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CHAPTER III
TIDAL COMPUTATIONS 1~NTD PREDICTIONS
PAUL SCH L-REMAN
U. S. Coa.st and Geodetic Survey
TREAT)IEN T OF TIDE :ElECORDS
Since the advent of the automatic tide ¢,au~e, the records of tide obser-
vations usually consist of continuous curves graphically representing, the
rise and fall of the tide, or of height reaclin~,s recorded mechanically at
regular intervals such as at eatery quarter of an hour or at any other
interval that may be clesired. These records are usually tabulated in two
forms. In one form the times and heights of the high and low waters
are given, and in the other form the heights at each hour are tabulated.
In both forms the heights are referred to some uniform datum, which
may be arbitrarily selected or which may be based on tidal planes pre-
viously determinecl.
There are two general classes of methods of treatment of the tidal data
which cli-ffer raclicall.v from each other, one including the non-harmonic
methods and the other the harmonic analysis. Son-harmonic methods
have been in use since tides were first studiecl, but the application of the
harmonic analysis to the reducton of tides began about 1867, when a
committee under the auspices of the British Association for the Aclvance-
ment of Science unclertool: a special study of the matter.
Vor~-1'arrno~ic reduc-t~on.s. The high and low waters form the prin-
cipal basis for the non-l~armonic treatment of the tides, although the
hourly heights are also used when the purpose is a study of sea-level
charges. The results sought i~clucle the mean high and low water heights,
the range of tide and the lunitidal intervals, together with the variations
due to changes in the phase, parallax and declination of the moon and
sun. 1( luniticlal interval is the time that elapses between the transit of
the moon and the following, high or low water. In some countries the high
water luniticlal intervals at the times of new and lull moon are selected
lol: use in maritime publications and are denominated " high water, full
and change," or " establishment of the port." The mean high and low
water intervals for the entire month, which in general are about 10 to 15
minutes less than the corresponding intervals at lull and change, are
used in publications oft the IJ. S. Coast and Geodetic Survey-. In estab-
lishin~, tide planes and studs in;, long-period changes in the earth's crust,
40

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TIDAL COMPUTATIONS AND PREDICTIONS 41
the variations in the sea-level and the relation of the same to fixed points
known as bench marls are clesi.red.
In regard to clet.ails, the methods of the non-harmonic reductions may
differ in dil-ferent countries. In the United States Coast and Geodetic
Survey, the calendar month is taller as the unit of' reduction for obtaining,
mean heights, ran,~,es and luniticlal intervals at. the principal or primary
tide stations; that is to say, separate averages are obtained for each
calendar month and yearly averages are then obtained from the monthly
averat,es. Unless the series covers 19 years or more, corrections depending,
upon the lon~,it.ude of the moon's node are applied for the inclination of'
the moon's orbit to the equator in order to reduce the results. to mean
values. In reducing, short series of observations at subordinate stations
comparisons are made with simultaneous observations at one of the
primary tide stations.
To obtain the spring and Reap titles, which depend upon the phase
ci the moon, the perit,ean and apogean. titles, which depend upon the
parallax of' the moon, and the tropic titles, which depend upon the declina-
t.ion of the moon, the high and low waters are selected and grouped in
accordance with the astronomical conditions upon which these tides
depend and averages are then obtained. The non-harmonic reductions
as carried on in the U. S. Coast and Geodetic Survey are described in
some detail in Part III of' Harris' NIanual of Tides, which was published
as an appendix to the Annual Report of the IT. S. Coast and Geodetic
Survey in 1894, in Special Publication No. 13a of the same Bureau, and
in instructions publi.shecl from time to time for the use of field officers
o:t' the U. S. Coast and Geocletic Survey. Instructions f'or Analysis, Tidal
Observations, published in 1928 by order of' the Lords Commissioners of
the Admiraltv, London, En~lancl, include instructions for computing
non-harmonic constants. Lists of' the principal non-harmonic constants
for different parts of' the world as derived from reductions of' high and
low waters are included in the tide tables published by the principal
maritime nations.
I-Ic~rmon~e c~r~alys~.s. The harmonic method of' reducing tidal observa-
tions is entirely different from the non-harmonic methods. In the
harmonic analysis no special attention is given to the incliviclu.al high
and low waters but the entire cor~tinuous record is consiclered. For
practical convenience, however, this continuous record is represented in
the calculations by the tabulated hourly heights. The harmonic analysis
is based upon an assumption that the continuous motion of the water in
the alternate ri.s;.n¢, and falling of' the title may he represented by the
combination of' a number of' single harmonic motions, each of' which
may be attributed to some periodic cause which is mainly astronomical.
.,
.J

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~2
FI G URE OF THE EAR TH
The purpose of the analysis its to separate the observed tide into the
simple elements which are called components or constituents, each element
being represented by an amplitude and an epoch, and when these elements
have once been obtained they may be used in classifying the tides and
also for predicting, for any future time.
The application of the harmonic analysis to the reduction of' tides was
placed upon a practical basis about the ::ear 1867 through the work of
Sir William Thomson. This scientist was a member of a committee
appointed on his suggestion by the British Association for the Advance-
ment of Science for the purpose of promoting the extension, improvement
n.ncl harmonic analysis of tidal observations. The progress of the world
v
1 ~7
of the committee was reported in 1868 and subsequent rears in the
Reports of the British Association for the Advancement of Science. Owe
years later a committee consisting of Professor G. H. Darwin and J. C.
Adapts drew up a full report on the subject which was published in
1883 in the Report of the British Association for the Aclvancen~ent of
Science, and since that time reductions by harmonic analysis have been
based largely upon Darwin's work.
In minor details, the uncork as carried on by different authorities mail
vary, but in the main it is the same. The periods of the components or
constituents sought are fixed in advance by astronomical data. The period
of the principal lunar component ^~ is the semilunar day, which is about
12.42 solar hours, and the period of the principal solar component So
is the half of the solar day, or 12 hours. The number of components
which theoretically compose the tide is unlimited, but the number o:t
. ,.
components which are o:t' practical importance in tidal work depends
upon the precision requirecl. The components are grouped into several
classes according to their period.
Components with period approximating the half day are called semi-
clillrn~l onmnnno.nt.~ n.nd in most harts of the world this group includes
the largest and most important components. Second in importance to
the semidiurnal components are the diurnal components with periods
approximating the whole day. These components give rise to the in-
equalities in the times and heights of the two high and two low waters
of each day and in some localities may predominate to such an extent
that only a single high and low tide will occur each day. Lon~-periocl
components with periods extending, over a number of days or months
affect the height of the water from day to day. Components with periods
less than one half day are due primarily to the configuration of a harbor
or river and are I'reque~-~tly callecl shallow-water components.
In the first step of harmonic analysis o:t' the tides, the hourly heights
are grouped in accordance with the period of each component sought.

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TIDAL COMPUTATIONS AND PREDICTIONS 43
The heights are then summed a.ncl 24 means obtained, which represent
approximately the component sought. A treatment based on the method
of least squares is ther1 applied and for each component there are obtained
an amplitude representi.n~, the magni.tucl.e, and an epoch representing,
the time of the maxima of the component. The epochs are referred to
certain astronomical events, and corrections are applied to both ampli-
tudes and epochs in order to eliminate the effects of other components
and to reduce to mean values.
The following are the principal publications giving in detail the
methods of harmonic reduction as carried on by different authorities:
Report of a Committee for the Harmonic Analysis of Tidal Observa-
tions, prepared by G. H. Darwin and originally published in 1883 in
Report of the British Association for the Advancement of Science, and
afterwarcls. in a Collection of Scientific Papers, by G. H. Darwin.
Borgen's method of harmonic analysis as given by E. Hessen in
Annalen cler Elydrographie uncl Maritimen Meteorolo~,ie, in 1920.
A Manual of Harmonic Analysis and Prediction of Tides, by Paul
Schureman, Special Publication No. )8 of the U. S. Coast and Geodetic
Survey, published in 1924.
The Analysis of Tidal Observation, bit A. T. Doocl.son of the Ticlal
Institute, University of Liverpool, published in Philosophical T.rans-
actions of the Royal Society of' London, Series A, vol. 227.
PREDICTION OF TIDES
Tidal predictions are oracle ancl published annually by the principal
maritime nations of the worlcl. The methods used for these predictions
may be classified as harmonic and non-harmonic. In some cases a combi-
r~ation of' both methods is used. The non-harmonic methods were used
first, but since the development of the harmonic analysis the harmonic
method is being more extensively aclopted.
iNor~-ha~mo?~ic predictions. The simplest :form of the non-harmonic
predictions consists in the application of the mean lunit.idal intervals
to the times of' the moon's transits. The results obtained by this simple
method may be considered only as rough approximations to the true times
of the title, which are subject to various. inequalities clependin~ upon
changes in the relative positions of' the earth, moon anal sun. In order
to take account of these inequalities a more elaborate system is necessary.
One of the earliest methods~used was that introduced by Sir John
Lubbock in about the year 1830, and described in the Philosophical Trans-
actions for 1836. In this method, tables show-in, the lunitidal intervals
and heights of high and low hater for cliff:'erent times of the moon's
transit, were calculated from tidal observations at each port for which

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44
FIGURE OF THE EAR TH
predictions were to be made, these intervals and heights being corrected
for actual declination and parallax by tables calculated in accordance
with Bernoulli's theory.
~ later system of non-harmonic predictions, called the Equation
Method, is described in considerable detail ire the British Admiralty
Tide Tables. In this method the following inequalities affecting the
mean lunar se~nidiurnal wave are considered: 1) The phase inequality
with a period of one half synodic month; 2) the lunar tropical diurnal
and semidiurnal inequalities depending, upon changes in the moon's
declination and having, periods of one tropical month, and one-half
tropical month, respectively; 3) the lunar anomalistic semidiurnal ille-
qua,lity depending upon changes ill the moods parallax and having a
period of one anomalistic month; 4) the solar tropic diurnal and semi-
diurnal inequalities depending upon changes in the sun's cleclination
and halving, periods of one solar year and one-hall' solar year, respective!:;
b) the solar anomalistic semidiurrral inequality depending, upon chan~,es
in the sun's parallax and having a period of one anomalistic year; 6)
the annual variation in height due primarily to seasonal meteorolo~,;.cal
changes.
Summary of observations or mean results extending over a, year or
more are plotted on section paper, the diagram showing the time of the
moon's transit, lunitid.al intervals and heights, and moon's declination
and parallax. NIean curves of' intervals and heights following the upper
and lower transits of the moon are then drawn. Tables are prepa,recl
in which the observations are grouped according to the hour of the
moon's transits. Variations of individual values from the mean of' each
group are obtained and from these are calculated separate tables of
corrections for moons declination, for moon's parallax and for time of
year. Predictions are made by combining the several corrections from
these tables to the mear1 times and heights corresponding to the appro-
priat,e hour of the moon's transit.
IJarrr~on~c prediction of tides. The harmonic method of' predicting
tides consists in re-combining the tidal components as represented by
the harmonic constants which have been obtained by an analysis of the
tidal observations at the station for which the predictions are desired.
lTor these predictions the following formula is used:
h=I-Io+>fE cos Scot+ (V0+u')-K]
in which
h=~-~ei~,ht of the tide at arty time (t).
90= n~ean height of water level above datum adopted for pre-
dictions.

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TIDAL COMPUTATIONS AND PREDICTIONS 45
0= mean amplitude of any component A.
f = factor for reducing mean amplitude E to year of prediction.
a= speed of component A.
i= time reckoned from some initial epoch, such as the begin-
ning of year of predictions.
(VO+~) = value of equilibrium argument of component A when t=o.
K=epoch of component A.
In the above formula the height of the tide (h) is expressed as a
function of time (t), the other quantities being considered as constant
for any particular year and place. Each term in the summation repre-
sents one of the tidal components and the number of such terms to be
included in the calculation will depend upon circumstances. In actual
practice the number will usually vary from ten to twenty, but a larger
number is often used. By means of this formula successive values of (h)
for different values of (~) may be computed. These values when plotted
will give a curve representing the rise and fall of the tide, and from the
curve it would be possible to scale off the times and heights of the high
and low waters.
Since the computation of each single value of (h) requires the sum-
mation of a number of terms consisting, of products involving the cosines
of angles, it is apparent that a direct computation of the thousands of
values of (h) which would be necessary in predicting the tides for a
single year at one station would be exceedingly tedious and time-consum-
in~, and render this method of predictions impracticable. Through the
use of a tide-predicting machine, originally designed by Sir William
Thomson, this summation is made mechanically and the resulting values
of (h) are graphically represented by a curve drawn by the machine.
The first tide-predicting machine was built about 1873 and since that
time a number of other predicting machines have been constructed.
The predicting machine used by the U. S. Coast and Geodetic Survey
was designed and constructed in that office and completed in 1910. This
machine, in addition to evaluating the formula given above, also evaluates
the first derivative of the equation and thereby indicates the exact times of
the maxima and minima or high and low waters. This machine not only
traces a tide curve similar to the curves traced by other machines, but it
also indicates the times and heights on dials from which the curves may
be tabulated directly. A predicting machine more recently constructed
in Germany also accomplishes these same purposes.
The same principle is used for direct reading in other tide-predicting
machines; for instance, at the Tidal Institute of the University of Liver-
pool, the machine is first set up and the times of high and low waters are
tabulated direct from the machine. The machine is then re-set and the

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46
FI G ~ RE OF THE EAR TH
heights of the high and low waters are tabulated direct, no curse being,
required except under very special circumstances.
Through the use of tide-preclicting machines the harmonic method
becomes the most convenient and accurate system of preclictin~ tides
letdown Although the machines now in use provide for the summations
of all the principal tidal components, it is probable that there are other
components which knight add to the accuracy of the preclictions, and it
is quite reasonable to believe that any required degree of precision can be
obtained in this method by including a sufficient number of components
in the work. Tide-predictin<, machines are now in use in the following
countries: Argentina, Brazil, France, Germany, Great Britain, India,
Japan, Portugal, and the IJnited States. Descriptions of the various
ticle-preclictin~, machines are given in Special Publication No. 13 of the
International I-I: dro~raphic Bureau.
Tildes In s1rc`1low water. Mullen a tide -me created by- astronomical
conditions runs into shallow water, tl~e trou~,l: of the wave is retarded
snore than its crest and there results considerable departure from a
simple harmo-nic-iorn~. In order to represent this modified form of the
tide wave, it is necessary to introduce a series of components of short
periocls. Such components with speeds which are exact multiples of the
fundamental astronomical tides are called overtides. Components with
speeds equal to the sum of di-~lere~ce of tl~e speeds of the -funclamenta1
tides are called compow:lcl tides.
Nlthou~,h a number of the shallow-water con poets al e regularly
sought in the harmonic analysis and used in the harmonic preclictions
of the tides, there are undoubtedly mall:: other short period co~pollellts
which on later d~velop~er~ts will greatly- improve the preclictio:~s for
shallow bodies of water.
I?E1~ERENCES
18&~. Thomson, Sir William. Report of Committee for the Pt rpose of Promoting
the Extension, Improvement, and Harmonic Analysis of Tidal Observa
tions. Brit. Assoc. Advancement Sci., Rept., p. 489-505. Supplementary
report by E. Roberts, Brit. Assoc. Advancement Sci., Rept., p. 505-510.
18~0. - - . Ibid. Brit. Assoc. Advancement Sci., Rept., p. 120-151.
1871. Roberts, E. Ibid. Brit. Assoc. Advancement Sci., Rept., p. 201-207.
1872. . Ibid. Brit. Assoc. Ads ancement Sci., Rcpt., p. 355-395.
1874. Ferrel, NY. Tidal researches. U. S. Coast Survey, Separate Pubp. 1-~68.
1876. Thomson, Sir William. Report of Committee for the Purpose of Promoting
the Extension, Improvement, and Harmortic Analysis of Tidal Observa
tions. Brit. Assoc. Advancement Sci., Rept., p. 275-307.
18~8. Ferrel, Hi. Discussion of tides in Pcnobscot Bay, Maine; general principles
of the harmonic analysis and discussion of tide observations. IT. S. Coast
and Geodetic Survey Rept., Appendix 11, p 268-304.

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TIDAL COMPUTATIONS AND PREDICTIONS 47
Evans, Captain, and Thomson, Sir William. On the tides of the Southern
Hemisphere and of the Mediterranean. Brit. Assoc. Advancement Sci.,
Rept., p. 477-481.
Thomson, Sir William. Harmonic analyser. Proc. Roy. Soc., 27: 371-3~3.
1881. . The tide gauge, tidal harmonic analyser and tide predictor. Proc.
Inst. Civil Engrs., 65: 2-25.
1882. Darwin, G. H. On the method of harmonic analysis used in deducting the
numerical values of the tides of long period. Brit. Assoc. Advancement
Sci., Rept., p. 319-327.
:Ferrel, W. Discussion of the tides of the Pacific Coast of the United States.
U. S. Coast and Geodetic Survey Rept., Appendix 17, p. 437~50.
1883. Darwin, G. H., and Adams, J. C. Report of a Committee for the Harmonic
Analysis of Tidal Observations. Brit. Assoc. Advancement Sci., Rept.,
p. 49~-118. (Reprinted in 190~7 in Scientific Papers, by G. H. Darwin,
1: 1-69.)
Ferrel, W. Report on the harmonic analysis of tides at Sandy Hook, New
Jersey. U. S. Coast and Geodetic Survey Rept., Appendix 9, p. 247-251.
Description of a maxima and minima tide-predicting machine.
U. S. Coast and Geodetic Survey Rept., Appendix 10, p. 253-272.
1884. Borgen, C. Die harmonische Analyse der Gezeitenbcobachtungen. Ann.
Hydrographie. p. 3015-312, 387-399, 438-449, 499-5101, 558-566, 615-622,
664-6176.
Darwin, G. H. Second report of a Committee for the Harmonic Analysis of
Tidal Observations. Brit. Assoc. Advancement Sci., Rept., p. 33-35.
18&5. . Third report of a Committee for the Harmonic Analysis of Tidal
Observations. Brit. Assoc. Advancement Sci., Rept., p. 35-60. (Reprinted
in 1907 in Scientific Papers by Sir G. H. Darwin, 1: 70~96.)
Baird, A. W., and Darwin, G. H. Results of harmonic analysis of tidal ob
servations. Proc. Roy. Soc., 39: 135-207.
Ferrel, W. On the harmonic analysis of the tides at Governors Island, New
Yorl: Harbor. U. S. Coast and Geodetic Survey Rept., Appendix 13,
p. 489-493.
1886. Darwin, G. H. Fourth report of Committee for the Harmonic Analysis of
Tidal Observations. Brit. Assoc. Advancement Sci., Rept., p. 41-56. (Re-
printed in 1907 in Scientific Papers, by G. H. Darwin, 1: 98-116.)
- . Tides, admiralty scientific manual, p. 53-91. (Reprinted in 1907 in
Scientific Papers, by G. H. Darwin, 1: 119-156.)
. On the dynamical theory of the tides of long period. Proc. Roy.
Soc., 41: 337-342. (Reprinted in 19~07 in Scientific Papers, by G.. H.
Darwin, 1: 366-371.)
1889. . Second series of the results of the harmonic analysis of tidal ob
servations. Proc. Roy. Soc., 45: 556-01 1 .
18DO. . On the harmonic analysis of tidal observations of high and low
water, Proc. Roy. Soc., 48: 278-340. (Reprinted in 1907 in Scientific
Papers, by G. H. Darwin, 1: 157-215.)
1892. . On an apparatus for facilitating the reduction of tidal observations,
Proc. Roy. Soc., 52: 345-389. (Reprinted in 1907 in Scientific Papers, by
Sir G. H. Darwin, 1: 216-257.)
1893. Stencils for harmonic analysis devised by L. P. Shidy. Rept. of IJ. S. Coast
and Geodetic Survey, p. 108.
4

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48
FI GURE OF THE EAR TH
1894. Borgen, C. Ober eine neue Methode, die harmonischen Konstanten der
Gezeiten abzuleiten. Ann. Hydrographie, p. 219-232, 256~270, 295-310~.
1897. Harris, R. A. Tidal observation, equilibrium theory, and harmonic analysis.
Manual of Tides, Part II. U. S. Coast and Geodetic Survey Rept. Ap-
pendix 9, p. 471-618.
Van der Stok, J. P. Wind and weather' currents, tides and tidal streams
in the East Indian Archipelago.
190~1. Eccles, J. Details of tidal observations in India. 1873-1892. Great trigono-
metrical survey of India, vol. 16. Additional results from the harmonic
analysis for tide stations in India have been published in subsequent
reports of the Survey of India.
. Constantes harmoniques d'un certain nombre de ports calculees par
le Service des Marees du Service Hydrographique. Ann. Hydrographiques
Tome 23.
1905. de ['Isle, R. Observation etude et prediction des marees, Service Hydro-
graphique de la Marine. Publication No. 870~.
1906. Van Beresteyn, M. H. Getijconstanten voor Plaatsen fangs de Kusten en
Benedenrivieren in Nederland, Berel~end nit de ~aterstanden.
1910. Van der Stok, J. P. Elementaire theorie der getijden Getijconstanten in den
Indischen Archipel. Mededeelingen en Verhandelingen, Koninl~lij k Neder-
landsch Meteorologisch Institute, No. 102.
1911. Hirayama, S. Results of the harmonic analysis of tidal observations made at
various ports of Japan. J. College Sci. Imp. Univ. Tokyo.
1919. v. Sterneck, Dr. Robert Daublebsky. Die Ge%eitenerscheinungen in der
Adria. Die theoretische Erl~larung der Beobachtungstatsachen. Denl~--
schr. Akad. NViss. Wien, Math.-naturw. Klasse, 96 Band.
1920. Doodson, A. T. Report on harmonic prediction of tides. Brit. Assoc. Ad-
vancement Sci., Rept. Report on state of science a. 321-323.
.
Hessen, K. Die Borgensche Methode der harmonischen Analyse der
Meeresgezeiten. Ann. Eydrographie, p. 1-1S, 73-93, 123-136, 177-186.
-- . Uber eine neue Methode die harmonischen Konstanten der lang-
periodischen Tiden der Meeresgezeiten ab%uleiten. Ann. Hydrographie,
48: 441-455.
Proudman, J. Report on harmonic analysis of tidal observations in the
British Empire Brit. Assoc. Advancement Sci., Rept. Report on state
of science, p. 323-345.
1921. Doodson, A. T. Intensive analysis of tidal observations. Brit. Assoc. Ad-
vancement Sci., Rept., p. 217-243.
. The harmonic development of the tide-generating potential. Proc.
Roy. Soc., A, 100: 305-328.
1922. Sternecl<, R. Harmonische Analyse und Theorie der Mittelmeergezeiten.
Sitzb. Aloud. Wiss. Wien, Abt. IIa, 131. Band, ID, Heft, 1922.
1923. Doodson, A. T., and Proudman, J. Notes on tidal researches. Brit. Assoc.
Advancement Sci., Report of Committee on Tides, p. 299-304.
Sternecl<, R. Zur Praxis der harmonischen Analyse der Gezeitenbeobacht-
ungen. Ann. Hydrographie, p. 39-45.
1924. Doodson, A. T. Perturbations of harmonic tidal constants. Proc. Roy. Soc.,
A, 106: 513-526.
Rauschelbach, H. Harmonische Analyse der Gezeiten des Meeres. Pub-
lished by the Deutsche Secwarte.

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TIDAL COMPUTATIONS AND PREDICTIONS 49
Schureman, Paul. A manual of the harmonic analysis and prediction of
tides. Special Publication No. 98, U. S. Coast and Geodetic Survey, 1924.
Includes an extended lisl; of harmonic constants for the world.
1925. Rauschelbach, H. Mittlung von harmonischen Konstanten und Berechoung
ihrer mittleren Fehler. Ann. Hydrographie, p. 86-94.
1926. I>ha~, Rear Admiral. Investigation of harmonic constants, prediction of
tides and currents and their description by means of constants, supple-
mented by an extended list of harmonic constants for the world. Inter-
national Hydrographic Bureau, Monaco, Special Publication No. 12.
- . Tide predicting machines. International Hydrographic Bureau,
Special Publication No. 13.
1928. Instructions for analysing tidal observations. Published by order of the
Lords Commissioners of the Admiralty, London, England, for the Hydro-
graphic Department, Admiralty.
Doodson, A. T. The analysis of tidal observations, Phil. Trans. Roy. Soc.,
A, 227: 223-279.