon the first and principal inequality. Where the term depending upon E
has a sensible effect, a second condition is necessary, which may depend either upon the second or
third inequalities. Laplace, in his last tidal investigations in the fifth volume of the Mécanique
Celeste, used the first and third inequalities. Airy, in the discussion of the tides of Ireland, formed
conditions from the first and second inequalities, but obtained a mass very much too great. When
the terms depending upon F in the preceding conditions have a sensible value, it is readily seen
that these two sets of conditions must give very different results, and that the conditions based
upon the first and third inequalities must be much better than those based upon the first and

DISCUSSION OF TIDES IN BOSTON HARBOR. 39

second, since the effects of the two terms depending upon E and F are in the same direction in the
former conditions, and somewhat in the same proportion, while in the latter they are in contrary
directions.

When the terms depending upon F are sensible, there are three quantities to be determined,
and consequently three conditions are necessary, and two of these conditions, if the heights alone
are used, have to depend upon the small parallactic and declination inequalities belonging to the
moon. Where circumstances are such as to make the terms depending upon EK and F small, as at
Brest, the conditions are sufficient to determine them with adequate precision to get a pretty accu-
rate determination of the moon’s mass; but in the case of the Boston tides these become terms of a
first order, which, in a great measure, destroy the first and third inequalities, upon which the
determination mainly depends, so that the problem becomes nearly indeterminate, and the condi-
tions are not sufficient to give a reliable determination of the moon’s mass; for the small semi-
monthly inequality observed is not wholly due to the solar tide, which is a kind of base in the
determination, but a considerable part of it belongs to that term in the moon’s parallax depending
upon the argument of variation, and which gives no advantage in forming the conditions.

By the preceding method, in which conditions from the times are taken in, which our tidal
expressions enable us to do, the great weight of the determination which rests upon the small
declination inequality, which, in the case of the Boston tides, is reduced to 1.3 inches, is thrown upon
the principal semi-monthly inequality of the lunitidal intervals, which is 22™.6, and which can be
observed with much greater accuracy, proportionately, than 1.3 inches in the heights. The two other
conditions from the times also give some little additional weight to the determination. The accu-
racy of the determination, of course, depends very much upon the accuracy of the tidal expressions,
and the probable error-may be greater than that which we have obtained from so few conditions,
but still I think the determination is entitled to considerable weight. But it is very evident, from
what has been stated, that the Boston tides are not at all favorable for an accurate determination
of the moon’s mass, and that the same magnitude of errors in theory or observation makes the
probable error of the determination several times greater than in the case of tides, as those of
Brest, in which the magnitudes of the inequalities differ but little from what the equilibrium theory
would give.

COMPARISONS WITH THE DYNAMIC THEORY.

41. In the oscillations of the first kind, which are oscillations of long period, the terms depend-
ing upon BH, F, and G vanish, and the dynamic corresponds with the equilibrium theory, with which
comparisons have already been made. The diurnal tides of Boston being very small, only the con-
stants of the principal term depending upon the moon’s longitude have been determined from the
observations; and, as it requires at least two conditions to determine the constants in the expres-
sions (25) and (27) for each kind of oscillation, we have no means of making any comparisons of
results obtained from the observations with those given by these expressions. The constants E and
F having been determined for the semi-diurnal tide, (90), the first of (25) should give the value of
R, for each of the inequalities of which we know the value of P;; and K, being known we then have
K, R;, the coefficients of the inequalities. The third of (27) should likewise give M;—=5, cos(H,—4;).
Three of the values of R; and M,, obtained from observation, belonging to the three principal ine-
qualities, have been used in the six conditions by which the constants have been determined, and,
from the smallness of the residuals, it is seen that, so far as these three inequalities are concerned,
the observations are well represented by theory.

The values of z, obtained from (28), with the angles of epoch a;, should all be equal, according
to theory, taken in terms of a first order only. But, we see from (64), (68), and (71), these values
differ considerably at Boston. Neglected terms of a second order, which are very large at Boston,
are, no doubt, sensible in this case. It is difficult, also, to obtain the angles of epoch of small ine-
qualities of long period with much accuracy from the observations, but in this case the differences
seem to be rather great to be attributed to errors of observation. The other inequalities are too
small to give a reliable value of c.

The values of the angles of epoch E,, in the inequalities of the intervals, should be given by the
second of (27), but no value for EF’ in the expression of N; can be obtained which will represent them

40 DISCUSSION OF TIDES IN BOSTON HARROR.

accurately, and there are evidently some neglected sensible terms which affect this expression.
But these angles depend upon very small quantities, since the coefficients are mostly very small,
and consequently a very small effect throws them very much out.

In the fourth and fifth inequalities of the heights depending upon the sun’s parallax and de-
clination, the observed inequalities are both quite small, less than a half-inch, as theory requires ;
but here also there are some disturbing influences not represented in the theory, for the coefficients
have the contrary sign, and the angles of the epoch, which in this case should be sensibly 0, since
Di, and D;7; are insensible in the expression (25), are quite large.

These disturbing effects belong mostly to the fourth inequality having an annual period, and
are, no doubt, due in part to the varying effects of friction, caused by annual variations in the
velocities and positions of ocean-currents, as the Gulf Stream; for such variations, depending upon
the changes of the seasons and of temperature, must have an annual period. We have seen that
in the oscillations of mean level also there is a very considerable annual inequality not indicated by
theory, which we have supposed to be due to influences of the same kind, (§ 36). Now, any amount
of change of mean level, from whatever cause, must also produce a slight corresponding effect upon
the range of the tidal oscillations and also upon the time, which, in very shallow seas and harbors,
may be quite sensible to observation. According to theory, the value of B, (85) should be 0; hence
we have an annual inequality in the times with a coefficient of four minutes, which must be due to
the same causes, as the angle of the epoch, corresponding very nearly with that in the inequality of
the heights, seems also to indicate. These seeming deviations from theory are merely the effects of
slight disturbing influences not taken into account in the theory, and should be regarded as very
important in the investigation of the subject of tidal friction in connection with ocean-currents
varying with the seasons.

We come now to the sixth inequality, depending upon the moon’s node. In this case U, in the
first of (25) being sensibly nothing, we should have (1+ F) R,=P;, or substituting the value of Ps
(18), and the observed value of Rg (85), we should have .0235 (1 +4 F)==.0375. This gives the value
of F, as deduced from this small inequality alone, equal to nearly .6, whichis larger than the value
before obtained, from conditions from all the principal inequalities. With the value .401 before
obtained, the preceding equation gives Rs ——.0268, and consequently the tidal coefficient given by
the tidal expression is too great by —(.0268— .0235) x 60 inches, or about one-fifth of an inch. If
F—0, then the observed value of RK, should be equal to P,, the value of R, belonging to static
equilibrium. But the difference is nearly an inch, which is entirely too great to be attributed to
errors of observation; and hence the comparison of observation with theory, in the case of this
small inequality alone, shows that F must have a sensible value, and that all the terms depending
upon it which have never before been taken into account in any tidal theory, must, in the port of
Boston at least, have very sensible values.

The value of Bs =—2™.5 (85) in this case must depend upon EF” in the expression of Ng (27),
since all the other terms in the expression vanish in this case. This indicates that F should have
a positive value, as is also required in the first inequality. The effect of terms depending upon F!
then is to cause the lunitidal intervals of larger tides to be a little greater than those of smaller
ones, and consequently neglecting terms of a third order, to introduce small inequalities into the
intervals proportional to the inequalities in the heights.

All the remaining coefficients of the inequalities are quite small and unimportant, and, in the
comparison of them with the tidal expressions as here given, there is not a very nice agreement,
some of the residuals being nearly an inch. But the correct tidal expressions for these inequalities
of a second order depend upon so much development that the more simple expressions, as given in
the preceding pages, in which many terms are necessarily neglected, cannot be expected to give
accurately these small inequalities in this case on account of the neglected terms, which, although
in most cases insensible in the Boston tides, on account of the large value of E, must be quite sen-
sible. The coefficients of the remaining inequalities in the times, as given by observation, agree
very well with those given by the tidal expressions, none of the residuals being more than 0™.5.

42. All solutions of the tidal problem, extended to the cases of different and varying motions
of the disturbing bodies in right ascension assume, as Laplace substantially expresses it, that if
the tidal coefficient changes with any change of the motion of the disturbing body in right ascen-

DISCUSSION OF TIDES IN BOSTON HARBOR. 41

sion, the corresponding changes when small may be regarded as exactly proportional. This is
equivalent to supposing that in the development of the tidal coefficient K, which is a function of 0s
by Taylor’s theorem, only the first term depending upon 47; of which the coefficient is D; K or E, is
sensible, and that all the others, depending upon the square and higher powers of 4%, may be
neglected. ‘This is, no doubt, true of Brest, and of most Huropean ports, but we are hardly safe in
assuming that it is strictly true in the case of the Boston tides; for we have seen that the effect
of the first term, that depending upon BH, is a quantity of the first order of the inequalities, and in
the ease of the semi-monthly inequality amounts to more than a fourth part of the whole tidal
coefficient, and hence it can hardly be supposed that the second term depending upon (di)? is
entirely insensible. If such terms are sensible they must cause slight deviations of observation
from theory, and also affeet the determination of the moon’s mass. The smallness of the residuals,
however, do not indicate that the effect of such terms, if at all sensible, can be of much consequence.

PREDICTION FORMULA AND TABLES.

43. With the preceding tidal expressions the heights and times of the tides may be computed
for any given time; but although these expressions are in a form most suitable for the discussion
of the tide observations and comparisons with theory, and for the study and investigation of the
tidal theory, yet on account of the great number of arguments which would have to be used, and
number of terms taken in, resulting from the developments, the whole result can be put into a
more suitable form for prediction, containing but few arguments. For this purpose we shall
determine from the constants belonging to the resultant tides of the moon and sun, which have been
obtained directly from comparisons of observation with the tidal expressions, certain constants
belonging to the lunar and solar tides considered separately, and then combine these separate tides
into a form of expression similar to that of the resultant expression of the potentials of the
disturbing forces of the moon and sun, (5) and (6).

By neglecting the effect of friction depending upon F, which decreases the larger tides a little
more in proportion than the smaller ones, the first of (25) may be used, with the value of E which
we have obtained, in determining the relative magnitudes of the lanar and solar tides. In this
case the quantity corresponding to U; is D; 7,=.426, twice the difference of the velocities of the
moon and sun in right ascension. » If we therefore put e/ = the ratio between the solar and lunar
tide, since ¢ (8) is the ratio between the solar and lunar disturbing forces, we shall have

(92) ¢ =e (1—.426 E)—0.180
using the values of e, ov, and E in (10) and (90). Hence the-solar tide is decreased in the ratio of 1
to 1—.426.E, or of 1 to .401, in consequence of the sun’s having a slower motion in right ascension
than the moon.

44, If we now consider the lunar tide alone, omitting that part of the effect of friction which
depends upon F, since in this case the inequalities in the disturbing force depend upon the moon’s
parallax and declination only, we shall have in (17)

(93) 3 P; cos n=O(14"? a — sin’ vy)
in which C is a constant and p is the mean parallax, 0p is the excess of the parallax above the
mean parallax, and v is the moon’s declination, as heretofore defined. In this case the values of
P; in the first member, and the corresponding values of Uj, are different from those already given
in the case of the resultants of the potentials, and are found to be

P,=.0250, C—O Os}
(94) P,==.1638, U,—=— .0500
P;=.0860, Cj 0397

omitting the terms depending upon the moon’s node and other small secondary terms. With these
values the first of (25) gives, omitting F,
R,=.1638 + .0503 x 1.408 —=.2342

(99) R3—=.0860—.0397 x 1.408—=.0302

42 DISCUSSION OF TIDES IN BOSTON HARBOR.

Hence R,, and consequently the coefficient of the corresponding tidal inequality, is increased
in the ratio of .1638 to .2342, or of 1 to 1.429, in consequence of the moon’s excess of motion at
perigee over its mean motion. Also R;, and the corresponding tidal coefficient, are decreased in
the ratio of .0860 to .0302, or of 1 to .851, in consequence of the moon’s decreased motion in right
ascension when on the equator, on account of the obliquity of the ecliptic. It should be under-
stood that the preceding ratios are not the ratios of increase or decrease compared with the whole
tidal coefficient, but the ratio of increase or decrease of the whole tidal coefficient compared with
the corresponding inequality having the same argument. ‘The ratio of increase of the mean tidal
coefficient at the maximum is, in the former case, 14.0500 x 1.408=1.070, and of decrease in the
latter, 1—.0397 x 1.408=.944.

Since the increase of the moon’s angular motion is proportional to the increase of the parallax,
for any increase of the parallax above the mean parallax the ratio of increase of the tidal coefficient
of the inequality or variation is 1.429 times greater. If we also compare the moon’s motion in right
ascension with that motion when on the equator, the increase of motion may be assumed to be as
sivy. This is not strictly true, especially with regard to the effects depending upon the moon’s
node, but the error, as applied to this small inequality, is insensible even in the Boston tides. For
any decrease, therefore, of the coefficient of the disturbing force due to declination, the ratio of
decrease of the corresponding tidal coefficient is .351. !

What has been stated with regard to the lunar tide is also true of the solar tide, except that
the relative ratios of increase or of decrease are different.

45, If we therefore put, supposing F=0,

M=the value of A», (33), in the case of the moon,
S=its value in the case of the sun,
we shall have, without sensible error,

op 3 “Cin 2ov.
M= K(1+m < (1—n sin? v)
(96), ;

yn! 3
S=e' K G +m! =) (1L—’ sin? v’)
)

in which the accented letters denote the same with regard to the sun which the same letters with-
out an accent do with regard to the moon, and in which p and y must be taken at the times z and
t; earlier. In the case of the moon we have found m=1.429, n=.351. “On account of the sun’s
slow motion in right ascension, m/ and 2’ may be put equal to unity without sensible error.

If we combine the separate expressions of the lunar and solar tides, as given by (33), as in the
case of the potentials of the disturbing forces (5), we get for the combined semi-diurnal tides,

(97) Yo= VM?+S8?+2 MS cos (y7;— a1) cos (2 ep—1—')=Q cos (2 p—I—F’)
in which

S sin (71—41)
M-++458 cos (7; —.«1)
These expressions do not include certain small corrections belonging to inequalities of a second

order due to changes of the moon’s motion in right ascension, but these are very small and of no
practical importance.

(98) tan f/=

The lunitidal interval in the preceding expression, in solar time, is
gl .
~ pt
99 =1.03>

em ; D;(2e—I—F’)
and is equivalent to (27), putting 3; Q;=1.035 4’, and using the preceding values of P;, belonging to
the moon only, in the rest of the expression.

The great advantage of the preceding forms of expressions is that we dispense with all develop-
ment in the computation of the tidal coefficient, and the trouble of taking into account a very great
number of small terms, in the development with arguments dependi3g upon the sums and differences
of the principal arguments, and have as arguments merely the time of the moon’s transit and the
parallaxes and declinations of the moon and sun. The same is true also of the part of the lunitidal

DISCUSSION OF TIDES IN BOSTON HARBOR. -43

interval depending upon /’, which is the principal part, the remainder being generally quite small,
so that it need be applied as a correction only to a few of the principal terms.

46. If we develop the preceding expressions, as in the case of the potential of the disturbing
force, substituting for p, v, p’, and v’, their expressions depending upon the angles 7;, we should
obtain expressions of the resultant of the lunar and solar tides similar to those in the preceding
pages, (25) and (27), obtained from the resultant of lunar and solar disturbing forces, of which the
constants should be equivalent, and the coefficients of the inequalities of the latter, divided by
(1+F), should be equal to the corresponding ones in the former.

In the comparison of the constants we get

(100) V1+e” 956 K=K,

The preceding must be used as a condition to determine K in (96).

In the comparison of the coefficients of the semi-monthly inequalities in the development of
(97) and (98), using the value of e/ (92) with those given by observation and by (25) and (27), it is
found that the coefficient of the inequality in heights is too great by 0™.7, and that of the lunitidal
intervals is too small by 1™.5. The theory with regard to this last expression and development
seems to be in error by these amounts. But as our object here is merely to get the most convenient
expression which will represent the observations with sufficient accuracy for practical purposes,
this can be ‘obtained by changing a little the constants as obtained from the former conditions. If
we take

(101) f= 205, m=1.530, n=.410, F=.500

with these values in (96), (97) will give the principal inequalities in the heights, except the effects
of F, within 0™.2, and all the others with about the same accuracy as (25), and (97) will give such
a value of (/ as, substituted in (99), will give the coefficients of the intervals within 0™.5, except
the few discrepancies already mentioned in the comparisons with the expression of (27). The
preceding constants, however, in this case do not quite have the relations to one another required
by theory in the other developments.

With the preceding value of e’, and the value of K, (60), (100) gives for the constant in (96),

(102) : K=5.004 feet.

47. Having obtained M and § from (96), the coefficient of the tide Q and the value of (7 in
(97) and (98) are readily obtained by construction as follows:

Take A B equal to M, and BC equal to S, making an angle with AB, bp
produced to D, equal to 7,—«, which is twice y—y’ at a time x, previous to
the time of high or low water, and then join AC. The latter is Q (97), or lp S<
the coefficient of the tide, neglecting the effect of F. The angle B A C also BS
is the value of 4’. Of course the same are readily obtained by a trig- \
onometrical calculation. One-half 5’, reduced to time and increased by 5th,
is the part of the lunitidal interval in solar time depending upon /’.

The values of M and § (96) contain only the parallaxes and declinations
of the moon and sun as arguments, and very simple and convenient tables f
may be constructed giving their values for any given arguments; and then
by a very simple construction, or trigonometrical calculation, we get the
coefficient of the tide and the principal part of the interval, and thus take
in completely all the numerous terms arising from any developed form of
expression. These terms do not consist only of the terms corresponding to -
the terms given in the development of the disturbing forces in (12) and = |—
(18), but a great many others of the same order as many given there. The
coefficient of the tide thus obtained must be diminished by one-third of the
inequality from the mean tide for the effect of friction depending upon
F, which is the same as dividing the inequality by (1+-F).

It now remains to determine the part of L depending upon / in (99), 4
which is the lunitidal interval of the lunar tide. The constant of this is. B, determined by

44. DISCUSSION OF TIDES IN BOSTON HARBOR.

observation (56), and the inequalities are determined by (27) omitting Q, and using the preceding
values of P; (94) belonging to the lunar tide only. This gives for the lunar part,

(103) M,=1".7, M,=5™8, M;—6™1, &e.

The first of these belongs to the small term in the moon’s parallax depending upon variation, and
is added to the value of the first inequality depending upon f’, which is —24™.2, to give the
whole semi-monthly inequality —22™.5. The second is the whole value of B,, there being no part
depending upon f/’. The part of the third inequality depending upon #’ is —1™.0, and hence the
whole value of B; is 5™.1.

48. For the sake of convenience in computation we can put in (27),in the case of the lunar tide,

(104) »; M; sin 75=C D; p+C/ sin 2 v Di v
in which
D; p=the hourly change of parallax in seconds,
D,v=the change of declination in seconds for one minute of time.

For the principal term of parallax of which the coefficient is 186.5; the hourly change is
1.79 sin 72; also, the change of sin 2 v D; v inseconds for one minute of time is 5.6 sin 73. Hence,
the constants in the preceding expression are determined by the following conditions, using 5".3
given by observation for the value of M; instead of 5".8 given by theory, (103): ‘

= 1.79 C =52.3

ee) } 5.6 O'=6,1

These conditions give C=3 very nearly, and C’=1.1. With these constants, (104), using see-
onds of are as minutes of time, gives the sum of all the terms depending upon D, 7;, independently
of any developed expression, directly from the hourly differences of parallax, and the differences of
declination for one minute, taken from the Nautical Almanac. :

In addition to the preceding we have the terms 3; N; cos 7; (27) depending upon friction and
other disturbing causes, of which it is only necessary to take account of the following, in which the
coefficients given by observation are used, being reduced from transit D to the transit occurring at
the time zr before the time of the tide, using the correction (34) for changing from one transit to
another :

Ny, cos 7;= 4.0 cos 7,
Nz Cos 72—=— 6™.0 cos 72
N; CoS 73= 1.5 cos 73
N, cos 7x—= + 4.0 Cos 74
Ng COS 7s—= + 2™.5 COS 75

All the other terms of this form are included in the terms depending upon / in (99).

The first three and the last of these terms are embraced in Table III, the fourth one in Table II.

To both the times and heights must be added, also, the effect of the term depending upon the
fourth power of the moon’s distance, given in (69) and (70). These inequalities are given in the last
two columns of Table IV.

The summation of the preceding effects gives the values of A», (24) or (33), and Ip, (26),

The value of Ho, the height of mean level, neglecting the very small inequalities given by
observation as of no practical importance, is given in Table III.

The value of A, added to Hy, gives the height of high water, and, subtracted from Hy, gives
that of low water.

To both the heights and times must then be added the effects of the lunar and solar diurnal
tides to obtain the complete height and time of the tide. The effects of the lunar tide upon both
the height and time of the tide are contained in Table IV, and those of the solar diurnal tide in
Tables VUI, IX, X, and XI.

COMPUTATION OF A TIDAL EPHEMERIS.

49, The method of using the preceding formule and results in the computation of a tidal
ephemeris is most conveniently explained by a reference to the example given at the end:

DISCUSSION OF TIDES IN BOSTON HARBOR. 45

A is the mean time of the moon’s upper transit over the meridian preceding the Washington
meridian 2.4 hours, and is obtained from page 332 of the American Ephemeris by interpo-
lation by means of the hourly differences.

ais the equation of time to be subtracted from mean time.

B contains the hours and minutes of the apparent time of the moon’s transit over the meridian
above stated.

Cis the moon’s horizontal parallax for a time 18 hours before the time of the Washington
transit, or about 6 hours after the preceding upper transit, taken from page 339 of the
American Ephemeris, by interpolation by means of the hourly differences.

c is the corresponding hourly difference.

D is the moon’s declination 7 hours preceding the time of Washington transit, or 2 hours pre-
ceding the time of the Greenwich transit, taken from page 6 of the American Hphemeris.

dis the corresponding hourly difference of declination,

I’ is the part of L (99) depending upon /’, or upon the solar tide, plus a constant of 30 minutes.

The part independent of the constant, and also Q, can be readily obtained, with sufficient accu-
racy for practical purposes, by construction or by trigonometry, as has been explained in ($47).
But the method by trigonometry does not answer well near the times of the conjunctions or quad-
ratures, where the angles are very small, unless these angles are determined with great accuracy.
If L/ and Q are determined by computation it is best to compute Q and f’, (97) and (98), directly
from the expressions, as in the last part of the example at the end, in which

M is taken from Table I, with the arguments C and D, and

S from Table II, with the date as the argument.

Log sin 2 B and log cos 2 B are taken from Table VII, which is so arranged, with the sine and
cosine adjacent to each other, that they can be taken out at the same time, using B instead of 2B
as an argument. The remaining steps in the example, to obtain tan /’, need no explanation. With
tan #’ as an argument, the variable part of L’ is taken from Table VI, to which the constant, 30
minutes, is added to make all the values positive.

eis equal to three times ¢ with the sign changed, calling seconds of are minutes of time, plus a
constant of 10 minutes.

J is taken from Table V, with D and d as arguments, a constant of 10 minutes being also
added.

g is taken from Table III, with the time of transit, B, as an argument.

his taken from the same, with the parallax, C, as an argument.

i is also taken from the same, with the declination, D, as an argument.

j is taken from the last part of Table IV, using the declination, D, one day in advance, as taken

out above.

By is taken from Table I, with the date as the argument, the day and hours not being written
in the example, but borne in mind.

H=A+L/-te+f+g+h+itj+By

The value of A being taken from the ephemeris in astronomical time, the constant, Bo, in the
table has been increased 12 hours in order to give E in civil time. When the apparent time of
high water is required, B should be used instead of A in the preceding expression of E.

4’ and 4? are the first and second differences of E belonging to the upper transits, used as a

check, and also for interpolating the intermediate numbers belonging to the lower transits.
The minutes only of 4' are written in the example, the 24 hours being understood. The
intermediate numbers, in smaller type, are $ 4! and £4’, used in the interpolation.

o' are the differences after interpofition, used as a check, and also in interpolating to low water,
the 12 hours understood not being written.

k is taken from the last part of Table IV, with D as an argument taken one day in advance.

U is the effect of the solar diurnal tide upon the time of high water, taken from Table VIII, with
B and the day of the year as arguments.

m=k-+1 is the effect of the whole diurnal tide upon the time of high water. The values belong-
ing to the lower transit are readily obtained by interpolation, and must be used with a
contrary sign.

46

DISCUSSION OF TIDES IN BOSTON HARBOR.

t.h. w=E-+m are the times of high water; the days and hours, being the same as in E, are not
written. . .

Q, the coefficient of the semi-diurnal tide, independent of the effect of F, is obtained either by
construction, as has been explained, or from log Q in the latter part of the example, when
obtained by computation. The different steps in the example, by which log Q is obtained,
need no explanation. _ :

n—=Q—4.90 ft.; that is, it is the excess of Q above its mean value. One-third of this sub-
tracted from Q gives A,=4.90+ =
affected by F.

4}, 4? are the first and second differences of A,, used as a check, and also in interpolating for
the lower transits.

o' are the first differences of A, after interpolation, used as a check, and for interpolating to
low water.

Ho+ Ag, in which H) is taken from Table II, with the day of the year as an argument, is the
height of high water of the semi-diurnal tide.

p is taken from the first part of Table IV, with the declination, D, one day in advance, and is
the effect of the lunar diurnal tide upon the height of high water.

q is taken from Table X, with B and the day of the year as arguments, and is the effect of the
solar diurnal tide upon the height of high water above the zero of the tide-gauge.

r=p+q is the effect of the whole diurnal tide upon the height of high water. The values of r
for the lower transits are readily obtained by interpolation, and must be used with the
contrary sign.

h.h. w=H)+A.+r is the height of high water.

HY’ is obtained from Ei by interpolation to the time of low water by means of 61, using only the
first differences.

k’ is taken from the first part of Table IV for low water, using the argument D one day in
advance.

lis taken from Table LX, with the arguments B and the day of the year.

m =k'+V is the whole effect of the diurnal tide upon the times of losy water. The values of m
for lower transits are obtained by interpolation and must be used with the contrary sign.

t.1.w=E+m’ are the times of low water, the days and hours being the same as in Bi’.

A’, is obtained from A, by interpolation to the time of low water.

H,— A”, is the height of the iow water of the semi-diurnal tide.

pis taken from the first part of Table IV for low water, using the argument D one day in
advance, and is the effect of the lunar diurnal tide upon the height of low water.

q is taken from Table XII, with B and the day of the year as arguments, and is the effect of
the solar diurnal tide upon the height of low water.

7’ =p'+q/ is the effect of the whole diurnal tide upon the height of low water. The values of r
for lower transits are obtained by interpolation, and must have the contrary sign.

h.l.w=H,)—A’,-+7 is the height of low water.

, or the coefficient of the semi-diurnal tide as

CONCLUSION.

50. In the preceding discussion all the results have been brought out which can be of much

interest to any one in any theoretical tidal investigations: and much attention has been given to
the arranging and presenting of the whole matter in as systematic and concise a manner as pos-
sible, and with a convenient and appropriate notation. These results must be found to be the more
interesting to investigators on account of the great peculiarities due to local circumstances which
make them differ so much in many respects from the results obtained in most of European ports.
A comparison of these results has also been made with both the equilibrium and the dynamical
theories of the tides, so far as it could be done, where it is neither convenient nor proper to enter
very thoroughly into the discussion of tidal theories; and the great defects of the equilibrium
theory have been shown, and also various discrepancies of a small order between the results and

DISCUSSION OF TIDES IN BOSTON HARBOR. 47

our tidal expressions based: upon the dynamic theory have been pointed out. These are due, no
doubt, mostly to friction in connection with ocean currents, and to the peculiarities arising from
local circumstances, which cause many small terms, which are necessarily neglected in the tidal
expressions, and which in most ports are insensible, to be quite large in the Boston tides.

Much study and care has also been given to the formation, from these results, of the most
convenient formule possible for the prediction of the times and heights of the tides, and by means
of various auxiliary tables, to render the labor of their computation as small as possible. An
example has also been given of the most convenient method of carrying out the computations, from
which it may be seen that they can be made with great facility and also with great accuracy.

In the comparison of individual tides as computed with the formule and tables, with obser-
vation, considerable discrepancies are often found in both the times and: the heights, as is to be
expected, on account of the many abnormal disturbances arising from the changes in the forces and
directions of the wind and in the barometric pressure, and these discrepancies are especially found
during the winter and spring, when these changes are the greatest; but still it is thought that the
computation gives very accurately the true normal tide. From the discussion of these residuals
with reference to the winds and the barometric pressure, some interesting results may yet be
obtained with regard to their effects upon the tides. The computation of the tides for a portion of
the series is now being made for this purpose, the results of which must be the subject of a future
report.

In the prosecution of the preceding discussions I have to acknowledge the receipt of much
valuable and very satisfactory aid from the Misses Lane, in the Coast Survey service.

Very respectfully, yours,
WM. FERREL.

Professor BENJAMIN PEIRCE,

Les Superintendent United States Coast Survey.

Nore.—The following tables are added by way of appendix to the preceding discussion of the Boston tides. They
were prepared by Mr. Ferrel, and show the application of the theory.

48

APPENDIX TO THE DISCUSSION OF THE BOSTON TIDES, BY MR. W. FERREL.

DISCUSSION

OF TIDES

IN BOSTON

HARBOR.

TABLE I—Showing the value of M for every 10” of the moon’s parallax, and for every 2° of its declination.

Moon's declination.
Moon's
parallax.

0 3 22) 4o 6° g° 10° 125 14° 16° 18° 20° 220 249° 26° 28° 30°
eh Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft. Ft.
54 0 3. 88 3. 87 3. 86 3. 85 3. 83 3.81 3. 79 3. 76 3.73 3. 70 3. 66 3. 62 3.58 3.53 3.48

10 3.94 3.93 3. 92 3. 90 3. 88 3. 86 3. 84 3. 81 3. 78 3.75 3.71 3. 67 3. 63 3. 58 3.53
20 4. 00 3.99 3. 98 3.96 3. 94 3. 92 3. 90 3. 87 3. 84 3. 81 3.77 3.73 3. 69 3. G4 3. 58
30 4. 06 4,05 4. 04 4, 02 4.00 3.98 3.96 3. 93 3. 90 3. 36 3, 82 3. 78 3. 74 3. 69 3. 64
40 4.12 4.11 4.10 4. 08 4. 06 4. 04 4.02 3.99 3. 96 3. 91 3. 87 3. 83 3.79 3. 74 3. 69
50 4.18 4.17 4.16 4.14 4.12 4.10 4.08 4.05 4.01 3. 97 3. 93 3. 89 3. 85 3. 80 3. 74
55 (0 4,24 4.23 4 22 4.20 4.17 4.15 4.13 4.10 4. 07 4.03 3.99 3.95 3. 90 3. 85 3.79
10 4. 30 4.29 4, 28 4.26 4, 23 4.21 4.19 4.16 4.13 4.09 4,05 4.00 3. 95 3. 90 3. 85
20 4, 36 4.35 4, 34 4. 32 4.29 4.27 4,25 4, 22 4.18 4.14 4.10 4.05 4.00 3.95, 3. 90
30 4,42 4.41 4. 40 4. 38 4. 36 4. 34 4.31 4,28 4. 24 4, 20 4.16 4.11 4. 06 4.01 3. 96
40 4. 48 4.47 4, 46 4. 44 4, 42 4.40 4, 37 4.34 4.30 4.26 4.22 4.17 4,12 4.07 4.01
50 4.54 4.53 4, 52 4.50 4. 48 4.46 4,43 4.40 4. 36 4, 32 4, 28 4. 23 4.17 4,12 4. 06
56 0 4. 60 4.59 4, 58 4.56 4, 54 4,52 4,49 4. 46 4, 42 4.37 4,32 4,27 4.22 4.17 4,12
10 4. 67 4. 66 4.65 4. 63 4, 60 4,58 4.55 4.51 4.47 4. 43 4.38 4. 33 4, 28 4, 23 4.17
20 4,74 4.73 4.72 4.70 4, 67 4. 65 4, 62 4.58 4, 54 4,50 4.45 4. 40 4, 35 4. 29 4.23
30 4.81 4.80 4.79 4.77 4.74 4,72 4.69 4. 65 4. 61 4.56 4.51 4.46 4. 41 4, 35 4.29
40 4. 87 4. 86 4. 85 4, 83 4.81 4.79 4.76 4,72 4. 67 4. 62 4. 57 4. 52 4.47 4.41 4.35
50 4. 93 4. 92 4, 91 4.89 4.87 4.85 4, 82 4.78 4.73 4. 68 4. 63 4. 58 4, 53 4.47 4, 41
ov (0 5. 00 4.99 4.98 4.96 4.94 4. 92 4.89 4. 85 4. 80 4,75 4,70 4.65 4.59 4,53 4,47
10 5. 07 5. 06 5. 05 5. 03 5. 00 4,98 4.95 4.91 4, 86 4.81 4.76 4.71 4. 66 4. 60 4. 54
20 5.14 5.13 5. 12 5.10 5.07 5.05 5. 02 4.98 4.93 4. 87 4, 82 4.77 4.712 4. 66 4. 60
30 5. 21 5. 20 5.19 3. 17 5. 14 9. 12 5. 09 5.05 5. 00 4.94 4.89 4, 84 4.79 4.73%) 4. 67
40 5. 28 5. 27 5. 26 5. 24 9. 21 5.19 5.16 5.11 5. 06 5. 01 4.96 4.91 4. 85 4.79 4.73
50 5. 35 5. 34 5. 33 5. 30 5. 27 5. 25 0: 22 5.18 5,13 5. 08 5. 03 4,97 4. 91 4, 85 4.79
oo, 0 5. 41 5. 40 5. 39 Syl 5. 34 5. 32 9. 29 5. 25 5. 20 5.14 5. 09 5, 03 4.97 4,91 4.85
10 5. 48 5.47 5. 46 5. 44 5. 41 5. 39 5. 36 9. 32 o.20 3,21 5.16 5.11 5. 05 4.98 4, 91
20 95.55 5.54 5.53 5. 51 5. 49 5.47 5. 44 5. 40 5. 34 5. 28 5. 23 5.17 5. 11 5. 05 4.98
30 5. 63 5. 62 5. GL 5. 59 5. 56 5.54 On 0L 5. 46 5, 41 5.35 0.29 5. 23 5.17 o. 11 5. 04
40 5. 70 5. 69 5. 68 5. 67 5. 64 5. 61 5.07 5. 52 5. 47 5. 42 5. 37 5. 31 5. 24 5.17 5. 10
50 5. 77 5. 76 5. 75 5. 73 5. T1 5. 68 5. 65 5. 61 5.55 5. 49 5. 43 5. 37 5. 31 5, 24 5.17
59 (0 5. 85 5. 84 5. 83 5. 81 5. 78 5. 75 5. 72 5. 68 5. 62 5. 56 5. 50 5. 44 5, 37 5. 30 5. 23
10 5. 93 5. 92 5. 90 5. 88 5. 85 5. 82 5. 78 9. 13 5. 68 5. 63 5. 57 5. 51 5. 44 5. 37 5. 30
20 6. 00 5. 99 5. 97 5.95 5. 92 5. 89 5. 85 5. 81 5. 76 5. 71 5, 65 5. 58 5. 51 5. 44 5. 37
30 6. 08 6. 07 6. 06 6. 03 6. 00 5.97 5. 93 5. 89 5. 84 5.7 5. 72 5. 65 9. 08 5.51 5. 43
40 6.15 6.14 6.13 6710 6. 07 6. 04 6.00 5.95 5. 90 5. 85 5.79 5. 72 5. 65 5. 58 5. 50
50 6. 23 6.22 6.20 6.17 6.14 6.11 6. 07 6. 02 5. 97 5. 92 5, 86 5. 79 5. 72 5. 64 5. 56
60 0 6. 30 6.29 6. 28 6. 25 6. 22 6.19 6.15 6.10 6.05 5. 99 5: 92 5. 85 5. 78 5.71 5. 63
10 6. 38 6.37 6. 36 6. 33 6. 30 6. 27 6. 23 6.18 6.12 6. 06 5. 99 5. 92 5. 85 5. 78 5. 70
20 6. 46 6. 45, 6. 43 6. 41 6. 38 6.35 6.31 6. 26 6. 20 6.14 6. 07 6.00 5. 93 5. 85 5. 77
30 6. 54 6.53 6. 51 6.49 6. 46 6. 43, 6. 39 6.34 6..28 6. 22 6.15 6, 08 6. 00 5. 92 5. 84
40 6. 62 6. 61 6. 60 6,57 6. 54 6.51 6.47 6. 42 6, 36 6. 30 6. 23 6.16 6. 08 6. 00 5. 92
50 6. 70 6. 69 6, 67 6. 65 6, 62 6, 59 6.55 6.50 6. 44 6. 37 6, 30 6. 23 6.15 6. 07 5. 99
61 0 6.79 6.78 6.76 6.73 6.70 6. 67 6. 63 6. 58 6, 52 6.45 6.38 6. 31 6.23 6.15 6. 07
10 6. 88 6.87 6.85 6. 82 6. 78 6.75 6.71 6. 66 6. 60 6. 53 6. 46 6. 39 6. 31 6, 23 6.15
20 6.97 6.96 6. 94 6. 91 6. 87 6. 84 6. 80 6.75 6. 68 6.61 6.54 6.47 6. 39 6.31 6. 23
30 7.05 7. 04 7. 02 6.99 6.95 6.91 6. 87 6. 82 6. 76 6. 70 6. 63 6.55 6.47 6.39 6.37

DISCUSSION

TaBLE I1—Showing the values of S and Ho
for each third of a month, and also of Bo,
as affected by the annual inequality, and
the constants of the equations.

Date. SS) Hy Bo
Ft. It. d.h. m.
Jan. 1 0. 92 20. 09 123 9.0
cut 0. 93 20. 07 9.1
21 0.95 20. 04 9.3
Feb. 1 0.97 20. 00 9.6
WW 1.00 19. 96 10.0
21 1.02 19. 96 10.5
Mar. 1 1.03 19. 98 11.0
11 1.03 20. 01 11.6
a1 1. 02 20. 03 12.3
|
April 1 1.01 20. 06 13.0 |
11 1.00 20. 08 13.7
Q1 0.97 20.11 14.4
May 1 0.93 20. 14 15.0
1 0.90 20. 15 15.5
a1 0. 87 20.16 15.9
r
June 1 0.84 | 20.16 16.3
ol 0. 83 20.16 16.6 |
| a1| O81 | 20.16 16.9
|
| July 1 0. 82 20.17 17.0
} U1 0. 83 20.17 16.9
21 0. 86 20. 18 16.6
Aug. 1 0. 89 20. 20 16.3
11 0. 92 20. 21 15.9
21 0.95 20. 21 15.5
Sept. 1 0.98 20. 22 15.0
11 . 00 20. 23 14.4
21 1.01 20. 25 1387
Oct. 1 1.01 20. 27 13.0
11 1.01 20. 29 12.3
21 1.00 20. 30 11.6
Nov. 1 0.98 20. 31 11.0
maT 0.96 20. 31 10.5
21 0.94 20. 30 10.0
Dec. 1 0. 92 20. 27 9.6
11 0.91 20. 22 9.3
21 0. 92 20. 16 123 9.1

Nore.—For the value of Hy above mean low

water, subtract 15. 26 feet.

7*

OF TIDES IN

TABLE II1—Showing the inequalities resulting from friction and
other causes, and depending upon the moon’s transit, parallax, and

BOSTON

HARBOR.

declination.
D's transit. | “Equation. yee Equation. | D’sdec. | Equation.
| |
h. m. m | & @ mM. © m.
0 00 80), |) 54" 00 15.0 0 0.0
30 | 78 | 30 14.0 2 0.1
1 00 500) 13.0 4 0.2
30 69 | 30 12.0 6 0.4
2 00 6.0 56 00 11.0 8 0.6
30 5.0 30 10.0 10 0.9
3 00 4.0 57 00 9.0 12 1.2
30 3.0 30 8.0 14 1.6
4 00 OXON al e589 00. 7.0 16 2.0
30 ii | 30 6.0 18 2.5
| 8 @ 0.5 59 00 5.0 20 3.0
30 0.2 30 4.0 22 3.6
6 00 0.0 60 00 3.0 24 4.3
30 0.2 30 9.0 26 5.1
7 00 0.5 61 00 1-9 28 6.0
30 it || 30 0-9 30 7.0
8 00 20 |
30 30 |
9 00 4.0
30 5.0
| 10 00 6.0
30 6.9 | Constant ..-.9m.0 Constant. ... 2m.0
11 00 Tha”
30 7.8
Constant ....4m. 0

TaBLE 1V—Showing the effect of the moon’s diwrnal tide upon the
‘times and heights, and also of the term depending upon the fourth .
power of the moon’s distance, contained in the last columns.

argument is D, taken one day in advance.

The

D's dee. Equations of high | Equations of low Equations of
water. water. semi-diurnal tide.
° mM. Ft. mM. It. m. Ft.
+30 3.0 +0. 63 13.0 —0. 37 0.0 0. 06
20 25 0. 53 2.5 0. 32 0.3 0. 06
20 2.0 0. 43 220) 0. 26 0.8 0. 05
15 1.5 0. 32 1.5 0. 20 1.4 0. 05
10 1.0 0, 22 1.0 0.13 1.9 0. 04
+5 —0.5 +0. 11 +0.5 —0. 07 2.5 0. 04
0 0.0 0. 00 0.0 0. 00 3.0 0. 03
—5 +0.5 —0.11 —0.5 +0. 07 3.5 0. 02
10 1.0 0.22 1.0 0.13 4.1 0. 02
15 Aso) 0. 32 1.5 0. 20 4.6 0. 01
20 2.0 0. 43 2.0 0. 26 5. 2 0. 01
20 2.5 0. 53 255) 0, 32 5.7 0. 00
—30 +3. 0 —0. 63 —3.0 +0. 37 6.0 0. 00

Notre.—For lower transits the signs of the diurnal tide must be reversed,

49

DISCUSSION OF TIDES IN BOSTON HARBOR.

TaBLe V—Showing the value of C’ sin 2 v Dt v (104) in tenths of minutes for each degree of declination
and for each second of the value of Dt v.

Values of Dev.
Dec. | | |
) qr | 2” 31 4 5 6” via gY | gu 10” 11” | 12” 13” 14" | 15” 16” 1
| z| | Ne |
o | | | |
Lee | 3 | 8 4 EON BS 5 5 6 6 7
| 6 7 8 | 9 | to ak |! ee) Tey |. 783
3 | 1 ON sQ | th) 1 | | es ay | Te) Te [Png
4 | | | | 12 | 13 | 15 | 16 | 18 | 20 | Qt | 23 | 25 | 26
5 | 15} 16 | 19: || 205) 22) || 25°) 26) | 29) au} ae
6 | UGH) 185/20) ||) 23) 1) 25RD Tan SON 32ie |p esd feeszin lindo)
| i) |) BIL PRY ey | Pid) Bl) SBS I see ay |) Bh | ae
8 | : Oy, GA 1 bye By | 8B | Se} cM | Ze | A | AO | aD
| | 20 | 24 | 27 | 3 Bye) Be |) cil |] ee | ee |) a |] a
10 | | 22 | 26 | 30. | 33 7 | 41°] 46 | 49 || 53 | 57 || 60
leet ee ay ion so |) 6o || Be | Sh | 6B | Be | Ay) 2 | aD |) ee |) ae | Ge Ge
12) | BOOT |) estes e386) | d0l| (440) 4g ies4e |) Seee| mesielG zie)
13 | 24 | 29 | 34 | 39 | 43 | 47 | 53 | 58 | G3 | G8 | 72 | 78
14 | a) || Ga |) Sl} S68 | 2) 25 |) Bl |) Be I Ge) Get we ll wa
15 | 16 | 22 | 27 | 33 | 38 | 44 | 49 | 55 | 60 | c6 | m1 | 7 | 82
1B. |} oo 12 | iy |) GE |) Os |) hy at ear |) Be) BEY RN a) aa GR || ee 4
7 Gj) 1m | Te || Bs |) By |) Be i) 2 I ao | 6) GL | Ge | we |) eo | ea
18 | 6 | 13 | 19 | 26 | 32 | 30 | 45 | 52 | 58 | 64 | 71 | 78 | e4 | 90
19 69/13) |) 20) |) sez) 73373) 40) || ee sb GO) |) (67) ozs es. | 86
20 || 1 1) Ol || BE | 35 | 42 | 49 | 57 | G3 | 70 | 78 | 86 | 90
21 | 7 | 15 | 22 | 29 | 36 | 44 | 51 | 59 | Go | 73 | 81 | 90 |
22 G |) ie | BS |e | ge | Aa || See GT |G aa |) GB |
23 GY) S|) 1 yk ay eae ey GRY Ne Fe)
24] 8 | 16 | 24 | 32 | 40 | 49 TGS ial e73) jee |
OB i ht ai! | OG; | BES |) ae | Bi) |) Bae |) |
| 26 | 8 | 17 | 26 | 34 | 43 | se | ot | 69 | | .|
QT 9 | 18 | 27 | 35 | 44 | 53 | 63 | |
28 ay} aie | Bee BIN Zar ea os |
EDP | TO | PRB AB noe I) oot |} co || a0 Ea :
| i | i

Norr.—When the arguments have the same signs, the quantities are positive; when different signs, negative.

TaBLE VI—Showing the value of L', and the part of L (99) depending upon 3', corresponding to any given
value of the logarithmic tangent of 3’ (

] ] ] ,
L! Om. 1m. | 2m. 3m. | 4m. 5m. 6m. 7m. 8m. 9m.
| |
m. |
Seer 6. 926 T. 227 7.403 | 7528 | 7.615 7. 704 7.771 7. 829 7. 380 |
1 7. 926 7. 967 8.005 | 8.040 | 8. 072 8.101 8. 130 8.156 8. 181 8. 204
2 8. 227 8. 248 8. 268 8. 238 | 8. 306 8. 324 8. 341 8. 357 8. 373 8. 388
0 oO. 000 7. 926 8.227 | 8.403 | 8.528 &. 625 8.704 | 8.771 8. 830 8. 881
10 8. 927 8. 969 9.006 | 9.042 | 9. 074 9. 104 9. 133 9.159 9.185 9. 209
20 9. 231 97253 95274 it 95293 | 9.312 9. 333 9, 348 9. 365 9. 382 9. 398
30 9. 413 9, 428 9, 442 | 9. 457 | 9. 470 | 9. 484 9. 497 9.510 9. 522 9.534 |

Notr.—In the first division of the table 0m., 1m., 2m., &c., must be taken as tenths of a minute.

TABLE VII—Showing the logarithmic sine and cosine of 2(—w') to three places for each minute of (4)—w').

DISCUSSION

OF

TIDES IN BOSTON HARBOR.

(w—w') Sine. Cosine. | (w—w’) f (w—w’') Sine. Cosine. | (¥—w’) | w/) Sine. Cosine. | (’—J’)
h. m. h. am. ™m. | h. m. m™. h. m.
0 0 oo 08 10. 000 5 60 0 9. 699 | 9.938 | 4 60 0 9. 938 9. 699 3 60
1 7. 941 000 59 | 705 935 59 1 940 692 59
2 8. 242 000 58 2 712 | 933 58 2 942 686 58
3 418 000 57 3 718 931 57 3 944 679 57
4 543 020 56 4 724 928 56 4 946 672 56
5 640 000 55 5 cei 926 55 5 948 G64 55
6 719 9, 299 54 6 736 924 54 6 950 657 54
ri 786 999 53 7 742 921 53 Ul 952 650 53
8 844 999 og 8 748 919 52 8 954 642 52
9 895 999 5 51 9 753 MG) 2 wl 9 955 (i3y al ey §
0 10 8.940 9. 998 50 10 |, 748 9.913 50 10 9. 957 9. 626 50
11 982 998 49 11 | 764 911 49 11 959 618 49
12 9.019 993 48 12 | TY) 908 48 12 961 609 48
13 054 997 47 13 | 114 | 205 47 13 962 651 47
14 086 997 46 14 | 779 | 902 46 14 o64 592 46
15 116 996 45 ie} |} 724 | e899 45 15 £66 583 45
16 144 996 44 16 | 789 | 897 44 16 867 574 44
17 | 170 995 43 17 794 | 804 43 17 969 564 43
18 194 995 42 18 | 799 | 890 42 18 970 554 42
19 218 9o4 5 41 19 | £04 | 887 4 41 19 972 544 | 3 41
0 20 9. 240 40 20 | 9. 208 | 9. 884 40 2) 9, 973 40
21 260 39 Qt | 813 | Bel 39 at 974 39
22 281 38 22 | 817 | 878 33 22 976 38
23 300 37 23 | 821 | 874 37 23 977 37
24 318 36 24 826 e71 36 24 978 36
29 335 35 25 830 868 35 25 979 35
26 352 34 26 834 | 864 34 26 981 34
21, 368 33 Pt | 833 861 33 QT 982 33
22 384 32 28 842 857 32 28 983 32
29 399 by BH 29 846 853 4 31 29 084 3 31
0 30 9. 413 9. 985 30 30 9. 849 9. 849 30 30 9. 985 9. 413 3)
31 427 984 29 31 853 246 29 3L 986 399 29
32 440 983 23 32 yt 242 23 32 QR7 384 28
33 453 982 QT 33 861 838 Q7 33 9x8 368 Q7
34 466 981 26 3 864 | 834 Q 34 989 352 26
35, 478 979 25 35 | 868 | 830 25 35 990 335 25
36 490 978 24 36 | a7 | 826 24 36 980 318 Q4
37 50L 977 23 37 874 821 23 37 991 300 23
33 513 976 22 38 | 878 | 817 22 38 992 Q81 22
39 523, 974 | 5 2Q1 39 | 881 813 4 21 39 993, 260 3 21
0 40 9. 534 9. 973 20 40 9. 884 9. 808 20 40 9. 993 9. 240 20
41 544 972 19 41 887 804 19 41 994 218 1y
42 554 970 18 42 | 890 | 799 18 42 995 194 18
43 564 969 7 43 | 804 | 794 17 43 995 170 17
44 574 967 16 44 | 897 739 16 44 996 144 16
45 583 966 15 45 899 734 15 45 996 116 15
46 592 964 14 46 $02 779 14 46 997 086 14
47 601 962 13 47 905 774 13 Aq 997 054 13
48 609 961 12 48 208 769 12 48 998 019 12
49 618 959 5 11 49 911 764 4 11 49 998 8.982) | 3) 11
0 50 9. 626 9.957, 10 50 9. 913 9. 759 10 50 9, 998 8. 940 10
51 634 955 9 51 916 753 9 51 999 895 9
52 642 954 8 52 919 748 8 52 999 844 8
53 650 952 a 53 921 742 7 53 999 786 ral
54 657 950 6 54 024 736 6 54 999 719 6
55 664 048 5 55 926 | 730 5 55 10. 000 640 5
56 672 946 4 56 §28 | 724 4 56 0°70 543 4
57 679 044 3 57 931 713 3 57 000 418 3
58 686 942 2 58 933 12 2 58 000 242 2
59 692 940 1 59) | 935 705 1 59 000 7.941 1
60 9. 699 9. 938 5 0 60 | 9.928 | 9.699 4 0 60 10. 000 SIO)

51

52 DISCUSSION OF TIDES IN BOSTON HARBOR.

Tapie VIII—Showing the effect of the solar diurnal tide upon the time of high water for every hour of the moon’s transit, and
for the first of each month, expressed in tenths of a minute.

| HOURS OF MOON’S TRANSIT IN ASTRONOMICAL TIME.
Mouth. | 7 ,
0 1 2 3 4 5 6 7 8 9 10 11 12) || 13) 14 15 | 16 17 18 19D 20M ee 22 | 23
OftiNecses +13 |414 +15 |4+15 |4+15 |414 |--13 |+11 |+ 6 0 |— 6 |—11 |—13 |—14 !|—15 |—15 |—15 |—14 |—13 |—11 |— 6 0 |+ 6 |+11
mebeee-- 10 20) 2) 12 12 12 10 8 4 0 4 8 10 12) 12 12 125 | 2 10 8 5 0 5 8
Mar +4]+5/+5)/4+ 5)/+ 5/4 5/+ 4/+ 3 |4 2 0|\—2\|—3'—4|/—5 5 5|—5 5) 4 3/— 2 0/4 2\+3
Apres —2}-3\-3|-3-3—3|-2)-2-1 (0) fe al ee) fae Oy IES} jak Sy [ise By EB} IE B} LE) eo} ee aT | @) al I
May Heol eeLON | ON sen ee) 9 " 6 3 0 3 6 7 Qi W} iQ}) tO] io 9 7 4 0 4 7
June. 12} 14 14 14 14 13 12) 10 5 0 oD |) 20 12 13 14 14 14 14 12 10 6 0 6 10
July -- 13 15} 15 15 15; 15 14 bE} 0 6 11 14 15 15 15 15 | 15 13 li 6 0 |} abt
Aug .-- 9 12 | 12 12 12 12 11 8 4 0 4 8| i1 12} 12) 12") 12 12 10 8 5 0 4 8
Sept ----|— 5 1— 5 5 i 5 4 ay 0) 0/+ 2/4 3/4 4\)+5)/4+ 5/4+ 5/4 5\|4+ 5)/4+ 4/4 3 |4 2 0|/—2\—3
Octi--=- £2|+ 3/4 3 3+ 3+ 3/4 2|4 2 ]+ 1 0/—1/—2 |— 2 |-— 3 — 3 |— 3 |— 3 — 3 |— 2 | 2 — 1 0 |+ 1/+ 2
INOW == 9) 10 10 10 10 10 8 6 3 0 3 6 8 10 10 10 10 10 9 7 4 0 4 7
Dece---4 +12 14 j+l4 14 |+14 [+13 |+12 |+10 |+ 5 0 |— 5 |—10 |—12 |—13 —14 |—14 |—14 ;13 —12 |—10 oa 6} 0Oj|-+ 6 )+10

Nore.—For lower transits the signs must be reversed.

TABLE IX—Showing the effect of the solar diurnal tide upon the time of low water for every hour of the moows transit, and for
the first of each month, expressed in tenths of a minute.

HOURS OF MOON’S TRANSIT IN ASTRONOMICAL TIME.
Month. |
O | 2 Oy) 8 | 2) Be le |) Sh ih OY) seo fp stay) ey fp as) af Ss | at | ae |) te} |] |) Bo |) eh | Be || a
shee a
Janeeee |-13 |—11 |— 6 | 0 J+ G J4-41 [413 |414 J415 |415 |415 [494] 413 |411 |+ 6] 0 |—G |—11 |—13 |-14 |—15 |-15 |—15 |—14
Uses) Ol Sy Ze |) i i) OI) ey) Ge ee a Ne) | A a) DI] a sea te |) a
Mar....\—3|—3/—2) 0|]+2|/4+3/43/4+4/45/4+5/+5/45]/+4|/4+ 3/42] o| 2) 3/4} 5/-5|-5|_5|-4
NyMilcete OEE OIE || OM) Wa pi gies pe aaa tl OTe OL Ok Sie Bike aide BLE @
Meyoccd) OF H] 8) OF Sl Bil ol ol sol lao} ol] el 7) Bi oO} Bl a) Ss] Oo} wm] tm] mm) 9
SAUNT SN SO I at ey SEE SES SEN Sey SE SGI Gy) Syl RENE ae} ae} aN) aie) Tee) ai)
| July -.--| iG} |) id Gi) O 6) WW) 13) 14] 15) 15] 15) 14) 13) 12 GQ) oO) Gl) si) wey) 2) a) aes) a) ae
jAngeess! O) El 2) Ol 4) BY ON) re) sey ol) Oy Bl 2il Oi) 2) Bi) oi) co] ey ce) eI) a0
| Sept.---|4: 31 3 [+ 2 | W) BB sh et 5 | 5/5) 5 4\|— 3 |—2 0|+ 2+ 3 Si al |e, Gy te Bye ay [tk |e,
Oct .-.-— 2|— 2} -1 | OfF1)+ 2/2) 3 i+ 3) 3/43 35 8) |RE B) [26 P} |/6 a Qe te BiB BiH Hh. 8 |e B
Worbere| Oy) eS Ol Bl 2h Ol Oil wl wl Ol ol ol ei] sl ws) al. ol) Ol a} aol) a! “s
Meese: 12 | — 6] 0+ 6 |441 14-12 |419 |414 [424 |4a4 [413 |492 }420|4 5] 0}—5 |-10 |-12 [13 |-14 |-14 [14 [42

Note.—For lower transits the signs must be reversed.

Taste X—Showing the effect of the solar diurnal tide upon the height of high water for every hour of the moon’s transit, and for
the first of each month, expressed in hundredths of a foot.

| HOURS OF MOON'S TRANSIT IN ASTRONOMICAL TIME.
| Month. | eT as ad 5 7 7
OM || 2863 Nese oem as | EY |] aah fe) Yh) ee || as |) aes fae |) at | a) || atD |] ed |) 2. |) a
| —6| 0]+ 7|+13 ris |422 [104 +04 |+o4 | +22 [418 [413/46] 0 |— 7 |-13 18 [92 [94 |_24 |_o4 |_92
| 4] 0] 5] 10) 13} 36) 18] 19] 18) 16] 13) 10) 4) 0} 5] 10] 13) 16/ 18) 19] 18) 46
I—2) O]+ 2)4 4]4 6]4 7/4 8]+ 8l4 7/4 7]+ 644/42) ol el 4—6/-7/-8\—8|_7/-7
| 1 rn ey Ly En) Ee Offi leo aie aie ole 5 i+ 4 l+ 4
May .--.| 4| 0 a GQ) |) 2) es |) | Te) ae) 2 OI A) BN Te ae es | aes) |) a)
June....| 16] 11] 5); 0] 6] 12] 16] 20] 22] 293] 22] a1) 16/ 1) 5] 0] 6| 12] 16] 20] 22] 23] 22]
dilysse-e| 18) |) 13) 96") 0 7| 13| 18| 22| 24] 24| 24] 22| 18| 13| 6/ o| 7| 13| 18| 22] 24| 24] 23| 2
Aug....| 13] 10/ 4]: 0] 5| 10| 13| 16| 18| 19] 18] 16] 13] 10/ 4] ©] 5| 10] 13] 16] 18] 19] 18] 46
Sept [pr © ie teas eee Os a ah 8 ae Os 2+ 4 le 6+ Ti+ BI el T+ 7
Oe sce S821] OF Ape seal cle slp aie ale si], 211) oH tee 445i 5/4 4
Nov HW) 8a) os a) on) a | a) |) 5) 13 | mi) a) 2 Of 2) oi) me tay ws) a) 1a 1
Dec j-16 [11 |— 5 | 0 f+ 6 +92 416 | 420 [+92 | p23 | 422 | 421 |416 [411 45] 0 —6 L492 |46 20 |-92 |-23 |—22 |—a1

DISCUSSION OF TIDES IN BOSTON IARBOR. 53

TABLE XI—Showing the effect of the solar diurnal tide upon the height of low water for every hour of the moon's transit, and for
the first of each month, expressed in hundredths of a foot.

|
HOURS OF MOON’S TRANSIT IN ASTRONOMICAL TIME.
Month. - 7 -
0 1 2 3 4 5 6 7 8 9 10) | 11 |/-12) | 135) 14) 15 | 16) 27 | 18 | 19) || 20 |) 21 ) 22) |) 23
| | en ee ee eee
dam..--. +11 |4+13 |+14 |4+14 |+13 |411 |+ 9 |+ 6 |+ 3 0 |— 4 |— 8 |--11 |—13 |—14 |—14 13 |\—11 |— 9 6 |— 3 O;+ 4/48
HebEess= |} ) )) wie || hh) tO) 8 6 4 2 0 2 6 By 1) ghey ghey al) 8 6 4 2 0 2 6
Mar. .--|+ 3 |+ 4 |4 5/+ 4|+ 3/4 3/4 2/41 0 1 3 /— 3 4 5/—5\—4/—3|— 3 —2/-1 0|+1)\+ 3
April ...|— 2 |— 2 — 2 |— 2 |—- 2 — 2 2/—1 0 0 J+ 0 [+ 1 |+ 2 J+ 2 ]+ 2+ 2 }4+ 2/4 2)/4-2)4 1/41 0—1\—1
May --- 6 8 9 9 8 6 5 3 il 0 2 5 8 9 9) 8 6 5 3 2 0 2 5
dune.---| 10} 12) 13} 13) 12) 10 8 5 3 0 3 7} 10) 12} 13) 13] 12) 10 8 5 3 0 3 7
July --- yl |) aly) ME) aE Sig} |) all 9 6 3 0 4 8 | 12) 135) Way Way) 1s) a 9 6 3 0 4 8
Aug --- 8] 10; 11} 11) 10 6 4 2 0 2 6 jf a) Gh) bh |) ala) 8 6 4 2 0 2 6
Sept esq 8 = S =o = 8 | —3|—3|—2 1 Oj 1/4 3/4 3/4 414+ 5/4 5)/4 4/4 3)4+ 3/42 )/41 Yt =8
Oct .---. + 2/4 2\+ 2)/+ 2/4 2/4 2)+ 2/4 1/40 0 0 1j\—2 2\—2 2 2\—2 2\—-1\—-1 O jar ab ae a
Nov 6 8 | 9 9) 6 5 3 1 0 2 5 6 8 9 9 8 6 5 3 2 0 2) 5
WEG sss05 {710 +12 pris +13 |+12 |+10 |+ 8/+ 5 |4 3 0—3\|-7 f 10 j12 13 |—13 j-le } 10 |— 8 — 5 | 3 0 | 3 | 7

Nore.—For lower transits the signs must be reversed.

54 ; DISCUSSION OF TIDES IN BOSTON HARBOR.

Example of the computation of a tidal ephemeris for the first part of January, 1871.

A a B | Cc c D d | ity e af g i Mf 2 j Bo
d. h.m. ™m. h. m. hu ut ° | Ma ™ ™ ™ mM. ™. m
Dec. 30 6 55.0 +2. 7 6 52.3 54 47 | —170 +3) 411.6 44.4 13.0 14.3 0.1 2.8 9.0
3t 7 36.5 3.2 4 33.3 54 22 0.7 8 10..9 53. 6 12.1 14.6 0.6 2.2 9.0
dan. 1 818.8 3.7 8 15.1 54. 9|) —0.3 12 959) o8. 3. 10.9 14.8 12 1.7 9.0
2 9) 207 4.2 8 58.5 54.5 0.0 16 8.5 58. 4 10.0 14.7 2.0 1.3 9.0
3 9 42.9 4.7 9 44.2 54 10 +0.3 F 19 6.6 54.3 9.1 14.3 2.8 0.8 9.0
410 37.2 5.1 10 32.1 54 23 0.6 22 4.4 47.1 8.3 13.5 3.5 0.6 9.0
3 1! 27.5 5.6 11 21.9 54 42 0.8 23 +1.7 37.4 7.6 12:7 4.0 0.5 Ghil
6 1219.2 6.0 12/13: 9 55° 6 1.0 23) —1.2 Qin 7.0 11.8 4.0 0.5 9.1
E Al a2 ol k 1 m \th 7 Q | n dn | Ao Al Hy)+A. p qd r \|hhew
d. h. m. ™m mM. m. m. Mm. | m.| ml Ft. | Ft Ft. Ft. Ft. Ft. Ft. It. Ft.
Jan. 1 7 30.3 | 52.9 56) Bish) .8 }+1.3 |+05 | 31 § 3.35 ie 55 |—. 52 | 3.87 |+.03 23.96 |4+.18 |+.22 |+.40 | 24.32
19 57.2 | 26.4 26.0 +0.1 57 | 3, 87 ol 23.95. .45 | 23.50
2 8 23.2) 48.7 |—4.2 | 24.87 1.2] 0.9 |—0.3} 23] 3.40) 1.50 50 | 3.90 14 23. 99 26) .24 -00 | 24.49
20 48.0 | 24.3 0.5 | 23.9 0.8 49 | 3.96 07 24.05 .o4 | 23.51
3) 9) 11-19))| 45:12) | 355) 1 2350 1.6 |+0.4 1159 11 f 3. 61| 1.29 43 | 4.04 19 24.13 34) .24 58 | 24.71
21 34.9 | 22.6 0.4 | 22,2 1.6| 36 | 4,12 +09 24.21 - 62 | 23.59
4 9.57.1 | 42.1 3.1 | 21.4 1.9] 0.0 1.9] 55/3 89) 1.01 34 | 4.23 | .20 24. 32 41 24 65 | 24.97
2218.5] 21.0 0.4 | 20.7 | 2.3 | Q1 | 4, 33 10 24. 42 . 68 | 23.74
5 10039.2 | 39.9] 2.2 | 20.3) 2.2 |—0.5 | 2.7 | 3714.19) 0.71 24 | 4.43 19 24. 52 47 | .24 71 ) 25. 23
22 59.5 | 20.0 0.3 | 19.6 3.0 | 62 | 4.52 09 24, 61 TI) OBE GH |)
(Fabb abbal eehye |) abe) akeoi)) Bssyil (hh) 3.2] 16 4 4. 48) 0.42 14 | 4.62 17 24, 71 49 - 23 12) | 20143
| 23 38.6) 19.3)° 0.2 | 19.2 | 3.4 42 | 4.70 - 08 24. 79 71 | 24.08
| 711 57.8 | 38.0 |—0.7 | 19.1 2.3 1.2] 3.5 | 544 4.74 —0.16 |—.05 | 4.79 |+.12 24.288) .49 20 69 | 25.57
| 8 016.9] 190 0.1) 189 3.9} 20 | 4. 86 + 06 24.95 66 | 24.29
12) 35. 8 19.0} 2.2 |-1.3) 3.5 32 f 4.92)+0.02 |+.01 | 4.91 25.00 |+. 47 |+.16 |+.63 | 25.63
| | | | S cos 2B
rE | Ke uv m' |\t.wh A’, | Hy—A‘| pl q | » | blew} M s hee: Lice:
j | a Ne sin 2 B | cos 2 B.
Log. No
| d.h. m. | m m m m Ft. Ft. It Ht. Ft. Ft Tt, Ft.
| Jan. 1 13 43.7 |+0.8 |+1.4 |42.2 46 3. 86 | 1.23 |4.11 |4-.08 |+.19 16. 42 | 4.15 | 0.92 9. 644, 9, 9537 9. 9177} —O. 83
2 210.2 | 2.5 8 3. 88 16.21 «2h 16. 01
14535565) 152 EGY ||| Pal 38 3.93 16.16 16 06} .22 16. 38 | 3.97% | 0.92 | 9. 862 | 9. 837 9. 801 0.63 |
| 3 2.0.0} | 2.9| 57} 4.00 16.09 | | 23 | 15.36 | |
| 15 23.4 1.6 Uabyi| ebal 26 4.08 16. 01 21 }4.03 - 24 16.25 | 3.86 | 0.92 9. 966 9. 582 9.548 0.35
| 4 3 46.0 3.2 43 4.17 15. 92 «20 15. 67 r
LG 785) 1-19) Mya} || She 11 4, 28 15. 81 25] .00) |) .25)) 16:06 )— 3:79 | 0.92 10. C00 8. 0907 8. 054n} —0. OL
5 428.8 | 3.6 25 4.38 15. 71 | ~25 | 15.46
16 49.3 | 252 )- 155) 3.7 53 4.48 15.61} .28 |—.03 a20. 15. 86 | 3.76 | 0.92 9. 967 | 9. S76p 9.540p) +0. 35
| G 8 O88} 3.7 6 4.57 15. 52 | .24 15. 28
17 28.8 2.3) 1.4] 3.7 32 4. 66 15. 43 30 06 24 | 15.67 § 3.78 | 0.92 | 9. 842 9. 857 9. 821 0. 66
eo 4852 3.7 44 4,75 15. 34 23 15. 11 |
18 7.3) 2.3 Je3)|) 36 11 4.23 | 15. 26 30 . 09 21 15. 47 | 3.86 | 0.93 | 9. 5017; OTT, 9. 941 0. 87
8 6 26.4 | 3.5 23 4.289 15. 20 sil) 15. OL |
| 18 45.3 |-+2.2 +11 43.3 49 4.93 | 15.16 |+.28 \—11 4.17 15.33 } 4.00 | 0.93 | 9. 083p) 9. 99%p 9.965p) +0. 92
lees ee a )
| M-+S cos2B | 2MScos2B |M?24+S?4+2MScos2B
ie'—Continued. ~——]; |, ae B ab Log. M Log. M2] M2 |M?+S2 = Log. Q
No. Log. | Log. No. No. Log.
d.h. m Ft. |
| Jan. 1 13 43.7 3. 32 0. 521 9. E08p 9. O87p} 0. 618 | 1. 236 17, 22 18. 07 0. 8367) —6. 85 11. 22 1. 050 0. 525
Pda (5) | |
14 35.6 3.34 0. 524 9. 826 9.302 } 0.589 | 1.198 15. 78 16. 63 - 701 5. 02 11. 61 1.065 | 0.532
3 2 0.0 |
15 23.4 Srol 0.545 9. 930 9.385 f 0.587 | 1.174 14.93 15. 78 - 436 2.713 13. 05 1.116 | 0.558
4 3 46.0 |
| 16 7.8 3.78 0.578 9"965 9.388 § 0.579 1.158 14.39 15. 24 8.934n, —0. 09 15.15 ~ 1.180 | 0.590
> 428.8 ¥ |
16 49.3 4.11 0. 614 9.933 9.319 § 0.575 | 1.150 14.13 14,99 9. 416p) +2. 61 17. 60 1.245 | 0.622
@ Gy) 8h} |
| 17 28.8 4.44 0. 647 9. 809 9.162 # 0.577 | 1.154 14. 26 15.12 0. 699 5.00 20:12 1.303 | 0.651
| 7 5 48.2 |
18 7.3 4.73 | 0. 675 9. 462} 8.7924 0. 587 | 1.174 14.93 iby YA) 0.829 | 6.74 | 22.53 1.392 0. 676
® 6 26.4 | | |
| 18 45.3 4°92) | 0. 692 9. 051n 1. 204 16.00) 16.86 | 0.86% 4-7. 38 24, 24 1. 385 0. 692
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