U. S. DEPARTMENT OF COMMERCE COAST AND GEODETIC SURVEY MANUAL OF HARMONIC ANALYSIS _ AND PREDICTION OF TIDES SPECIAL PUBLICATION No. 98 REVISED (1940) EDITION WED tS (| DOCU MENT | SEESTION / ier cas ee ee a See ee ad 1 ¢heebUl TDEo Oo WAAC IOHM/T8IN CREE nn U. S. DEPARTMENT OF COMMERCE JESSE H. JONES, Secretary COAST AND GEODETIC SURVEY LEO OTIS COLBERT, Director Special Publication No. 98 Revised (1940) Edition (Reprinted 1958 with corrections) MANUAL OF HARMONIC ANALYSIS AND PREDICTION OF TIDES BY PAUL SCHUREMAN Senior Mathematician ae —— — =~ y a an = \ J : ju Pe at yi Reprinted October 1971 UNITED STATES GOVERNMENT PRINTING OFFICE WASHINGTON : 1958 PREFACE This volume was designed primarily as a working manual for use in the United States Coast and Geodetic Survey and describes the procedure used in this office for the harmonic analysis and prediction of tides and tidal currents. It is based largely upon the works of Sir William Thomson, Prof. George H. Darwin, and Dr. Rollin A. Harris. In recent years there also has been considerable work done on this subject by Dr. A. T. Doodson, of the Tidal Institute of the University of Liverpool. The first edition of the present work was published in 1924. In this revised edition there has been a rearrangement of the material in the first part of the volume to bring out more clearly the development of the tidal forces. Tables of astronomical data and other tables to facilitate the computations have been retained with a few revisions and additions and there has been added a list of symbols used in the work. The collection of tidal harmonic constants for the world that appeared in the earlier edition has been omitted altogether because the work of maintaining such a list has now been taken over by the International Hydrographic Bureau at Monaco. These constants are now published in International Hydrographic Bureau Special Publication No. 26, which consists of a collection of loose sheets which permit the addition of new constants as they become available. Special acknowledgment is due Walter B. Zerbe, associate mathe- matician of the Division of Tides and Currents, who reviewed the manuscript of this edition and offered many valuable suggestions. II ra 5 cated s ~ —s CONTENTS MMerOAUCtIONM ee ie. ae ee ee Dh ape ee DG pele Ek MORN a wel Se A istoricaliistatemve mites reas see gh as See SR eek General explanation of tidal movement__-____--------------------- Harmonic treatment of tidal data_________________-__-_-_---_----- AStroromieall Gl Ge ese rete yet rey ee eg SI Degree. ofapproximalionusstesn 42 Hanes eee Se St ae Wevelopmentof tide-producine: forces Shy s2 {wee ts ae eee ee hundanrentalsformulasiss 22 2s eee ee eee Sh ee ae Merticalicomponent offorcesse= see. = oa eee Horizont alvcomponentsyondtorcess 8) 22s) Sees saa a Ze ee cape HFRS NT ab wei Ta VL en ne Ba est, ee Terms involving 4th power of moon’s parallax__________-___-_------ Sew stid ets= PATA METH ae Dies clare. Ya een eons Sree eee Be HS UITATS@ Lear eyes UNL PRG Gl COS mead pa poe gree ere ey I Meteorological and shallow-water tides_______________-_-_-_------ Amalvsis ot observations ise 22 SR Oo Ba Res Sie Sea die se ee Larmonicrconstants ss. 2b Mae Ws et eae eset, Bld Wem eee Olosenvatiomneallyclata lowes. Upc oRinee Me faces ely Ski OP ke oa ae SUMMA TIONS LOM ANAIVSIS oauals Spence Sd ce ae ee i ee SUE T Cll Sees eee erent aco ty ln. ieee NE eae oh a Naess ees Le Second anyyst emcee oj ei yak) Ue ae AIG RN EE OS Beal ae ee ROUT ETSSE TI Sie cae ake ie laa nnd par tele SR see 5 he ml ape a PasewlageOr CpOChisetsaaty sn seven 2: Sea Lie Pea ae nes Oe ee linferencevor:constantss ok A ee es ee FEAPTVT TVG TO eRe AR Ns ee ES aU ee A eos sera onesperiod constituents ae ee es een Amal SIS.Ol hiehvanGd lowe WaAteISe se aan oa yh Ne ees nana eee Horms used tor analysisson tides. =. scm ose oe ee eneee AMA sistotstidalycunrentses= ee a oe nye) Eley ee 2 pee ee IBTECiGGiOnKOfebiGeser (ene tle Nee yee 8 be at Ces le 2 RES i ae See LEE ere OGY aes SO STH OVeX0 beedigpn yey ofan eR em pig oP EET rth ORM ya ay Cong ke dhide=predictine, machines sree ts fe ees See Ue Seale Forms used with tide-predicting machine_-_-_---._----_----------- Rrediction,of tidal™ Currents ee atte ee de eer een en era PATER La) os mpeg dcr ee ese RR I a G8 sas a I ee cee er eae Rea a Explanationoritabless 22.) seesaw iad = es ae a ee ee 1. Fundamental astronomical data____.__________________-______- De LErMoniCecOnsticuentss= = 4s ee ee ee Eh Cy pee Mean longitude of lunar and solar elements_________-_-------- Differences to adapt table 4 to any month, day, and hour_------ . Values of I, v, —, v’, and 2y’’ for each degree of N___----------- . Values of log R, for amplitude of constituent Ly.____----------- . Values of R for argument of constituent Ly______-_------------ . Values of log Q, for amplitude of constituent M,_--_----------- . Values of Q for argument of constituent M,___-_-------------- 11. Values of u of equilibrium arguments for each degree of N_-_-_--- 12. Values of log factor F for each tenth degree of /__-_------------ 13. Values of u and log factor F for constituents L. and M, for the Wears: 1OOO CoOeZOQO segs icees craw) fs AI ee RE eae AERO Lia ae eet 14. Node factor f for middle of each year 1850 to 1999_____-------- 15. pauilibr den argument V,+u for beginning of each vear 1850 to —_ SOWONID CB co Iv CONTENTS Tables—Continued. 16. Differences to adapt table 15 to beginning of each calendar month_ 17. Differences to adapt table 15 to beginning of each day of month__ 18. Differences to adapt table 15 to beginning of each hour of day___ 1OSProductsitor Worm G4 yee ee se iy a RNR ea 20) -Auementime hactoOrs esac 2 aie 2S Oy ep 21. Acceleration in epoch of K; due to P,_________________________ 22. Ratio of increase in amplitude of K,; due to Py_________________ 23. Acceleration in epoch of S, due to Ky_________________________ 24. Ratio of increase in amplitude of S, due to Ky_________________ 254 Acceleration inyiepoch ofeS> duelto iho =a = eae ee 26. Resultant amplitude 'S, due to Woe 22) 44.) ey ee 2, Criticallogarithms) for Horm) 24 5 ee pe pene ee 28. Constituent speed differences (6—a)e"e 145.) 45 Sau) a ee 29), Klimination factors 2.2222 5-545 22 se oe ee eye ee 315) Hor constructionrof primary Sten ell saa meee aren anne 32. Divisors) for primary? stem ei (sum se ee ee 33. Hor construction ofisecondany stencils sas = aa a ee 34. Assignment of daily page sums for long-period constituents______ 35. Products: for Horm 44452 22228 Sea Se se eee 36) Angle differences for Horm 45 eee a eee 37. Coast and Geodetic Survey tide-predicting machine No. 2—gen- eral. gearsae. 2.25.20. | Ae Oe ee ee 38. Coast and Geodetic Survey tide-predicting machine No.2—con- stituent gears. 2. =. 2s ee = eee 39) Synodic periodslof i constiGwemisme eee eee eee re eer 40. Day of common year corresponding to day of month___________ 41. Values of h in formula h=(1+7?+27 cos x)?___________________ 42. Values of k in formula k=tan-) —="* Hxplanation, of symbols. + 225-2) S350 See ee ee os Index. 22202-26224 1g eee ee oe ee ee OND OUR OO NO ILLUSTRATIONS EHeliptic; celestial equator, and moon's orbit=— 222-5 = — a ee Tide=producing force 32 3= _- 2 ee 2 ee @elestialsphere. 2224 5-224 2652225 52 ee ee Longitude relations'.-s..24 2282) _). 2. 2 eae ee Hquilibriumitide withymcon on iequators 55a . Equilibrium tide with moon at maximum declination-__---_--------_- Constituent: tide Curven. i =.=. 32 2 a ee wiPhase relationss =o. See ee Ss ee eee Rorm 362: hourly heightses 4-22 222 54 ee See trove Crornsuniu@ing Wl Se oe Sn eee te ae . Application of"steneil. =... 2. 3 ee ee », Rorm. 142 stencil sums. 2220222 = 222 = a ere 5 Conmjomlieionl Or INoniMby WIENS aa eee oe see Se Sees aS wHorm 244° computationy or Vigil e . Form 244a, log F and arguments for elimination_-___-_----_-----=- Pl Noracn levee ne aaornnks Hos | = ee ese ae . Form 452, R, x, and ¢ from analysis and inference, diurnal tides_--_- . Form 452, R, «, and ¢ from analysis and inference, semidiurnal tides_ iy Porm 245, elimination. = 322-6 = 25. 2) See eee 1) Horm) 723, currents. harmonic) companisones == eyes en . Coast and Geodetic Survey tide-predicting machine_-_-------------- ‘) Tide-predicting machine, time side |. = Nees ee . Tide-predicting machine, recording devices____-------------------- / Nide-predictins: machine} driving; wears ss) =e ee . Tide-predicting machine, dial case from height side___-_------------ . Tide-predicting machine, dial case from time side_--__-------------- . Tide-predicting machine, vertical driving shaft of middle section-_--_-- . Tide-predicting machine, forward driving shaft of rear section- - ----- }) Tide-predicting machine, rear enG@_= == 5 2 ee ee . Tide-predicting machine, details of releasable gear______------------ . Tide-predicting machine, details of constituent crank_____---------- . Form 444, standard harmonie constants for predictions __-_---------- . Form 445, settings for tide-predicting machine__-____--------------- . Graphic solution of formulas (470) and (471)__-._.___------------ Page MANUAL OF HARMONIC ANALYSIS AND PREDICTION OF TIDES INTRODUCTION HISTORICAL STATEMENT 1. Sir William Thomson (Lord Kelvin) devised the method of reduction of tides by harmonic analysis about the year 1867. The principle upon which the system is based—which is that any peri- odic motion or oscillation can always be resolved into the sum of a series of simple harmonic motions—is said to have been discovered by Eudoxas as early as 356 B. C., when he explained the apparently irregular motions of the planets by combinations of uniform. circu- lar motions. In the early part of the nineteenth century Laplace recognized the existence of partial tides that might be expressed by the cosine of an angle increasing uniformly with the time, and also applied the essential principles of the harmonic analysis to the reduc- tion of high and low waters. Dr. Thomas Young suggested the importance of observing and analyzing the entire tidal curve rather than the high and low waters only. Sir George B. Airy also had an important part in laying the foundation for the harmonic analysis of the tides. To Sir William Thomson, however, we may give the credit for having placed the analysis on a practical basis. 2. In 1867 the British Association for the Advancement of Science appointed a committee for the purpose of promoting the extension, improvement, and harmonic analysis of tidal observations. The report on the subject was prepared by Sir William Thomson and was published in the Report of the British Association for the Advance- ment of Science in 1868. Supplementary reports were made from time to time by the tidal committee and published in subsequent reports of the British association. A few years later a committe, consisting of Profs. G. H. Darwin and J. C. Adams, drew up a very full report on the subject, which was published in the Report of the British Association for the Advancement of Science in 1883. 3. Among the American mathematicians who have had an important part in the development of this subject may be named Prof. William Ferrel and Dr. Rollin A. Harris, both of whom were associated with the U. S. Coast and Geodetic Survey. The Tidal Researches, by Professor Ferrel, was published in 1874, and additional articles on the harmonic analysis by the same author appeared from time to time in the annual reports of the Superintendent of the Coast and Geodetic Survey. The best known work of Doctor Harris is his Manual of Tides, which was published in several parts as appendices to the annual reports of the Superintendent of the Coast and Geo- detic Survey. The subject of the harmonic analysis was treated principally in Part II of the Manual which appeared in 1897. 1 Nautical Science, p. 279, by Charles Lane Poor. 2 U. S. COAST AND GEODETIC SURVEY GENERAL EXPLANATION OF TIDAL MOVEMENT 4, That the tidal movement results from the gravitational attraction of the moon and sun acting upon the rotating earth is now a well- _ established scientific fact. 'The movement includes both the vertical rise and fall of the tide and the horizontal flow of the tidal currents. It will be shown later that the tide-producing force due to this attrac- tion, when taken in connection with the attraction between the par- ticles of matter which constitute the earth, can be expressed by mathe- matical formulas based upon the well-known laws of gravitation. 5. Although the acting forces are well understood, the resultant tidal movement is exceedingly complicated because of the irregular distribution of land and water on the earth and the retarding effects of friction and inertia. Contrary to the popular idea of a progressive tidal wave following the moon around the earth, the basic tidal movement asevidenced by observations at numerous points along the shores of the oceans consists of a number of oscillating areas, the move- ment being somewhat similar to that in a pan of water that has been tilted. Such oscillations are technically known as stationary waves. The complex nature of the movement can be appreciated when con- sideration is given to the fact that such stationary waves may overlap or be superimposed upon each other and may be accompanied by a progressive wave movement. 6. Any basin of water has its natural free period of oscillation de- pending upon its size and depth. The usual formula for the period of oscillation in a rectangular tank of uniform depth is 2Z/-/gd, in which L is the length and d the depth of the tank and g is the accelera- tion of gravity. When a disturbing force is applied periodically at intervals corresponding to the free period of a body of water, it tends to build up an oscillation of much greater magnitude than would be possible with a single application of the force. The major tidal Geallations have periods approximating the half and the whole lunar we HARMONIC TREATMENT OF TIDAL DATA 7. The harmonic analysis of tides is based upon an assumption that the rise and fall of the tide in any locality can be expressed mathe- matically by the sum of a series of harmonic terms having certain relations to astronomical conditions. A simple harmonic function is a quantity that varies as the cosine of an angle that increases uniformly with time. In the equation y=A cos at, y is an harmonic function of the angle at in which a is a constant and ¢ represents time as measured from some initial epoch. The general equation for the height (h) of the tide at any time (¢) may be written h=H)+A cos (att+a)+B cos (bt+6)+C cos (ct+y)+ ete. (1) in which H is the height of the mean water level above the datum used. Other symbols are explained in the following paragraph. 8. Each cosine term in equation (1) is known as a constituent or component tide. The coefficients A, B, O, etc. are the amplitudes of the constituents and are derived from observed tidal data in each locality. The expression in parentheses is a uniformly-varying angle and its value at any time is called its phase. Any constituent term has its maximum positive value when the phase of the angle is zero and a maximum negative value when the phase equals 180°, and the => HARMONIC ANALYSIS’ AND PREDICTION OF TIDES 3 term becomes zero when the phase equals 90° or 270°. The coefficient of ¢ represents the rate of change in the phase and is called the speed of the constituent and is usually expressed in degrees per hour. The time required for a constituent to pass through a complete cycle is known as its period and may be obtained by dividing 360° by its speed. The periods and corresponding speeds of the constituents are derived from astronomical data and are independent of the locality of the tide station. The symbols a, 8, y, etc. refer to the initial phases of the constituent angles at the time when ¢ equals zero. The initial phases depend upon locality as well as the instant from which the time is reckoned and their values are derived from tidal observations. Harmonic analysis as applied to tides is the process by which the observed tidal data at any place are separated into a number of harmonic constituents. The quantities sought are known as harmonic constants and consist of the amplitudes and certain phase relations which will be more fully explained later. Harmonic prediction is accomplished by reuniting the elementary constituents in accordance with astronomical relations prevailing at the time for which the predictions are being made. ASTRONOMICAL DATA 9. In tidal work the only celestial bodies that need be considered are the moon and sun. Although every other celestial body whose gravitational influence reaches the earth creates a theoretical tide- producing force, the greater distance or smaller size of such body renders negligible any effect of this force upon the tides of the earth. In deriving mathematical expressions for the tide-producing forces of the moon and sun, the principal factors to be taken into consideration are the rotation of the earth, the revolution of the moon around the earth, the revolution of the earth around the sun, the inclination of the moon’s orbit to the earth’s equator, and the obliquity of the ecliptic. Numerical values pertaining to these factors will be found in table 1. 10. The earth rotates on its axis once each day. There are, how- ever, several kinds of days—the sidereal day, the solar day, the lunar day, and the constituent day—depending upon the object used as a reference for the rotation. The sidereal day is defined by astronomers as the time required for the rotation of the earth with respect to the vernal equinox. Because of the precession of the equinox, this day differs slightly from the time of rotation with respect to a fixed star, the difference being less than the hundredth part of a second. The solar day and lunar day are respectively the times required for rotation with respect to the sun and moon. Since the motions of the earth and moon in their orbits are not uniform, the solar and lunar days vary a little in length and their average or mean values are taken as standard units of time. A constituent day is the time of the rotation of the earth with respect to a fictitious satellite representing one of the periodic elements in the tidal forces. It approximates in length the lunar or solar day and corresponds to the period of a diurnal constituent or twice the period of a semidiurnal constituent. 11. A calendar day is a mean solar day commencing at midnight. Such a calendar day is known also as a civil day to distinguish it from the astronomical day which commences at noon of the same date. 4 U. S. COAST AND GEODETIC SURVEY Prior to the year 1925, the astronomical day was in general use by astronomers for the recording of astronomical data, but beginning with the Ephemeris and Nautical Almanac published in 1925 the civil day has been adopted for the calculations. Each day of what- ever kind may be divided into 24 equal parts known as hours which are qualified by the name of the kind of day of which they are a part, as sidereal hour, solar hour, lunar hour, or constituent hour. 12. The moon revolves around the earth in an elliptical orbit. Al- though the average eccentricity of this orbit remains approximately constant for long periods of time, there are a number of perturbations in the moon’s motion due, primarily, to the attractive force of the sun. Besides the revolution of the line of apsides and the regression of the nodes which take place more or less slowly, the principal inequalities in the moon’s motion which affect the tides are the evection and variation. The evection depends upon the alternate increase and decrease of the eccentricity of the moon’s orbit, which is always a maximum when the sun is passing the moon’s line of apsides, and a minimum when the sun is at right angles to it. The variation ine- quality is due mainly to the tangential component of the disturbing - force. The period of the revolution of the moon around the earth is called a month. The month is designated as sidereal, tropical, anomalistic, nodical, or synodical, according to whether the revolution is relative to a fixed star, the vernal equinox, the perigee, the ascend- ing node, or the sun. The calendar month is a rough approximation to the synodical month. 13. It is customary to refer to the revolution of the earth around the sun, although it may be more accurately stated that they both revolve around their common center of gravity; but if we imagine the earth as fixed, the sun will describe an apparent path around the earth which is the same in size and form as the orbit of the earth around the sun, and the effect upon the tides would be the same. This orbit is an ellipse with an eccentricity that changes so slowly that it may be considered as practically constant. The period of the revolution of the earth around the sun is a year, but there are several kinds of years. The sidereal year is a revolution with respect to a fixed star, the tropical year is a revolution with respect to the vernal equinox, the eclipse year is a revolution with respect to the moon’s ascending node, and the anomalistic year is a revolution with respect to the solar perigee. 14. A calendar year consists of an integral number of mean solar days and may be a common year of 365 days or a leap year of 366 days, these years being selected according to the calendars described below so that the average length will agree as nearly as practicable with the length of the tropical year which fixes the periodic changes in the seasons. The average length of the calendar year by the Julian calendar is exactly 365.25 days and by the Gregorian calendar 365.2425 days and these may be designated respectively as a Julian year and a Gregorian year. 15. The two principal kinds of calendars in use by most of the civilized world since the beginning of the Christian era are the Julian and the Gregorian calendars, the latter being the modern calendar in which the dates are sometimes referred to as ‘‘new style” to dis- tinguish them from the dates of the older calendars. Prior to the year 45 B. C. there was more or less confusion in the calendars, inter- HARMONIC ANALYSIS! AND PREDICTION OF TIDES 5 calations of months and days being arbitrarily made by the priesthood and magistrates to bring the calendar into accord with the seasons and for other purposes. 16. The Julian calendar received its name from Julius Cesar, who introduced it in the year 45 B. C. This calendar provided that the common year should consist of 365 days and every fourth year of 366 days, each year to begin on January 1. As proposed by Julius Cesar, the 12 months beginning with January were to be alternately 31 days and 30 days in length with the exception that February should have only 29 days in the common years. When Augustus succeeded Julius Cesar a few years later, he slightly modified this arrangement ‘by transferring one day from February to the month of Sextilis, or August as it was then renamed, and also transferred the 31st day of September and November to October and December to avoid having three 31-day months in succession. 17. The Gregorian calendar received its name from Pope Gregory, who introduced it in the year 1582. It was immediately adopted by the Catholic countries but was not accepted by England until 1752. This calendar differs from the Julian calendar in having the century years not exactly divisible by 400 to consist of only 365 days, while in the Julian calendar every century year as well as every other year divisible by 4 is taken as a leap year with 366 days. For dates before Christ the year number must be diminished by 1 before testing its divisibility by 4 or 400 since the year 1 B. C. corresponds to the year 0 A. D. The Gregorian calendar will gain on the Julian calendar three days in each 400 years. When originally adopted, in order to adjust the Gregorian calendar so that the vernal equinox should fall upon March 21, as it had at the time of the Council of Nice in 325 A. D., 10 days were dropped and it was ordered that the day following October 4, 1582 of the Julian calendar should be designated as October 15, 1582 of the Gregorian calendar. This difference of 10 days between the dates of the two calendars continued until 1700, which was a leap year according to the Julian calendar and a com- mon year by the Gregorian calendar. The difference between the two then became 11 days and in 1800 was increased to 12 days. Since 1900 the difference has been 13 days and will remain the same until the year 2100. 18. Dates of the Christian era prior to October 4, 1582, will, in general, conform to the Julian calendar. Since that time both cal- endars have been used. The Gregorian calendar was adopted in England by an act of Parliament passed in 1751, which provided that the day following September 2, 1752, should be called September 14, 1752, and also that the year 1752 and subsequent years should commence on the Ist day of January. Previous to this the legal year in England commenced on March 25. Except for this arbitrary beginning of the year, the old English calendar was the same as the Julian calendar. When Alaska was purchased from Russia by the United States, its calendar was altered by 11 days, one of these days being necessary because of the difference between the Asiatic and American dates when compared across the one hundred and eightieth meridian. Dates in the tables at the back of this volume refer to the Gregorian calendar. 19. The three great circles formed by the intersections of the planes of the earth’s equator, the evliptic, and the moon’s orbit with the 6 U. S. COAST AND GEODETIC SURVEY celestial sphere are represented in figure 1. These circles intersect in six points, three of them being marked by symbols in the figure, namely, the vernal equinox T at the intersection of the celestial equator and ecliptic, the ascending lunar node & at the intersection cf the ecliptic and the projection of the moon’s orbit, and the lunar inter- section A at the intersection of the celestial equator and the projection of the moon’s orbit. For brevity these three points are sometimes called respectively “‘the equinox,” “the node,’”’ and “the intersection.” The vernal equinox, although subject to a slow westward motion of about 50’’ per year, is generally taken as a fixed point of reference for the motion of other parts of the solar system. The moon’s node has a westward motion of about 19° a year, which is sufficient to carry it entirely around a great circle in a little less than 19 years. 20. The angle w between the ecliptic and the celestial equator is known as the obliquity of the ecliptic and has a nearly constant value of 233°. The angle 7 between the ecliptic and the plane of the moon’s orbit is also constant with a value of about 5°. FIGURE 1. The angle J which measures the inclination of the moon’s orbit to the celestial equator might appropriately be called the obliquity of the moon’s orbit. Its magnitude changes with the position of the moon’s node. When the moon’s ascending node coincides with the vernal equinox, the angle J equals the sum of w and 7, or about 281°, and when the descending node coincides with the vernal equinox, the angle I equals the difference between w and 7, or about 184°. This variation in the obliquity of the moon’s orbit with its period of approximately 18.6 years introduces an important inequality in the tidal movement which must be taken into account. 21. In the celestial sphere the terms “‘latitude’”’ and “longitude” apply especially to measurements referred to the ecliptic and vernal equinox, but the terms may with propriety also be applied to meas- urements referred to other great circles and origins, provided they are sufficiently well defined to prevent any ambiguity. For example, we may say “‘longitude in the moon’s orbit measured from the moon’s HARMONIC ANALYSIS AND PREDICTION OF TIDES a node.” Celestial longitude is always understood to be measured toward the east entirely around the circle. Longitude in the celestial equator reckoned from the vernal equinox is called right ascension, and the angular distance north or south of the celestial equator is called declination. 22. The true longitude of any point referred to any great circle in the celestial sphere may be defined as the arc of that circle intercepted between the accepted origin and the projection of the point on the circle, the measurement being always eastward from the origin to the projection of the point. The true longitude of any point will generally be different when referred to different circles, although reckoned from a common origin; and the longitude of a body moving at a uniform rate _ of speed in one great circle will not have a uniform rate of change when referred to another great circle. 23. The mean longitude of a body moving in a closed orbit and referred to any great circle may be defined as the longitude that would be attained by a point moving uniformly in the circle of reference at the same average angular velocity as that of the body and with the initial position of the point so taken that its mean longitude would be the same as the true longitude of the body at a certain selected position of that body in its orbit. With a common initial point, the mean longitude of a moving body will be the same in whatever circle it may be reckoned. Longitude in the ecliptic and in the celestial equator are usually reckoned from the vernal equinox Y, which is common to both circles. In order to have an equivalent origin in the moon’s orbit, we may lay off an arc & 1’ (fig. 1) in the moon’s orbit equal to £& Y in the ecliptic and for convenience call the point Y’ the referred equinox. ‘The mean longitude of any body, if reckoned from either the equinox or the referred equinox, will be the same in any of the three orbits represented. This will, of course, not be the case for the true longitude. 24. Let us now examine more closely the spherical triangle 2 T A in figure 1. The angles w and 7 are very nearly constant for long periods of time and have already been explained. The side 27, usually designated by JN, is the longitude of the moon’s node and is undergoing a constant and practically uniform change due to the regression of the moon’s nodes. This westward movement of the node, by which it is carried completely around the ecliptic in a period of approximately 18.6 years, causes a constant change in the form of the triangle, the elements of which are of considerable im- portance in the present discussion. The value of the angle J, the supplement of the angle A 7, has an important effect upon both the range and time of the tide, which will be noted later. The side A 7, designated by »v, is the right ascension or longitude in the celestial equator of the intersection A. The arc designated by & is equal to the side 2 Y—side & A and is the longitude in the moon’s orbit of the intersection A. Since the angles 2 and w are assumed to be constant, the values of J, v, and & will depend directly upon NV, the longitude of the moon’s node, and may be readily obtained by the ordinary solution of the spherical triangle QT A. Table 6 give the values of J, v, and é for each degree of N. In the computation of this table the value of w for the beginning of the twentieth century was used. However, the secular change in the obliquity of the ecliptic is so slow that a difference of a century in 8 U. S, COAST AND GEODETIC SURVEY the epoch taken as*the basis of the computation would have resulted in differences of less than 0.02 of a degree in the tabular values. The table may therefore be used without material error for reductions pertaining to any modern time. ~ 25. Looking again at figure 1, it will be noted that when the longitude of the moon’s node is zero the value of the inclination J will equal the sum of w and 2 and will be at its maximum. In this position the northern portion of the moon’s orbit will be north of the ecliptic. When the longitude of the moon’s node is 180°, the moon’s orbit will be between the Equator and ecliptic, and the angle J will be equal to angle w—angle 7. The angle J will be always positive and will vary from w—7 to w+7i. When the longitude of the moon’s node equals zero or 180°, the values of v and & will each be zero. For all positions of the moon’s node north of the Equator as its longitude changes from 180 to 0°, v and & will have positive values, as indi- cated in the figure, these ares being considered as positive when reckoned eastward from Y and Y’, respectively. For all positions of the node south of the Equator, as the longitude changes from 360 to 180°, v and é will each be negative, since the intersection A will then lay to the westward of T and T’ DEGREE OF APPROXIMATION 26. The problem of finding expressions for tidal forces and the equilibrium. height of the tide in terms of time and place does not admit of a strict solution, but approximate expressions can be ob- tained which may be carried to as high an order of precision as desired. In ordinary numerical computations exact results are seldom ob- tained, the degree of precision depending upon the number of decimal places used in the computations, which, in turn, will be determined largely by the magnitude of the quantity sought. In general, the degree of approximation to the value of,any quantity expressed numerically will be determined by the number of significant figures used. With a quantity represented by a single significant figure, the error may be as great as 33) percent of the quantity itself, while the use of two significant figures will reduce the maximum. error to less than 5 percent of the true value of the quantity. The large possible error in the first case renders it of little value, but in the latter case the approximation is sufficiently close to be useful when only rough results are necessary. The distance of the sun from the ae is popularly expressed by two significant figures as 93,000,000 miles. 27. With three or four significant figures fairly satisfactory approxi- mations may be represented, and with a greater number very precise results may be expressed. For theoretical purposes the highest at- tainable precision is desirable, but for practical purposes, because of the increase in the labor without a corresponding increase in util- ity, it will be usually found advantageous to limit the degree of precision in accordance with the prevailing conditions. 28. Frequently a quantity that is to be used as a factor in an expres- sion may be expanded into a series of terms. If the approximate value of such a series is near unity, terms which would affect the third decimal place, if expressed numerically, should usually be re- tained. The retention of the smaller terms will depend to some ex- HARMONIC ANALYSIS AND PREDICTION OF TIDES 9 tent upon the labor involved since their rejection would not seriously affect the final results. 29. The formulas for the moon’s true longitude and parallax on pages 19-20 are said to be given to the second order of approximation, a fraction of the first order being considered as one having an approxi- mate value of 1/20 or 0.05, a fraction of the second order having an approximate value of (0.05)? or 0.0025, a fraction of third order having an approximate value of (0.05)* or 0.000125, etc. As these formulas provide important factors in the development of the equations repre- senting the tide-producing forces, they determine to a large extent the degrees of precision to be expected in the results. DEVELOPMENT OF TIDE-PRODUCING FORCE FUNDAMENTAL FORMULAS 30. The tide-producing forces exerted by the moon and sun are similar in their action and mathematical expressions obtained for one may therefore by proper substitutions be adapted to the other. Be- cause of the greater importance of the moon in its tide-producing effects, the following development will apply primarily to that body, the necessary changes to represent the solar tides being afterwards indicated. 31. The tide-producing force of the moon is that portion of its gravitational attraction which is effective in changing the water level on the earth’s surface. This effective force is the difference between the attraction for the earth as a whole and the attraction for the differ- ent particles which constitute the yielding part of the earth’s sur- face; or, if the entire earth were considered to be a plastic mass, the tide-producing force at any point within the mass would be the force that tended to change the position of a particle at that point relative to a particle at the center of the earth. That part of the earth’s surface which is directly under the moon is nearer to that body than is the center of the earth and is therefore more strongly attracted since the force of gravity varies inversely as the square of the dis- tance. For the same reason the center of the earth is more strongly attracted by the moon than is that part of the earth’s surface which is turned away from the moon. 32. The tide-producing force, being the difference between the attraction for particles situated relatively near together, is small com- pared with the attraction itself. It may be interesting to note that, although the sun’s attraction on the earth is nearly 200 times as great as that of the moon, its tide-producing force is less than one-half that of the moon. If the forces acting upon each particle of the earth were equal and parallel, no matter how great those forces might be, there would be no tendency to change the relative posi- tions of those particles, and consequently there would be no tide- producing force. 33. The tide-producing force may be graphically represented as in figure 2. Let O=the center of the earth, C=the center of the moon, P=any point within or on the surface of the earth. Then OC will represent the direction of the attractive force of the moon upon a particle at the center of the earth and PC the direction of the attractive force of the moon upon a particle at P. Now, let the magnitude of the moon’s attraction at P be represented by the length of the line PC. Then, since the attraction of gravitation varies inversely as the square of the distance, it is necessary, in order to represent the attraction at O on the same scale, to take a line CQ of such length that CQ : CP=CP? : CO’. 10 HARMONIC ANALYSIS AND PREDICTION OF TIDES 11 34. The line PQ, joining P and Q, will then represent the direction and magnitude of the resultant force that tends to disturb the posi- tion of P relative to O, for it represents the difference between the force PC and a force through P equal and parallel to the force QC which acts upon 0. This last statement may be a little clearer to the reader if he will consider the force PC as being resolved into a force PD equal and parallel to QC, and the force PQ. The force PD, acting upon the particle at P, being equal and parallel to the force QC, acting upon a particle at O, will have no tendency to change the position of P relative to O. The remaining force PQ will tend to alter the position of P relative to O and is the tide-producing force of the moon at P. The force PQ may be resolved into a vertical component PR, which tends to raise the water at P, and the hori- zontal component PT’, which tends to move the water horizontally. FIGURE 2. 35. If the point P’ is taken so that the distance CP’ is greater than the distance CO, the tide-producing force P’Q’ will be directed away from the moon. While at first sight this may appear paradoxical, it will be noted that the moon tends to separate O from P’, but as O is taken as the point of reference, this resulting force that tends to separate the points is considered as being applied at the point P’ only. 36. To express the tide-producing force by mathematica] equations, refer to figure 2 and let r= OP =distance of particle P from center of earth, b= PC =distance of particle P from center of moon, d= OC =distance from center of earth to center of moon, 2=COP=angle at center of earth between OP and OC. Also let M=mass of moon, E=mass of earth, a=mean radius of earth, u=attraction of gravitation between unit masses at unit distance. g=mean acceleration of gravity on earth’s surface. Since the force of gravitation varies directly as the mass and inversely as the square of the distance, Attraction of moon for unit mass at point O in direction oc" (2) 12 U. S. COAST AND GEODETIC SURVEY Attraction of moon for unit mass at point P in direction po=e (3) 37. Let each of these forces be resolved into a vertical component along the radius OP and a horizontal component perpendicular to the same in the plane OPC, and consider the direction from O toward P as positive for the vertical component and the direction corresponding to the azimuth of the moon as positive for the horizontal component. We then have from (2) and (8) Attraction at O in direction O to p= COS 2 (4) Attraction at O perpendicular to op=" sin 2 (5) Attraction at P in direction O to pate cos CPR (6) Attraction at P perpendicular to op="= sin CPR (7) 38. The tide-producing force of the moon at any point P is measured by the difference between the attraction at P and at the center of the earth. Letting F,=vertical component of tide-producing force, and F,=horizontal component in azimuth of moon, and taking the differences between (6) and (4) and between (7) and (5), we obtain the following expressions for these component forces in terms of the unit p: F, ju= (°° a *) (3) Pe [n= M(t") (9) 39. From the plane triangle COP the following relations may be obtained: b=r?+ @2—2rd cos z=d*[1—2(r/d) cos z+ (r/d)?| (10) : ; sin z sin CPR=sin CPO= (d/b) sin 2=—9G/d) eee (11) bad coe cos z—r/d cos CPR= (1—sin OGD i= =r mae GED. (12) 40. In figure 2 it will be noted that the value of z, being reckoned in any plane from the line OC, may vary from zero to 180°, and also that the angle CPR increases as z increases within the same limits. Sin z and sin CPR will therefore always be positive. As the angle OCP is always very small, the angle CPF will differ by only a very small amount from the angle z and wil] usually be in the same quad- rant. In obtaining the square root for the numerator of (12) it was therefore necessary to use only that sign which would preserve this HARMONIC ANALYSIS! AND PREDICTION OF TIDES 13 relationship. The denominators of (11) and (12) are to be consid- ered as positive. 41. Substituting in equations (8) and (9) the equivalents for b, sin CPR, and cos CPR from equations (10) to (12), the following basic formulas are obtained for the vertical and horizontal components of the tide-producing force at any point P at r distance from the center of the earth: me cos z—7/d Fr, (Sop i 1—2(r/d) cos 2+ (r/d)*}3 ays z| (13) M sin 2 : Felv=-3| aay cos eT GFT ~™ *| 42. To express these forces in their relation to the mean accelera- tion of gravity on the earth’s surface, represented by the symbol g, we have glu=E/a*, or pu/g=a'/E (15) in which F is the mass and a is the mean radius of the earth. Sub- stituting the above in formulas (13) and (14), we may write cos z—r/d F, [9= (MIB) (ala)? | ppp Oe acon 2] (16) sin 2 P, [g= (MIB) "| peg eae gem | 0 43. Formulas (16) and (17) represent completely the vertical and horizontal components of the lunar tide-producing force at any point in the earth. If 7 is taken equal to the mean radius a, the formulas will involve the constant ratio M/E and two variable quantities— the angle z which is the moon’s zenith distance, and the ratio a/d which is the sine of the moon’s horizontal parallax in respect to the mean radius of the earth. Because of the smallness of the ratio a/d it may also be taken as the parallax itself expressed as a fraction of a radian. The parallax is largest when the moon is in perigee and at this time the tide-producing force will reach its greatest magnitude. A more rapid change in the tidal force at any point on the earth’s surface is caused by the continuous change in the zenith distance of the moon resulting from the earth’s rotation. The vertical com- ponent attains its maximum value when z equals zero, and the hori- zontal component has its maximum value when 2 is a little less than 45°. Substituting numerical values in formulas (16) and (17) and in similar formulas for the tide-producing force of the sun, the fol- lowing are obtained as the approximate extreme component forces when the moon and sun are nearest the earth: Greatest F’, /g=.14410~° for moon, or .054 107° for sun (18) Greatest F, /g=.107 107° for moon, or .04110~° for sun (19) The horizontal component of the tide-producing force may be meas- ured by its deflection of the plumb line, the relation of this component to gravity as expressed by the above formula being the tangent of the angle of deflection. Under the most favorable conditions the 14 U. S. GOAST AND GEODETIC SURVEY greatest deflection due to the moon is about 0.022”’ and the greatest deflection due to the sun is less than 0.009’’ of arc. 44, To simplify the preceding formulas, the quantity involving the fractional exponent may be developed by Maclaurin’s theorem into a series arranged according to the ascending powers of r/d, this being a small fraction with an approximate maximum value of 0.018. Thus 1 {1—2(r/d) cos z+ (r/d)?}? =1+3 cos z (r/d) +3/2 (5 cos? z—1)(r/d)? +5/2 (7 cos? z—3 cos z)(r/d)?+ ete. (20) 45, Substituting (20) in formulas (16) and (17) and neglecting the higher powers of r/d, we obtain the following formulas: F, /g=3 (M/E) (a/d)’ (cos’ z—1/3) (r/d) +3/2 (M/E) (a/d)? (5 cos? z—3 cos 2) (r/d)? (21) F, /g=3/2 (M/E) (a/d)? (sin 2 z) (7/d) +3/2 (M/E) (a/d)? sin z (5 cos? z—1) (r/d)? (22) 46. If r, which represents the distance of the point of observation for the center of the earth, is replaced by the mean radius a, it will be noted that the first term of each of the above formulas involves the cube of the ratio a/d while the second term involves the fourth power of this quantity. This ratio is essentially the moon’s parallax ex- pressed in the radian unit. These terms may now be written as sepa- rate formulas and for convenience of identification the digits 3” and “4”? will be annexed to the formula symbol to represent respectively the terms involving the cube and fourth power of the parallax. Thus F,; /|g=3 (M/E) (a/d)? (cos? z—1/3) (23) Fy, [g=3/2 (M/E) (a/d)*(5 cos’ z—3 cos 2) (24) F,;, [g=3/2 (M/E) (a/d)? sin 22 (25) F 4 /g=3/2 (M/E) (a/d)* sin 2 (5 cos? z—1) (26) Formulas (23) and (25) involving the cube of the parallax represent the principal part of the tide-producing force. For the moon this is about 98 per cent of the whole and for the sun a higher percentage. The part of the tide-producing force represented by formulas (24) and (26) and involving the fourth power of the parallax is of very little practical importance but as a matter of theoretical interest will be later given further attention. 47. An examination of formulas (23) and (25) shows that the prin- cipal part of the tide-producing force is symmetrically distributed over the earth’s surface with respect to a plane through the center of the earth and perpendicular to a line joining the centers of the earth and moon. The vertical component (23) has a maximum positive value when the zenith distance z=0 or 180° and a maximum negative value when z=90°, the maximum negative value being one-half as great as the maximum positive value. The vertical component be- HARMONIC ANALYSIS! AND PREDICTION OF TIDES 15 comes zero when z=cos-!+-/1/3 (approx. 54.74° and 125.26°). The horizontal component (25) has its maximum value when z=45° and an equal maximum negative value when z=135°. The horizontal component becomes zero when z=0, 90°, or 180. 48. If numerical values applicable to the mean parallax of the moon are substituted in (23) and (25), these component forces may be written F'., /g at mean parallax=0.000,000,167 (cos? z—1/3) (27) F.3 /g at mean parallax=0.000,000,084 sin 22 (28) For the corresponding components of the solar tide-producing force, the numerical coefficients will be 0.46 times as great as those in the above formulas. 49. For the extreme values of the components represented by (23) and (25), with the moon and sun nearest the earth, the following may be obtained by suitable substitutions: Greatest F’,3 /g=.140<10~* for moon, or .05410~° for sun (29) Greatest F,3 /g=.10510~* for moon, or .04110~° for sun (30) Comparing the above with (18) and (19), it will be noted that the maximum values of the lunar components involving the cube of the moon’s parallax are only slightly less than the corresponding maxi- mum values for the entire lunar force, while for the solar components the differences are too small to be shown with the number of decimal places used. VERTICAL COMPONENT OF FORCE 50. It is now proposed to expand into a series of harmonic terms formula (23) which represents the principal vertical component of the lunar tide-producing force. In figure 3 let O represent the ee of the earth and let projections on the celestial sphere be as follows: C, the north pole I M’ P’, the earth’s equator I M, the moon’s orbit M, the position of the moon IP the place of observation CM M’, the hour circle of the moon CP P’, the meridian of place of observation I, the intersection of moon’s orbit and equator Also let I =angle M J M’=inclination of moon’s orbit to earth’s equator t =are P’ M’ or angle PCM=hour angle of moon X=IP’=\longitude of P measured in celestial equator from intersection J j =1M=longitude of moon in orbit reckoned from intersection J z =PM=zenith distance of moon D=M’M=declination of moon ee ett cen ee a 16 U. S. COAST AND GEODETIC SURVEY The solution of a number of the spherical triangles represented in figure 3 will provide certain relations needed in the development of the formulas for the tide-producing force. 51. In spherical triangle MCP, the angle C equals ¢ and the sides MC and PC are the complements of D and Y, respectively. We may therefore write cos z=sin Y sin D+ cos Y cos D cos t (31) Substituting this value in formula (23), we obtain Fy /g=3/2 (M/E) (a/d)?(1/2—3/2 sin? Y)(2/3—2 sin’? D)___- Fis /g +3/2 (M/E) (a/d)? sin 2Y sin 2D cos t____ Fy 31 /g +3/2 (M/E) (a/d)’ cos? Y cos? D cos 2t____Fy39 | g (32) Cc FIGURE 3. 52. In formula (32) the vertical component of the tide-producing force has been separated into three parts. The first term is inde- pendent of the rotation of the earth but is subject to variations aris- ing from changes in declination and distance of the moon. It in- cludes what are known as the long-period constituents, that is to say, constituents with periods somewhat longer than a day and in general a half month or longer. The second term involves the cosine of the hour angle (¢) of the moon and this includes the diurnal constituents with periods approximating the lunar day. The last term involves the cosine of twice the hour angle of the moon and includes the semidiurnal constituents with periods approximating the half lunar day. The grouping of the tidal constituents according to their approximate periods affords an important classification in the further development of the tidal forces and these groups will be called classes or species. Symbols pertaining to a particular species are often identified by a subscript indicating the number of periods in a day, HARMONIC ANALYSIS AND PREDICTION OF TIDES 7 the subscript o being used for the long-period constituents. In formula (32) the individual terms are identified by the annexation of the species subscript to the general symbol for the formula. 53. As written, all of the three terms of formula (32) have the same coefficient 3/2 (M/E) (a/d)?. In each case the latitude (Y) factor has a maximum value of unity, this maximum being negative for the first term. For the long-period term (F'2) /g), the latitude factor has a maximum positive value of }4 at the equator, becomes zero in latitude 35.26° (approximately), and reaches a maximum negative value of—1 at the poles, the factor being the same for corresponding latitudes in both northern and southern hemispheres. For the diurnal term (F’,3; /g), the latitude factor is positive for the northern hemisphere and negative for the southern hemisphere. It has a maximum value of unity in latitude 45° and is zero at the equator and poles. For the semidiurnal terms (F’,3. /g), the latitude factor is always posi- tive and has a maximum value of unity at the equator and equals zero at the poles. 54. For extreme values attainable for the declinational (D) factors, consideration must be given to the greatest declination which can be reached by the tide-producing body. The periodic maximum decli- nation reached by the moon in its 18.6 year node-cycle is 28.6° but this may be slightly increased by other inequalities in the moon’s motion. The maximum declination for the sun, taken the same as the obliquity of the ecliptic, is 23.45°. The declination factor of the long-period term (F',39 /g) has a maximum value of 2/3 when the decli- nation is zero. It diminishes with increasing north or south declina- tion but must always remain positive because of the limits of the declination. For the diurnal term (F,3, /g) the declinational factor has its greatest value when the declination is greatest. For the moon the maximum value of this factor is approximately 0.841 and for the sun 0.730. This factor is positive for the north declination and negative for the south declination. For the semidiurnal term (F532 /g) the declinational factor for both moon and sun is always positive and has a maximum value of unity at zero declination. 55. The greatest numerical values for the several terms of the vertical component of the tide-producing force as represented by formula (32) and applicable to the time when the moon and sun are nearest the earth, are as follows: Greatest F',3) /g=—.070X 10-8 for moon, or —.027X10-$ for sun (33) Greatest F 3, /g= +.08810-* for moon, or +.03010-° for sun (34) Greatest F432 /g=+.105<10-§ for moon, or +.041X107° for sun (35) For the long-period term (33) the greatest value applies to either pole and is negative. For the diurnal term (34) the greatest value applies in latitude 45° and may be positive or negative according to whether the latitude and declinational factors have the same or opposite signs. For the semidiurnal term (35) the greatest value applies to the equator and is positive. 56. Referring to formula (32), let a/c equal the mean value of parallax a/d. Then a/d may be replaced by its equivalent (a/c) (c/d), in which the fraction c/d expresses the relation between the true and, the mean parallax. Also let U=(M/E) (a/c)?, the numerical value of which will be found in table 1. Expressing separately the three terms of formula (32), we then have 18 U. S. COAST AND GEODETIC SURVEY Fray [g=3/2 U (e/d)® (1/2—3/2 sin? ¥)(2/3—2 sin? D) —86) F 3, /g=3/2 U (c/d)? sin 2Y sin 2D cos t (37) Fy. /g=3/2 U (c/d)? cos? Y cos? D cos 2t (38) 57. Referring to figure 3, the following relations may be obtained from the right spherical triangles 1/7’ and MP’M’ and the oblique spherical triangle MP’T: sin D=sin IJ sin j (39) cos D cos t=cos MP’ (40) cos MP’=cos X cosj7+sin X sin j cos I (41) cos D cos t=cos X cosj7+sin X sin j cos I =cos’ 4] cos (X—7)+sin? $7 cos (X¥+7) (42) 58. Replacing the functions of D and ¢ in formulas (36) to (88) by their equivalents derived from equations (39) and (42), there are obtained the following: Py /g=3/2 U(c/d)?(1/2—3/2 sin’ Y) X [2/3—sin? [-+sin? I cos 24] (43) Te gs) 2Ui(c/d) simi2¥ x [sin I cos? 4 cos (X¥+90°—27) +1/2 sin 27 cos (X—90°) +sin J sin? 4] cos (X¥—90°+27)] (44) ei gG—ol2) Ueld)*eos? YX [cos* 41 cos (2X—27) +1/2 sin? I cos 2X +sin* 4J cos (2X+2))] (45) The above formulas involve the moon’s actual distance d and its true longitude 7 as measured in its orbit from the intersection. While these are functions of time, they do not vary uniformly because of certain inequalities in the motion of the moon, and it is now desired to replace these quantities by elements that do change uniformly. 59. Referring to paragraphs 23-24 and to figure 1, it will be noted that longitude measured from intersection A in the moon’s orbit equals the longitude measured from the referred equinox 7’ less are £, and longitude measured from intersection A in the celestial equator equals the longitude measured from the equinox 7 less are v. Now let s’=true longitude of moon in orbit referred to equinox s =mean longitude of moon referred to equinox k =difference (s’—s) Then j=s'—t=s—E+k (46) 60. In figure 4 let S’ and P’ be the points where the hour circles of the mean sun and place of observation intersect the celestial equator, Y the vernal equinox, and J the lunar intersection. Then X will equal the arc P’I and »y the arc IY. Now let h=mean longitude of sun T=hour angle of mean sun X=T+h—p (47) Then HARMONIC ANALYSIS AND PREDICTION OF TIDES 19 61. Substituting the values of 7 and X from (46) and (47) m formu- las (43) to (45), these may be written F 39 /g==3/2 U(1/2—3/2 sin? Y) X [(c/d)?(2/3—sin? I) + (c/d)® sin? I cos (2s—2&+2k)] (48) F331 /g=3/2 U sin Dix [(c/d)? sin I cos? 41 cos (T—2s+h+2&—v+90°—2k) +1/2 (c/d)? sin 2I cos (T+h—v—90°) + (e/d)* sin I sin? 47 cos (T+2s+h—2é—v—90°+ 2k)] (49) F 32 /[g=3/2 U cos? YX [(c/d)* cos* 41 cos (2T—2s+2h+2&—2y—2k) +1/2 (c/d)® sin? I cos (2T-+2h—27) + (e/d)? sin 42 cos (2T7+2s+2h—2é—27+ 2k) (50) Cc FIGURE 4. Disregarding at this time the slow change in the function of J, the variable part of each term of the above formulas may be expressed in one of the following forms—(e/d)3, (c/d)? cos A, (e/d)* cos (A+2k), or (c/d)? cos (A—2k), in which A includes all the elements of the variable angular function excepting the multiple of k. 62. The following equations for the motion of the moon were adapted from Godtray’s Elementary Treatise on the Lunar Theory: s’=true longitude of moon (in radians) ee Ba Ray ay aly RR NY nn, ein (mean longitude) +2e sin (s—p)+5/4 e? sin 2(s—p)-_----- (elliptic inequality) +15/4 me sin(s—2h+p)___-------- (evectional inequality) +11/8 m? sin 2(s—h)_______-- (variational inequality) (51) 20 U. S. COAST AND GEODETIC SURVEY c/d= (true parallax of moon)/(mean parallax of moon) =unity +e cos (s—p)+e? cos 2(s—p)___-_____- (elliptic inequality) +15/8 me cos (s—2h+p)___--_____- (evectional inequality) =m? COSi 248A). 2. ean (variational inequality) (52) in which s’=true longitude of moon in orbit (referred to equinox) s=mean longitude of moon h=mean longitude of sun p=mean longitude of lunar perigee e=eccentricity of moon’s orbit=0.0549 m=ratio of mean motion of sun to that of moon=0.0748 The clements e and m are small fractions of the first order and the square of either or the product of both may be considered as being of the second order. In the following development the higher powers of these elements will be omitted. 63. Since k has been taken as the difference between the true and the mean longitude of the moon, we may obtain from (51) je—2 Sina (S—/p) 10) 40€- SUS 7p) +15/4 me sin (s—2h+p)+11/8 m? sin 2(s—h) (53) The value of k is always small, its maximum value being about 0.137 radian. It may therefore be assumed without material error that the sine of k or the sine of 2k is equal to the angle itself. Then sin 2k=2k=4e sin (s—p)+5/2 e? sin 2(s—>p) +15/2 me sin (s—2h+p)+11/4 m? sin 2(s—h) (54) cos 2k=1—2 sin? k=1—2k? = 1—4e?+ 4e? cos 2(s—p) (55) terms smaller than those of the second order being omitted. 64. Cubing (52) and neglecting the smaller terms, we obtain (c/d)?=1+3/2 e?+3e cos (s—p)+9/2 e cos 2(s—p) +45/8 me cos (s—2h+p)+3 m? cos 2(s—h) (56) Multiplying (54) and (55) by (56) (c/d)* sin 2k=4e sin (s—p)+17/2 e? sin 2(s—p) +15/2 me sin (s—2h+p)+11/4 m? sin 2(s—h) (57) (c/d)? cos 2k=1—5/2 e?+3 e cos (s—p)+17/2 e? cos 2(s—p) +45/8 me cos (s—2h+p)+3 m? cos 2(s—h) (58) 65. From (56), (57), and (58), we may obtain the following general expressions applicable to the further development of formulas (48) to (50). Negative coefficients have been avoided by the introduction of 180° in the angle when necessary. (c/d)? cos (A—2k) = (ce/d)? cos 2k cos A+ (e/d)? sin 2k sin A = (1—5/2 e?) cos A +7/2 e cos (A—s+p)+1/2e cos (A+s—p+180°) +17/2 e cos (A—2s+2p) +105/16 me cos (git p)+ 15/16 me cos (A+s—2h+p+180°) +23/8 m? cos (A— 38+ 2h) 11/8 m? cos (A+2s-—2h) (59) HARMONIC ANALYSIS! AND PREDICTION OF TIDES Pit (c/d)? cos A=(1+3/2 e?) cos A +3/2 € cos (A—s+p)+3/2 e cos (A+s—p) +9/4 e? cos (A—2s+2p)+9/4 e cos (A+2s—2p) 145/16 | me cos (A—s+2h— p)+45/16 me cos (A+s—2h-+>p) +3/2 m? cos (A—2s+2h)+3/2 m? cos (A+2s—2h) (60): (c/d)® cos (A+2k) = (c/d)’ cos 2k cos A— (e/d)* sin 2k sin A =(1—5/2 e?) cos A +7/2 € cos (A+s—p)+1/2 e cos (A—s+p+180°) +17/2 e? cos (A+2s—2>p) +105/16 sme cos (At+s—2h+p) + 15/16 me cos (A—s+2h— py Ae +23/8 m? cos (A+2s—2h)+1/8 m? cos (A—2s+2h) (61) 66. After suitable substitutions for A have been made in the three preceding equations they are immediately applicable to the final expansion of the several terms in formulas (48) to (50), excepting the first term of (48) for which formula (56) may be used directly. Hach term in the expanded formulas given below represents a constituent of the lunar tide-producing force and for convenience of reference is designated by the letter A with a subscript. There are also given the generally recognized symbols for the principal constituents, and when such a symbol is enclosed in brackets it signifies that the term given only partially represents the constituent so named. 7. Formula for long-period constituents of vertical component of principal lunar tide-producing force: F439 /[g=3/2 U(/2—3/2 sin? Y) X Can) H(2/3 sincere 3/2 eae be WEY on ee Td permanent term Ay) ae ORCL COS (Shp) eae hee 7). Cus tues 4.1 Wine ain Mm (A3) +9/2 e? cos (2s—2p) (A,) +45/8 me cos (s—2h+ 7) (As) aS? COS, 28a Oh RRL e ey cima) vrei in MSf (An) =|-1sin el 5/202") teos Qs = 2 )e EP os Web Mf (A,) +7/2 € cos (8s—p—2é) (As) +1/2 e cos (s+p+180°—2é) (Ay) +17/2 e? cos (4s—2p—2é) (Ajo) +105/16 me cos (8s—2h-+ p—2é) (Aj;) +15/16 me cos (s+2h—p+180°—28) (Aj) +23/8 m? cos (4s—2h—28) (Aj) +1/8 m? cos (2h—2€)}] (62) 68. Formula for diurnal constituents of vertical component of principal lunar tide-producing force: Tan (9 —3)/ 2 Oesime2 vax (Ay) [sin I cos? 1/2T {(1—5/2 e?) cos (T—2s+h+90°+2¢—v)____ O, (A,5) +7/2 e cos (T—38s+h+p+90°+2é—»)____ Q, (Ajg) +1/2 e cos (T—s+th—p—90°+2é—»y)_____ [M,| (Aj7) +17/2 e cos (T—4s+h+2p4+90°+2é—y)__ 2Q, (Ajs) +105/16 me cos(T-—3s+3h—p+90°+2é—»p)_ pi (Aj) +15/16 me cos (T—s—h+p—90° + 2é— 1) (Ago) +23/8 m? cos (T—4s+3h+90°+2&— 1) ____ O71 (Aoi) +1/8 m? cos (T—h+90°+2¢—») } (Formula continued next page) 22 U. §. COAST AND GEODETIC SURVEY (Ag.) -+sin 27 { (1/2+3/4 e?) cos (T+h—90°—7)_______- [Ky] (Ag3) +3/4 e cos (T—s+h+p—90°—7)_____-__- [Mi] (Ao,4) +3/4 e cos (T+st+h—p—90°—y)________- a (Ags) +9/8 e? cos (T—2s+h+2p—90°—p) (Age) +9/8 e? cos (T-+2s+h—2p—90°— 1) (Ao7) +45/32 me cos (T—s+3h—p—90°—p)____- X1 (Ags) +45/32 me cos (T+s—h+p—90°—y) _____ 6, (Ago) +3/4 m? cos (T—2s+3h—90°—v)________- MP, (Azo) +3/4 m? cos (T+2s—h—90°—») }___------ SO; +sin I sin? 4J (A31) { (1—5/2 e?) cos (T+2s+h—90°—2¢—v)___ OO; (Ago) = f)/2€°Cos (LoS p90 2 2c 1) ayy (Ags) +1/2 e cos (T+s+h+p+90°—2é—v) (A3,) +17/2 e? cos (T+4s+h—2p—90° —2é—v) (A335) +105/16 me cos (T+3s—h+p—90°—2é—v) (Ase) +15/16 me cos (7-+s+3h—p+90°—2é—»p) (Az7) +23/8 m? cos (T+ 4s—h—90°—2é—p) (Ags) +1/8 m? cos (T+3h—90°—2é—») }] (63) 69. Formula for semidiurnal constituents of vertical component of principal lunar tide-producing force: Jia) |= BP) (ORCORMICN< (As9) [cost 41 { (1—5/2 e?) cos (2T—2s+2h42&—27) __-. Maz {Ao +7/2 e cos (2T—3s+2h+p+2——2y) ._-_- No (Ay) +1/2 ecos (2T—s+2h—p+180°+2~—27) _ [L] (Ax) +17/2 e& cos (27T—4s+2h-+2p+2&—2v) _.. 2N, (Ags) +-105/16 me cos (2T—3s+4h—p+2é—2y) - V2 (Ags) +15/16 me cos (2T—s+p+180°+2——2y)_ ge (Ajs) +23/8 m? cos (2T7—4s+4h+2—&—2r) ____- Me (Aye) +1/8 m? cos (27-+2&—2y)} (Ay) +sin?I{ (1/2+3/4 e?) cos (2T+2h—2y) ____-------- [K3] (Ays) +3/4 e cos (2T—s+2h+p—2v) ___--.---- [Lo] (Ag) +3/4 e cos (27-+s+2h—p—2y) _-.------ KJ2 (A50) +9/8 e? cos (2T—2s+2h+2p—2v) (As1) +9/8 e? cos (27+2s+2h—2p—2y) (Asp) +45/32 me cos (2T—s+4h—p—2yp) (A53) +45/32 me cos (2T+s+p—27) (A54) +3/4 m? cos (2T7—2s+4h—2yr) (Ass) +3/4 m? cos (27+2s—2y7)} (Ase) --sint LI{ (1—5/2 e2) cos (27-+2s+2h—2&—27) (As7) +7/2 e cos (27+3s+2h—p—2é—27) (Ass) +1/2 e cos (27-+s+2h+p+180°—2£—27) (Asp) +17/2 e& cos (27+4s+2h—2p—2é—2p) (Ago) +105/16 me cos (2T7+3s+p—2é—2?) (Agi) +15/16 me cos (27+s+4h—p+180° —2&—2y) (Ago) +23/8 m? cos (27-+4s—2&—2y) (Ass) -+1/8 m? cos (27+4h—2£—27) }] (64) 70. Arguments —Except for the slow changes in the values of J, &, and v which result from the revolution of the moon’s node, each term other than the permanent one in the three preceding formulas is an harmonic function of an angle that changes uniformly with time. This angle is known as the argument of the constituent, also as the equilibrium argument when obtained in connection with the develop- HARMONIC ANALYSIS AND PREDICTION OF TIDES 5 23 ment of the equilibrium tide. By analogy, the argument of the per- manent term may be considered as zero, the cosine of zero being unity. 71. The argument serves to identify the constituent by determining its speed and period and fixing the times of the maxima and minima of the corresponding tidal force. It usually consists of two parts represented by the symbols V and wu. When referring to a particular instant of time such as the beginning of a series of observations, the V is written with a subscript as Vj). The first part of the argument in- cludes any constant and multiples of one or more of the following astronomical elements—T, the hour angle of the mean sun at the place of observation; s, the mean longitude of the moon; h the mean longitude of the sun; and 7, the longitude of the lunar perigee. The second part wu includes multiples of one or both of the elements é and », which are functions of the longitude of the moon’s node and vary slowly between small positive and negative limits throughout a 19-year _ cycle. In a series of observations covering a year or less they are treated as constants with values pertaining to the middle of the series. They do not affect the average speed or period of the constituent. Their values corresponding to each degree of N, the longitude of the moon’s node, are included in table 6, formulas for their computation being given on p. 156. 72. The hourly speed of a constituent may be obtained by adding the hourly speeds of the elements included in the V of the argument. These elementary speeds will be found in table 1. The period of a constituent is obtained by dividing 360° by its speed. The approxi- mate period is determined by the element of greatest speed contained in the argument. Thus, the hour angle T has a speed of 15° per mean solar hour and all constituents with a single 7 in their argu- ments have periods approximating one day, while constituents with arguments containing the multiple 27 have periods approximating the half day. Next to J, the element of greatest speed is s the mean longitude of the moon, and long-period constituents with a single s in their arguments will have periods approximating the month and with any multiple of s the corresponding fraction of a month. The arguments and speeds of the constituents are listed in table 2. Numerical values of the arguments for the beginning of each calendar year from 1850 to 2000 are given in table 15 for con- | stituents used in the Coast and Geodetic Survey tide-predicting | machine. Tables 16 to 18 provide differences for referring these arguments to any day and hour of the year. 73. In order to visualize the arguments of the constituents depend- ing primarily upon the rotation of the earth, some have found it convenient to conceive of a system of fictitious stars, or ‘‘astres fictifs”’ as they are sometimes called, which move at a uniform rate in the celestial equator, each constituent being represented by a separate star. Thus, for the principal lunar constituent we have the mean moon and for the principal solar constituent the mean sun, while the various inequalities in the motions of these bodies are served by imaginary stars which reach the meridian of the place of observation at times corresponding to the zero value of the constituent argument. For the diurnal constituents the argument equals the hour angle of the star but for the semidiurnal constituents the argument is double the hour angle of the star. 24 U. S. COAST AND GEODETIC SURVEY 74. Coefficients—-The complete coefficient of each term of formulas (62) to (64) includes several important factors. First, the basic factor U, which equals the ratio of the mass of the moon to that of the earth multiplied by the cube of the mean parallax of the moon, is common to all of the terms. This together with the common numerical coeffi- cient may be designated as the general coefficient. Next, the function involving the latitude Y is known as the /atitude factor, each formula having a different latitude factor. Following the latitude factor is a function of J, the inclination of the moon’s orbit to the plane of the earth’s equator, which may appropriately be called the obliquity factor, each factor applying to a group of terms. Lastly, we have an indi- vidual term. coefficient which includes a numerical factor and involves the quantity e or m. Since these factors are derived from the equa- tions of elliptic motion, they will here be referred to as elliptic factors. The product of the elliptic factor by the mean value of the obliquity factor is known as the mean constituent coefficient (C). Numerical values for these coefficients are given in table 2. Since all terms in any one of the formulas have the same general coefficient and latitude factor, their relative magnitudes will be proportional to their constitu- ent coefficients. Terms of different formulas, however, have different latitude factors and their constituent coefficients are not directly comparable without taking into account the latitude of the place of observation. 75. The obliquity factors are subject to variations throughout an 18.6-year cycle because of the revolution of the moon’s node. Dur- ing this period the value of J varies between the limits of w—7 and w+i, or from 18.3° to 28.6° approximately, and the functions of J change accordingly. In order that tidal data pertaining to different years may be made comparable, it is necessary to adopt certain stand- ard mean values for the obliquity factors to which results for different years may be reduced. While there are several systems of means which would serve equally well as standard values, the system adopted by Darwin in the early development of the harmonic analysis of tides has the sanction of long usage and is therefore followed. By the Darwin method, the mean for the obliquity factor is obtained from the product of the obliquity factor and the cosine of the elements £ and y appearing in the argument. This may be expressed as the mean value of the product J cos u, in which J is the function of J in the coefficient and wu the function of € and v in the argument. Since u is relatively small and its cosine differs little from unity, the result- ing mean will not differ greatly from the mean of J alone or from the function of J when given its mean value. 76. Using Darwin’s system as described in section 6 of his paper on the Harmonic Analysis of Tidal Observations published in volume I of his collection of Scientific Papers (also in Report of the British Association for the Advancement of Science in 1883), the following mean values are obtained for the obliquity factors in formulas (62) to (64). These values were used in the computation of the corresponding constituent coefficients in table 2. The subscript , is here used to indicate the mean value of the function. For terms A, to A; in formula (62) [2/3—sin? I= (2/3—sin? w) (1—3/2 sin? 1) =0.5021 (65) For terms A, to Aj; in formula (62) [sin? I cos 2é])>=sin? w cos! 44=0.1578 (66) HARMONIC ANALYSIS AND PREDICTION OF TIDES 25 For terms A,, to A,, in formula (63) [sin J cos? 4] cos (2—v)]p>=sin w cos’ tw cos* 44=0.3800 (67) For terms Ay»: to Ajo in formula (63) [sin 27 cos y]>=sin 2a (1—3/2 sin? 2) =0.7214 (68) For terms A3, to A; in formula (63) | [sin J sin? 4] cos (2+ y)]p>=sin w sin’ 4w cos* 42=0.0164 (69) For terms Aj) to Ax, in formula (64) [cos 4] cos (2&—2v)]o>=cos* 4m cos* 41=0.9154 (70) For terms Ay, to A;; in formula (64) [sin? J cos 2v]>=sin? w (1—3/2 sin? 7) =0.1565 (71) For terms A;_ to Ags in formula (64) [sin* 47 cos (2+2y)],>=sin* $w cos* 31 =0.0017 (72) 77. The ratio obtained by dividing the true obliquity factor for any value of J by its mean value may be called a node factor since it is a function of the longitude of the moon’s node. The symbol generally used for the node factor is the small f. The node factor may be used with a mean constituent coefficient to obtain the true coefficient corresponding to a given longitude of the moon’s node. Node factors for the several terms of formulas (62) to (64) may be expressed by the following ratios: f(A)) to f(A;) =f(Mm) = (2/3—sin? 1)/0.5021 (73) f(As) to f(A:3) =f(Mf) =sin? I /0.1578 (74) F(Ays) to f(A) =f(O1) =sin I cos? J /0.3800 (75) f(Asy) to f( Ag) =f(J:) =sin 2T /0.7214 (76) f(Az;) to f(Ass) =f(OO,)=sin J sin? 4] /0.0164 (77) (Aso) to f(Asg) =f(M2) =cos! 41 /0.9154 (78) f(Ayz) to f(A55) sin? I /0.1565 (79) f(Age) to f(Ags)=sin! 4 I /0.0017 (80) Node factors for the middle of each calendar year from 1850 to 1999 are given in table 14 for the constituents used in the Coast and Geodetic Survey tide-predicting machine. These include all the factors above excepting formulas (79) and (80). However, since formula (79) represents an inerease of only about one per cent over formula (74), the tabular values for the latter are readily adapted to formula (79). Node factors change slowly and interpolations can be made in table 14 for any desired part of the year. For practical purposes, however, the values for the middle of the year are generally taken as constant for the entire year. 78. The reciprocal of the node factor is called the reduction factor and is usually represented by the capital F. Applied to tidal coeffi- cients pertaining to any particular year, the reduction factors serve to reduce them to a uniform standard in order that they may be comparable. Logarithms of the reduction factors for every tenth of a degree of J are given in table 12 for the constituents used on the tide-predicting machine of this office. 79. Formulas (62), (63), and (64), for the long-period, diurnal, and semidiurnal constituents of the vertical component of the tide-pro- ducing force may now be summarized as follows: Let H=constituent argument from table 2 C=mean constituent coefficient from table 2 f =node factor from table 14 26 U. S. COAST AND GEODETIC SURVEY Then F439 /g=3/2 UA/2—3/2 sin?Y) 2 fC cos E (81) Fi [g=3/2 U sn 2Y > fC cos FE (82) F 32 [g=3/2 U cos’?Y 2 fC cos E (83) Latitude factors for each degree of Y are given in table 3. The column symbol in this table is Y with annexed letter and digits corre- sponding to those in the designation of the tidal forces. Thus, Yo9 represents the latitude factor to be used with force Fy, its value being equal to the function (1/2—3/2 sin?7Y). Taking the numerical value for the basic factor U from table 1, the general coefficient 3/2 U is found to be 0.8373 X 107". HORIZONTAL COMPONENTS OF FORCE 80. The horizontal component of the principal part of the tide- producing force as expressed by formula (25), page 14, is in the direc- tion of the azimuth of the tide-producing body. This component may be further resolved into a north-and-south and an east-and-west direction. In the following discussion the south and west will be considered as the positive directions for these components. Now let F 3 /g=south component of principal tide-producing force F 3 /g=west component of principal tide-producing force =azimuth of moon reckoned from the south through the west. From formula (25), we then have F 3 /g=3/2 (M/E) (a/d)? sin 22 cos A (84) F 3 /g=3/2 (M/E) (a/d)? sin 22 sin A (85) 81. Referring to figure 3, page 16, the angle P’PM equals A, the azimuth of the moon. Now, keeping in mind that the angle MPC is the supplement of A, the angle PCM equals t, and the arcs MC and PC are the respective complements of D and Y, we may obtain from the spherical triangle 1/PC the following relations: sin zg cos A=—cos Y sin D+ sin Y cos D cos t (86) sin z sin A=cos D sin t (87) Multiplying each of the above equations by the value of cos z from formula (31), the following equations may be derived: sin 2z cos A=2 sin z cos z cos A =3/4 sin 2Y (2/3—2 sin’D) —cos 2Y sin 2D cos t +1/2 sin 2Y cos?D cos 2¢ (88) sin 22 sin A=2 sin z cos z sin A =sin Y sin 2D sin t¢ +eos Y cos*D sin 2¢ (89) 82. Substituting in (84) and (85) the quantities from equations (88) and (89), we have F 3 /g =9/8 (M/E) (a/d)* sin 2Y (2/83—2 sin?D)_______- F 39 /g —3/2 (M/E) (a/d)? cos 2Y sin 2D cos ¢______---- Fy |g +3/4 (M/E) (a/d)? sin 2Y cos?D cos 2¢_______-_-- F302 |g (90) Fis /g—3/2, (ME) \(G/d)esimp acme 2 Dicimuias 2 ee Fs: |g +3/2 (M/E) (a/d)* cos Y cos?D sin 2t________---- Fuso [g (91) HARMONIC ANALYSIS AND PREDICTION OF TIDES 7 The south component is expressed by three terms representing respec- tively the long-period, diurnal, and semidiurnal constituents. For the west component there are only two terms—the diurnal and semidiur- nal, there being no long-period constituents in the west component. Each term has been marked separately by a symbol with annexed digits analogous to those used for the vertical component to indicate the class to which the term belongs. 83. Comparing formula (90) for the south component with formula (32) for the vertical component, it will be noted that the same functions of D and ¢ are involved in the corresponding terms of both formulas, and that the terms differ only in their numerical coefficient and the latitude factor. Allowing for these differences, summarized formulas analogous to those given for the vertical component (page 26) may be readily formed. In order to eliminate the negative sign of the coefficient of the middle term, 180° will be applied to the arguments of that term. With all symbols as before, we then have F349 /g=9/8 U sn 2Y 2 fC cos EF (92) F 3, /g=3/2 U cos 2Y 2 fC cos (E+180°) (93) Fx. [g=3/4 U sin 2Y 2 fC cos E (94) 84. Comparing the two terms in formula (91) for the west com- ponent with the corresponding terms in formula (32) for the vertical component, it will be noted that the D functions are the same but that in (91) the sine replaces the cosine for the functions of ¢. It may be shown that the corresponding development of these terms will be the same as for the vertical component except that in the developed series each argument will be represented by its sine instead of cosine. In order that the summarized formulas may be expressed in cosine functions, 90° will be subtracted from each argument. With the same. symbols as before and allowing for differences in the latitude factors, we obtain F 3, /g=3/2 Usin Y 2 fC cos (E—90°) (95) FPy32 /[g=3/2 U cos Y & fC cos (E—90°) (96) 85. Formulas for the horizontal component of tide-producing force in any given direction may be derived as follows: Let A equal the azimuth (measured from south through west) of given direction, and let Faso /g, Fos: /g, and Fras. /g, respectively, represent the long-period, diurnal, and semidiurnal terms of the component in this direction. Then F 30 /|g=F sa) |g X cos A (97) Fis: /g=F 31 /gXcos A+ Fys: /gXsin A (98) ie lg= F 29 /gXcos A+ Fy 39 /gXsin A (99) As the long-period term has no west component, the summarized formula for the azimuth A may be derived by simply introducing the factor cos A into the coefficient of formula (92). For the diurnal and semidiurnal terms it is necessary to combine the resolved clements from the south and west components. 86. Referring to formulas (93) to (96) and considering a single constituent in each species we. obtain the following: 28 U. 8S. COAST AND GEODETIC SURVEY Diurnal constituent, 3/2 UfC [cos 2Y cos A cos (H+180°)+sin Y sin A cos (H—90°)] =3/2 UfC (—cos 2Y cos A cos H-+sin Y sin A sin E) =3/2 UfC P, cos (E—X,) (100) in which P,=(cos? 2Y cos? A+sin? Y sin? A)} (101) ea sin Y sin A (102) —cos 2Y cos A Semidiurnal constituent, 3/2 UfC [sin Y cos Y cos A cos E-+cos Y sin A cos (H—90°)] =3/2 UfC cos Y (sin Y cos A cos E-+sin A sin EF) =3/2 UfC P, cos (H—X2) (103) in which P,=cos Y (sin? Y cos? A+sin? A)? (104) eae (105) sin Y cos A 87. Summarized formulas for the horizontal component of the tide-producing force in any direction A may now be written as follows: F 3) /g=9/8 U sin 2Y cos A 2 fC cos EF (106) Fon, /g—3/2 UP, = fC cos (H—X,) (107) Fz. /[g=3/2 UP, = fC cos (H— X2) (108) the values for P,, P:, X; and X, being obtained by formulas in the preceding paragraph. P, and P, are to be taken as positive and the following table will be found convenient in determining the proper quadrant for X, and X). North latitude South latitude IN —_——— ———<——— —- quadrant Xi Xp Xi Xs quadrant quadrant quadrant quadrant 1 2 2 1 or 2 3 4or3 4 3 For the X; quadrant the first value of each pair is applicable when the latitude does not exceed 45° north orsouth. Otherwise the second value is applicable. EQUILIBRIUM TIDE 88. The equilibrium theory of the tides is a hypothesis under which it is assumed that the waters covering the face of the earth instantly respond to the tide-producing forces of the moon and the sun and form a surface of equilibrium under the action of these forces. The theory disregards fricticn and inertia and the irregular distribution of the land masses of the earth. Although the actual tidal movement 29 HARMONIC ANALYSIS AND PREDICTION OF TIDES POLE oc fe) Ee < 5 Gg Ww FIGURE 5. a 2 < =) og Wl FIGURE 6. 30 U. S. COAST AND GEODETIC SURVEY of nature does not even approximate to that which might be expected under the assumed conditions, the theory is of value as an aid in visualizing the distribution of the tidal forces over the surface of the earth. The theoretical tide formed under these conditions is known as the equilibrium tide, and sometimes as the astronomical or gravita- tional tide. 89. Under the equilibrium theory, the moon would tend to draw the earth into the shape of a prolate spheroid with the longest axis in line with the moon, thus producing one high water directly under the moon and another one on the opposite side of the earth with a low water belt extending entirely around the earth in a great circle midway between the high water points. It may be shown mathe- matically, however, that the total effect of the moon at its mean dis- tance would be to raise the high water points about 14 inches above the mean surface of the earth and depress the low water belt about 7 inches below this surface, giving a maximum range of tide of about 21 inches. The corresponding range due to the sun is about 10 inches. Figures 5 and 6 illustrate on an exaggerated scale the theoretical disturbing effect of the moon on the earth. In the first figure the moon is assumed to be directly over the equator and in the last figure the moon is approximately at its greatest north declination. 90. With the moon over the equator (fig. 5), the range of the equi- librium tide will be at a maximum at the equator and diminish to zero at the poles and at any point there wil] be two high and low waters of equal range with each rotation of the earth. With the moon north or south of the equator (fig. 6), a declinational inequality is introduced and the two high and low waters of the day for any given latitude would no longer be equal except at the equator. This inequality would increase with the latitude and near the poles only one high and low water would occur with each rotation of the earth. Although latitude is an important factor in determining the range of the equilibrium tide, it is to be kept in mind that in the actual tide of nature the latitude of a place has no direct effect upon the rise and fall of the water. 91. A surface of equilibrium is a surface at every point of which the sum of the potentials of all the forces is a constant. On such a surface the resultant of all the forces at each point must be in the direction of the normal to the surface at that point. If the earth were a homogeneous mass with gravity as the only force acting, the surface of equilibrium would be that of a sphere. Each additional force will tend to disturb this spherical surface, and the total deforma- tion will be represented by the sum of the disturbances of each of the forces acting separately. In the following investigation we need not be especially concerned with the more or less permanent deformation due to the centrifugal force of the earth’s rotation, since we may assume that the disturbances of this spheriodal surface due to the tidal forces will not differ materially from the disturbances in a true spherical surface due to the same cause. 92. The potential at any point due to a force is the amount of work that would be required to move a unit of matter from that point, against the action of the force, to a position where the force is zero. This amount of work will be independent of the path along which the unit of matter is moved. If the force being considered is the gravity of the earth the potential at any point will be the amount HARMONIC ANALYSIS AND PREDICTION OF TIDES 31 of work required to move a unit mass against the force of gravity from the point to an infinite distance from the earth’s center. For the tide-producing force, the potential at any point will be measured by the amount of work necessary to move the unit of mass to the earth’s center where this force is zero. 93. Referring to formula (21) for the vertical component of the tide- producing force, if the unit g is replaced by the unit » from equation (15), the formula may be written as follows: Sul a ese z—1/3)r+—— odF (5 cos*? z—3 cos 2)r? (109) 94. Considering separately the tide-producing potential due to the two terms in the above formula, let the potential for the first term involving the cube of the moon’s distance be represented by V3 and the potential for the second term involving the 4th power of the moon’s distance by Vy. In each case the work required to move a unit mass against the force through an infinitesimal distance —dr toward the center of the earth is the product of the force by —dr, and the potential or total work required to move the particle to the center of the carth may be obtained by integrating between the limits rand zero. Thus ‘== ee (cos? 21/3) |r dr =e (cos? z—1/3)r (110) Ve= ag cos’ z—3 cos 2) ie dr mG cos’ z—3 cos z)r’ (111) 95. At any instant of time the tide-producing potential at different points on the earth’s surface will depend upon the zenith distance (2) of the moon and may be either positive or negative. It will now be shown that the average tide-producing potential for all points on the earth’s surface, assuming it to be a sphere, is zero. Assume a series ‘of right conical surfaces with common apex at center of earth and axis coinciding with the line joining centers of earth and moon, the angle between the generating line and the axis being z. These conical surfaces separated by infinitesimal angle dz will cut the surface of the sphere into a series of equipotential rings, the surface area of any ring being equal toa 2 mr’ sin zdz. The average potential for the entire spherical surface may then be obtained by summing the products of the ring areas and corresponding potentials and dividing the sum by the total surface area of the sphere. Thus Average Vin 28 | "(cost z—1/3) sin z dz 0 pe T a | -13 cos’ z+1/3 cos z[=0 (112) 32 U. S. COAST AND GEODETIC SURVEY 4 _ pM’ 7” 2GR Average V,= a aS iMG cos’ z—3 cos 2) sin z dz | 5/4 cos‘ 2+3/2 cos? z[=0 (113) 96. Let V, represent the potential due to gravity at any point on the earth’s surface. Since the force of gravity at any point on or above the earth’s surface equals »E/r’, the corresponding potential becomes V,=Lil. = =— (114) If the earth is assumed to be a sphere with radius a, the gravitational potential at each point will equal n/a, which may be taken as the average gravitational potential over the surface of the earth. 97. For a surface of equilibrium under the combined action of gravity and that part of the tide-producing force involving the cube of the moon’s distance the sum of the corresponding potentials must be a constant, and since the average tide-producing potential for the entire surface of the earth is zero (par. 95), the constant will be the average gravitational potential or pwH/a. Then from (110) and (114) we have 3uM EF wk Vit Vi= ogre (cos 2 18)r (115) Transposing and omitting common factor », we may obtain as 2 @— OT _ 3/2 (M/E) (ald)*(cos* 2-1/3) (116) Let r=ath (117) so that A represents the height of the equilibrium surface as referred to the undisturbed spherical surface of an equivalent sphere. Then (—a)a ha? r (ath)? As fraction h/a is very small, its greatest value being less than 0.000001, the powers above the first may be neglected. Substituting in (116) and writing A with subscript 3 to identify it with the prin- cipal tide-producing force, we have hg [a=3/2 (M/E) (a/d)3 (cos? z—1/3) (119) 98. Similarly, for a surface of equilibrium under the combined action of gravity and the part of the tide-producing force involving the 4th power of the moon’s distance, we have from (111) and (114) =h/a—3(h/a)?+6(h/a)?— ete. (118) Vit V,= 2a (5 cos? z—3 cos zp He ve (120) HARMONIC ANALYSIS AND PREDICTION OF TIDES 33 = (M/E) (a/d)*(5 cos? z—3 cos 2) (121) Letting 7=a-+A, and expanding the first member of the above formula, it becomes equal to A, /a after the rejection of the higher powers of this small fraction. The formula may then be written hy /a=1/2 (M/E) (a/d)*(5 cos? z—3 cos 2) (122) 99. Formulas (119) and (122) involving the cube and 4th power of the moon’s parallax, respectively, represent the equilibrium heights of the tide due to the corresponding forces, the heights being expressed in respect to the mean radius (a) of the earth as the unit. In deriving these formulas the centrifugal force of the earth’s rotation was dis- regarded and the resulting heights represent the disturbances in a true spherical surface due to the action of the tide-producing force. It may be inferred that in a condition of equilibrium the tidal forces would produce like disturbances in the spheroidal surface of the earth and the A of the formulas may therefore be taken as being referred to the earth’s surface as defined by the mean level of the sea. 100. The extreme limits of the equilibrium tide, applicable to the time when the tide-producing body is nearest the earth, may be obtained by substituting the proper numerical values in formulas (119) and (122). They are given below for both moon and sun. From formula (119) involving the cube of parallax— Greatest rise =1.46 feet for moon, or 0.57 foot for sun (123) Lowest fall =0.73 foot for moon, or 0.28 foot for sun (124) Extreme range=2.19 feet for moon, or 0.85 foot for sun. (125) From formula (122) involving the 4th power of parallax— Greatest rise =0.026 foot for moon, or 0.000025 foot for sun (126) Lowest fall | =0.026 foot for moon, or 0.000025 foot for sun Gi2 7) Extreme range=0.052 foot for moon, or 0.00005 foot for sun. (128) 101. A comparison of formulas (23) and (119), the first expressing the relation of the vertical component of the principal tide-producing force to the acceleration of gravity (g) and the other the relation of the height of the corresponding equilibrium tide to the mean radius (a) of the earth, will show that they are identical with the single excep- tion that the coefficient of the height formula is one-half that of the force formula. Therefore the development of the force formula into a series of harmonic constituents is immediately applicable in obtain- ing similar expressions for the equilibrium height of the tide. Using a notation for the height terms corresponding to that used for the force terms, let hg /a, hz; /a, and hg, /a represent, respectively, the long- period, diurnal, and semidiurnal terms of the equilibrium tide involv- ing the cube of the moon’s parallax. Then referring to formulas (81) to (83) we may write hz /a=3/4 UA/2—3/2 sin? Y) 2 fC cos E (129) hs, /4=3/4 U sin 2Y 2 fC cos E (130) hz [a=3/4 U cos? Y 2 fC cos E (131) the symbols having the same significance as in the preceding discussion of the tidal forces. 34 U. S. COAST AND GEODETIC SURVEY TERMS INVOLVING 4TH POWER OF MOON’S PARALLAX 102. Formulas (24) and (26) represent the vertical and horizontal components of the part of the tide-producing force involving the 4th power of the moon’s parallax. This part of the force constitutes only about 2 percent of the total tide-producing force of the moon and for brevity will be called the lesser force to distinguish it from the principal or primary part involving the cube of the parallax. The vertical component F,,4 /g has its maximum value when 2 equals zero and, if numerical values pertaining to the moon and sun when nearest the earth are substituted in formula (24), the extreme values for this component are found to be 0.371078 for the moon and 0.385107" for the sun. The horizontal component F,,4 /g has its greatest value when 2 equals about 31.09° and the substitution of numerical values in formula (26) gives the extreme value of this component as 0.26 107° for the moon or 0.24107" for the sun. 103. Substituting in (24) the value of cos z from (31), the vertical component of the lesser force is expanded into four terms as follows: Fy, /g=15/4 (M/E) (a/d)* sin Y (cos? Y—2/5) sin D(5 cos? D—2)_ Fyao /g +45/8 (M/E) (a/d)*cos Y (cos? Y—4/5) cosD (5cos?D—4) cost Fy /g +45/4 (M/E) (a/d)* sin Y cos? Y sin D cos? D cos 2t______- Frye |g “hols (MLE) (a/d) cos icos Dicosst: eee eae eee Pug |g (132) These four terms represent, respectively, long-period, diurnal, semi- diurnal, and terdiurnal constituents, according to the multiple of the hour angle ¢t involved in the term. Each term is followed by a symbol wach is analogous to those used in the development of the principal orce. 104. Each term in formula (132) may be further expanded by means of the relations given in formulas (39) and (42). Expressing these terms separately we hayve— Fy /g=15/4 (M/E) (a/d)* sin Y(cos? Y—2/5) X [3 (sin [—5/4 sin’ J) cos (7—90°) +5/4 sin? I cos (87—90°)] (133) Fy /g=45/8 (M/E) (a/d)* cos Y (cos? Y—4/5) X [5/4 sin? I cos? 41 cos (X—37) + (1—10 sin? 4/+ 15 sin‘ 3/7) cos? 4 cos (X—7) +(1—10 cos? 47+ 15 cos* 4/) sin? 47 cos (X+7) +5/4 sin? I sin? 4J cos (X+3)7)] (134) Fy» /g=45/8 (M/E) (a/d)* sin Y cos? YX [sin J cos* 47 cos (2X—3j)+ 90°) +3 (cos? 4/—2/3) sin I cos? 41 cos (2X—7—90°) +3 (cos? 4J—1/3) sin I sin? 4] cos (2X+7—90°) +sin I sin‘ 47 cos (2X+37—90°)] (135) Fug [g=15/8 (M/E) (a/d)* cos? YX [cos® 37 cos (3X—3)) +3 cos 4/ sin? $I cos (3.X—7) +3 cos’ 4/ sint 4/7 cos (8X+7) +sin® 4/7 cos (3X+37)] (136) 105. If the common factor (a/d)* in formulas (133) to (136) is replaced by its equivalent (a/c)* X (c/d)*, these formulas may be de- HARMONIC ANALYSIS AND PREDICTION OF TIDES 35 veloped into numerous constituent terms by a method similar to that already described in the development of the principal lunar force (paragraphs 59-69). In the following development constituents of very small magnitude are omitted. Those given are numbered con- secutively with the constituent terms of the principal lunar force. Fy /g=15/4 (M/E) (a/c)! sin Y (cos? Y—2/5) X (Ags) [(sin J—5/4 sin? [){3(1+2e?) cos (s—90°—€é) (Ags) +9e cos (2s—p—90°—€) (Ags) +3¢ cos (p—90°—£)} (Ag. +sin? [{5/4(1—6e?) cos (8s—90°—3é) (Ags) +25/4 e cos (4s—p—90° — 38) }] (137) F4, /[g=45/8 (M/E) (a/c)* cos Y (cos? Y—4/5) x (Ago) [sin? I cos? 4/{ 5/4(1—6e?) cos (T—3s+h+3é—1) (Azo) +25/4 e cos (T—4s+h+p+3é—y) } +(1—10 sin? 4J+15 sin* 4/) cos? 42 (An) ( (Cige2e2) CoS GI S0-Fitar ey) Seah et [M1] (Azo) +3e cos (7—2s+h+p+évp) (Azz) se cos (hp 6) + (1—10 cos? 41+15 cos* 41) sin? 4] (Ars) {(1-+2e?) cos (T+s+h—£—») (Ajs) +3e cos (T-+2s+h—p—é—y) }] (138) Fy, /[g=45/8 (M/E) (a/c)* sin Y cos? Y X (Aze) [sin I cos‘ 47{ (1—6e?) cos (2T—3s+2h490°+3é—27) (Azz) +5e cos (27—4s+2h+p+90° +3£—27) (Azs) sé cos 27 —2s--2h—p—90° + 3£—2y) } + (cos? 4/—2/3) sin I cos? 41 (Azo) {3(1+2e?) cos 2T—s+2h—90°+ é—2v) (Ago) +9e cos (2T—2s+2h+ p—90°-+ ——2yp) } -+ (cos? 4/—1/3) sin I sin? 4] (Asy) {3(1+2e?) cos (27-+s+2h—90°—é—27)}] (189) F'y3 /g=15/8 (M/E) (a/c)* cos’ YX (Ago) [cos® 7 { (1—6e?) cos (83T7—3s+3h+3E—3p) _____ M; (Ags) +5e cos (3T—4s+3h+p+3t—37) (Ags) +e cos (3T—2s+3h—p-180°+3£—37) (Ags) +127/8 e cos 8T—5s+3h+2p+3£—3r) (Ags) +75/8 me cos (83T—4s+5h—p+3é—37) } + cos* 4] sin? 41 (Ag7) {3(1+2e?) cos (3T—s+3h+£—37) (Ags) +9e cos (3T—2s+3h-+ p+é—37) }] (140) 106. All of the constituent terms in formulas (137) to (140) are relatively unimportant but they are listed in table 1 because of their theoretical interest. The only one of these terms now used in the prediction of tides is (Ag:.) representing the constituent M; which has a speed exactly three-halves that of the principal lunar constituent M;. Term (Az) is of interest in having a speed exactly one-half that of M, and is sometimes called the true M, to distinguish it from the composite M, which is used in the prediction of tides and which will be described later. 36 U. §. COAST AND GEODETIC SURVEY 107. For simplicity and the purposes of this publication, the mean values of the obliquity factors in the terms of the lesser tide-producing force will be taken as the values pertnining to the time when J equals w or 23.452°, excepting that for constituent M; and associated terms the mean has been obtained in accord with the system described in paragraph 75. The corresponding node factors (paragraph 77) may then be expressed by the following formulas in which the denominators are the accepted means of the obliquity factors: F(Ags) to f(Acs) = (sin J—5/4 sin?J)/0.3192 (141) F(Agz) to f( Ags) =sin®//0.0630 (142) F (Ass) to f(Ayo) =sin?l cos*41/0.1518 (143) F(An) to f(Az3) = (1—10 sin?37+15 sin*4/) cos’3I/0.5873 (144) f(A) to f(Azs) = (1—10 cos’4I+15 cos*4J) sin?47/0.2147 (145) Ff (Are) to f(Ars) =sin I cos*31/0.3658 (146) F(Ar) to f(Aso) = (cos?3/—2/3) sin I cos?37/0.1114 (147) Ft (Asi) = (cos’37—1/3) sin I sin?4//0.0103 (148) F(Age) to f(Ase) =f(M3) = cos®3I/0.8758 (149) f(As;) to f(Asg) =cos*3l sin?47/0.0380 (150) Comparing formulas (149) and (78), it will be noted that the node factor for M; is equal to the node factor for M, raised to the 3/2 power. Computed values applicable to terms Ag. to Ass are included in table 14 for years 1850 to 1999, inclusive. 108. For the tabulated constituent coefficients of the terms in formulas (137) to (140) there are included not only the elliptic and mean obliquity factors but also such other factors as may be necessary to permit the use of the general coefficient (8/2 U) of formulas (81) to (83) for the vertical component of the principal tide-producing force. The common coefficient ((4/E) (a/e)* of formulas (137) to (140) is equal to U multiplied by the parallax a/c, and the latter together with the necessary numerical factors is included in the constituent coefficients in table 2. Formulas (137) to (140) may then be summarized as follows: Fay [g=3/2 U sin Y(cos?-Y—2/5) 2 fC cos E (151) Fy. /g=3/2 U cos Y(cos*¥—4/5) 2 fC cos E (152) Fy, /g=3/2 Usin Y cos?¥ 2 fC cos E (153) Fa. (G3) 2. ico Ya C. cose Es, (154) 109. It is to be noted that in formulas (151), (152), and (153), the maximum value of the latitude factor in each is less than unity, being HARMONIC ANALYSIS AND PREDICTION OF TIDES if 0.4, 0.2754, and 0.3849, respectively, if the sign of the function is disregarded. In formula (154), as in the corresponding formulas for the principal tide-producing force, the maximum value of this factor is unity. In comparing the relative importance of the various con- stituents of the tide-producing force the latitude factor should be in- cluded with the mean coefficient. Attention is also called to the fact that the relative importance of the constituents mvolving the 4th power of the moon’s parallax is greater in respect to the vertical com- ponent of the tide-producing force than in respect to the height of the equilibrium tide. In table 2 the mean coefficients are taken com- parable in respect to the vertical component of the tide-producing force and the constituent coefficients pertaining to the lesser force are therefore 50 percent greater than they would be if taken comparable in respect to the equilibrium tide. 110. The south and west horizontal components of the lesser tide- producing force may be obtained by multiplying formula (26) by cos A and sin A, respectively. Using the same system of notation as before, we then have FP, /g=3/2 (M/E) (a/d)* sin 2 (5 cos? z—1) cos A (155) Fy, /g=3/2 (M/E) (a/d)* sin z (5 cos? z—1) sin A (156) 111. By means of the relations expressed in formulas (31), (86), and (87), the above component forces may be separated into long- period, diurnal, semidiurnal, and terdiurnal terms as follows: South component, Fs49 [= —15/4 (M/E) (a/d)* cos Y (cos?¥ —4/5) sin D(5 cos2?D—2) (157) Ps, [g=45/8 (M/E) (a/d)* sin Y (cos? ¥ —4/15) cos D(5 cos? D—4) cos t (158) Paz [g= —45/4 (M/E) (a/d)* cos Y (cos? ¥—2/3) sin D cos?D cos 2t (159) Faz [g=15/8 (M/E) (a/d)* sin Y cos?Y cos*D cos 3t (160) West component, Fy /g=15/8 (M/E) (a/d)*(cos?Y —4/5) cos D(5 cos?D—4) sint (161) Pi /g=15/4 (M/E) (a/d)* sin 2Y sin D cos?D sin 2¢ (162) Fx3 |g=15/8 (M/E) (a/d)* cos?Y cos?D sin 3¢ (163) 112. Comparing formulas (157) to (160) for the south component force with the corresponding terms of (132) for the vertical com- ponent, it will be noted that they differ only in the latitude factors and in sign for two of the terms. With adjustments for these dif- ferences the summarized formulas (151) to (154) are directly applicable ir expressing the corresponding terms in the south component. us Py) /g=3/2 U cos Y(cos*¥—4/5) 2 fC cos(H+180°) (164) Fy [g=3/2 U sin Y(cos?¥—4/15) = fC cos HE (165) F'42 [g=3/2 U cos Y(cos?¥ —2/3) = fC cos(H+180°) (166) F43 [g=3/2 U sin Y cos’Y 2 fC cos E (167) 113. For the west component there is no long-period term. Com- paring (161) to (163) with the corresponding terms of (132), it will be noted that the ¢-functions are expressed as sines instead of cosines but they may be changed to the latter by subtracting 90° from each 38 U. S. COAST AND GEODETIC SURVEY argument. With this change and allowing for differences in the latitude factors and numerical coefficients, the summarized formulas for the west component will be similar to those for the vertical com- ponent and may be written as follows: Fy, /g=1/2 U (cos?¥—4/5) 2 fC cos (E—90°) (168) Fy. /g=1/2 U sin 2Y 2 fC cos (H—90°) (169) Fy [g=3/2 U cos*Y 2 fC cos (E—90°) (170) 114. To obtain the horizontal component of the lesser force in any direction, the same procedure may be followed as was used for the principal tide-producing force (paragraphs 85 to 87). With the same system of notation we then have Fy /g=3/2 U cos Y (cos? ¥ —4/5) cos A 2 fC cos (H+180°) (171) Foy ig—3/2 U0 Py Df cos (=e) (172) Fug |g=3/2 U P, = fC cos (H—X,) Gis) Fyu3 /g=3/2 U P3 2 fC cos (H—X3) (174) in which P,=(sin’Y (cos? ¥Y —4/15)? cos?A+1/9(cos? Y —4/5)? sin?A]* (175) P.=cos Y|(cos?¥Y —2/3)? cos?A+4/9 sin?Y sin?A]” (176) P;=cos’Y (sin?Y cos?A-+sin?A)” (177) renk eh (cos? Y —4/5) sin A AG — tan 3 sin Y(cos?Y—4/15) cos A (178) s 2 simi sin Al oe en 23 (cae = 2/3) cos A a) XG tana ee (180) sin Y cos A The proper quadrants for Xj, X52, and X3 will be determined by the signs of the numerators and denominators in the above expressions, these signs being respectively the same as for the sine and cosine of the corresponding angles. 115. Comparing formula (122) for the equilibrium height of the tide due to the lesser tide-producing force with formula (24) for the vertical component of the force, it will be noted that they are the same with the exception that the numerical coefficient of the former is one-third that of the latter. With this change, the summarized formulas (151) to (154) for the vertical force may be used to express the corresponding equilibrium heights. Following the same system of notation as before, we have hy /a=1/2 U sin Y(cos?¥Y—2/5) = fC cos E (181) hy /a=1/2 U cos Y(cos*?Y—4/5) = fC cos E (182) hy /a=1/2 U sin Y cos?Y 2 fC cos EF (183) lug [a=1/2 U cos’¥ 2 fC cos E (184) It is to be noted that the equilibrium height of the tide due to the principal tide-producing force when measured by the mean radius of the earth as a unit is one-half as great as the corresponding vertical component force referred to the mean acceleration of gravity as a unit, while the equilibrium height due to the lesser tide producing force similarly expressed is only one-third as great as the corresponding force. In table 2, the coefficients (C) of the constituents derived HARMONIC ANALYSIS AND PREDICTION OF TIDES 39 from the lesser force are made comparable with the others in respect to the vertical component force rather than in respect to the equi- librium height. SOLAR TIDES 116. Since the tide-producing force of the sun is similar in action to that of the moon, the formulas derived for the latter are applicable, with suitable substitutions, to the solar forces. Referring to formulas (62), (63), and (64), let U be replaced by U; representing the product (S/E )(a@/e1)? in which S is the mass of the sun and (a@/c;) its mean parallax. Also replace e by e, the eccentricity of the earth’s orbit; IJ by w, the obliquity of the ecliptic; s by h, the mean longitude of the sun : and p by pi, the longitude of the solar perigee. For the solar forces the arcs € and v become zero and all terms representing the evectional and variational inequalities are omitted. 117. Making the changes indicated the solar constituents are now expressed in the following formulas. Each term is marked for iden- tification by the letter B with the same subscript used for the corre- sponding term in the lunar tide. The usual constituent symbol is also given for the more important terms. Using the same system of notation as before, Solar F',3) /g =3/2 U, (1/2—3/2 sin? Y)X (B;) [(2/3—sin? w){(1+3/2 e?,) -..-____- permanent term (B.) +3 e, cos (h—7p,) (Bs) +9/2 e?, cos (2h—2p,) } (Bs) STASI co (== 5)/2) Cry COSe2 Npt ates ben Ssa (Bz) +7/2 e, cos (3h—py) - (Bs) +1/2 e, cos (h+p:+180°) “(B)) +17/2 e, cos (4h—2>7,) }] (185) Solar F',3; /g=3/2 U, sin 2YX (By4) [sin w cos” 4w{ (1—5/2 e,;) cos (T—h+90°)____- P, (Bis) +7/2 e,cos (T—2h+ p,+90°)_______- TH (Bis) +1/2 e, cos (T—p;—90°) (B;7) +17/2 e, cos (T—3h+2p,+90°) } (Boo) +sin 2w{ (1/2+3/4 e,) cos (T+h—90°)______- [Ky] (Bos) -+3/4 e, cos (T-+p,—90°) (Bos) sro eeer Gos Gia woe OS eases V1 (Bos) +9/8 e1 cos (T—h+2p,—90°) (Bos) ok e”, cos (T+3h—2p,—90°) } (B31) +sin w sin? tw{ (1—5/2 e”,) cos (T+3h—90°)_- (B32) Tp e, cos (T+4h— p,— 90°) (Bs3) +1/2 e, cos (T+2h+p,+90°) (Bzx) +17/2 e, cos (T+5h—2p,—90°) }] (186) Solar F,3, /g=3/2 U, cos? YX (Bsy) [cost ta" 5)/ 2862) cos Oe 2 2. bas aoe S, (Bao) +7/2e,cos (2T—h+p,)__----------- i (Bux) +1/2e,cos (27+h—p,+180°)____._- R, (Biz) +17/2 e, cos (2T—2h+2p,)} (Biz) +sin? w{ (1/2+3/4 e7,) cos (2T+2h)__.__--__- [K] (Bis) +3/4 e, cos (27+h+7p)) (Bis) +3/4 e, cos (27+3h—/p,) (Formula continued on next page) 40 U. S. COAST AND GEODETIC SURVEY (Bro) +9/8 e?, cos (2T+2,) (Bs1) +9/8 e, cos (2T+4h—2p))} (Bs) +sin* 4w{ (1—5/2 e?,) cos (27'+4h) (Bs7) +7/2 e, cos (2T+5h—p;) (Bs) +1/2 e, cos 27+38h+,4+ 180°) (Bso) +17/2 e; cos (2T+6h—2>p)) }] (187) 118. The general coefficient for the solar tide-producing force differs from that of the lunar force in the basic factor. From the fundamental data in table 1, the ratio of U,/U is found to be 0.4602. This ratio, which will be designated as the solar factor with symbol 8’, represents the theoretical relation between the principal solar and lunar tide-producing forces. In computing the constituent coefficients of the solar terms for use in table 2, the solar factor was included in order that the same general coefficient may be applicable to both lunar and solar terms. All of the summarized formulas involving the coefficients and arguments of table 2 are therefore applicable to both lunar and solar constituents. For the solar constituents, however, the node factor (f) is always unity since w, the obliquity of the ecliptic, may be considered as a constant. 119. By substituting solar elements in formulas (137) to (140) the corresponding solar constituents pertaining to the 4th power of the sun’s parallax are readily obtained. Since the theoretical magnitude of the lesser solar tide-producing force is less than 0.00002 part of the total tide-producing force of moon and sun, it is usually disregarded altogether. However, certain interest is attached to three of the constituents which are considered in connection with shallow water and meteorological tides (p. 46). These are constituents Sa, S$,, and S3, corresponding respectively to terms Ags, A7;, and Age of the lunar series. They are listed in table 2 with reference letter B and cor- responding subscripts. Sa has a speed one-half that of constituent Ssa represented by term B, of formula (185). Its theoretical argu- ment as derived from term A,, contains the constant 90°, but being considered as a meteorological rather than an astronomical consti- tuent, this constant is omitted from the argument. Constituents S, and S; have speeds respectively one-half and three-halves that of the principal solar constituent S. 120. The arguments of a number of the solar constituents include the element p, which represents the longitude of the solar perigee. As this changes less than 2° in a century, it may be considered as practically constant for the entire century. Referring to table 4 it will be noted that p, changes from 281.22° in 1900 to 282.94° in 2000. The value of 282° may therefore be adopted without material error for all work relating to the present century. With p, taken as a constant, it will be found that a number of terms in table 2 have the same speeds and may therefore be expected to merge into single constituents. Thus, constituents receiving contributions from more than one term are as follows: Sa from terms B:, B;, and Begs; Ssa from terms B; and B;; P, from terms B,, and B,;; S; from terms Byjg, B.3, and Bz; ¥, from terms By, and B33; ¢, from terms B., and B3;; S. from terms By) and Bs; and Re from terms By and By. A few other solar terms also merge. HARMONIC ANALYSIS AND PREDICTION OF TIDES 41 THE M: TIDE 121. The separation of constituents from each other by the process of the analysis depends upon the differences in their speeds. Constit- uents with nearly equal speeds are not readily separated unless the analysis covers a very long series of observations but they tend to merge and form a single composite constituent. In formula (63), terms A;,and A;; have nearly equal speeds, one being a little less and the other a little greater than one-half the speed of the principal lunar constituent M,. These two terms are usually considered as a single constituent and represented by the symbol M;. Neglecting for the present the general coefficient and common latitude factor, the two terms may be written as follows: term A;;=1/2 e sin I cos’ 3J cos (T—s+h—p—90°-+ 2—») (188) term A.3;=3/2 e sin I cos I cos (T—s+h+p—90°—y) (189) The latter term, having a coefficient nearly three times as great as that of the first term, will predominate and determine the speed and period of the composite tide while the first term introduces certain inequalities in the coefficient and argument. 122. For brevity, let A and B represent the respective coefficients of terms Aj, and A; and let 6=T—s+h+p—90°—»p (190) Also let P equal the mean longitude of the lunar perigee reckoned from the lunar intersection. Then Bee (191) We then have term A,,=A cos (@—2P) =A cos 2P cos 6+A sin 2P sin @ (192) term A.3;=B cos 6 (193) M,=A,,+ A.3,= (A cos 2P+B) cos 0+ A sin 2P sin 6 A sin 2P _( A2 2)4 Reo) Aen ee i Geis a as 2P+B’)? cos E tan ne SDE==B a Ee Ecos (T—s+h+p—90°—v—Q,) (194) in which P cos cos’ i 1/Q.=| 1/4+3/2 £25 00s 2P+9/4 (195) Q,=tan! ues (196) 3 cos I/cos? 4/+cos 2P If J is given its mean value corresponding to w, formula (195) may be reduced to the form 1/Q.= (2.310-+1.435 cos 2P)} (197) Values of log Q, for each degree of P based upon formula (197) are given in table 9. 42 U. S. COAST AND GEODETIC SURVEY 123. The period of the composite constituent M, is very nearly an exact multiple of the period of the principal lunar constituent Mz, and for this reason the summations which are necessary for the analysis of the latter may be conveniently adapted to the analysis of the former. With other symbols as before, let Qe Ts hone tee (198) Terms A,, and A.; may then be combined as follows: term A,,—A cos (6—P) =A cos P cos 6+A sin P sin 6 (199). term A .,;—B cos (6+ P) =B cos P cos 6—B sin P sin 0 (200) M.=Ay.+:A.3= (A+B) cos P cos 6+ (A—B) sin P sin 6 = (A?+2AB cos 2P+B’)* cos | o—tan-"( 5 tan P)| : 2 a2 Sn OSE cos(T—s-+h—00° +89 + Q) (201) in which —tan-( 2 cos L—1 Q=tan 7 cos Tani *20 P) (202) If J is given its mean value corresponding to w, formula (202) may be reduced to the following form which was used for computing the values of Q in table 10. tan Q=0.483 tan P (203) 124. Formulas (194) and (201) are the same except in the method of representing the argument. The elements +p —Q, in the first formula are replaced by +£+Q in the latter, but it may be shown from (196) and (202) that Q,+Q=P=p-—t (204) p—Q=E+Q (205) The complete arguments are therefore equal but in formula (201) the uniformly varying element p has been transferred from the V of the argument and included in the value of Q where it is treated as a constant for a series of observations being analyzed. The speed of the argument as determined by the remaining part of the V is then exactly one-half that of the principal constituent M, and with this assumption the summations for the latter may be adapted to the analysis of the former. It is to be noted, however, that the wu in this case has a progressive forward change of nearly 41° each year. The true average speed of this constituent is determined by the V of formula (194) which includes the element p. 125. The obliquity factor for the composite M, constituent may be expressed by the formula sin J cos*4JX<1/Q,. According to the work of Darwin (Scientific Papers by Sir George H. Darwin, vol. 1, p. 39) the HARMONIC ANALYSIS AND PREDICTION OF TIDES 43 mean value of this factor is represented by the product sin w cos? 4 w cos‘ 44 2.307, which equals 0.38001.52, or 0.5776. When deriving the node-factor formula for M,, Darwin inadvertently omitted the factor 2.307 and obtained the approximate equivalent of the following: sin I cos*t sin I cos*}1 ae I sin w cost w cos*d2 x 1/Q.=—9 3800 — X 1/Q2 (206) J(M)) = Comparing the above with formula (75), it will be noted that (Mi) =f(O1) X1/Qa (207) Factors pertaining to constituent M, in tables 13 and 14 are based upon the above formulas. 126. Because of the omission of the factor +/2.307 from formula (206), the node factors for M, which have been in general use since this system of tidal reductions was adopted are about 50 percent ereater than was originally intended, while the reciprocal reduction factors are correspondingly too small. This constituent is relatively unimportant and no practical difficulties have resulted from the omis- sion. The M, amplitudes as reduced from the observational data are comparable among themselves but should be increased by 50 percent to be on the same basis as the amplitudes of other constituents. The predicted tides have not been affected in the least since the node factors and reduction factors are reciprocal and compensating. The theoretical mean coefficient for this constituent with the factor +/2.307 included is 0.0317; but in order that this coefficient may be adapted for use with the tabular node factors when computing tidal forces or the equilibrium height of the tide, the coefficient 0.0209 with the factor 2.307 excluded should be used. 127. Although M, is one of the relatively unimportant constituents and the error in the node factor has caused no serious difficulties, it may be questionable whether it should be perpetuated. It is obvious, however, that any change in the present procedure would lead to much confusion unless undertaken by general agreement among all the principal organizations engaged in tidal work. By making any change applicable to the analysis of all series of observations beginning after a certain specified date it would be possible to interpret the results on the basis of the period covered by the observations without the neces- sity of revising all previously published amplitudes for this constituent. THE L2 TIDE 128. The composite L, constituent is formed by combining terms A, and A, of formula (64). Neglecting the general coefficient and common latitude factor these terms may be written term A,,=1/2 e cos* J cos (2T—s+2h—p+180°+2&—2y) (208) term Ay=3/4 e sin? J cos (27—s+2h-+ p—2p) (209) A reference to table 2 will show that the mean coefficient of the first term is about four times as great as that of the latter term. The first . 44 U. S. GOAST AND GEODETIC SURVEY term will therefore predominate and determine the speed of the composite constituent. 129. With other symbols as before, let A and B represent the respective coefficients of the two terms and @ the argument of the first term. We then have Ay=A cos 6 (210) Aw=B cos (6+2P—180°)=—B cos (0+2P) (211) L,=Ay-+ A= (A—B cos 2P) cos 6+ B sin 2P sin 6 B sin 2P lg Abies 2) 4 = fan = (A?—2AB cos 2P+B’)? cos E tan Teas OP =1/2e = 2Teos (2T—s+2h—p+180°+2t—2»—R) (212) in which i 1/R,=(1—12 tan? 47 cos 2P+36 tan‘ 4)? (213) se St sin 2P (214) 1/6 cot? 4J—cos 2P Values of log R, and R computed from the above formulas are given in tables 7 and 8, respectively. 130. The obliquity factor for the composite L, constituent may be expressed by the formula cost $/1/R,. The mean value of 1/R, is approximately unity, and in accord with the Darwinian system the mean for the entire obliquity factor is taken as the product cos* 1 cost 4i, which equals 0.9154 and is the same as the mean value of the obliquity factor for the principal constituent M;. Multiplying this by the elliptic factor 4e gives 0.0251 as the mean constituent coefficient. 131. The node factor formula for constituent L, based upon the above mean for the obliquity factor is as follows: cost 4] fl) = Fes X VR. =f(Me) X UR, (215) Node factors for constituent L. based upon the above formula are included in table 14 for the middle of each year from 1850 to 1999, inclusive. The logarithms of the reciprocal reduction factors covering the period 1900 to 2000 are contained in table 13. LUNISOLAR Ki AND K2 TIDES 132. Lunar diurnal term A, of formula (63) and solar diurnal term B,, of formula (186) have the same speed. Together they form the lunisolar K, constituent. Also, lunar semidiurnal term Ay, of formula (64) and solar semidiurnal term By; of formula (187) have speeds exactly twice that of constituent K, and together form the lunisolar K, constituent. In order that the solar terms may have the same general coefficient as the lunar terms, the solar factor U,/U, which will be designated by the symbol S’, will be transferred from the general coefficient of the solar terms and included in the constituent coefficients. Then, neglecting the general coefficient and HARMONIC ANALYSIS AND PREDICTION OF TIDES 45 latitude factors common to the terms combined, we have the following formulas in which numerical values from table 1 have been sub- stituted for constant quantities. term A= (1/2+3/4e?) sin 27 cos (T+h—90° —r) = 0.5023 sin 2/7 cos (T+h—90°—1) (216): term By.=(1/2+3/4e?)S’ sin 2 w cos (7-+h—90°) —0.1681 cos (T-+h—90°) (217) term Ay= (1/2+3/4e?) sin’J cos (27+2h—2y7) =0.5023 sin’J cos (27'-+2h—2p) (218) term B= (1/2+3/4e?)S’ sin? w cos (2T+2h) =0.0365 cos (T+2h) (219) - 133. Taking first the diurnal terms, let A represent the lunar co- efficient 0.5023 sin 2/ and let B represent the solar coefficient 0.1681. We then have An=A COS (T+h—90°— vp) =A cos vy cos (T+h—90°)+A sin » sin (7'-+h—90°) (220) By,=B cos (T+h—90°) (221) K,=(A cos v+ B) cos (7 +h—90°) +A sin v sin (T+ h—90°) =(A’?+2AB cos y+ B’)” cos | 7+h—90°—tan? 2 SOP, Bree a (222) in which C,= (A?+2AB cos EB)? = (0.2523 sin? 27+0.1689 sin 2/ cos »-+0.0283)2 (223) Yate a PBS Spey SRT (224) Values of v’ for each degree of N, which is the longitude of the moon’s node, are included in table 6. 134. The obliquity factor for K, will be taken to include the entire coefficient (A?+2AB cosv+ B’)? and its mean value will be taken as the mean of the product (A?+2AB cos v +B’)? cos v’. From (224) we may obtain cos v’=(A cos v+ B)/(A?+2AB cos v+ B?)? (225) Then for mean value of coefficient of K, [(A?+2AB cos v+ B?)? cos v’|}p>=[A cos v+ B]) =[0.5023 sin 27 cos v+0.1681],>=0.5305 (226) the numerical mean for sin 2I cos y being obtained from formula (68). For the node factor of K, divide the coefficient of (222) by its mean value and obtain f (K,) = (0.2523 sin? 27-+0.1689 sin 27 cos v+-0.0283)?/0.5305 = (0.8965 sin? 27+0.6001 sin 27 cos v+0.1006)? (227) 46 U. S. COAST AND GEODETIC SURVEY The node factors for the middle of each year 1850 to 1999 are included in table 14. Logarithms of the reciprocal reduction factors for each tenth of a degree of J are given in table 12. 135. The semidiurnal terms Ay and By may be combined in a similar manner. Letting A represent the lunar coefficient 0.5023 sin?/ and B the solar coefficient 0.0365, we have Ay=A cos (2T+2h—2p) =A cos 2v cos (27+2h)+A sin 2 sin (27+ 2h) (228) By=B cos (2T+2h) (229) K,=(A cos 27+ B) cos (27+2h)+A sin 2p sin (27+ 2h) A sin 2p nee 2 2) 4 BSS =I =(A’+2AB cos 2v+B’)? cos | 27 +-2h tan Aviary eB =(C, cos (2T-+2h—2v’’) (230) in which C,= (A?+2AB cos 2v+ B?)t = (0.2523 sin* J+ 0.0367 sin? J cos 2v+0.0013): (231) Bi Manes A sin 2» ee sin *J sin 2y te Anos 27358 + 8 sin ecost27 BOT (232) Values for 2v’’ for each degree of N are included in table 6. 136. The obliquity factor for K, will be taken to include the entire coefficient (A?+2AB cos 2v+ 5?) and its mean value will be taken as the mean of the product (A?+2AB cos 2v+B?)* cos 2y’’, From (232) cos 2v”=(A cos 2v+ B)/(A?+2AB cos 2v+ B?)i (233) Then for the mean value of coefficient of K, [(A?+2AB cos 2v+ B’)? cos 2v”|>=[A cos 27+ B]) = [0.5023 sin? J cos 2v-+0.0365])>=0.1151 (234) the numerical mean for sin? J cos 2v being obtained from formula (71). For the node factor of Ky divide the coefficient of (230) by its mean value and obtain f (Ky) = (0.2523 sin* [+0.0367 sin? J cos 2v+0.0013)? /0°1151 = (19.0444 sin* J+-2.7702 sin? J cos 2v+-0.0981): (235) See table 14 for node factors and table 12 for reciprocal reduction factors. METEOROLOGICAL AND SHALLOW-WATER TIDES 137. In addition to the elementary constituents obtained from the development of the tide-producing forces of the moon and the sun, there are a number of harmonic terms that have their origin in meteorological changes or in shallow-water conditions. Variations in temperature, barometric pressure, and in the direction and force of the wind may be expected to cause fluctuations in the water level. Although in general such fluctuations are very irregular, there are some seasonal and daily variations which occur with a rough periodic- ity that admit of being expressed by harmonic terms. The meteoro- logical constituents usually taken into account in the tidal analysis are HARMONIC ANALYSIS AND PREDICTION OF TIDES 47 Sa, Ssa, and S, with periods corresponding respectively to the tropical year, the half tropical year, and the solar day. These constituents are represented also by terms in the development of the tide-producing force of the sun but they are considered of greater importance as meteorological tides. Ssa occurs in the development of the principal solar force while Sa and S, would appear in a development involving the 4th power of the solar parallax (par. 119). In the analysis of tide observations both Sa and Ssa are usually found to have an appreciable affect on the water level. Constituent S, is relatively of little im- portance in its effect on the height of the tide but has been more noticeable in the velocity of off-shore tidal currents, probably as a result of periodic land and sea breezes. 138. The shallow-water constituents result from the fact that when a wave runs into shallow water its trough is retarded more than its crest and the wave loses its simple harmonic form. The shallow-water constituents are classified as overtides and compound tides, the over- tide having a speed that is an exact multiple of one of the elementary constituents and the compound tide a speed that equals the sum or difference of the speeds of two or more elementary constituents. 139. The overtides were so named because of their analogy to the overtones in musical sounds and they may be considered as the higher harmonics of the fundamental tides. The only overtides usually taken into account in tidal work are the harmonics of the principal lunar and solar semidiurnal constituents M, and S,, the lunar series being designated by the symbols My, Me, and Mg, and the solar series by Sy, Ss, and Ss. The subscript indicates the number of periods in the constituent day. These overtides with their argu- ments and speeds are included in table 2a, the arguments and speeds being taken as exact multiples of those of the fundamental con- stituent. There are no theoretical expressions for the coefficients of the overtides but it is assumed that the amplitudes of the lunar series undergo variations due to changes in the longitude of the moon’s node which are analogous to those in the fundamental tide. The node factors for M,, Ms, and Msg, respectively, are taken as the square, the cube, and the fourth power of the corresponding factor for M,. For the solar terms this factor is always zero. 140. Compound tides were suggested by Helmholtz’s theory of sound waves. Innumerable combinations are possible but the prin- cipal elementary constituents involved are Ms, S:, No, Ky, and QO. Table 2a includes the compound tides listed in International Hydro- eraphic Bureau Special Publication No. 26, which is a compilation of the tidal harmonic constants for the world. The argument of a compound tide equals the sum or difference of the arguments of the elementary constituents of which it is compounded. The node factor is taken as the product of the node factors of the same con- stituents. Table 2a contains the arguments, speeds, and node factors of these tides. 141. Omitted from table 2a are a number of compound tides which have the same speeds as elementary constituents included in table 2. Thus, 2MS,, compounded by formula 2M,—S,, has the same speed as constituent uw, represented by term A,; of formula (64). Considered as a compound tide there would be a small difference in the wu of the argument and also in the node factor. Since there is no practical way of separating the elementary constituent from the compound A8 U. S. COAST AND GEODETIC SURVEY tide of the same speed, this has been treated solely as an elementary constituent. Constituent MSf represented by term A; of formula (62) has the same speed as a compound tide of formula S,—M>. This con- stituent is relatively unimportant and it makes little difference whether treated as an elementary or a compound tide. Following the pre- vious practice in this office it is treated in the harmonic analysis as a compound tide with corresponding argument and node factor. When included in the computation of tidal forces, however, the argument and node factor indicated in table 2 should be used. ANALYSIS OF OBSERVATIONS HARMONIC CONSTANTS 142. In the preceding discussion it has been shown that under the equilibrium theory the height of a theoretical tide at any place can be expressed mathematically by the sum of a number of harmonic terms involving certain astronomical data and the location of the place. It has also been pointed out that for obvious reasons the actual tide of nature does not conform to the theoretical equilibrium tide. How- ever, the tide of nature can be conceived as being composed of the sum of a number of harmonic constituents having the same periods as those found in the tide-producing force. Although the complexity of the tidal movement is too great to permit a theoretical computation ‘based upon astronomical conditions only, it is possible through the analysis of observational data at any place to obtain certain constants which can be introduced into the theoretical formulas and thus adapt them for the computation of the tide for any desired time. 143. In the formulas obtained for the height of the equilibrium tide each constituent term consists of the product of a coefficient by the cosine of an argument. For corresponding formulas expressing the actual height of the tide at any place, the entire theoretical coeffi- cient including the latitude factor and the common general coefficient is replaced by a coefficient determined from an analysis of observa- tional data for the station. This tidal coefficient, which is known as the amplitude of the constituent, is assumed to be subject to the same variations arising from changes in the longitude of the moon’s node as the coefficient of the corresponding term in the equilibrium tide. The amplitude pertaining to any particular year is usually designated by the symbol R while its mean value for an entire node period is represented by the symbol H. Amplitudes derived directly from an analysis of a limited series of observations must be multiplied by the reduction factor F (par. 78) to obtain the mean amplitudes of the harmonic constants. For the prediction of tides, the mean ampli- tudes must be multiplied by the node factor f (par. 77) to obtain the amplitudes pertaining to the year for which the predictions are to be made. 144. The phases of the constituents of the actual tide do not in general coincide with the phases of the corresponding constituents of the equilibrium tide but there may be lags varying from 0 to 360°. The interval between the high water phase of an equilibrium con- stituent and the following high water of the corresponding constituent in the actual tide is known as the phase lag or epoch of the constituent and is represented by the symbol « (kappa) which is expressed in angular measure. The amplitudes and epochs together are called harmonic constants and are the quantities sought in the harmonic analysis of tides. Each locality has a separate set of harmonic con- stants which can be derived only from observational data but which remain the same over a long period of time provided there are no 49 50 U. S. COAST AND GEODETIC SURVEY physical changes in the region that might affect the tidal conditions. 145. If we let y,; equal the height of one of the tidal constituents as referred to mean sea level, it may be represented by the following formula: y¥i=fH cos (HE—«)=fH cos (V+u—x) (236) The combination symbol V+ is the equivalent of / and represents the argument or phase of the equilibrium constituent. 146. Formula (236) is illustrated graphically in figure 7 by a cosine curve with amplitude fH. The horizontal line represents mean sea level and the vertical line through 7 may be taken to indi- cate any instant of time under consideration. If the point M/ repre- sents the time when the constituent argument equals zero, the interval from M to the following high water of the constituent will be the epoch «x. The interval from the preceding high water to M is measured by the explement of x which may be expressed as —x. The phase of the constituent argument at time TJ is reckoned from M and is expressed by the symbol (V+w). The phase of the constit- mM HW (-K)i FIGURE 7. uent itself at this time is reckoned from the preceding high water and therefore equals (V-+u—xk). OBSERVATIONAL DATA 147. The most satisfactory observational data for the harmonic analysis are from the record of an automatic tide gage that traces a continuous curve from which the height of the tide may be scaled at any desired interval of time. This record is usually tabulated to give the height of the tide at each solar hour of the series in the kind of time normally used at the place. It is important, however, that the time should be accurate and that the same system be used for the entire series of observations regardless of the fact that daylight saving tume may have been adopted temporarily for other purposes during a portion of the year. When the continuous record from an automatic gage is not available, hourly heights of the tide as observed by other methods may be used. The record should be complete with each hour of the series represented. If a part of the record has been lost, the hiatus may be filled by interpolated values; or, if the gap 1s very extensive, the record may be broken up into shorter series which do not include the defective portion. HARMONIC* ANALYSIS! AND PREDICTION OF TIDES 51 148. If hourly heights have not been observed but a record of high and low waters is available, an approximate evaluation of the more important constituents may be obtained by a special treatment. The results, however, are not nearly as satisfactory as those obtained from the hourly heights. 149. Although the hourly interval for the tabulated heights of the tide has usually been adopted as most convenient and practicable for the purposes of the harmonic analysis, a greater or less interval might. be used. A shorter interval would cause a considerable increase in the amount of work without materially increasing the accuracy of the results for the constituents usually sought. However, if an attempt were made to analyze for the short period seiches a closer interval would be necessary. An interval greater than one hour would lessen the work of the analysis but would not be sufficient for the satisfactory development of the overtides. 150. In selecting the length of series of observations for the purpose of the analysis, consideration has been given to the fact that the pro- cedure is most effective in separating two constituents from each other when the length of series is an exact multiple of the synodic period of these constituents. By synodic period is meant the interval between two consecutive conjunctions of like phases. Thus, if the speeds of the two constituents in degrees per solar hour are represented by a and b, the synodic period will equal 360°/ (a~b) hours. If there were only two constituents in the tide the best length of series could be easily fixed, but in the actual tide there are many constituents and the length of series most effective in one case may not be best adapted to: another case. It is therefore necessary to adopt a length that is a compromise of the synodic periods involved, consideration being given to the relative importance of the different constituents. 151. Fortunately, the exact length of series is not of essential im- portance and for convenience all series may be taken to include an integral number of days. Theoretically, different lengths of series should be used in seeking different constituents, but practically it is more convenient to use the same length for all constituents, an excep- tion being made in the case of a very short series. The longer the series of observations the less important is its exact length. Also the greater the number of synodic periods of any two constituents the more nearly complete will be their separation from each other. Con- stituents like S, and K, which have nearly equal speeds and a synodic period of about 6 months will require a series of not less than 6 months for a satisfactory separation. On the other hand, two constituents differing greatly in speed such as a diurnal and a semidiurnal con- stituent may have a synodic period that will not greatly exceed a day, and a moderately short series of observations will include a relatively large number of synodic periods. For this reason, when selecting the length of series no special consideration need be given to the effect of a diurnal and a semidiurnal constituent upon each other. 152. The following lengths of series have been selected as conform- ing approximately to multiples of synodic periods involving the more important constituents—14, 15, 29, 58, 87, 105, 134, 163, 192, 221, 250, 279, 297, 326, 355, and 369 days. The 369-day series is considered as a standard length to be used for the analysis whenever observations covering this period are available. This length conforms very closely with multiples of the synodic periods of practically all of the short- 52 U. S. COAST AND GEODETIC SURVEY period constituents and is well adapted for the elimination of seasonal meteorological effects. When observations at any station are avail- able for a number of years, it is desirable to have separate analyses made for different years in order that the results may be compared and serve as a check on each other. Although not essential, there are certain conveniences in having each such series commence on January 1 of the year, regardless of the fact that series of consecutive years may overlap by several days because the length of series is a little longer than the calendar year. 153. If the available observations cover a period less than 369 days, the next longest series listed above which is fully covered by the observations will usually be taken, any extra days of observations being rejected. However, if the observations lack only a few hours of being equal to the next greater length, it may be advantageous to extrapolate additional hourly heights to complete the larger series. The 29-day series is usually considered as a minimum standard for short series of observations. This is a little shorter than the synodical month and a little longer than the nodical, tropical, and anomalistic months. It is the minimum length for a satisfactory development of the more important constituents. 154. For observations of less than 29 days, but more than 14 days, provisions are made for an analysis of a 14-day series for the diurnal constituents and a 15-day series for the semidiurnal constituents, the first conforming to the synodic period of constituents K, and O,, and the latter to the synodic period of Mz, and S;. Through special treatment involving a comparison with another station, it is possible to utilize even shorter series of observations. This treatment is rarely required in case of tide observations but is useful in connection with tidal currents where observations may be limited to only a few days. SUMMATIONS FOR ANALYSIS 155. The first approximate separation of the constituents of the observed tide is accomplished by a system of summations, separate summations being made for all constituents with incommensurable periods. Designating the constituent sought by A, assume that the entire series of observations is divided into periods equal to the period of A and each period is subdivided into a convenient number of equal parts, the subdivisions of each period being numbered consecutively beginning with zero at the initial mstant of each period. All subdivi- sions of like numbers will then include the same phase of constituent A but different phases for all other constituents with incommensurable speeds. The subdivisions will also include irregular variations arising from meteorological causes. By summing and averaging separately all heights corresponding to each of the numbered subdivisions over a sufficient length of time, the effects of constituents with incommensur- able periods as well as the meteorological variations will be averaged out leaving intact constituent A with its overtides. 156. The principle just described for separating constituent A from the rest of the tide is applicable if the original periods into which the series of observations is divided are taken as some multiple of constituent A period. In general practice, that multiple of the constituent period which is most nearly equal to the solar day is taken as the unit. This is the constituent day and includes one or more HARMONIC ANALYSIS!) AND PREDICTION OF TIDES 53 periods according to whether the constituent is diurnal, semidiurnal, etc. The constituent day is divided into 24 equal parts, the beginning of each part being numbered consecutively from 0 to 23 and these are known as constituent hours. 157. To carry out strictly the plan described above would require separate tabulations of the heights of the tide at different intervals for all constituents of incommensurable periods, a procedure involving an enormous amount of work. In actual practice the tabulated solar hourly heights are used for all of the summations, these heights being assigned to the nearest constituent hour. Corrections are afterwards applied to take account of any systematic error in this approximation. 158. There are two systems for the distribution and assignment of the solar hourly heights which differ slightly in detail. In the system ordinarily used and which is sometimes called the standard system, each solar hourly height is used once, and once only, by being assigned to its nearest constituent hour. By this system some constituent hours will be assigned two consecutive solar hourly heights or receive no assignment according to whether the constituent day is longer or shorter than the solar day. In the other system of distribution, each constituent hour receives one and only one solar hourly height neces- sitating the occasional rejection or double assignment of a solar hourly height. The difference in the results obtained from the two systems is practically negligible but the first system is generally used as it affords a quick method of checking the summations. STENCILS 159. The distribution of the tabulated solar hourly heights of the tide for the purpose of the harmonic analysis is conveniently accom- plished by a system of stencils (fig. 10) which were devised by L. P. Shidy of the Coast and Geodetic Survey early in 1885 (Report of U.S. Coast and Geodetic Survey, 1893, vol. I, p. 108). Although the original construction of the stencils involves considerable work, they are serviceable for many years and have resulted in a very great saving of labor. These stencils are cut from the same forms which are used for the tabulation of the hourly heights of the tide and 106 sheets are required for the summation of a 369-day series of observa- tions for a single constituent. Separate sets are provided for different constituents. Constituents with commensurable periods are included in a single summation and no stencils are required for constituents, eh So, on etc. 160. The use of the stencils makes a standardized form for the tabulation of the hourly heights essential. This form (fig. 9) is a sheet 8 by 10/4 inches, with spaces arranged for the tabulation of the 24 hourly heichte of each day in a vertical column, with 7 days of record on each page. The hours of the day are numbered consecu- tively from 0° at midnight to 23" at 11 p.m. When the tabulated heights are entered, each day is indicated by its calendar date and also by a serial number commencing with 1 as the first day of series. The days on the stencil sheets are numbered serially to correspond with the tabulation sheets and may be used for any series regardless of the calendar dates. 161. The openings in the stencils are numbered to indicate the constituent hours that correspond most closely with the times of the 54 U. S. COAST AND GEODETIC SURVEY height values showing through the openings when the stencil is applied to the tabulations. Openings applying to the same constituent hour are connected by a ruled line which clearly indicates to the eye the tabular heights which are to be summed together. For convenience in construction two stencil sheets are prepared for each page of tabulations, one sheet providing for the even constituent hours and the other sheet for the odd constituent hours. 162. The stencils are adapted for use with tabulations made in any kind of time provided the time used is uniform for the entire series of observations. For convenience the tabulations are usually made in the standard time of the place. The series to be analyzed, however, must commence with the zero hour of the day and this is also taken as the zero constituent hour for each constituent. Suc- cessive solar hours will fall either earlier or later than the correspond- ing constituent hour according to whether the constituent day is longer or shorter than the solar day. 163. For the construction of the stencils it is necessary to calculate the constituent hour that most nearly coincides with each solar hour of the series. Let a=speed or rate of change in argument of constituent sought in degrees per solar hour. p=number of constituent periods in constituent day; 1 for diurnal tides, 2 for semidiurnal tides, ete. sh=number of solar hour reckoned from 0 at beginning of each solar day. shs=number of solar hour reckoned from 0 at beginning of series. dos=day of series counting from 1 as the first day. ch=number of constituent hour reckoned from 0 at beginning of each constituent day. chs=number of constituent hour reckoned from 0 at beginning of series. Then ‘ : 360 1 constituent period=—— solar hours. (237) 1 constituent day = solar hours. (238) : 15p 1 constituent hour oa solar hours. (239) 1 solar hour Er constituent hours. (240) Therefore, a a = = as 241 (chs)=]7 eae) 7 apes (dos) —1}+(sh)] (241) 164. The above formula gives the constituent hour of the series (chs) corresponding to any solar hour of the series (shs). The observed heights of the tide being tabulated for the exact solar hours of the day, the (shs) with which we are concerned will represent successive integers counting from 0 at the beginning of the series. The (chs) as derived from the formula will generally be a mixed number. As HARMONIC ANALYSIS! AND PREDICTION OF TIDES 55 it is desired to obtain the integral constituent hour corresponding most nearly with each solar hour, the (chs) should be taken to the nearest integer by rejecting a fraction less than 0.5, or counting as an extra hour a fraction greater than 0.5, or adopting the usual rule for computations if the fraction is exactly 0.5. The constituent hour of the constituent day (ch) required for the construction of the stencils may be obtained by rejecting multiples of 24 from the (chs). 165. In the application of the above formula it will be found that the integral constituent hour will differ from the corresponding solar hour by a constant for a succession of solar hours, and then, with the difference changed by one, it will continue as a constant for an- other group of solar hours, etc. This fact is an aid in the prepara- tion of a table of constituent hours corresponding to the solar hours ef the series, as it renders it unnecessary to make an independent calculation for each hour. Instead of using the above formuta for each value the time when the difference between the solar and con- stituent hours changes may be determined. The application of the differences to the solar hours will then give the desired constituent hours. 166. Formula (241) is true for any value of (shs), whether integral or fractional. It represents the constituent time of any instant in the series of observations in terms of the solar time of that same instant, both kinds of time being reckoned from the beginning of the series as the zero hour. The difference between the constituent and the solar time of any instant may therefore be expressed by the following formula: a~ldp 15p 167. If the constituent day is shorter than the solar day, the speed a will be greater than 15p, and the constituent hour as reckoned from the beginning of the series will be greater than the solar hour of the same instant. If the constituent day is longer than the solar day the constituent hour at any instant will be less than the solar hour of the same instant. At the beginning of the series the difference between the constituent and solar time will be zero, but the difference will increase uniformly with the time of the series. As long as the difference does not exceed 0.5 of an hour the integral constituent hours will be designated by the same ordinals as the integral solar hours with which they most nearly coincide. Differences between 0.5 and 1.5 will be represented by the integer 1, differences between 1.5 and 2.5 by the integer 2, etc. If we let d represent the integral difference, the time when the difference changes from (d—1) to d, will be the time when the difference derived from formula (242) equals (d—0.5). Substituting this in the formula, we may obtain 15p a~15p Difference=75- (shs) ~ (shs) = (shs) (242) (shs) = (d—0.5) (243) in which (shs) represents the solar time when the integral difference between the constituent and solar time will change by one hour from (d—1) tod. By substituting successively the integers 1, 2, 3, etc., for d in the formula (243) the time of each change throughout the series may be obtained. The value of (shs) thus obtained will 56 “U. S. COAST AND GEODETIC SURVEY generally be a mixed number; that is to say, the times of the changes will usually come between integral solar hours. The first integral solar hour after the change will be the one to which the new difference will apply if the usual system of distribution is to be adopted. In this case we are not concerned with the exact value of the fractional part of (shs) but need note only the integral hours between which this value falls. 168. If, however, the second system of distribution should be desired, it should be noted whether the fractional part of (shs) is greater or less than 0.5 hour. With a constituent day shorter than the solar day and the differences of formula (242) increasing positively, the application of the differences to the consecutive solar hours will result in the jumping or omission of a constituent hour at each change of difference. Under the second system of distribution each of the hours must be represented, and it will therefore be necessary in this case to apply two consecutive differences to the same solar hour to represent two consecutive constituent hours. The solar hour selected for this double use will be the one occurring nearest to the time of change of differences. If the fractional part of the (shs) in (248) is less than 0.5 hour, the old and new differences will both be applied to the preceding integral solar hour; but if the fraction is greater than 0.5 hour the old and new differences will be applied to the integral solar hour following the change. 169. With a constituent day longer than the solar day and the differ- ences of formula (242) increasing negatively, the application of the differences to the consecutive solar hours will result in two solar hours being assigned to the same constituent hour at each change of differences. Under the second system of distribution this must be avoided by the rejection of one of the solar hours. In this case the integral solar hour nearest the time of change will be rejected, since at the time of change the difference between the integral and the true difference is a maximum. Thus, if the fractional part of the (shs), is less than 0.5 hour, the preceding solar hour will be rejected; but if the fraction is greater than 0.5 hour the next following solar hour will be rejected. 170. Table 31, computed from formula (243), gives the first solar hour of the group to which each difference applies when the usual system of distribution is adopted. Multiples of 24 have been rejected from the differences, since we are concerned only with the constituent hour of the constituent day rather than with the constituent hour of the series, and these differences may be applied directly to the solar hours of the day. For convenience equivalent positive and negative differences are given. By using the negative difference when it does not exceed the solar hour to which it is to be applied, and at other times using the positive difference, the necessity for adding or rejecting multiples of 24 hours from the results is avoided. 171. The tabulated solar hour is the integer hour that immediately follows the value for the (shs) is formula (243). An asterisk (*) indicates that the fractional part of the (shs) exceeds 0.5, and that the tabular hour is therefore the one nearest the exact value of (shs). If the second system for the distribution of the hourly heights is adopted, the solar hours marked with the asterisk will be used with both old and new difference to represent two constituent hours, or will be rejected altogether according to whether the constituent day HARMONIC ANALYSIS’ AND PREDICTION OF TIDES 57 is shorter or longer than the solar day. If the tabular hour is un- marked, the same rule of double use or rejection will apply to the untabulated solar hour immediately preceding the tabular unmarked hour. For the ordinary stencils no attention need be given to the asterisks. By the formula constituents with commensurable periods will have the same tabular values, and no distinction is made in the construction of the stencils. Thus, stencils for constituent M serve not only for M, but also for M3, My, Mg, ete. 172. For the construction of a set of stencils for any constituent a preliminary set of the hourly height forms is prepared with days of series numbered consecutively beginning with 1 and each hourly height space numbered with its constituent hour as derived by the differences in table 31. The even and odd constituent hours are then transferred to separate sets of forms and the marked spaces cut out. In the Coast and Geodetic Survey this is done by a small machine with a punch operated by a hand lever. Spaces corresponding to the same constituent hour are connected by ruled lines which are num- bered the same as the hours represented. Black ruling with red numbering is recommended, the red emphasizing the distinction between these numbers and the tabulated hourly heights which are to be summed. 173. When in use the stencils are placed one at a time on the sheets of tabulated heights, with days of series on stencils matching those on the tabulations, and all heights on the page corresponding to each constituent hour are then summed separately. For constituent S no stencils are necessary as the constituent hours in this case are identical with the solar hours. For constituents K, P, R, and T with speeds differing little from that of S, the lines joining the hourly spaces frequently become horizontal and the marginal sum previously ob- tained for constituent S becomes immediately available for the sum- mation at hand. In these cases a hole in the margin of the stencil for the sum replaces the holes for the individual heights covered by the sum. SECONDARY STENCILS 174. After the sums for certain principal constituents have been ob- tained by the stencils described in the preceding section, which for con- venience will be called the primary stencils, the summations for other constituents may be abbreviated by the use of secondary sten- cils which are designed to regroup the hourly page sums already ob- tained for one constituent into new combinations conforming to the periods of other constituents. Certain irregularities are introduced by the process, but in a long series, such as 369 days, these are for the most part eliminated, and the resulting values for the harmonic con- stants compare favorably with those obtained by use of the primary stencils directly, the differences in the results obtained by the two methods being negligible. For short series the irregularities are less likely to be eliminated, and since the labor of summing for such a series is relatively small, the abbreviated form of summing is not recommended. As the length of series increases the saving in labor by the use of the secondary stencils increases, while the irregularities due to the short process tend to disappear. It is believed that the use of the secondary stencils will be found advantageous for all series more than 6 months in length. 58 U. S. COAST AND GEODETIC SURVEY 175. In the primary summations there are obtained 24 sums for each page of tabulations, representing the 24 constituent hours of a con- stituent day. In general each sum will include 7 hourly heights, and the average interval between the first and last heights will be 6 con- stituent days. A few of the sums may, however, include a greater or less number of hourly heights within limits which may be a day greater or less than 6 constituent days. 176. Let the constituent for which summations have been made by use of the primary stencils be designated as A and the con- stituent which is to be obtained by use of the secondary stencils as B. For convenience let it be _ first assumed that the heights included in the sums for constituent A refer to the exact A-hours. This assumption is true for constituent S but only approximately true for the other constituents. It is now pro- posed to assign each hourly page sum obtained for constituent A to the integral B-hour with which it most nearly coincides. Constituent A and constituent B-hours separate at a uniform rate, and the proposed assignment will depend upon the relation of the hours on the middle day of each page of tabulations. The tabulated hourly heights on each full page of record run from zero (0) solar hour on the first day to the 23d solar hour on the seventh or last day of the page. The middle of the record on each such page is therefore at 11.5 solar hours on the fourth day, or 83.5 solar hours from the beginning of the page of record. 177. Let a and 6 represent the hourly speeds of the constituents A and B, respectively, and p and p, their respective subscripts, and let nm equal the number of the page of tabulation under consideration, beginning with number one as the first page. The middle of page m will then be [168(n—1) 4-83.5] or (168n—84.5) solar hours (244) from the beginning of the series. Since one solar hour equals a/15p constituent A-hours (formula 240), the middle of page n will also correspond to (168 — 84.5) ;5 constituent A-hours (245) from the beginning of the series. As there are 24 constituent hours in each constituent day, the middle constituent A-day of each page will commence 12 constituent A-hours earlier than the time represented by the middle of the page, or at [1682 — as 12] constituent A-hours (246) from the beginning of the series. 178. The 24 integral constituent A-hours of the middle constituent day of the page will therefore be the integral constituent A-hours which immediately follow the time indicated by the last formula. The numerical value of this formula will usually be a mixed number. Let f equal the fractional part, and let m be an integer representing the number of any integral constituent hour according to its order in the middle constituent day of each page. For each page m will have HARMONIC ANALYSIS AND PREDICTION OF TIDES 59 successive values from 1 to 24. The integral constituent A-hours falling within the middle constituent day of each page of tabulations will then be represented by the general formula. [(168n—84.5) 735 — 12—f+m] constituent A-hours (247) from the beginning of the series. ; 179. The relation of the lengths of the constituent A- and constit- uent B-hours is given by the formula 1 constituent A-hour= Pi eonstituent B-hours (248) 1 The constituent B-hour corresponding to the integral constituent A-hour of formula (247) is therefore pice De yi ature Le ee : [(168n—84.5)> ae 12 tigate egos Une B-hours (249) from the beginning of the series. The last formula will, in general, represent a mixed number. The integral constituent B-hour to which the sum for the constituent A- hour is to be assigned will be the nearest integral number represented by this formula. Let g be a fraction not greater than 0.5, which, applied either positively or negatively to the formula, will render it. an integer. 180. The assignment of the hourly page sums for constituent A- hours to the constituent B-hours may now be represented as follows, multiples of 24 hours being rejected: [(168n—84.5) 5 —12—f-+-m—multiple of 24] constituent A-hour (250) sum to be assigned to [{ (168n—84.5)-2 —12—f-+-m}”° + q—multiple of 24] constituent B- hour. P Pi (251) The difference between the constituent A-hour and the constituent -B-hour to-which the A-hour sum is-to be assigned is [f (168n—84.5);55—12—f-+m}| Po —1| +9 —multiple of 24] (252) By means of the above formula table 33 has been prepared, giving the differences to be applied to the constituent A-hours of each page to obtain the constituent B-hours with which they most nearly coincide. 181. For the construction of secondary stencils the forms designated for the compilation of the stencil sums from the primary summations may be used Because of the practical difficulties of constructing stencils with openings in adjacent line spaces it is desirable that the original compilation of the primary sums should be made so that cach alternate line in the form for stencil sums is left vacant. As with the 60 U. §. COAST AND GEODETIC SURVEY primary stencils, it will generally be found convenient to use two stencils for each page of the compiled primary sums, although in some cases it may be found desirable to use more than two stencils in order to separate more clearly the groups to be summed. The actual construction of the secondary stencils is similar to that of the primary stencils. A preliminary set of forms is filled out with constituent B- hours as derived by differences from table 33 applied to the constit- uent A-hours. The odd and even constituent B-hours are then transferred to separate forms and the spaces indicated cut out. The openings corresponding to the same constituent b-hour are connected with ruled lines and numbered to accord with the constituent hour represented. The page numbering corresponding to the page num- bering on the compiled primary sums and referring to the pages of the original tabulated hourly heights is to be entered in the column provided near the left margin of the stencil. 182. In using the stencils each sheet is to be applied to the page of compiled primary sums having the same page numbering in the left- hand column as is given on the stencil. The primary sums applying to the same constituent B-hour are added and the results brought together in a stencil sum form, where the totals and means are ob- tained. A table of divisors for obtaining the means may be readily derived as follows: In a set of stencil sum forms corresponding to those used for the compilation of constituent A primary sums the number of hourly heights included in each primary sum is entered in the space corresponding to that used for such primary sum. The secondary stencils for constituent B are then applied and the sums of the numbers obtained and compiled in the same manner as that in which the constituent B height sums are obtained. The divisors having been once obtained are applicable for all series of the same length. 183. In the analysis the means obtained by use of the secondary stencils may be treated as though obtained directly by the primary summations except that a special augmenting factor, to be discussed later, must be applied. The closeness of the agreement between the hourly means obtained by use of the secondary stencils and those obtained directly by use of primary stencils will depend to a large extent upon the relation of the speeds of constituents A and B. The smaller the difference in the speeds the closer will be the agreement. 184. To determine the extreme difference in the time of an indi- vidual hourly height and of the B-hour to which it is assigned by the secondary stencils, let an assumed case be first considered in which the tabulated heights coincide exactly with the integral A-hours, and that on the middle day of the page of tabulated hourly heights one of the integral B-hours coincides exactly with an A-hour. At the corresponding A-hour, one A-day later, the B-hour will have increased by 24 we constituent B-hours. Rejecting a multiple of 24 hours, 1 this becomes n4(P"—1), so that at the end of one A-day after the 1 coincidence of integral hours of constituents A and B the constituent A hourly height will differ in time from the integral constituent B-hour to which it is to be assigned by 2a( -1) constituent B-hours. 1 HARMONIC ANALYSIS AND PREDICTION OF TIDES 61 At the end of the third A-day this difference becomes 72 -1) 1 constituent B-hours. The same difference with opposite sign will apply to the third constituent day before the middle day of the page. Now, taking account of the fact that the B-hour on the middle day of the page may differ by an amount as great 0.5 of a B-hour from the integral A-hour, and that the integral A-hour may differ as much as 0.5 of a constituent A, or 0.5 pb/p,a of a constituent B hour from the time of the actual observation of the solar hourly height, the extreme difference between the time of observation of an hourly height and the time represented by the B-hour with which this height is grouped by the secondary stencils may be represented by the formula pb ) pb ] : —w~w] Bil = - é + | 722 +0 aa 1 ) constituent B-hours. (253) The differences may be either positive or negative, and in a Jong series it may reasonably be expected that the number of positive and negative values will be approximately equal. 185. The above formula for the extreme difference furnishes a criterion by which to judge, to some extent, the reliability of the method. Testing the following schedule of constituents for which it is proposed to use the secondary stencils, the extreme differences as indicated are obtained. The differences are expressed in con- stituent B-hours and also in constituent B-degrees. It will be noted that one constituent hour is equivalent to a change of 15° in the phase of a diurnal constituent, 30° in the phase of a semidiurnal constituent, etc. Constituent Ab: peers ss J 8 Constituent Bae wee ee 0O 28M Ky Ke Re T: Pi Difference in hours_____________ 3. 58 1.36 1.20 1.20 1.10 1.10 1. 20 Difference in degrees__________- 54 41 18 36 33 33 18 Constitwent: At: seer Se ee ee se L 2MK ConstitwentpBese see Ds Pe SST ee i ee MS da MK MN vo Ne Differences MOuTSy eee ee ee 1.09 1.18 1.43 1.24 1. 26 1.45 Difference in degrees_____-_---___--------- 65 35 64 74 | 38 44 WOnSEUGTE TEARS See ee ae ee ea eee oO Constitventyi nest See sels AeA eet Saas eee [t) 2N pl Q 2Q Mifferencenunghourse =. ee eke CERNE eRe ey ihe al 1. 02 3.42 3. 79 6. 58 i 51 57 99 MD ifferencerinidesrees <2 ie Bs 36 31 186. In the ordinary primary summation the extreme difference between the time of the observation of a solar hourly height and the intregal constituent hour to which it is assigned is one-half of a con- stituent hour and, represented by constituent degrees, it is 7.5° for diurnal, 15° for semidiurnal, 22.5° for terdiurnal, 30° for quarter 62 U. S. COAST AND GEODETIC SURVEY diurnal, 45° for sixth-diurnal, and 60° for eighth-diurnal constituents. By the above schedule it will be noted that the extreme difference exceeds 60° in only a few cases. The largest difference is 99° for constituent 2Q when based upon the primary summations for O. This is asmall and unimportant constituent, and heretofore no analysis has been made for it, the value of its harmonic constants being in- ferred from those of constituent O. Although theoretically too small to justify a primary summation in general practice, the lesser work involved in the secondary summations may produce constants for this constituent which will be more satisfactory than the inferred constants. FOURIER SERIES 187. A series involving only sines and cosines of whole multiples of a varying angle is generally known as the Fourier series. Such a series is of the form h=H,+C, cos 6+ ©, cos 26+ 0; cos 36+ ______ +S, sin 6+ S, sin 20+S; sin 36+______ It can be shown that by taking a sufficient number of terms the Fourier series may be made to represent any periodic function of 6. This series may be written also in the following form: h=H)+ A; cos (8+ a;)+ Az cos (20+ a,)+ Az cos (80+ a3) +__-._ (255) in which (254) An=[Cn?+Sn’|? and an=-—tan! = m being the subscript of any term. 188. From the summations for any constituent 24 hourly means are obtained, these means being the approximate heights of the constituent tide at given intervals of time. These mean constituent hourly heights, together with the intermediate heights, may be represented by the Fourier series, in which Hy=mean value of the function corresponding to the height of mean sea level above the adopted datum. é=an angle that changes uniformly with time and completes a cycle of 360° in one constituent day. The values of 6 corresponding to the 24 hourly means will be 0°, 15°, 30°, _ _ _ - 330°, and 345°. Formula (254), or its equivalent (255), is the equation of a curve with the values of @ as the abscisse and the corresponding values of h as the ordinates. If the 24 constituent hourly means are plotted as ordinates corresponding to the values of 0°, 15°, 30°, _ _ _ — for @, it is possible to find values for Hy, Cn, and Sm, which when substituted in (255) will give the equation of a curve that will pass exactly through each of the 24 points representing these means. 189. In order to make the following discussion more general, let it be assumed that the period of 6 has been divided into n equal parts, and that the ordinate or value of A pertaining to the beginning of each of those parts is known. Let w equal the interval between these ordi- nates, then NM U=27, or 360° (256) Let the given ordinates be ho, hi, hy ---- h «1 corresponding to the abscissae 0, u, 2u ____ (n—1) u, respectively. HARMONIC ANALYSIS AND PREDICTION OF TIDES 63 It is now proposed to show that the curve represented by the following Fourier series will pass through the n points of which the ordinates are given: ‘h=H+C, cos 0+ OC, cos 26+ -_-_________- CO, cos k 6 + §, sin 6+ S, sin 20+ _____ Le S, sin l 6 m=k m=/ =HM)+ >)5 Cn cos m6+ >) Sp sin mé (257) m=1 m=1 ; , See i. ‘ n—1. : in which the limit k=5 if n is an even number, or k= 5 if n is an odd number; and the limit l=5-1 if m is even, Of ae Dat m 1s odd, 190. By substituting successively the coordinates of the n given points in (257) we may obtain n equations of the form m=k m=! ha=Ho+ >) Cn cos mau+ >) S,, sin mau (258) m=1 m=1 in which a represents successively the integers 0 to (n—1). By the solution of these n equations the values of n unknown quantities may be obtained, including H, and the (n—1) values for Cy and Sy. It will be noted that the sum of the limits k and 1 of (257) or (258) equals (n—1) for both even and odd values of n. 191. The reason for these limits is as follows: A continued series 2 C,, cos m a u may be written C, cosautC, cos2au+___.+C, cosnau + Cas cos (n+1) aUutCas., cos (n+2) au+___._+C,,cos2nau + Cent) COS (2n+1) @ U+C en+12, Cos (2n-+2) a U+ ___- +03, cos3 NAU Ee See NN eR pgm le oa! ah aye RIAL. ett (259) Since n w=27 and a is an integer, the above may be written [C+ City + Cent + eassescs | cosau aes, Con42) + Cent2) + sececos< J cos2au (Cee Oe eCae tc. Jeos(n—1)au +[(C,+Cy+Cm+-------- |cosnau (260) Since cos n a u=cos 2a r=1; cos (n—1) au=cos (2a r—a4 U)=COS AU; cos (n—2) a u=cos 2 a u; ete., (260) mav be written (C+ Con + Con I | COS 0 (Ona Cia Cepan ans aoe: =e Sr Olin OL Ca Opel OCC i SS55 S555 ] cos au a [C; = Cnt) at Cons) =P BSeSSs= os + Cn —2) + C en-2) + Cian—2) + ee |cos2au Hl Creer Cipanyar Cease as = 222s Ca-1 + Cony + Cen-1 + --e- eee ] cos k au (261) The first term of the above is a constant which will be included with the H, in the solution of (258). From an examination of (261) it is evident ee the cosine terms will be completely represented when == — oy one a? according to whether 7 is even or odd. ee the continued series = S,, sin m a u may be written 64 U. S. COAST AND GEODETIC SURVEY iSeate Son San+ SSSR 2e5 | sin 0 +[Si +S ei) tSenpy+ Sa aoocse : er Se ae (Ol me Gal) aan eee ] sin a u sh [So-+ S riz) + S en+2) a= -------- ; — @—2) — @n=2) S Gn=2) a = - = -- == ]sin2au +[Si+Sesryp+tSenty+ ; 1D) a (on—l) — Sign— 1) — Ab Lads ae ] snlau (262) The first term in the above equals zero. The remaining terms will take complete account of the series = S,, sin m a u, if a when n is even, or oe when n is odd. From the foregoing it is evident that the limit of m will not exceed 5 192. If we let u and a represent any angles with fixed values, m and p any integers with fixed values, and a an integer having successive values from 0 to (n—1), it may be shown that a=(n—1) , nsnmu . >> sin (am tee = sin [3 (n—1) m uta] (263) a=0 2 a=(n—1) inl m ; >) cos (am a-+-a)= 2, — cos [4 (n—1) mu+al] (264) oe sin 4 m u a=(n—1) int ‘y—m , i ae, pea >> smavusinam Ge pe ee eee aD i) bs cost Ati ieee a—o sin 2 (p—m) WU ts sin 3 2 (pm) L COs 3 (n—1) (p+m) u (265) sin 5 (p—m) u a=(n—l) sin 3 n (p—m) u cos 3(n—1) (p—m) uw beats 2 2 Za cosa pucosamu=%z ann (ee 1 sin 3.7 (p+m) u cos 3 (n—1) (P+m) u 3 sin 4 (p+m) u RD) a ine ol) ee sin 4 n (p—m) u sin 4 (n—1) (p—m) u Balas 2 \ 2 2a sn apucosamu=%z Ss ee u sin 4” (pm) i sin 3°(n—1) (ptm) u (267) sin $ (p—m) wu 193. If we let a=0 and we", or 7 u=27, then formulas (263) to (267) may be written as follows: a=(n—1) | > sin a m U= =O Sat SIO SS 15 Nn sin m 7m sin (1 1" r) ———— (268) F m sin mm cos{ m r—— a=(n-1) 5 ( n ) >) cos am Y= (269) 73 on th ane sin — + n HARMONIC ANALYSIS!) AND PREDICTION OF TIDES 65 sin (p—m) 7 cos | w—m) 7 r| a. ' BS 4 sindpusinamu=i ae fg CL T n sin (p-++m) 7m cos [ +m) aay aE r| —} == " (270) ane n s es sin (p—m) 7 cos | om) —— r| COS @ PU COS @ MW AANA ________-~ nel sin 2—™ 5 nN sin (p+m) 7 cos [ @+m) n—Pim | Sie epee a marae (271) sin T pean) sin (p—m) 7 sin | em) i r| De SCO ce ee ea a7 sin 2 T sin (p-+m) = sin | o+m) n—Pe™ | i sin 1 n 194. If p and m are unequal integers and neither exceeds > the above (268) to (272) become equal to zero. Thus, a=(n—1) sina m u=0 a=o a=(n—1) >) cosam u=0 a=(n—1) x : >) sinapusinam u=0 (273) ase) cosapucosam u=0 ee >) sna pu cosa m u=0 a=0 195. If p and m are equal integers and do not exceed > formulas (270), (271), and (272) will contain the indeterminate quantity Bas => and also when p and m each equal > the indetermin- Tv sin sin (p+m)m_ 0 ate quantity = sin oe p 66 U. §, COAST AND GEODETIC SURVEY Evaluating these quantities we have sin (p—m)r __& COS (p—m)x @—m)=—0 7 19 \@—m)—o sin tm | _@ cos ta, | om SUN Sag Ploy aay COST aa. “Guiness In (275) it will be noted that when the integers p and m each equal , p=m sin ee n and > m must be an even number, and therefore cos nz is positive, while COS 7 1S negative. 196. Assuming the condition that p and m are equal integers, each less than ~, we have by substituting (274) in (270), (271), and (272), Oy) a=(02=1) : a=(n—1) | 4 De sin @ pwn Mm a— >) sina mi w— 4 (276) a=o0 a=0 a=(n—1) a=(n—1) Dy) cos ar pw costa m a— — >a cost aim u— sin (277) a=0 a=0 ae (n—1) ) a=(n-—1) , by sin @ p wicos am w—— >) sin am Wicostarmar—O0 (278) a=0 a=90 197. Assuming the condition that p and m are each equal to 3 we have by substituting (274) and (275) in (270), (271), and (272), a=(n-—1) sin? a m u=4n+3n cos r=0 (279) a=o a=(n—1) 2 cos? a m u=4 n—}4 nN COS T=N (280) a=0 a=(a—1) sin am u cos am u=0 (281) a=0 198. Returning now to the solution of (258), by substituting the successive values of a from 0 to (n—1), we have h=H,+C; cos 0+ C, cos 04-_______- +0; cos 0 +S, sin 0+8S, sin 0+ _______- +S; sin 0 h,=H,+C, cos u+C, cos 2u+_______- +0, cos ku +S, sin ut S, sin 2u+______-- +S, sin lu hz =H,+ C0, cos 2u+C, cos 4u+_____-_- +0, cos 2ku JESh sin PHB SS sin Zs +§, sin 2lu/ (282) hea-y=H,+C; cos (n—1)u+C, cos 2(n—1)u+ ____-- +0, cos (n—1)ku +S, sin (n—1)u+ 8) sin 2(n—1)u-+-------- +S; sin (n—1)lu HARMONIC ANALYSIS! AND PREDICTION OF TIDES 67 199. To obtain value of H,, add above equations a=(n—1) See —n A, a=0 co (n—1) a=(n—1) 2s as GOL >) 60520 TE Sion ds de gee +C, >) cosaku a=0 a=(n—1) a=(n—1) | Sy sin a w+, > sin 2 qu pert oP +S, >>} snalu : a=0 a= (n— = a=(2—}) ==) H.+> 0 S00 am >> Sn 25 sin amu (283) a —1) =(n—1) From (273), > cos am u and >) sin am u each equals zero, a=0 since neither k nor J, the maximum values of m exceeds 5 Therefore a=(n—1) pp. —ele (284) and y oe 285 One 7 a= a ( ) 200. To obtain the value of any coefficient C, such as C,, multiply each equation of (282) by cosa pu. Then hy cos O=H) cos 0 +0, cos 0+C, cos 0+ _______- +0; cos 0 +S, sin 0+8, sin 0+ _______- +S, sin 0 h, cos p u=H cos p u +C, cos u cos p u+C, cos 2u cos p u+__~--- +C;,, cos k u cos p wu +8, sin u cos p u+S, sin 2u cos p u+ -------- +S, sinl wu cos pu hz cos 2p u= Hh cos 2p u +(C, cos 2u cos 2p u+C; cos 4u cos 2p u+__------ +C;, cos 2k u cos 2p wu +; sin 2u cos 2p u+S, sin 4u cos 2p u+___----- +S; sin 21 u cos 2p u Die ve cos (n—1) p u=H, cos (n—1) p +C, cos (aon, u cos (n—1) p HG cos (n—1) ucos (n—1) put +C;, cos (n—1) k u cos (n—1) p +S; sin (n—1) uw cos (n—1) p noted sin 2 (n—1) ucos (n—1) pu+_- +S, sin (n—1)lucos (n—1) pu (286) Summing the above equations a=(n—1) a=(n—1) DI cosa — ea Cos a pew a=0 a=0 a=(n—1) a=(n—1) | >) cosaucosapu+tS,; >) smaucosapu a=o (Formula continued next page) 68 U. S. COAST AND GEODETIC SURVEY a=(n—1) a=(n—1) +C, >) cos2aucosapu+sS, >) sin 2aucosa pu = a=o0 3 a=0 a=(n—1) a=(n—1) +C, >5 cosak u cosa put+S; >) snalucosapu a=0 a=0 a=(n—1) m=k a=(n—1) —H, >> cosa p u+ Cn >) cosamucosapu a=0 In=1 a=0 m=l1 a=(n—1) ; -{- Sn D>) Shamucosapyu (287) m=1 a=o 5 201. Examining the limits of (287), it will be noted by a reference to page 63 that k, the maximum value of m for the C terms is - when n is even and 1 when n is odd; also, that / has a value of 57! when n is even and —— d when n is odd. The limits of p, which is a partic- ular value of m, will, of course, be the same as those of m. =(n—1) By (273) the quantity = cos a p u becomes zero for all the 3 a=(n—1) values of p, and the quantity >} cosa mucosa p u becomes zero a=0 for all values of m and p except when p equals m. By (273), (278) a=(n—1) and (281) the quantity 24 sin a mu cos a p u becomes zero for all values of m and p Formula (287) may therefore be reduced to the form a=(n—1) a=(n—1) >> hs cosa pu=C,. >3 cos* a pu (288) a=o0 a=0 For any value of p less than 5 a=(n—1) De costa pu — ne ae) a=0o but when P=5) this quantity becomes equal to n (280). Therefore for all values of p less than a 2 2 a=(n—1) Q=- h, cosa pu (289) NM a=0 but when 7 is exactly > 1 a=(n—1) Oy >> hacosapu (290) a=0 Since in tidal work 7 is always taken less than ~, we are not especially 9? concerned with the latter formula. HARMONIC ANALYSIS!) AND PREDICTION OF TIDES 69 202. To obtain the value of any coefficient S, such as S,, multiply each equation of (282) by sina p wu. obtain a=(n—1) >S h.snapw =A a=0 aS wie By (278), (278), and (281) the quantities a=(n—1) 2 a=o a=(n—1) and") Ss, a=0 and p except when m and p are equal. Sm a=(n—1) Pas a=o0 Cn sn ap u a=(n—1) : >> cosamusinapu a=0 a=(n—1) ; >) sinamusinapu a=(n—1) Bs a=0 a= 2 m and » is less than 5 = and by (276), the quantity a=0 and S,= Therefore, Aas (291) reduces to the form a=(n—1) h, sina p u=4 nS) 2 x te GEO oD a=0 Sum the resulting equations and (291) sin a p wu and cos am wu sin a p u are zero for all the values of m and p; sin @ m usin a p u becomes zero for all the values of m In this case the limit of / for (n—1) | sin? a p u a=0 (292) (293) 203. By substituting (285), (289), (290), and (293) in (257), the following equation of a curve, which will pass through the n given points, will be obtained © Il a B I ey ahs) ) cos a w| cos 6 9 ll ° ~o ll ol), 3 h, sin a w | sin 6 p © Ml —-~ Oo B —1) h, cos 2 au|cos 20 » © tol B (=) —1) Sip sin2au|sin 20 —1) h, coska | oO =(n—1) ; ha sin law |sinJ@ a=0 1 *If n is even and k= > this fraction is — instead of 2. n (294) 70 U. S. COAST AND GEODETIC SURVEY 204. Although by taking a sufficient number of terms the Fourier series may thus be made to represent a curve which will be exactly satisfied by the n given ordinates, this is, in general, neither necessary nor desirable in tidal work, since it is known that the mean ordinates obtained from the summations of the hourly heights of the tide in- clude many irregularities due to the imperfect elimination of the me- teorological effects and also residual effects of constituents having periods incommensurable with that of the constituent sought. It is desirable to include only the terms of the series which represent the true periodic elements of the constituent. With series of observations of sufficient length, the coefficient of the other terms, if sought, will be found to approximate to zero. 205. The short-period constituents as derived from the equilibrium theory are, in general, either diurnal or semidiurnal. If the period of 6 in formula (257) is taken to correspond to the constituent day, the diurnal constituents will be represented by the terms with coefficient CO, and S,, and the semidiurnal constituents by the terms with co- efficients C, and S;. For the long-period constituents, the period of 6 may be taken to correspond to the constituent month or to the constituent year, in which case the coefficients C, and S, wil] refer to the monthly or annual constituents and the coefficients C, and S, to the semimonthly or semiannual constituents. For most of the constituents the coefficients C,, S,, C2, and S, will be the only ones required, but for the tides depending upon the fourth power of the moon’s parallax and for the overtides and the compound tides, other coefficients will be required. Terms beyond those with coefficients OC, and S;, for the overtides of the principal lunar constituent are not generally used in tidal work. 206. When it is known that certain periodic elements exist in a constituent tide and that the mean ordinates obtained from obser- vations include accidental errors that are not periodic, it may be readily shown by the method known as the least square adjustment, using the observational equations represented by (258), that the most probable values of the constant H, and the coefficients C, and S, are the same as those given by formulas (285), (289), and (293), respectively. 207. Since in tidal work the value of H,, which is the elevation of mean sea level above the datum of observations, is generally deter- mined directly from the original tabulation of hourly heights, formula (285) is unnecessary except for checking purposes. Formulas (289) and (293) are used for obtaining the most probable values of the coefficients C, and S, from the hourly means obtained from the summations. 208. When 24 hourly means are used n=24 and u=—15°, and the formulas may be written ] 8=23 O,=— >5 ha cos li ap (295) 12 a= | 8=23 F Ss==5 >) Aasin 15 ap (296) 12 a= in which the angles are expressed in degrees. If only 12 means are used, the formulas become a=l11 C=. >> haz cos 30 ap (297) a=0 HARMONIC ANALYSIS AND PREDICTION OF TIDES all | a= Sh ne sis mae (298) a=0 209. The upper part of Form 194 (fig. 16) is designed for the compu- tation of the coefficients C, and S, in accordance with formulas (295) and (296) to take account of the 24 constituent hourly means. It is now desired to express each constituent in the form y=A cos (p 6+ a) (299) or using a more specialized notation by y=A cos (p 6—¢) (300) By trigonometry A cos (p 6—¢)=A cos ¢ cos p 9+ A sin ¢ sin p 0 (301) =C, cos p 6+S8, sin p 6 in which C5_ tA cost and=2:S)—AlsiniG (302) Therefore, _S» tan CSG (303) and = p= VOTE (304) cos MsiniG Substituting in formulas (303) and (304) the values of C, and S, from formulas (295) and (296), the corresponding values for A and ¢ may be obtained. Substituted in formula (300), these furnish an ap- proximate representation of one of the tidal constituents, but a further processing is necessary in order to obtain the mean amplitude and epoch of the constituent. AUGMENTING FACTORS 210. In the usual summations with the primary stencils for all the short period constituents, except constituent S, the hourly ordinates which are summed in any single group are scattered more or less uniformly over a period from one-half of a constituent hour before to one-half of a constituent hour after the exact constituent hour which the group represents. Because of this the resulting mean will differ a little from the true mean ordinate that would be obtained if all the ordinates included were read on the exact constituent hour, as with constituent S, and the amplitude obtained will be less than the true amplitude of the constituent. The factor necessary to take account of this fact is called the augmenting factor. 211. Let any constituent be represented by the curve y=A cos (at+a) (305) in which A=the true amplitude of the constituent a=the speed of the constituent (degrees per solar hours) t=variable time (expressed in solar hours) a=any constant. ee, U. S. COAST AND GEODETIC SURVEY The mean value of y for a group of consecutive ordinates from 7/2 hours before to 7/2 hours after any given time ¢, 7 being the number of solar hours covered by the group, is Ali i cos (at-+-oat—| 2° 2 sin (a tea) | _ 2 *| sin Gas (atta-¥ )| 360 A 360 aa eign COS (at+ a) sin 5=78 sin FA cos (at+ a) (306) 212. Since the true value of y at any time ¢, is equal to A cos (at+a) by (305), it is evident that the relation of this true value to the mean value (306) for the group 7 hours in length is A cos (at+a) rr TAT am sin 5 — A cos (at+ a) 360 sin 5 Gu The quantity — StS isthe augmenting factor which is to be 360 sin—,— applied to the mean ordinate to obtain the true ordinate. In the use of this factor it is assumed that all the consecutive ordinates within the time 7/2 hours before to 7/2 hours after the given time have been used in obtaining the mean. This assumption is, of course, only approximately realized in the summation for any constituent, but the longer the series of observations the more nearly to the ‘truth it approaches. 213. According to the usual summations with the primary stencils, the hourly heights included in a single group may be distributed over an interval from one-half of a constitutent hour before to one-half of a constituent hour after the hour to be represented. In this case 7 é 15, equals one constituent hour, or = solar hours. Substituting this in (307), the Tp . 1ldp which is the formula generally adopted for the short-period constituents and is the one used in the calculation of the augmenting factors in Form 194. For the long-period constituents special factors are necessary which will be explained later. 214. If the second system of distribution of the hourly heights as described on page 53 is adopted, 7 equals one solar hour and formula (307) becomes augmenting factor= Ta 360 sin 5 augmenting factor= (309) HARMONIC ANALYSIS AND PREDICTION OF TIDES: Ve It will be noted that formula (308) depends upon the value of p and therefore will be the same for all short period constituents (S excepted) with like subscripts. Formula (809) depends upon the speed a of the constituent and will therefore be different for each constituent. 215. When the secondary stencils are used, the grouping of the ordinates is less simple than that provided by the primary stencils only. Let it be assumed that the series is of sufficient length so that the distribution of the ordinates is more or less uniform in accordance with the system adopted. Suppose the original primary summations have been made for con- stituent A with speed @ and that the secondary stencils have been used for constituent B with speed 6. Then let p and p’ represent the subscripts of constituents A and B, respectively. The equation for constituent B may be written y=B cos (bt+ B) (310) 216. In the primary summation for constituent A, the group of ordi- nates included in a single sum covers a period of one constituent A hour or P solar hours. Expressed in time t, midway of this interval and representing the exact integral constituent A hour to which the group applied, the average value of the B ordinates included in such a group may be written oe a a 5p fe cos (b§-+ B) dt LO Gh Bet 4 LS DON io 3a /h 15pb = rap B | sin (bt+8+ ae )=sin (01+ 6-52) | a 15pb -(= A sin pee) B cos (bt+ B) =F, B cos (bt+ B) (311) 24a . 15pb In which F;, for brevity, is substituted for the coefficient = un and gives the relation of the average B ordinate included in the A grouping to the true B ordinate for the time ¢ represented by that group. The reciprocal of this coefficient will be that part of the augmenting factor necessary to take account of this primary grouping. If the primary summing has been for the constituent 5, this coefficient may be taken as unity since the original 5 sums refer to the exact S hour. 217. When the secondary stencils are applied to the constituent A group sums, the groups applying to an exact constituent A hour at any time ¢ and represented by that time, will be distributed over an “oP : solar hours. interval of a constituent B hour, or For an integral constituent B hour at any time ¢t within the middle day represented by a seven-day page of original tabulations the limits , le of this interval will be (SF) and (1+4P). For the same page 74 U. S. COAST AND GEODETIC SURVEY of tabulations, letting ¢ represent the same time in the middle day, the limits of the group interval for the day following the middle one, are (Pe a nd (1+ 972 +53 NEES p) If we let n=—3, —2, —1, 0, +1, +2, +3, respectively, for ne teed successive days represented by a single page of original tabulations, the limits of the group interval for any day of the page may be represented by 360pn__15p’’ | 360pn =i 4 woke and (14-7 +—- 218. Formula (311) gives the mean value of the B ordinate for grouping of the A summations. The mean value of (311) obtained by combining the groups falling in any particular day of page of tabulations in the limits indicated above is b Se isp? PB cos (bt+ B) dt 360pn _ 15p’ a OD asf OY : FB) sin(bt+ 6+ 2° mw 5p’ aoe hon sy auf 24 ye : Lop 360pn =( aa FB cos (bet epee ) =F\F)B cos Ee pp *) (312) OUI a ) 1 15p’ ; if we put R= a sin aie for brevity. 219. Formula (312) represents the mean value of the B ordinate for a particular day of the page record. The average value for the 7 days may be written n=+3 1F\F,B >) cos (4+ + 00Pt ) n=—3 = B | cos (bt +8) cos (—3 SOP.) sin (bt+ 8) sin (— 3 oe) +cos (bt+ B) cos (—2 se) sin (bp+ B) sin (— 2 20re ) +cos (bf+ 8) cos (-1 sone) — sin (bt+ 8) sin (-1 sO") +cos (bt-+ 8) cos 0—sin (b¢-+ 8) sin 0 -+cos (bt+ 8) cos (See) — sin (bt+ 8) sin sr ) (Formula continued next page) HARMONIC ANALYSIS AND PREDICTION OF TIDES 765) +eos (b¢+ 8) cos = oe sin (b¢+ 8) sin eG sayy +cos (bt+ B) cos (3 ore) — sin (bt+ 8) sin (3 ne) 360bp a +2 co s 2 200? + 9 cos uly cos (bt-+ 8) == On ible) E +2 cos sin 2 Belle cos OU 2 =1F\F,B\ 2 sia AE a ae cos (bt-+ 8) sin -——— i 1260bp a a 220. Replacing the equivalents of F; and F, in (313), the average value of the B ordinate as obtained by the secondary summations may be written sin 126060 24a : Fin OD |” cos (bt+ 8) (814) mpb~ —o 7 sin Since the true ordinate of constituent B at any time ¢ is equal to B cos (bt+ 8), the reciprocal of the bracketed coefficient will be the augmenting factor necessary to reduce the B ordinate as obtained from the summations to their true values. This augmenting factor may be written 180bp arbp wp’ [7 sin ; 24a sin uu 24 sin — | sin 126060 (315) a The first factor of the above is to be omitted if the primary sum- mations are for constituent S. It will be noted that the middle factor is the same as the augmenting factor that would be used if constituent B had been subjected to the primary summations. PHASE LAG OR EPOCH 221. The phase lag or epoch of a tidal constituent, which is repre- sented by the Greek kappa (x), is the difference between the phase of the observed constituent and the phase of its argument at the same time. This difference remains approximately constant for any con- stituent in a particular locality. The phase of a constituent argument for any time may be obtained from the argument formula in table 2 by making suitable substitutions for the astronomical elements. The argument itself is represented’ by the general symbol (V-+-u) or E and - 76 U. S. COAST AND GEODETIC SURVEY its phase or value pertaining to an initial instant of time, such as the beginning of a series of observations, is expressed by (V,+u). Refer- ring to formula (300), since @ is reckoned from the beginning of the series, the angular quantity (—{) is the corresponding phase of the observed constituent at this time. The phase lag may therefore be expressed by the following general formula: k= Votu—(—-p)=Votute (316) 222. Since the argument formulas of all short-period constituents contain some multiple of the hour angle (7) of the mean sun, the arguments themselves will have different values in different longitudes at the same instant of time. If p equals the coefficient_of 7 or the subscript_of the constituent-and LZ equals the longitude of the place “in degrees reckoned west from Greenwich, L being considered as nega- tive for east longitude, the relation between the local and Greenwich argument for any constituent may be expressed as follows: local (V-++-uv)=Greenwich (V+u)—pL (317) 223. Also, since the absolute time of the beginning of a day or the beginning of a year depends upon the time meridian used in the locality, the mitial instant taken for the beginning of a series of obser- vations may differ in different localities even though expressed in the same clock time of the same calendar day. If we let S equal the longitude of the time meridian in degrees, positive for west and nega- tive for east, the same meridian expressed in hours becomes S/15. Letting a equal the speed or hourly rate of change in the constituent argument, the difference in argument due to the difference in the absolute beginning of the series becomes a@S/15, and the relation between the local and Greenwich argument due to this difference may be expressed as follows: local (V,+u)=Greenwich (V,+u)—pL-+aS/15 (318) In the above formula the local and Greenwich (V,+«) pertain to the same clock time but not the same absolute time unless both clocks are set for the meridian of Greenwich. 224. Values of (V.+wu) for the meridian of Greenwich at the beginning of each calendar year 1850 to 2000 are given in table 15 for all constituents represented in the Coast and Geodetic Survey tide-predicting machine. Tables 16 to 18 provide differences for referring the arguments to other days and hours of the year. In the preparation of table 15 that portion of the argument included in the u was treated as a constant with a value pertaining to the middle of the calendar year. If the Greenwich (V,-++v) with its corrections is sub- stituted for the local (_V,+ wu) in formula (316), we obtain x=Greenwich (V,+u)—pL+aS/15+¢ (319) 225. The phase lag designated by « is sometimes called the local epoch to distinguish it from certain modified forms which may be used for special purposes. In the preparation of the harmonic constants for predictions it is convenient to combine the longitude and time meridian corrections with the local epoch to form a modified epoch HARMONIC ANALYSIS) AND PREDICTION OF TIDES Hi designated by k’ or by the small g._ The relation of the modified epoch to the local epoch may then be expressed by the following formula: x’ or g=x«+pL—aS/15=Greenwich (V,+u)+¢ (320) 226. The phases of the same tidal constituent in different parts of the world are not directly comparable through their local epochs since these involve the longitude of the locality. For such a comparison it. is desirable to have a Greenwich epoch that is independent of both longitude and time meridian. Such an epoch may be designated by the capital G and its relation to the corresponding local epoch ex- pressed as follows: Greenwich epoch (@)=x+pL=Greenwich (V,+u)+aS/15+¢ (321) 227. The angle x may be graphically represented by figures 7 and 8. In figure 7, we have a simple representation of a single con- is 8 ne) = ; S c - € fy 40 2 re) n . £ = g = o ae? 2 = oO ® 2¢ ie os = e E € © ‘ ae ee o 8 e os 5.0 = © a Ss £ ££ On toy} ie 2 S = Denes ed Ee == # ee rs 25 os SE ome c 0 a O re = De are) S c c (a © c= S) o O Oo o> oD om OO Ee ms = SS ~ 0 Oo 8 + vo — [= = [= Od —59) Y a2 Lt) rs pl ———»><« p(S-L) > < cL ><—¢(S-L) ><—— _ 5 —— > | <—____——_—_——.— Local V,+u wa cid NS i] cS — Local epoch (K) > r= Greenwich epoch (G) >| FIGURE 8. stituent. In this figure changes in the phase or angle are measured along the horizontal line, positive change toward the right and nega- tive change toward the left. The full vertical line indicates the beginning of the series, at which time the angle p 0, or at, equals 0. At the left of this vertical line, the symbol of a moon (M) indicates the zero value of the equilibrium argument that precedes the begin- ning of the series. For the principal lunar or solar constituent, this will be simultaneous with a transit of the mean moon (modified by longitude of moon’s node) or of the mean sun, and for other short- period constituents with the transit of a fictitious star representing such constituent (p. 23). At the point represented by this moon, the angle (V-++wu) has a value of zero. This angle increases to the right, and at the beginning of the series has a value represented by (V.+u), which may be readily computed for the beginning of any series. This interval from M to the time of occurrence of the first following constituent high water is the epoch x. This represents the lag or difference between the actual constituent high water at any . 78 U. S. COAST AND GEODETIC SURVEY place and the theoretical time as determined by the equilibrium theory. The distance from the beginning of the series to the follow- ing high water is the ¢ of formula (300), which is determined directly from the analysis of the observations. From the figure it is evident that the x is the sum of (V,+4) and ¢, and also that it is independent of the time of the beginning of the series. 228. Figure 8 gives a more detailed representation of the epoch of a constituent. In this figure the horizontal line represents changes in time. Distances along this line will be proportional to the changes in the angle of any single constituent, but since each constituent has a different speed equal distances along this line will not represent equal angles for different constituents. The time between the events may be converted into an equivalent constituent angle by multiplying by the speed of the constituent. The figure is to some extent self- explanatory. The word “‘transit”’ signifies the transit of the fictitious moon representing any constituent and also the time when the equili- brium argument of that constituent has a zero value. For all short- period constituents the time of such zero value will depend upon the longitude of the place of observation as well as upon absolute time. For long-period constituents the zero values are independent of the longitude of the place of observation, and the ‘‘transits’’ over the several meridians may be considered as occurring simultaneously, which is equivalent to taking the coefficient p equal to zero. The figure illustrates the relation between the Greenwich (V,+w) calcu- lated for the meridian of Greenwich and referring to standard Green- wich time and local (V,+) referring to the meridian of observation and the actual time of the beginning of the observations. INFERENCE OF CONSTANTS \ 229. Under the conditions assumed for the equilibrium theory the amplitudes of the constituents could be computed directly by means of the coefficient formulas without the necessity of securing tidal observations, and the phases would correspond with the equilibrium arguments of the constituents. Under the conditions that actually exist it has been found from observations that the amplitudes of the constituents of a similar type at any place, although differing greatly from their theoretical values, have a relation that, in genera], agrees fairly closely with the relations of their theoretical coefficients. It has also been ascertained from the results obtained from observations that the difference in the epochs or lags of the constituents have a relation conforming, in general, with the relation of the differences in their speeds. This last relation is based upon an assumption that the ages of the inequalities due to the disturbing influence of other constituents of a similar type are equal when expressed in time. 230. If the mean amplitudes, epochs, and speeds of several constit- uents A, B, C, are represented by H(A), H(B), H(C), x(A), «(B), «(C), and a, 6, c, respectively, the above relations may be expressed by the following formulas: mean coefficient of P mean coefficient of AHA) (322) eG) — HARMONIC ANALYSIS AND PREDICTION OF TIDES 79 K(C) —K(A) = 5 qle(B) —«(A)}] (323) or, «(C) =x(A) +5 —[e(B) —x(A)] (324) By formula (322) the amplitude of a constituent (B) may be inferred from the known amplitude of a constituent (A), and by formula (324) the epoch of a constituent (C) may be inferred from the known epochs of constituents (A) and (B). 231. These formulas have, however, certain limitations. They are not applicable to shallow water and meteorological constituents, nor are they adapted to the determination of a diurnal constituent from a semidiurnal constituent: or of a semidiurnal constituent from a diurnal constituent. The results obtained by the application of the formulas to tides of similar type may be considered only as rough approximations to the truth. They may, however, be preferable to the values obtained for certain constituents when the series of obser- vations is short. 232. By substituting the mean values of the coefficients and the speeds from table 2 the following special formulas may be derived from the general formulas (322) and (324) Diurnal constituents iG AO gaME (On ca) aK) 20.496 [x (KG)— «(,)) (325) H(OO)=0.043 H(O,); «(OO)=«(K,) +1.000 [«(K;) —«(O,)] (327) GE) =O.S31 IEG) e eB) =) OOS IC On) (328) A(Q,) =0.194 H(QO)); (Qi) =«(K,) —1.496 [«(K,) —«(O;)] (329) H(2Q) =0.026 H(O,); «(2Q) =«(K,) —1.992 [x(K,) —«(O;)] (330) F(p;) =0.038 H(O;); «(o:) =«(&i)—1.429 [«(Ky) —«(O;)] (331) Semidiurnal constituents =0.143 H(N,) ; =x(M2)+1.000 [e( Ma) —x(N) | (334) FI(N,) =0.194 H(M)); «(N2) =x(S.) —1.536 [xk(S.) —x(M.)] = (335) H(2N)=0.026 H(M.); «(2N)=«(S,) —2.072 [x(S)) —«(M,)] (336) =0.133 H(N;); = (Mz) —2.000 [x(M,) —«(N,)] (337) H(R2) =0.008 H(S2); «(R2) =«x(S2) +0.040 [k(S.) —x«(M,)] (338) ) =0.059 A(S,); «(T2) =«x(S.) —0.040 [x(S.) —«(M,)] (339) FAI(:) =0.007 H(M2); «(A2) =x«(S2) —0.536 [k(S.) —«(Mb2)] (340) Hu.) =0.024 H(M,); «(u2) =«(S:) —2.000 [«(S.) —«(M,)] (341) H(,) =0.038 H(M,); (v2) =«(So) —1.464 [x(S.) —«(M>)] (342) —0.194 H(N,); =«(M>»)—0.866 [«(M2) —x(N2)] = (343) 233. In order to test the reliability of the results obtained by infer- ence as above, 60 stations representing various types of tide in different parts of the world where the harmonic constants had been determined from observations were selected and a comparison was made between the values for certain constants as obtained by inference and by observations. The tests were applied to the diurnal constituents 80 U. S. COAST AND GEODETIC SURVEY M,, Py, and Q,, and to the semidiurnal constituents K, L:, and v2, and formulas (326), (328), (329), (332), (833), and (342) were used for the purpose. The following results were obtained for the differences between values as obtained from inference and from observations. The average gross difference is the average difference without regard to the signs of the individual items, and the average net difference takes into account these signs so that a positive difference may offset a negative difference in the mean. The last two lines in the table show the percentage of cases in which the differences were less than 0.05 and 0.10 foot, respectively, for the amplitudes, and less than 10° and 20°, respectively, for the epochs. Ft. Deg. Ft. Deg. Ft. Deg. Vila GieNeM CO eee eee ee ee es 0. 05 149 0. 27 49 0.05 105 Awverace eross\difteClen COssssner sean ene eee . 02 31 . 03 8 01 14 Mveracemeti@inleren Cet eee a= ae eae mee nee eee 01 1 01 3 . 00 0 % % % % % % Differences less than 0.05 foot or 10°.....-.....-.------ 93;|.o.1 B@thwer: 86/9 [76s| ee 96 ia es Differences less than 0.10 foot or 20°_____-__------------ 100 57 92 92 100 82 Ko Le v2 ampli Ke ampli- 12 ampli va Ft. Deg. Ft. Deg. Ft. Deg. INfeeeihaliron Gene a) 2 ee ee nate mee 0. 28 51 1.09 104 0. 28 53 Average gross differences. -=---222--a--2-- eae — naa . 02 9 . 09 25 . 04 14 iMcenase met qineremcCe ssa =o sane eee oe ea . 00 5 . 08 4 . 02 4 Ty. || Cox | | Gl\\S See) Geel ee Differences less than 0.05 foot or 10°_-_-_..------------- 87 65 58 20 71 48 Differences less than 0.10 fost or 20°___---.------------ 97 93 7 44 88 83 By using formulas (334) and (343) for L, and » the results are slightly improved, the average net differences for the amplitude and epoch of L, becoming 0.07 foot and 3°, respectively, the difference for the epoch of vy, becoming 2°, while the average net difference for the amplitude of ». remains unchanged. 234. Although there is a fairly good agreement indicated by the average differences, it is evident that the inferred constants, especially the epochs, cannot be depended upon for a high degree of refinement. It may be stated, however, that for constituents with very small amplitudes the epochs determined from actual observations may be equally unreliable. This becomes evident when results from different years of observations are compared. Fortunately, the large dis- crepancies in epochs are found only in constituents of small amplitude and are therefore of little practical importance. 235. Constituent 2 as determined by inference is relatively unim- portant. However, this constituent has the same period as_ the compound tide 2MS, and when obtained directly from the analysis of observations frequently differs considerably from the inferred py: both in amplitude and epoch. The inferred values for this constituent cannot therefore be considered as very satisfactory. 236. Prior to the elimination process described in the next section, certain preliminary corrections are applied to the amplitudes and HARMONIC ANALYSIS AND PREDICTION OF TIDES Sl epochs of constituents S, and K, because of the disturbing effects of K, and T, on the former and P, on the latter. In a short series of observations these effects may be considerable because of the small differences in the periods of the constituents involved. 237. Let yi=A cos (at+ a) (344) Y2=B cos (bt+ 8) (345) represent two constituents, the first being the principal or predomi- nating constituent and the latter a secondary constituent whose effect is to modify the amplitude and epoch of the principal constituent. The resultant tide will then be represented by y=yity=A cos (at+a)+B cos (bt+ 8) (346) Values of ¢ which will render (344) a maximum must satisfy the derived equation and Aa sin (at-+a)=0 (347) and the values of ¢ which will render (346) a maximum must satisfy the equation Aa sin (at+a)+ Bb sin (b¢+ 8) =0 (348) For a maximum of (344) jt te (349) in which n is any integer. 238. Let “=the acceleration in the principal constituent A due to the disturbing constituent B. Then for a maximum of (346) ;— 2h mat (350) This value of ¢t must satisfy equation (348), therefore we have Valein@nt 6) 48h ain | Fen 70-2) +6 | =— Aa sin 06+ Bb sin (2n 70-2) +p—a-0 =0 (3851) At the time of this maximum, when _2n z™—a—@ a the phase of constituent A will equal (2n r—a—0)+a and the phase of constituent B will equal - (2n r—a—O)+ 8B Let ¢=phase of constituent B—phase of constituent A at this time. Then t b] gt (2n r—a—O)+B—«a (352) 82 U. S. COAST AND GEODETIC SURVEY Substituting the above in (851) — Aa sin 6+ Bb sin (¢—8@) —— Aa sin 6+ Bb sin ¢ cos 6— Bb cos ¢ sin 6 =—(Aa+Bbd cos ¢) sin 6+ Bb sin ¢ cos 6=0 (353) Then i ee Sbysinge tan USGI GOS © (354) 239. For the resultant amplitude at the time of this maximum sub- stitute the values of ¢ from (850), in (846), and we have y=A cos (2n r—0)+8B cos E (2n 70-0) +8 =A cos 6+ B cos [= (2n x00) +6—a—0| =A cos 06+ B cos (¢—8) (855) =A cos 6+ B cos ¢ cos 6+B sin ¢ sin 8 =(A+B cos ¢) cos 6+B sin ¢ sin 6 bah =) ep sin = A?+ B?+2AB cos ¢ cos (o—tan : te 240. From (354) , . (ins 2 eee ee (356) A, +B cos ¢ Bet 68 ? A : sae In the special cases under consideration the ratio 7 is near unity, B sin $ A+B cos @ small, so that the cosine may be taken as unity. The resultant amplitude may therefore be expressed by and the difference between @ and tan7! is therefore very J A?+ B?+2AB cos b=Aq/1 4d = cos ¢ (357) The true amplitude of the constituent sought being A, the resultant amplitude must be divided by the factor y 1 go cos ¢ (358) in order to correct for the influence of the disturbing constituent. 241. The corrections for acceleration and amplitude as indicated by formulas (356) and (358) may to advantage be applied to the con- stants for constituent K, for an approximate elimination of the effects of constituent P, and to the constants for S, for an approximate elimination of the effects of constituents K, and T,. By taking the relations of the theoretical coefficients for the ratios s and the differ- ences in the equilibrium arguments as the approximate equivalents of the phase differences represented by ¢, tables may be prepared giving the acceleration and resultant amplitudes with the arguments referring to certain solar elements. Thus, from table 2, the following values may be obtained. HARMONIC ANALYSIS AND PREDICTION OF TIDES 83. B Aa A Bb % ESTE CTO feb One Kees ee eae a De See ata eee ee 0. 38086 3. 03904 —2h+v! 18)": TeECt Of KeKOntS ot) eee Pee ere els | SE ee ee, ee 0. 27213 3. 66469 | 2h—2p’ eC Gio OMS ob eto aa ae hs oe See eee ane 0. 05881 | 17. 02813 —h+p1. Substituting the above in (356) and (358) we have Effect of P,; on K, sin (2h—p’) 3.0390—cos (2h—v’) Resultant amplitude=0.813-/1.6767—cos (2h—v’) (360) Effect of K, on 8, Acceleration=tan7! (359) sin (2h—2v”) 3.6647-+ cos (2h—2v”) Resultant amplitude =0.738/1.9734+cos (2h—2v”) (362) Effect of T, on S, Acceleration= tan! (361) _ Ss (ep) 17.0281-++ cos (h—/) Resultant amplitude=0.343-/8.5318-+cos (h—p;) (364) Acceleration=tan“! (363) 242. The above formulas give the accelerations and resulting amplitudes for any individual high water. For the correction of the constants derived from a series covermg many high waters it is necessary to take averages covering the period of observations. Tables 21 to 26 give such average values for different lengths of series, the argument in each case referring to the beginning of the series. In the preceding formulas the mean values of the coefficients were taken to obtain the ratios a To take account of the longitude of the moon’s node, the node factor should be introduced. If the mean coefficients are indicated by the subscript 0, formulas (356) and (358) may be written Acceleration=tan— ae (365) FB)Bb te? Resultant amplitude=/1+( 1 + (Fey +: } Ly) HB Be cosy A, © oo) (366): 243. In the cases under consideration the ratio Te will not differ greatly from unity, the ratio a will be rather large compared with cos ¢, which can never exceed unity, and the acceleration itself is relatively small. Because of these conditions the following may be taken as the approximate equivalent of (365): 84 U. S$. COAST AND GEODETIC SURVEY Le Gnd Oy ye SED Aceeleration 77) tan TAB (367) Bp vee oy) Also because an these cases is small compared with unity, the following may be ‘taken as the approximate equivalent of (366): ae eee Resulting amplitude=1 +453} Re (32) +2 - cos ¢— 1 (368) . To allow for the effects of the longitude of the moon’s node, the tabular value of the acceleration should, therefore, be multiplied by the ratio — and the amount by which the resultant amplitude differs from unity by the same factor. In the particular cases under consideration the factor f, for constituents P, S:, and To, S unity for : 1 each. Therefore, for the effect of P; on Ki, ue TAO To FID = (K,), and for the effect of Ky upon S,, this ratio is f(K.). For the effect of T, upon S, the ratio is unity. ELIMINATION 244. Because of the limited length of a series of observations analyzed the amplitudes and epochs of the constituents as obtained by the processes already described are only approximately freed from the effects of each other. The separation of two constituents from each other might be satisfactorily accomplished by having the length of series equal to a multiple of the synodic period of the two con- stituents. To completely effect the separation of all the constituents from each other by the same process would require a series of such a length that it would contain an exact multiple of the period of each constituent. The length of such a series would be too great to be given practical consideration. In general, it is therefore desirable to apply certain corrections to the constants as directly obtained from the analysis in order to eliminate the residual effects of the constituent upon each other. 245. Let A be the designation of a constituent for which the true constants are sought and let B be the general designation for each of the other constituents in the tide, the effects of which are to be eliminated from constituent A. Let the original tide curve which has been analyzed be represented by the formula y=A cos (at+a)+z B cos (b¢+ 8) (369) in which y=the height of the tide above mean sea level at any time ¢. t=time reckoned in mean solar hours from the beginning of the series as the origin. A=R(A)=true amplitude of the constituent A for the time covered by series of observations. B=R(B)=true amplitude of constituent B for the time cov- ered by series of observations. a=—¢(A)=true initial phase of constituent A at beginning of series. HARMONIC ANALYSIS AND PREDICTION OF TIDES 85 B= —¢(B)=true initial phase of constituent B at beginning of series. a=speed of constituent A. b=speed of constituent B. 246. Formula (369) may be written y=A cos a cos at+ z B cos {(b—a)t+ B} cos at —A sin asin at—z B sin {(6—a)t+ 8} sin at =[A cos a+ > B cos { (b—a)t+B}] cos at —[A sin a+ B sin {(b—a)t+8}] sin at (370) The mean values of the coefficients of cos at and sin at of formula (370) correspond to the coefficients C, and S, of formulas (295) and (296) which are obtained from the summations for constituent A. 247. Let A’ and a’= the uneliminated amplitude and initial phase, respectively, of constituent A, as obtained directly from the analysis. The equation of the uneliminated constituent A tide may be written y=A’ cos (at+a’)=A’ cos a’ cos at—A’ sin a’ sinat (871) Comparing (370) and (371), it will be found that A’ cos a’=mean value of [A cos a+ = B cos {(b—a)t+6}] (372) A’ sin a’=mean value of [A sin a+ 2 Bsin {(b—a)t+8}] (373) 248. Let r=length of series in mean solar hours. Then the mean value of B cos {(b—a)t+8} within the limits t=0 and t=7, is 1s cos ((b—a)t+ pdt =a [sin {(b—a)r+6}—sin 8B] __180 sin 3(b—a) a 43(b—a)r “B cos {4(b—a)7+ B} (374) The mean value of B sin { (6—a)t+ 6} within the same limits is 1's sin {(b—a)t+ B}dt=— Papyles {(b—a)r+ 8}—cos 6] = 180 Se COB sin (30—a)r+- 8} (375) Substituting (374) and (375) in (372) and (373), and for brevity etting 180 sin 4(6—a) T we have A’ cos a’ =A cos a+ > F, cos {4(b6—a)t+ B} (377) A’ sin a’=Asin a+ F, sin {3(6—a)r+ 8B} (378) Transposing, A cos a=A’ cos a’—2 F, cos {4(6—a)7+ B} (379) Asin a=A’ sin a’— F, sin {4(b—a)7+ B} (380) Multiplying (379) and (380) by sin @’ and cos a’, respectively, 86 U. 8. COAST AND GEODETIC SURVEY A sin a’ cos a=A’ sin a’ cos a’— 2D F, cos {4(b—a)7+ B} sin a’ (381) A cos a’ sin a=A’ sin a’ cos a’— 2 Fy, sin {4(b—a)r+ 8} cos a’ (382) Subtracting (382) from (381) A sin (e’—a)=2z F, sin {$(b—a)r+ B—a’} (383) Multiplying (379) and (380) by cos a’ and sin a’, respectively, A cos a’ cos a=A’ cos’ a’ —z F, cos {4(b—a)7+ B} cos a’ (384) A sin a’ sin a=A’ sin? a’—z F, sin {3(b—a)7+ B} sin a’ (385) Taking the sum of (384) and (385) A cos (a’—a)=A’—2z F, cos {4(b—a)t+ B—a0’} (386) Dividing (383) by (886) > F, sin {4(6—a)7+ B—a’} — F, cos {4(b—a)r+ B—a’} tan (e!—a)=F From (386) (387) A’—® F, cos {4(b—a)r+ B—a’} cos (a’—a) 249. Substituting the value /, from (376) and the equivalents f(A), R(A), fy (CB) at CA) GCA) and 191023) for vA A, B, a’, a; and B, respectively, we have by (387) and (388) tan [¢(A)—¢/(A)]= sll sin 3(b—a)r © ¥b—a7 2B) sin {4(6-a)7—5(B)+4'(A)} B'(A)— 3 ah —ayr PB) 008 {46—a)r—0(B) +8'(A)} A= (388) 180 sin 4(b—a)r (389) Ria) — Se sn OO FRB) cos (4(b—a)7—1(B) +1"(A)} cos [¢ (A) —¢’(A)] R(A)= (390) 250. Formula (389) gives an expression for obtaining the difference to be applied to the uneliminated ¢’(A) in order to obtain the true ¢(A), and formula (390) gives an expression for obtaining the true amplitude R(A). These formulas cannot, however, be rigorously applied, because the true values of R(B) and ¢(B) of the disturbing constituents are, in general, not known, but very satisfactory results may be obtained by using the approximate values of R(B) and ¢(B) derived from the analysis or by inference. By a series of successive approximations, using each time in the formulas the newly climinated values for the disturbing constituents, any desired degree of refinement may be obtained, but the first approximation is usually sufficient and all that is justified because of the greater irregularities existing from other causes. 251. Form 245 (fig. 19) provides for the computations necessary in applying formulas (389) and (390). In these formulas the factors 180 sin 4(b—a)r represented by (b= aia and the angles represented by HARMONIC ANALYSIS AND PREDICTION OF TIDES 87 4(b—a)r will depend upon the length of series; but for any given length of series they will be constant for all times and places. Table 29 has been computed to give these quantities for different lengths of series. The factor as directly obtained may be either positive or negative, but for convenience the tabular values are all given as positive, and when the factor as directly obtained is negative the angle has been modified by +180° in order to compensate for the change of sign in the factor and permit the tabular values to be used directly in formulas (389) and (390). _ 252. An examination of formulas (889) and (390) will show that the disturbing effect of one constituent upon another will depend largely sin $(b—a)r 4(b—a)r not equal to a, this fraction and the disturbing effect it represents will = ° approach zero as the length of series 7 approaches in value a or upon the magnitude of the fraction Assuming that b is any multiple thereof, or, in other words, as 7 approaches in length any multiple of the synodic period of constituents A and B. Also, since the numerator of the fraction can never exceed unity, while the denominator may be increased indefinitely, the value of the fraction will, in general, be diminished by increasing the length of series and will approach zero as 7 approaches infinity. The greater the dif- ference (b—a) between the speeds of the two constituents the less will be their disburbing effects upon each other. For this reason the effects upon each other of the diurnal and semidiurnal constituents are usually considered as negligible. 253. The quantities R(B) and ¢(B) of formulas (389) and (390) refer to the true amplitudes and epochs of the disturbing constituents. These true values being in general unknown when the elimination process is to be applied, it is desirable that there should be used in the formulas the closest approximation to such values as are obtainable. If the series of observations covers a period of a year or more, the am- plitudes and epochs as directly obtained from the analysis may be considered sufficiently close approximations for use in the formulas. For short series of observations, however, the values as directly obtained for the amplitudes and epochs of some of the constituents may be so far from the true values that they are entirely unservice- able for use in the formulas. In such cases inferred values for the disturbing constituents should be used. LONG-PERIOD CONSTITUENTS 254. The preceding discussions have been especially applicable to the reduction of the short-period constituents—those having a period of a constituent day or less. They are the constituents that deter- mine the daily or semidaily rise and fall of the tide. Consideration will now be given to the long-period tides which affect the mean level of the water from day to day, but which have practically little or no effect upon the times of the high and low waters. There are five such long-period constituents that are usually treated in works on harmonic analysis—the lunar fortnightly Mf, the lunisolar synodic fortnightly MSf, the lunar monthly Mm, the solar semiannual Ssa, and the solar annual Sa. The first three are usually too small to be of practical importance, but the last two, depending largely upon 88 U. S. COAST AND GEODETIC SURVEY meteorological conditions, often have an appreciable effect upon the mean daily level of the water. 255. To obtain the long-period constituents, methods similar to those adopted for the short-period constituents with certain modifica- tions may be used. For the fortnightly and monthly constituents the constituent month may be divided into 24 equal parts, analogous to the 24 constituent hours of the day. Similarly, for the semiannual and annual constituents the constituent year may be divided into 24 equal parts, although it will often be found more convenient to divide the year into 12 parts to correspond approximately with the 12 calendar months. 256. Instead of distributing the individual hourly heights, as for the short-period constituents, a considerable amount of labor can be saved by using the daily sums of these heights. The mean of each sum is to be considered as applying to the middle instant of the period from 0 hour to 23d hour; that is, at the 11.5 hour of the day. If the constituent month or year is divided into 24 equal parts, the in- stants separating the groups may be numbered consecutively, like the hours, from 0 to 23, with the 0 instant of the first group taken at the exact beginning of the series. A table may now be prepared (table 34) which will show to which division each daily sum, or mean, of the series must be assigned. 257. Letting a=the hourly speed of any constituent, in degrees. p=1 when applied to a monthly or an annual constituent, and p=2 when applied to a fortnightly or a semiannual constituent. d=day of series. s=solar hour of day. Then 1 constituent period= > solar hours (391) and 1 constituent month="~2 solar hours (392) also 1 constituent year= FP solar hours (393) Dividing the constituent month or year into 24 equal parts, the length of 1 constituent division= "2? solar hours (394) Therefore, to express the time of any solar hour in units of the con- stituent divisions to which the solar hourly heights are to be assigned, the on hour should be multiplied by the factor a/15p. Thus, Constituent division=75- (solar hour of series) a Ayes dl a = 755 24(d—1) +115] (395) HARMONIC ANALYSIS AND PREDICTION OF TIDES 89 since in using the daily sums, the solar hour of the day to which each such sum applies will always be 11.5 hour. By substituting the speeds of the constituents from table 2 the a 15p° Mf, 0.036,601,10; MSf, 0.033,863,19; Mm, 0.036,291,65; Sa and Ssa, 0.002,737,91. By using the appropriate coefficient and substituting successively the numerals corresponding to the day of series (d), the corresponding value of the constituent division to which each daily sum is to be assigned may be readily obtained. The value of such division as obtained directly from the formula will usually be a mixed number. For table 34 the nearest integral number, less any multiple of 24, is used. 258. The distribution of the daily sums for the analysis of the long- period constituents may be conveniently accomplished by copying such sums in Form 142 (fig. 12), taking the constituent divisions as the equivalents of the constituent hours and using table 34 to deter- mine the division or hour to which each sum should be assigned. The total sum and mean for each division may then be readily ob- tained. These means can then be treated as the hourly means of the short-period tides according to the processes outlined in Form 194 (fig. 16) with such modifications as will now be described. 259. In using the daily means as ordinates of a long-period constitu- ent consideration must be given to the residual effects of any of the short-period constituents upon such means and steps taken to clear the means of these effects when necessary. Constituent S, with a period commensurate with the solar day, may be considered as being com- pletely eliminated from each daily mean. Constituents K, and K, are very nearly eliminated because the K day is very nearly equal to the solar day. Other short-period constituents may affect the daily means to a greater or less extent, dependmng largely upon their amplitudes. Of these the principal ones are constituents M2, No, and O,. In the distribution and grouping of the daily means for the analysis of the several long-period constituents the disturbing effects of the short-period constituents just enumerated, excepting the effects of M, upon MSf, will be greatly reduced, and in a series covering several years may be practically eliminated. Because the period of MSf is the same as the synodic period of M, and 8, there will always remain a residual effect of the constituent M, in the constituent MSf sums of the daily means, no matter how long the series, which must be removed by a special process. 260. Let the equation of one of the short-period constituents be y=A cos (at+a) (396) Letting d=day of series, the values of ¢ for the hours 0 to 23 of d day will be following numerical values are obtained for the coefficient DGD), PAGS), (Ses on Ss PGS) = 2s. Substituting these values for ¢ in (396) and designating the corre- sponding values of the ordinate y as Yo, Yi, Yo. » - - Yo3 the following are obtained: 90 U. 8. COAST AND GEODETIC SURVEY Yo =A cos [24(d—1)a+ a] y, =A cos [24(d—1l)a+a+a] Yo =A cos [24(d—1)a+a-+2a] (397) Yo3—=A cos [24(d—1)a+a+23a] Representing the mean of these 24 ordinates for d day by yz, we have Va=54 A cos {24(d—1)a+a} [1-++cos a+cos 2a+____+ cos 23a] a A sin {24(d—1)a+a}[sin a+sin 2a+___.+sin 23a] 1 >, ‘sm 12¢ 23 =54 A ae ra | cos {24(d—1)a+a} cos 9 @ 3 aD —sin {24(d—1)a+a} sin = a rt ily sin 12a 2455 sim sa 261. Formula (398), representing the average value of the constitu- ent A ordinates contained in the daily mean for d day, is the correction or clearance that must be subtracted from the mean for that day in order to eliminate the effects of A. It will be noted that if we let A represent any of the solar constituents, 5,, S:, $3, S,, ete., the factor sin 12a, and consequently the entire formula, becomes zero for all values of d. By formula (398) clearances for each of the disturbing short-period constituents for each day of series may be computed and these clearances then applied individually to the daily means, or, if first multiplied by the factor 24, to the daily sums. 262. The labor involved in making independent calculations for the clearance of the effect of each short-period constituent for each day of series would be considerable, but this may be avoided to a large extent by means of a tide-computing machine. If we let t=time reckoned in mean solar hours from the beginning of the series, then for any value of yg, which must apply to the 11.5 hour of d day, cos {24(d—1l)a+a+11.5a} (398) Pad = eeties and at=24 (d—1)a+11.5a (399) If the above equivalent is substituted in (398) and yq replaced by Ya, We have it sin 12a Ya=o4 Zak eine cos (at+a) (400) which represents a continuous function of ¢; and for any value of ¢ corresponding to the 11.5 hour of d day the corresponding value of y, Will be yg. This formula is the same as that for the short-period HARMONIC ANALYSIS AND PREDICTION OF TIDES Ol 1 2 Heine a anvans coefficient. The speed a is a known constant and the values of A and a are presumed to have already been determined from the har- monic analysis of the short-period constituents. Similarly, the dis- turbing effects of other short-period constituents may be represented by constituent A, except that it includes the factor sin 126 1 Yo og B sin 26 cos (bt+ 8) 1 _ sin 12¢ Yu 94 C sin 4¢ cos (ct+7) (401) cte. The combined disturbing effect of all the short-period constituents may, therefore, be represented by the equation sin 12a sin 4a Y=YatYotete.=s, 7A 1 i On B a cos (b¢t+ B)-+ etc. (402) cos (at+ a) 263. This formula is adapted to use on the tide-computing machine. With the constituent cranks set in accordance with the coefficients and initial epochs of the above formula, the machine will indicate the values of y corresponding to successive values of ¢. The values. of y desired for the clearances are those which correspond to ¢ at the 11.5 hour on each day. Thus, the clearance for each successive day of series may be read directly from the dials of the machine. In practice, it may be found more convenient to use the daily sums rather than the daily means for the analysis. In this case the co- efficients of the terms of (402) should be multiplied by the factor 24 before being used in the tide-computing machine. _ 264. Assuming that all the daily sums are used in the analysis, the augmenting facter represented by formula (308) which is used for the short-period constituent is also applicable to the long-period con- stituents, with p representing the number of constituent periods in a constituent month or year. Thus, for Mm and Sa, p equals 1, and for Mf, MSf, and Ssa, p equals 2. For the long- period constituents a further correction or augmenting factor is necessary, because the. mean or sum of the 24 hourly heights of the day is used to represent. the single ordinate at the 11.5 hour of the day. 265. If we let formula (396) be the equation of the long-period constituent sought, formula (400) will give the mean value of the 24 ordinates of the day which, in the grouping for the analysis, is taken. as representing the 11.5 hour of the day or the ¢a hour of the series. Since the true constituent ordinate for this hour should be A cos. sin 5a 7124 ust be applied to the mean ordinates as derived from ie: sum of the 24 hourly heights of the day in order to reduce the means to the 11.5 hour of each day. (atat+ a), itis evident that.an augmenting, factor of 24 - 92 U. S$. COAST AND GEODETIC SURVEY 266. The complete augmenting factor for the long-period constit- uents, the year or month being represented by 24 means, will be obtained by combining the above factor with that given in formula (308). Thus Tp 24 sin 3a Se =X pain “EP sin 12a augmenting factor= (403) If the year or month is represented by only 12 means as when monthly means are used in evaluating Sa and Ssa, the formula becomes Lap 24 sin 3a 12 sin 15p”~ sin 12a augmenting factor= (404) Values obtained from these formulas are given in table 20. 267. The following method of reducing the long-period tides, which conforms to the system outlined by Sir George H. Darwin, differs to some extent from that just described. In this discussion it is assumed that a series of 365 days is used. Let the entire tide due to the five long-period constituents already named be represented by the equation y=A cos (at+a)+B cos (bt+ 8)+C cos (at+y) (405) +D cos (dt+6)+E cos (et+e) 268. For convenience in this discussion let ¢ be reckoned from the 11.5th solar hour of the first day of series instead of the midnight beginning that day. Every value of ¢ to which the daily means refer will then be either 0 or a multiple of 24. Let' A’, B’, O7, D’, and E’, equal A cos a, B cos B, C cos y, D cos 6, and F cos e«, respectively, and JACOBY CX DO tendon Tequal —A sin a, —Bsin 8, —C sin y, —D sin 6, and —F sin ¢, respectively. (406) Then formula (405) may be written y= A’ cos at+ B’ cos bt+C’ cos ct+D’ cos dt+-K’ cos et +A’ sin at+B” sin bt+C” sin ct+D” sin dt+E” sin et (407) In the above equation there are 10 unknown quantities, A’, A’’, B’, B’’, etc., for which values are sought in order to obtain from them the amplitudes and epochs of the five long-period constituents. The most probable values of these quantities may be found by the least square adjustment. 269. Let 4, yo, . . - . Y3es represent the daily means for a 365 day series, as obtained from observations. If we let n be any day of the series, the value of ¢ to which that mean applies will be 24(n—1). By substituting in formula (407) the successive values of y and the values of ¢ to which they correspond, 365 observational equations are formed as follows: HARMONIC ANALYSIS AND PREDICTION OF TIDES 93 =A’ cos 04-B" cos 0-> . ale ’’ sin 0+ B”’ sin O+ . Y2—A’ cos 24a+B’ cos 24b-+- . + A’’ sin 24a+B” sin 246+ . (408) Yse5— A’ cos 24 364a-+B’ cos 24 3640+ . +A’’ sin 24364a+B” sin 2436464 . 270. A normal equation is now formed for each unknown quantity by multiplying each observational equation by the coefficient of the unknown quantity in that equation and adding the results. Thus, for the unknown quantity A’, we have yi: cos O= A’ cos’ 0+B’ cos 0 cos 0+ - +A” sin 0 cos 0+ B” sin 0 cos ogee Yy2 cos 24a= A’ cos” 24a+B’ cos 246 cos yi aa sin 24a cos 24a+ B” sin 246 cos Made Y365 COS (24 X364a) =A’ cos” (24 364a) +B’ cos (243646) cos (24 364a)4+ - +A” sin (24X364a) cos (24364a) +B” sin (243646) cos (24X364a)+ -+-.-:. Summing = Yn cos 24(n—1)a=A’ py cos’ 24(n—1)a +A” a sin 24(n—1l)a cos 24(n—1)a +B’ baie: 24(n—1)b cos 24(n—1)a +B" 5)sin 24(n—1)b cos 24(n—1)a +" "Sy cos 24(n—1)e cos 24(n—1)a + on 5) sin 24(n—1)e cos 24(n—1)a +)’ "00s 24(n—1)d cos 24(n—1)a +)” = sin 24(n—1)d cos 24(n—1)a + EH’ ot cos 24(n—1)e cos 24(n—1)a +E" >) sin 24(n—1)e cos 24(n—1)a (410) which is the normal equation for the unknown quantity A’. 271. In a similar manner we have for the normal equation for the quantity A’ 904 U. S. COAST AND GEODETIC SURVEY > yn sin 24(n—1)a =A’ > cos 24(n—1)a sin 24(n—1)a+A” 2 sin? 24(n—1)a +B’ > cos 24(n—1)6 sin 24(n—1)a+B” = sin 24(n—1)b sin 24(n—1)a +0’ > cos 24(n—1)e sin 24(n—1)a+C” 2 sin 24(n—1)e sin 24(n—1)a +D’ > cos 24(n—1)d sin 24(n—1)a+ D” & sin 24(n—1)d sin 24(n—1)a + E’ > cos 24(n—l1)e sin 24(n—1)a+ E” © sin 24(n—1)e sin 24(n—I1)a (411) the limits of n being the same as before. Normal equations of forms similar to (410) and (411) are easily obtained for the other unknown quantities. 272. By changing the notation of formulas (265) to (267) the fol- lowing relations may be derived: 365 sin 24na cos 24(n—1)a 2 1 bs cos? 24 (n—1)a=4n+4 COAG i 60a cos 8736a ze fr rinse 87 ; Ween: sin 24a 412) 1 sin 24na cos 24(n—1)a sin 24a sin 8760a cos 8736a Mi Got 2a SERS BON Cae Cen ONe Se sin 244 (413) oh sin? 24(n—1l)a=3n— Sone 24(n—1)b cos 24(n—l)a n=1 1 sin 12n(b— a) cos 12(n—1)(b—a) =s sin 12(b—a) es 12n(b+a) cos 12(n—1)(b+a) u sin 12(6-+a) __ ,sin 4380(b—a) cos 4368(b—a) 2 sin 12(b—a) as sin 4380 (6+) cos 4368(b-+-a) a sin 12(6-++a) (414) n=36 Sain 24(n—1)b sin 24(n—1)a nmi! sin 12n(b—a) cos 12(n—1)(b—a) sin 12(6—a) , sin 12n(6-+a) cos 12(n—1)(b+a) Te sin 12(6+a) __, Sin 4380(6—a) cos 4368(6—a) im sin 12(6—a) 2 a 4380 (b +4) cos 4368 (b+ a) sin 12(6+a) (415) 1 2 HARMONIC ANALYSIS AND PREDICTION OF TIDES 95 n=365 >} sin 24(n—1)b cos 24(n—I1)a n=1 sin 12n(b—a) sin 12(n—1) (b—a) eee Tee sin 12(b—a) La sin 12n(b+a@) sin 12(n—1) (6+) ? sin 12(6+4a) __, sin 4380(6—a) sin 4368(6—@) nite ¢ sin 12(6—a) , sin 4380(b-++@) sin 4368(6+-a) TRe sin 12(6+a) (416) 273. By substituting in (412) to (416) the numerical values of a, 6, ete., from table 2, the corresponding equivalents for these expressions are obtained. These, in turn, may be substituted in (410), (411), and similar equations for the other unknown quantities to obtain the 10 normal equations given below. In preparing these equations the symbols a, 6, c, d, and e are taken, respectively, as the speeds of constituents Mm, Mf, MSf, Sa, and Ssa. n=36 5 D>) Yn COS 24(n—1)a n=1 =183.05A’+0.72B’ +0.76C’ +4.88D’ + 4.96 E" +2.14A"+4.29B” +5.040” —0.34D”—0.70E” oes n=305 \ D5 Yn sin 24(n—1)a n=1 =? 14A’—4.15B’ —4.900’ + 3.80)! +3.88 FE" r +181.95A”+1.01B”+1.06C” +0.34D” +0.68 E” n=365 >) Yn COS 24(n—1)b n=1 =—(0.72A’+183.17B’ +0.56C’ —1.50D/ — 1.51 EF’ —4.15A”+0.88B” +0.92C” —0.09D” —0.18 EF” (417b) n=365 - 24 Yn sin 24(n—1)b =4.29A’+0.88B’+0.920’+3.05D’+3.06E’ if +1.01A”+181.83B” —0.80C” —0.08D” —0.17 EF” n=365 24 Yn COS 24(n—1)e =0.76A’ +0.56B’ +183.19C’ —1.68D’—1.70E” Ff —4.90A” +0.92B” +0.97C” —0.11D” — 0.21 FE” (417¢) n=365 z 24 Yn sin 24(n—1)e =5.04A’+0.92B’ +0.970’ +3.24D’+3.25E’ a +1.06A” —0.80B” + 181.810” —0.10D” — 0.20 h” / n=365 Ss Yn COS 24(n—1)d =4.88 A’ —1.50B’ —1.680’ + 182.38D’— 0.24 F’ i +3.80A”+3.05B” +3.24C” +0.00D” +0.01 F” (417d) n=365 Ds Yn sin 24(n—1)d = —0.34A’—0.09B’ —0.11C’ +0.00D’ +0.00 EF’ +0.34A” —0.08B” —0.10C” +182.62D”+0.00 EK” 96 U. S$. COAST AND GEODETIC SURVEY n=365 >) Yn COS 24(n—I1)e n=1 —=4,96A’ —1.51B’—1.70C’ —0.24D’ +182.38 LH’ +3.88A” +3.06B” +3.25C” +0.00D” +0.00 E” n=365 . dS Yo Sin 24(n—1)e n=1 =—0.70A’—0.18B’ —0.21C’+0.01D’+0.00H’ +0.68A” —0.17B” —0.20C” + 0.00D” + 182.62 EH” (417e) 274. The numerical value of the first member of each of the above normal equations is obtained from the observations by taking the sum of the product of each daily mean by the cosine or sine of the angle indicated. The solution of the equations give the values of A’, A", B’, B’’, etc., from which the corresponding values of quantities A and a, B and 8, etc., of formula (405) are readily obtained, since LA AG & A=, (A’)?+ (A’)? and a=tan7! In calculating the corrected epoch, it must be kept in mind that the t in this reduction is referred to the 11.5 hour of the first day of series instead of the preceding midnight. 275. Before solving equations (417), if the daily means have not already been cleared of the effects of the short-period constituents, it will be necessary to apply corrections to the first member of each of these equations in order to make the clearances. The disturbance in a single daily mean due to the presence of a short-period constituent is represented by equation (398). Intro- ducing the subscript s to distinguish the symbols pertaining to the short-period constituents, the disturbance in the daily mean of the n= day of series due to the presence of the short-period constituent A, may be written yslo=aqA —— cos {24(n—1)a,+11.5a,+a;} (418) ond 3 The disturbances in the products of the daily means by cos 24(n—1)a and sin 24(n—1)a may therefore be written [Ysln CoS 24(n—1)a 1 in 12a, oa A ee a 1 [eos {24(n—1) (a,ta)+11.5a,+ a5} +ecos {24(n—1) (a4,—a)+11.5a,+a5}] (419) and [Ysln Sin 24(rn—1)a 1 I 2 ‘ SA eee gin (28 Ge nae Wesoete| SIN 3@, —sin {24(n—1) (a,g—a)+11.5a,+a5}] (420) HARMONIC ANALYSIS AND PREDICTION OF TIDES 97 276. Then, referring to formulas (263) and (264) n=365 >) [ysln cos 24(n—1)a= n=1 a sin 12a, [ sin 12365(a,+a) 48 ~* gin da, sin 12(a,+a) in 12 365 BLS. en Tangy 008 U2XB6K(a—a) $11.50, 6} | 421) cos {12><364(a,+4@)+11.5a,+ ag} and n=365 : >> [Yeln Sin 24(n—1)a= n=1 1 sin 12a, [= 12X365(a,+a) ag “1s sin 12(a,-Fa) sin 3d, sin 12(a,+a) _ sin 12365 (a,—a@) sin 12(a,—a) Now let sin {12 364(a,+a)+11.5a,+ a,} ath (12364 (aa) +11.5ay+ 06} | (422) A’ =A, COS a, and (423) A’ .=—A, sil a, then (421) and (422) may be reduced as follows: n=365 > [ysln cos 24(n—1)a i | ee) 48 sin 3a; sin 12(a,+qa) sin 12365 (a,—a) sin 12(a,—a) sin 12a, sin 12° X365(@,+¢) = 9 > | et ~ sin 3; l ain IGesn) {12X364(a,+ a) +11.5a,} sin 12X365(a,—a@) . hoe py, sm 12(@,—a) °" (12364(a—«) +11.50,} |A ; (424) cos {12 364(a,+a)+11.5a,} + cos {12 364(a,—a) +11.54,} |4’, 1 +48 + and n=365 [ysln SIN 24(n—1)a n=1 1 sin 12a, [sin 12365(a,+4a) sin {12 X364(a,+ 4) +11.5a,} 48 sin 4a, sin 12(a,-+a) _ sin 12 365(a,—a) sin 12(a,—«a) _ 1 sin 12a, [ sin 12X365(a,+4a) 48 sin 3a, sin 12(a,+a) __sin 12X365(a,—a) sin 12(a,—a) sin {12364(a,—a) + 11.505) |’ cos {12 <364(a,+a) +11.5a,} cos (12364 (a,—a)-+11.5a,} |4”, (425) OS U. S. COAST AND GEODETIC SURVEY 277. Formulas (424) and (425) represent the clearances for any long-period constituent A due to any short-period constituent Ag. The first must be subtracted from terms corresponding to Yy,cos 24(n—1)a and the latter from terms corresponding to Ly, sin 24(n—1)a of formula (417) before solving the latter. 278. In (424) and (425) the coefficients of A’, and A’’,, which for brevity we may designate as C’, C’’, S’, and S”’, respectively, contain only values that are constant for all series and may therefore be computed once for all. Separate sets of such coefficients must, however, be computed for the effect of each short-period constituent upon each long-period constituent. In the usual reductions in which the effects of 3 short-period constituents upon 5 long-period con- stituents are considered, 15 sets of 4 coefficients each, or 60 coefficients in all, are required. The coefficients are given in the following table: * Long-period constituents | Mm Mf MSf Sa Ssa —_—_—— —_— — |! = = cca ahd Was ———$ — | —_ ——- —. W716} (CO) aes Se See See ete ook oe —0. 0556 +0. 0030 +5. 739 —0. 1041 —0. 1046 (CRE ARE Re. SOE eee eee —0. 1704 —0. 0377 —2. 923 —0. 0762 —0. 0755 OSD ilee eee ee Seer ee Bane eee ree —0. 1708 +0. 0417 —2. 840 —0. 0018 —0. 0035 (CS) Re at eee one oe me oe ea +0. 0441 | +0. 0105 —5. 727 -++0. 0048 -++0. 0096 INC (0/)) Se 5 ae Sinem —0.0588 | +0. 0368 +0.0294 | —0.0176 —0. 0176 COL) Fes ARO ee ee a | —0.0776 | —0. 2236 —0. 1938 +0. 0025 +0. 0025 (Sas ae eee ee oe oo ee ! —0.0206 | —0.1526 —0.1221 | +0. 0002 +0. 0004 (SE) a a oh ee Sr ee ae cae | +0. 1138 —0. 0854 —0. 0808 -++0. 0001 +0. 0002 ORC) ee see SR fee eee ce { —0.0648 | -+0.0166 +0. 0157 —0, 1924 —0. 1934 (COS) EE ih CE Eh Ee. | He apne fe —0. 3476 —0. 0778 —0. 0816 —0. 1826 —0. 1831 (Sak eo ee ee. eas ye ee ost —0.3452 | +0. 0841 +0. 0875 —0. 0046 —0. 0093 (Oi) eo Be ein aes eee eee | +0. 0405 -++0. 0338 -++0. 0331 -++0. 0090 +0. 0180 In the above table the sign is so taken that the values are to be applied to the sums directly as indicated. 279. After the clearances have been applied and the normal equa- tions (417) solved and the resulting amplitude and epoch obtained for each of the long-period constituents, the reductions will be completed in accordance with the processes already outlined, but it must be kept in mind that in this reduction the initial value of ¢ is taken to corre- spond to 11:30 a. m. on the first day of series. In obtaining the nu- merical values of such quantities as Dy, cos 24(n—1)a and 2y, sin 24(n—1)a, in order to avoid the labor of separate multiplications for each day, the following abbreviations have been proposed by the British authorities. The values of cos 24(n—1)a and of sin 24(n—l)a are divided into 11 groups according as they fall nearest 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, or 1.0. The daily values are then dis- tributed into 11 corresponding groups, so that all values in one group will be multiplied by 0, another group by 0.1, ete. The cos 24(n—l)a and sin 24(n—1)a include negative as well as positive values. The former are taken into account by changing the sign of the daily mean to which the negative values apply. 280. As a part of the routine reductions of the tidal records from the principal tide stations it is the practice of the office to obtain the mean sea level for each calendar month. It is therefore desirable to *From Scientific Papers by Sir George H. Darwin, Vol. I, p. 64. HARMONIC ANALYSIS AND PREDICTION OF TIDES 99 have a method of using these means directly in the analysis for the annual and semiannual constituents, thus avoiding any special sum- mation for the purpose. The period of the annual constituent is ap- proximately the length of the Julian year, that is, 365.25 days. If this period is divided into 12 equal groups and the mean of the hourly heights for each group taken, these means represent the approximate height of the combined annual and semiannual constituents for the middle of each group, and the middle of the first group will be the initial point from which the zeta (¢) as obtained by the usual process is referred. As each group represents 30° of motion for the annual constituent, or 60° for the semiannual constituent, to refer this ¢ to the actual beginning of the series of observations it will be necessary to apply a correction of 15° for the annual constituent or 30° for the semiannual constituent. 281. In obtaining the monthly means by calendar months the year is divided only approximately into 12 equal groups. The following table shows the difference between the middle of each group repre- senting a calendar month and the middle of the corresponding group obtained by dividing the Julian year into 12 equal parts. It is to be noted that the hourly heights included in a monthly sum extend from 0 hour on the first day of the month to the 23d hour on the last day. The middle of the group as reckoned from the beginning of the month will therefore be 13.98 days, 14.48 days, 14.98 days, or 15.48 days, respectively, according to whether the month has 28, 29, 30, or 31 days. Middle of group reckoned from beginning of year Differences Month ee Julian |Common| Leap }Common| Leap year year year year year Days Days Days Days 15. 48 15. 48 +0. 26 +0. 26 44,98 45. 48 —0. 68 —0.18 74. 48 75. 48 —1.61 —0. 61 104. 98 105. 98 —1.55 —0. 55 135. 48 136. 48 —1.49 —0. 49 165. 98 166. 98 —1. 43 —0. 43 196. 48 197. 48 —1. 36 —0. 36 227. 48 228. 48 —0. 80 +0. 20 257. 98 258. 98 —0. 74 +0. 26 288. 45 289. 48 —0. 68 +0. 32 318. 98 319. 98 —0. 61 +40. 39 349. 48 350. 48 —0. 55 +0. 45 aS be 8 AR le Sp a —11. 24 —0. 74 Bree Se | ew ee —0. 94 —0. 06 Speed of Sa constituent per day=0.9856°. © 2 Mean differences reduced to degrees of Sa________________________-_-_-___-_-_-_---_-- —0. 93 —0. 06 Gorrechiontorgot¢saewr ae etree ee eC lh et eee Le eee 14. 07 14. 94 Correction to ¢ of Ssa_______> Se reer ae ae ertee eee ae NE On See 28. 14 29. 88 282. From the above table it is evident that in the summation for the monthly means for a calendar year the middle of each group of a common vear is on an average 0.93° earlier than the middle of the corresponding group when the Julian year is equally subdivided and the middle of each group of a leap year is on an average 0.06° earlier. Subtracting these values from 15°, the interval between the beginning of the observations and the middle of the first group of an equal subdivision, we have 14.07° and 14.94°, for common and leap years, respectively, as a correction to be applied to the ¢ of Sa as 100 U. S. COAST AND GEODETIC SURVEY directly obtained, in order to refer the ¢ to the 0 hour of the Ist day of January. For Ssa the corrections will be twice as great. 283. If the year commences on the first day of any month other than January, the corrections will differ a little from the above. Calculated in a manner similar to that above, the following table gives the correction to be applied to the ¢ to refer to the first day of any month at which the series commences... The correction to the ¢ of Ssa will be twice the tabular value for Sa. Correction to ¢ of Sa Correction to ¢ of Sa to refer to begin- to refer to begin- ning of month ning of month Observations commence— Observations commence— Common Leap Common} Leap year year year year ° ° ° ° ReSriyy (SAO BL 2 OB Apres 14. 07 A OSE WO Vele eee ne ie ea 15. 56 15. 93 5X2) 0) ee ge a ee 13. 50 1 4e 45 al PAU CIES: ears Te TS Oe 14. 98 15. 43 Mars iia oeee fil ee oeen hee eee 15. 89 15893) || Sept): 3 see eee” 2 eed 14, 41 14, 94 ADT Se RSOeA ee N Ss Se Te 15. 31 5543} Oot. Wes se. ROLES SRR ea ee 14, 82 16. 43 Misys dere iar 2 ea 15. 72 1593) || NOV LEs-f. Sate oe eet eee 14, 24 14, 94 Juneyls 22. 8. cee Ee 15. 15 N5X435 | Dec) Leates oe eee 14. 65 15. 43 284. If the monthly means extend over many calendar years, it may be convenient to combine them for a single analysis. In: this case the (V,+4u) for January 1 may be taken as the average of the values for the beginning of each year included in the observations, and the correction to the ¢ to refer to the beginning of the year will be a mean of the values given above for common and leap years, weighted in accordance with the number of each kind of year included. If only a few years of observations are available, it is better to analyze each yeat separately in order that the results may serve as a check on each other. 285. The augmenting factors to be used for constituents Sa and Ssa when derived from the monthly sea level values are based upon for- mula (404) in paragraph 266 and are as follows: Sa 1.0115, logarithm 0.00497. Ssa 1.0472, logarithm 0.02003. ANALYSIS OF HIGH AND LOW WATERS 286. The automatic tide gage, which furnishes a continuous record of the rise and fall of the tide, now being in general use, it is seldom necessary to rely only upon the high and low waters for an analysis. It may happen, however, that a record of high and low water observa- tions is available for a more or less isolated locality where it has been impractical to secure continuous records. Such records, if they in- clude all the high and low waters for a month or more may be utilized in determining approximate values of the principal harmonic con- stants, but the results are not as satisfactory as those obtained from an analysis of the hourly heights. 287. An elaborate mode of analysis of the high and low waters is contained in volume 1 of Scientific Papers, by Sir George H. Darwin. Other methods are given by Dr. R. A. Harris in his Manual of Tides. The process outlined below follows to some extent one of the methods of Doctor Harris, extending his treatment for the K and O to other constituents. HARMONIC ANALYSIS AND PREDICTION OF TIDES 101 288. The lengths of series may be taken the same as the lengths used as the analysis of the hourly heights (see par. 152). It is some- times convenient to divide a series, whatever its length, into periods of 29 days each. This permits a uniform method of procedure, and a comparison of the results from different series affords a check on the reliableness of the work. 289. The first process in this analysis consists in making the usual high and low water reductions, including the computation of the lunitidal intervals. Form 138 provides for this reduction. The times and heights of the high and low waters, together with the times of the moon’s transits, are tabulated. For convenience the standard time of the place of observations may be used for the times of the high and low waters, and the Greenwich mean civil time of the moon’s transits over the meridian of Greenwich may be used for the moon’s transits. The interval between each transit and the following high and low water is then found, and the mean of all the high water intervals and the mean of all the low water intervals are then obtained separately. The true mean intervals between the time of the moon’s transit over the local meridian and the time of the following high and low waters being desired, the means as directly obtained must be corrected to allow for any difference in the kind of time used for the transit of the moon and the time of the tides and also for the difference in time between the transit of the moon over the local meridian and the transit over the meridian to which the tabular values refer. 290. If the tide is of the semidiurnal type, the approximate ampli- tude and epoch for M, may be obtained directly from this high and low water reduction. On account of the presence of the other con- stituents the mean range from the high and low waters will always be a little larger than twice the amplitude of M,. If the data are available for some other station in the general locality, the ratio of the M, amplitude to the mean range of tide at that station may be used in finding the M, amplitude from the mean range of tide at the station for which the results are sought. If this ratio cannot be ob- tained for any station in the general locality, the empirical ratio of 0.47 may be used with fairly satisfactory results. After the ampli- tude of M. has been thus obtained, it should be corrected for the longitude of the moon’s node by factor F' from table 12. 291. The epoch of M, may be obtained from the corrected high and low water lunitidal intervals HWI, LWI by the following formula: M°.=3(HWI+LWI) X28.984+90° (426) In the above formula HWI must be greater than LW/J, 12.42 hours being added, if necessary, to the HW as directly obtained from the high and low water reductions. 292. The difference between the average duration of rise and fall of the tide at any place, where the tide is of the semidiurnal type, de- pends largely upon the constituent My. It is possible to obtain from the high and low waters a constituent with the speed of M, which, when used in the harmonic prediction of the tides, will cause the mean duration of rise and fall to be the same as that at the station. The effect of M, upon the mean duration of rise will depend chiefly upon the relation of its amplitude and epoch to the amplitude and epoch of the principal constituent M,. By assuming an My, with epoch 102 U. S. COAST AND GEODETIC SURVEY such as to make the constituent symmetrically situated in regard to the maxima and minima of M,., the amplitude necessary to account for the mean duration of rise of the tide may be readily calculated. 293. Let DR=duration of rise of tide in hours as obtained from the lunitidal intervals, a=Hourly speed of M,.=28.°984. M,.=Amplitude of Mp. M.°=Epoch of M3. M,=Amplitude of M,. M,°=Epoch of Mg. Then, for M, to be symmetrically situated with respect to the maxima. and minima of M, M.°=2 M.°+90° (427) in which the upper or lower sign is to be used according to whether a(DR) is greater or less, respectively, than 180°. Multiples of 360° may be added or rejected to obtain the result as a positive angle less than 360°. The equations of the constituents M, and M, may be written Yi=M, cos (at+a) (428) Yo= My, cos (2at+ B) (429) and the resultant curve y=M, cos (at-+a)+M, cos (2at-+ 8) (430) 294. Values of ¢ which will render (428) a maximum must satisfy the derived equation aM, sin (at-+a)=0 (431) and for a maximum of (430) ¢ must satisfy the derived equation aM, sin (at-+a)+2aM, sin (2at+ 8)=0 (432) For a maximum of (428) jae (433) in which n is any integer. 295. Let = the acceleration in the high waters of M, due to the presence of M,. With the M, wave symmetrically situated with respect to the M, wave, . will also equal the retardation in the low 2, : water of M., due to the presence of My, and = will equal the total amount by which the duration of rise of the tide has been diminished by M,;. If the duration of rise has been increased, @ will be negative. Then, for a maximum of (430) __ 2nz—a—d rm a t (434) and this value of ¢ must satisfy equation (432). HARMONIC ANALYSIS AND PREDICTION OF TIDES 103 296. Substituting in (432), we have aM, sin (2n7—60)+2aM, sin (4n7—2a+ B—26) = —aM, sin 6—2aM, sin (26+2a— )=0 (435) But 2a— p=—2M.°+Me (436) From (427) = 2M MM — =: 00" 180° according to whether the duration of rise is greater or less than ’ or whether 6 is negative or positive. Then 2a— p= +90° (437) according to whether 6 is positive or negative. Substituting this in (435) —aM, sin 64+2aM, cos 26=0 (438) and M, | 1 sin 6 M, = 205 20 (439) the upper or lower sign being used according to whether 6 is positive or negative. As under the assumed conditions 6 must come within the limits +45°, the ratio of “¢ as derived from (439) will always be 2 positive. ao The duration of rise of tide due solely to the constituent Mg is 18 a The duration of rise as modified by the presence of the assumed 14, is pA 28 (440) a a Therefore 6=4(180°—aDR) (441) Substituting the above in (439) we have M,__ , :sin (90°—3aDR) __ — ,cos 3aDR (442) ~M, ~?cos (180°—aDR) — ? cosaDR ¢ and ue __ 00s 3aDE Mi T3 cos aDR M: >) M, must be positive, and the sign of the above coefficient will depend upon whether aDA is less or greater than 180°. 298. The approximate constants for S:, Ne, Ky, and O, may be obtained from the observed high and low waters as follows: Add to each low-water height the mean range of tide. Copy the high and modified low water heights into the form for hourly heights (form 362), always putting the values upon the nearest solar hour. Sum for the desired constituents, using the same stencils as are used for the regular 104 U. Ss. COAST AND GEODETIC SURVEY analysis of the hourly heights. Account should be taken of the num- ber of items entering into each sum and the mean for each constituent hour obtained. The 24 hourly means for each constituent are then to be analyzed in the usual manner. 299. The results obtained by this process are, of course, not as dependable as those obtained from a continuous record of hourly heights. The approximate results first obtained can, however, be im- proved by the following treatment if a tide-computing machine is available. Using the approximate constants as determined above for the principal constituents and inferred values for smaller constituents, set the machine for the beginning of the period of observations and find the predicted heights corresponding to the observed times of the high and low waters. Tabulate the differences between the observed and predicted heights for these times, using the hourly height form and entering the values according to the nearest solar hour. These differences are then to be summed and analyzed the same as the original observed heights. In this analysis of the residuals the con- stituent M, should be included. The results from the analysis of the residuals are then combined with the constants used for the setting of the predicting machine. 300. In making the combinations the following formulas may be used: Let A’ and x’ represent the first approximate values of the constants of any constituent. A”’ and x’’, the constants as obtained from the residuals. A and x, the resultant constants sought. Then A=+(A’ cos x’ A” cos x”)?+(A’ sin x’ +A” sin x”)? (444) and = A Sine sea Sn A’ cos x’ +A” cos x” c=tan (445) FORMS USED FOR ANALYSIS OCF TIDES 301. Forms used by the Coast and Geodetic Survey for the harmonic analysis of tide observations are shown in figures 9 to 19. A series of tide observations at Morro, California, covering the period February 13 to July 25, 1919, is taken as an example to illustrate the detail of the work. 302. Form 362, Hourly heights (fig. 9)—The hourly heights of the tide are first tabulated in form 362. Although the zero of the tide staff is usually taken as the height datum, any other fixed plane will serve this purpose. For practical convenience it is desirable that the datum be low enough to avoid negative tabulations but not so low as to cause the readings to be inconveniently large for summing. 303. The hours refer to mean solar time, which may be either local or standard, astronomical or civil, but standard civil time will generally be the most convenient to use. The series must commence with the zero (0) hour of the adopted time, and all vacancies in the record should be filled by interpolated values in order that each hour of the series may be represented by a tabulated height. It is the general practice to use brackets with interpolated values to distinguish them from the observed heights. The record for successive days of the series must be entered in successive columns of the form, and these HARMONIC ANALYSIS AND PREDICTION OF TIDES 105 columns are to be numbered consecutively, beginning with one (1) for the first day of the series. 304. The series analyzed should be one of the lengths indicated in paragraph 152. Series of observations very nearly equal to one of these standard lengths may be completed by the use of extrapolated hourly heights. If the observations cover a period of several years, the analysis for each year may be made separately, a comparison of the results affording an excellent check on the work. 305. The hourly heights on each page of form 362 are first summed horizontally and vertically. The total of the vertical sums must equal the total of the horizontal sums, and this page sum is entered in the lower right-hand corner of the page. Derantmenyoncommence TIDES; HOURLY HEIGHTS Station: __Morro, Califormia, ||| Nears elolon Chief of Party: ____E» Be Latham. Lat, 35° 22" Ne Long.120° 51" We Time Meridian; ___120_ Wa _ Tide Gauge No. 107 Scale 1:9. Reduced to Staff. \! Month || mo. amt d. d. d. d. i | al ot 4 | aren 4 ee | See ee Car | Feet. Feet Feet. Feet. Feet. 4.6 4.4 4.7 4.6 | 30.9 dale ele Tilak 2 4.9 26.6 (Ske Caee ERRE A 4 20.9 fay [ad [ed fed [eel [eel [sal | 6 : Ee EE a EL et ds Iannis sted fed [es [al [oa [eal [eal | | j 20 | 62 JEREREREREREREEE Noon) a9 2.8 4.5 TPEREREERER EERE ui] 2 4.3 3.6 i ‘ERE RE RENEE ae 2. i Sum, | 84 9 90. 6! 84 4 76 5| 75 6| 644 80 2 | Sum for 29 days, 1 to 29 of = Divisor=696; mean for 29 days= FIGURE 9. 106 U. S. COAST AND GEODETIC SURVEY 306. Stencils (figs. 10 and 11).—The first figure is a copy of the M stencil for the even hours of the first 7 days of the series, and the second figure illustrates the application of the same. This stencil being laid over the page of hourly heights shown in figure 9, the heights applying to each of the even constituent hours for this page show through the openings in the stencil, where they appear con- nected by diagonal lines, thus indicating each group to be summed. 307. For each constituent summation, excepting for S, there are provided two stencils for each page of tabulated hourly heights, one for the even constituent hours and the other for the odd constituent hours. Deranruentor commerce TIDES; HOURLY HEIGHTS Station: Year: Ba | ChiofiolPart ye ot Ss eee at Lone: Time Meridian; __________. Tide Gauge No. Scale L: Reduced to Staff. — d. 1 d. : d. d. = ___Divisor=696;_mean for 29 days= FIGURE 10. HARMONIC ANALYSIS AND PREDICTION OF TIDES 107 The stencils are numbered with the days of series to which they apply, and special care must be taken to see that the days of series on each stencil correspond with the days of series on the page of tabulations with which it is used. For constituent S no stencils are necessary, as the constituent hours correspond to the solar hours of the tabulations and the horizontal sums from form 362 may be taken directly as the constituent hour sums. 308. Form 142, Stencil sums (figs. 12 and 13).—The sums for each constituent hour are entered in form 142, one line of the form being used for each page of the original tabulations. The total of the hour Form 862 . Pe cusrmeoomcunn = LIDES: HOURLY HEIGHTS | Station: __ Stencil for component Year: ChiefiofeParty,) ate ONG Time Meridian: ____________ Tide Gauge No. Scale 1: Reduced to Staff, —————__ | Month | mo. a. d. a. a. d. | an Hori- Day. zontal um, ellen 4 6 7 » ; b Feet, o yo : fc es SEIS ce Pies abe ae ans Ee : Fi i ae B ‘i i fe zfs gfe 2 eae Sa enon pu e JIGS eget / ag ert para aws re a =a os J Ss Se (eal al a E rai IU Be: 3 ie acm a CH a : a, “ _ pas | ae) & lave EF -al s Coa) E2g. coe ivan’ | PS Se - a EE a: ‘oes 5 Geel L454 ay | | (3.8 |] (3 7) Coes] [| teabecl | “tezve] [Seareaeen =e Geen shoo fab al ete ae fl Sa Sum for 29 days, 1 to 29 of Divisor=696; mean for 29 days= dle ra FIGURE 1l. 108 U. S. COAST AND GEODETIC SURVEY sums in each line of the form must equal the corresponding page sum of the hourly heights in form 362, this serving as a check on the sum- mation. After the summing of all the pages of the series has been completed for any constituent the totals for each constituent hour are obtained, the divisors from table 32 entered, and the constituent hourly means computed (fig. 13). These means should be carefully checked before proceeding with the analysis. Large errors can usually be detected by plotting the means. 309. Form 244, Computation of (Vou) (fig. 14)—This form pro- vides for the computation of the equilibrium arguments for the beginning of the series of observations, the computation being in accordance with formulas given in table 2. For the most part the form is self-explanatory. The values of the mean longitude of the Form 142 o Cc ° WV a contranD exovenc svevty TIDES: STENCIL SUMS. Station: ,..c=Morros Gala formi@e 2 se Se ee Tat 1 eho nee tes Component: ..!"2""... Length of series: 168. Series begins: A9\9-Reb.-15-0__Long.: 120° 61" We Kind of time used: ......... NOM eeiscccce Comptted by Fred. As. Kummell,..De009,..1920. Page. o 1 2 3 + 5 8 7 8 i} 10 i 2403 2006 1769 1609 21.0 2320 28.0 3109 3902 B08 3109 2704 2 21.68 1765 1464 1366 lle2 12.5 14.6 2006 2167 23.9 2407 24.5 S 1967 16.9 11.0 966 1202 1767 2801 2409 27.4 27.6 29.8 21.1 4 26.4 16.0 1765 1763 2207 22.06 260.0 Bed 36.5 41.2 33.5 28.5 5B 2165 2164 1769 1862 1665 19.9 2409 2969 3707 S408 3302 2906 6 2063 16.8 1568 1201 1205 15.1 21.0 2204 23.6 24.8 25.0 2601 7 2301 1601 1363 13.1 1566 2307 2806 3309 3Oel 3409 2706 2306 B 2505 2300 21.64 2166 2008 2365 2762 2907 43.5 36.4 32.6 2705 9 20.9 16.5 16.2 12.1 1163 13.6 18.1 26.3 26.8 28.6 28.0 29.6 10 16.9 13.2 10.2 8.7 1165 1565 1603 21.5 2404 2563 2827 24.3 JL 18.6 15.0 1265 1567 1%.2 23.2 29.5 4101 36,7 34.4 27.8 24.0 12 2405 2505 2004 2067 21.0 2466 32.1 31.7 32e7 3605 3106 2506 13-2507 17,3 13.2 10.0 1169 1265 1662 2003 2463 30.2 3003 2404 140 1607 1266 805 827 965 1405 1960 -23.4 3002 2704 2702 2602 15 1960 1661 1663 1567 2001 2665 3707 3767 4002 39.3 3506 2966 16 29.6 22.8 2166 22.5 2567 3107 3109 3409 35.8 38el 28035 2703 17 «2209 18.6 14.5 llel 1065 12.5 15263 19.0 2504 2405 2409 2306 18 15.4 10.0 6:2 3.2 409 1002 16.0 2402 24.05 2501 2504 2706 19° 16.7 15.4 13.1 1563 19.8 29.4 318 35.8 3803 3709 38.7 283 2027.6 21,0 1928 20.4 2601 | 2929 31.5 36.1 36.4 39.9 26.9 22.8 tum. 43701 35605 30003 26605 323.8 40167 495.8 578.5 63504 64526 593.5 523.4 Page. 12 18 14 16 16 17 18 19 20 21 22 23 L 2207 2lel 1705 1365 1405 1765 1769 26.2 2602 2767 26.9 28.0 57606 2 2568 17.6 1764 1669 2105 2808 3201 35el 3605 35.9 3528 2606 552.5 S 1765 1705 1407 15.4 21.1 23565 2962 33,3 35.3 39.6 30.0 24.8 54708 4 23.2 20.8 12.9 9.0 7.1 709 1400 20.8 22.2 24.9 25.8 2503 638.0 5 2705 2002 1609 15.5 16.4 235.8 2565 Bod 2729 29.0 3.0 25.4 89705 6 23.8 2300 2000 2462 2501 2562 27.5 2808 39.4 38.2 2807 24.4 56208 7 1905 1565 1708 1466 1565 2002 29.8 31.4 33.7 3301 3006 20-0 574.3 6 22.4 19.4 12035 828 Te 1007 14.9 19.5 24.1 26.6 32.0 27.3 55801 9 2261 1801 1504 14.0 17el 1901 2403 2903 3800 2920 28.0 24.6 52802 10 23.2 2206 2603 22.0 2404 2609 2807 3308 2601 3007 2604 24065 53601 Tl 19.4 1802 11.8 12.5 13.1 16.8 2563 26.6 2902 2900 30.4 22.9 550.8 12 21.5 13.35 6.4 4.6 4.4 8e7 1500 21.0 2565 31.9 2801 27.0 53603 13 2167 1808 1665 1766 1766 2166 26.9 3601 3505 3604 3002 27.0 54202 14 2826 23.6 2704 2604 26.0 2808 3608 32.5 31.5 28.2 2820 25.0 56603 15 2005 1604 12.9 10.5 1466 2165 22.0 2507 27.8 3lel 2504 2206 58406 16 20.9 14.9 10.9 6.1 4.8 969 16.9 2502 3201 30.8 3006 28.9 560.2 17 2163 22.2 17%el 1668 1769 2607 2706 3600 33.5 36.5 3504 32el 54507 16 2202 2lel 2105 2306 3le2 S18 3004 3le7? 31.3 28.2 2706 21.l 51404 19° 19.6 15.5 12.5 11.3 10.0 12.2 1909 2601 2866 24.1 22.5 19.7 64205 20 __ 2004 1562 798 3.6 3,0 Jo2 17el 2202 2704 2908 Bed 2707 55801 ” 4306 37500 $1800 288.9 31007 386.8 <81.6 56906 61307 62006 59007 513.9 1109300 FIGURE 12, HARMONIC ANALYSIS AND PREDICTION OF TIDES 109 moon (s), of the lunar perigee (p), of the sun (h), of the solar perigee (p;), and of the moon’s ascending node (N), may be obtained from table 4 for the beginning of any year between 1800 and 2000. The values for any year beyond these limits may be readily obtained by taking into account the rate of change in these elements as given in table 1. The corrections necessary in order to refer the elements to any desired month, day, and hour are given in table 5. As the tables refer to Greenwich mean civil time, the argument used in entering them should refer also to this kind of time, and in the lines for the beginning and middle of the series at the head of the form space is therefore provided for entering the equivalent Greenwich hour. Any change in the day may be avoided by using a negative Greenwich hour when necessary. For example, 1922, January 1, 0 hour, in the standard time of the meridian 15° east of Greenwich, may be written as 1922, January 1,—1 hour in Greenwich time, instead of 1921, December 31, 23 hour, as would otherwise be necessary. If a negative argument is used in table 5, the corresponding tabular value must be taken with its sign reversed. For the middle of the series the nearest integral hour is sufficient. 310. The values of J, v, &, v’, and 2v’’ are obtained for the middle of the series from table 6, using N as the argument. If N is between 180° and 360°, each of the last four quantities will be negative, but I Form 149 CNetoumnemew: TIDES: STENCIL SUMS. Station sae Marr 0 es OTe rer A ag oer ees .. Lat.:..26°. 22! Me ye sea: 16 : wns: 1919 = Feb.=13-0 120° 51° W Component: {'3i" .... Length of series: Se Series begins: note etennee. Long. :-6L 92 We Kind of time used: ESAS ha PR See .. Computed by Prog Meume LT s_DOG23 s1920e—— Pos. = OY 1 2 3 + 6 6 7 a 26 10 VW 21 2509 18ol 14.8 1465 1006 11.1 1703 2308 2301 2404 2hel 2206 22 16.8 14.8 Te? S5e% 606 lel 1905 2302 2605 2706 30.8 24.9 23 1708 1607 1502 2001 2106 3007 3303 3703 3900 4208 35.9 2804 PA 7.2 6.8 6:2 6.1 645 8.0 _9e7 10.9 .}8.3 1201 11,0 Sums=21-24 6707 550% 4508 4604 4505 6009 7908 9502 10609 106.9 99.8 8503 m 1520 43701 356.3 300.3 28605 323.8 40207 49508 57805 63504 64526 593.5 52304 Sumse~ 50408 41107 B40 33209 369.1 46206 57506 67507 742035 75205 69303 60807 Divisore.- 164 163 162 165 164 4165 163 163 164 165 163 162 Neanse= 3.008 2.63 2012 2202 202d 284 3.53 4:13 4.53 $066 4025 3.76 21 23503 16.2 17.0 1703. 2303 2400 2807 32.9 3509 4201 Be? Slel 6558.8 22 2205 2007 2002 2600 2609 3le? 3602 34.0 40.0 31.3 2602 20.5 551% 23 2501 1665 1565 1166 12.9 1307 1966 2501 2606 2602 24.0 2405 57308 24 3.3 4.7 3.0 167 009 029 1.7 3.4 5.05 7.0 7.7 7.8 159.8 Sums 21-24 7202 5909 5507 5606 6300 70.3 8602 954 10800 10606 9206 8309 1843.8 m 1220 443.6 375-0 518.0 288.9 310.7 388.8 483.6 569.6 613-7 62006 590.7 5}3,9 11093.0 Sums e= 51508 45409 37307 34505 373507 45901 56708 66500 721.7 72702 68303 59708 1293608 Divisors.= 162 163 163 163 #162 4162 163 163 + 162 162 163 168 Means.= 3018 2267 2029 2012 2edl 2.83 5.48 4.08 4.45 4.49 4.19 367 FIGURE 13. 110 U. S. COAST AND GEODETIC SURVEY is always positive. Although table 6 is computed for the epoch, January 1, 1900, it is applicable without material error for any series of observations. 311. The values of u of L, and u of M,, may be obtained from table 13 for any date between 1900 and 2000, inclusive, using the value of N for interpolation. If the series falls beyond the limits of this table, the following formulas may be used: u of L,=2&—2v—R (par. 129) (446) u of M,y=é—v+ Q (par. 123) (447) The values of & and vy may be taken from form 244, the values of R and Q from tables 8 and 10, respectively, using the arguments J and P for the middle of the series. 246. F IREPARTMENT OF COMMERCE, TIDES: Computation of V,+u. &. 0. COAST 4.ND CHODETIO SURVEY. 120° 51' W. Station _._Horro, California Serres ae ae rae eecneace Lat. 35° 22" Ne Lemmons Long. .120.85 We. =L ye d. hr. t Grecnwich es) Beginning of series1919. Febe 15 ON Smee! Lengti: of series .....463._... Time mer. 120400 Wom vt. mo. d. Ar. ( t Greenwich a) Midite of series ....1919 May 5 de 20! Con:pute all values to two decimzl places. Table: in Harmonic Analysis and Prediction of the Tide. Q=s | ()=p (@O=N. | s Tsile 4, for jue 1 of ycar_.....---.| 268-04 bo benoit = wd reece SHEE ORME BETS 5i P2741 | bosace: raésnlaye Ll Tobie 5, correct:cn to Ist of morth...- Fd bce coe --2Q_ 21. ee See 13-37.. Paar Table 5, correction to day of month ..--| .----- 498+ 12 coher accdts “&.-} Taovle 5, currscticn to Orevnwich br..-- Pee Che i a1 (y= 119, 02 | ee ono an mascotcs : aR M, R, (= 1(TableG) —-----Ab276.) 3 act ey . (a)=v (Tables) —:----La68_. Spe + penne V.tu=2(M)= 140-52. ow Ess = yil7Gl| Se TaREORGOH (RECESS e geen SOROS as ee (10)= 0! (Table 6) ==. 28d. | AA aca Bre MEE A sk ML (11)=20""(Table6)=----- 167.63. _ Sy Kaae el aie ana a a = asi oo fl Se i oe (17) metres 51-4701 +a-=_..... 46 84 | vtue csi ie pcodearc Waal = (10) BESS = (1s) — BEBE | Loe Gene Bienes alana | Vetere Oe ea = (Table 13) ST | cee = eee | Vitue (as) = 998. 30_ K; s (13 pe SOL. ennennn ORD.) 4 (20) =. B22 94 | 4 =... 320-06 (19)=(3)+(5) 21-47) _ aye Teel Ma =26-78_ Vtue2Gy=_.3 96 60. a diywcoyator =. DEAT | pe 299257.) Views. 233-28 Se % oe ee At wlll eas oo F ms 354.90 | _ (18)=(1)—(2) - Sere SSN IL, 0% ae <2 a ee ra 19) AT} +18) aeee ABEE ZR! oye 131-68.) + on =... $34.15 | + @y-_.....358-30 |. = 170-6 | — wo) = -351.46 | ~ (2). -40-77 (20)=2 (16) ED HN ) ev ©) (21)=(}+ (2) onan LDL 26._ ; = soa eee ae 24. (aye (20) (21) ee a SEY | sanaoe So -- = Buseeeeaeeesee ee ee Oe 202. 45 | + @wr..---202 45 sana... 334-15 | (ysis0° =... 180.24 | nae +e 123-6 |.@n- 351.46 |_«ag_- _ 86.78 ne es a varus $26.05 | vue (25)—(22)+(24) === (26 )=2 (25) ono anna RON | ae ra see SS .-- ees (27)= (25) + (26) = Vitue +(26) = 46-84 | asy¢z =... 269-15 | 4a eo .. 46-84.) vt ue sim = 261-46. (28)=(1)—(2) =. 0 SS OR ae) = 9322-32 | -32 | + (35) =___ 46-60 _ PMG (20)—2 (28)... Votu=__. ON 24 restores 93-44 Goje(2s)eece =... A 46 | - Silene Sheets + Spe. ee. B82 BO. | M, 2 - M,=__=46 84 | G1) 06) (8) jis aan | Vatu (27) + 180° SO eee ee IG PGE feet (ate eo an Vise ececacce ee (32)=(3)—(4) --=----- oe fel we. -- 102 26..| = 018) = =86-78 |4(8)=-___ 86-78 |—— PPpEee Ree eeeeereee (23)=2 (15)... at al Ol Votum..... 255.2.91..) Vitu= (Hye Q)~(1) = 293+ 30 | M (2Q), (35)=2 (33)..--- = cog tte dekes | cocnsncaeresee seen (36)= (25) —(35) = (Q2)=@)=(2) = o. mee (38)=2 (37)...--=.....-. 22 16. eee Se OT _| vitun+2()= 284 64 | ¢ Greenwich hour= original bour+(8°+15). * Positive for West longitude; negative for East longitude. FIGURE 14. HARMONIC ANALYSIS AND PREDICTION OF TIDES 111 312. In finding the difference between the longitude of the time meridian (S) and the longitude of the place (Z) consider west longitude as positive and east longitude as negative. In the ordinary use of form 244 it is assumed that civil time has been used in the tabulations of the observations. If, however, the original hourly heights as tabulated in form 362 are in accordance with astronomical time in which the 0 hour represents the noon of the corresponding civil day and the 12th hour the following midnight, form 244 will still be applicable if the longitude of the time meridian (S) is taken equal to the civil time meridian plus 180°. For example, if tabulations have been made in astronomical time for a locality where the civil time is based upon the meridian 15° E., the value for S should be taken as — 15+ 180, or 165°. If tabulations have been in Greenwich astronomical time, S should be taken as 180°. 313. Form 244a, Log F and arguments for elimination (fig. 15).— Items (1) to (11) are compiled here for convenience of reference for Form 244a DEPARTMENT OF COMMERCE U. S. COAST AND GEODETIC SURVEY TIDES: Log F and Arguments for Elimination Station -....... Morras iC at fo rrid a: at SNe oe a Sie SIRT ie oS Ne ae ad RR Ey apo as yr. nm. d. b. Length of series __._.... 163. days. Series begins ................-.----------0--. 1919... Feb. 13. O. Component Log F Component Log F Component Log F (4 dec ) (4 dec.) (4 dec.) TA gieee er erae dee ek 0.0201 Whe 6 6 eo otal chore MKic. nvssvceesers 0.0092... Go ono oee lo dled 0-0160 Nz, 2N _...9° 9952 2M Kerem “cnreuns 0.0023... Ks 0:0472 Oh oo o foo soles QO. 0264 INOS) g) Gila §5 ad 9 9863... L,=Log F(M:)+(7) |...9:9589 OO i aise tess oa 30-0929... MS,2SM . . .« |_.9.:9932___._.... M,=Log F(0,)+(8) |..9:8856 | Phusiinaybaueenestes 0.0000 NE 65 6 6 o LO EES i] Maj se 9 9952 eel Qi2q) ys ey mONOR6d ES mst... . «(99982 Mine fre ner tan [OS 9.9897 | R:, 8), 82, S4, Se,T: | 0.0000 Mine xsresnusuee er | = 9.9772... M..... .4|..9:9863_ Move sw se | 9 99S2 Sa,Ssa . . « -| 0.0000 SY ge SE Sok eal Hie 9-9794 pL 0- 0264 (1) = N=item (6) from Form 244=...245° 011. (2 dec.) (2) = 7=item (7) from Form 244 =......21.:.76_........ (2 dec.) (3) =P=item (12) from Form 244= ...55..903 (2 dec.) (4) =(h—4p’) =item (3) —} item (10), from Form 244=..........327° . (0 dec.) (5) =(h—r’’) =item (3) —} item (11), from Form 244 =..........331 (0 dec.) (6) = (h—p,) =item (3) —item (4), from Form 244= _......... Gal (0 dec.) (7) =Log R, from Table 7 =......9e9657..... (4 dec.) (8) =Log Q, from Table 9=_.....9e8592....._ (4 dec.) {9) = Natural number from Log F(K,) =...16038.....__.. (3 dec.) (10) =Log f(K,) =10—Log F(K,) = 9¢9528......... (4 dec.) (11) = Natural number f(K,) from (10) =..0.897......... (3 dec.) ExpLanaTIon.—For all tables sco Special Publication No. 98. First fill in items (1) to (8). Then obtain values of log F for all components excepting L, and M, from Table 12. Log F(L,)=log F(M,)+ log R,, and log F(M,) =log F(O,)+log Q,. Items (9) to (11) are obtained after the rest of the form has been filled out. FIGURE 15. 112 U. S. COAST AND GEODETIC SURVEY this and form 452. Items (1) to (6) are obtained from values given in form 244. Item (7) is obtained from table 7, using items (2) and (3) as arguments, and item (8) is obtained from table 9, using item (3) as argument. Items (9) to (11) are obtained after the rest of the form has been filled out. 314. The log F for each of the listed constituents, except L, and M, and those for which the logarithm is given as zero, may be obtained from table 12, using item (2) as the argument. For constituents L, and M, Log F(L,) =log F(M,)+item (7) (448) Log F(M,)=log F(O,) +item (8) (449) If the tidal series analyzed was observed between the years 1900 and 2000, the log F(L2) and log F(M,) may be taken directly from bene r orm TIDES: HARMONIC ANALYSIS COAST AND GEODETIC SURVEY Tae aa vane Station Herre, Galiformia for, Ce AOA a LO Component... Series begins L919 Febs 13 0 rength of series 6S ee Tene Meryl? ae (Days’ Hourly Means from Form 142. | (4) Hours oto11.....|_ 3-08. 2:88|.2.02.| 2.02). 2:28| 2.04) 888 | #18) 8288 | 4:86 4253-76 (2) Moursiztom....|_3.18| 2.67 | 2 29 | 2.12] 2.31| 2.83! 3 48 | .4.08| 4.45| 4.49] 419 3.67 @-~-~@___ | 70.10 ]=0 14 |-0 17 | -0, 10) -0.06) 0.01) 0 05 | 0. 05} 0.08| 0.07/ © 06 009 Ces 0.09! 006 | 0.07! 0.08! 0.05! 0.00 | @-or@ eat Peete eee 4.56) 5.67| 7 01] 8.21) 8.98| 9.05! & 44 7,437 (6)~Last half of (5)..|_ 7.01] 8.21 | 8-98 | 9,05] 8.44| 7.43 (1) (8) (9) (10) (11) (12) (13) (14) (15) (19) (20) (21) Sin. (3)+(4) (Dx(8) Cos. (3)-(4) (10)x1)_| gin. ~ (8)X(13) Cos. : (7) X(S) (18) % (20) SX152 ae eng bra NX159 [ere | Ext) 123 m<4s° <30° _70: 10 | ~ 9.000] 1.000 2100) -000} 0-000} "1.000 0.013) . (32) (33) (34) | (35) (36) (37) cos. | OSX) | +6) MX90° | Jorg | Ast hal. (28)x(31) | (26)+027) | gin. 124 Rar 0080) 26.46 (33) X(34) | Gog, | 63)X@5) 125, |7X120°1 20 ae) 1,000 20 000 B66 —.866 22.871 _ so ae | l2m= ft’ isin Ist quadrant when we have +@ and +¢. tis in 3d quadrant when we have —s and —c. ¢ is in 2d quadront when we have +s and —c. tv isin 4th quadrant when we have —s and +¢. (38) =log. 128 (39) «log. 12¢_--..--------. = (40) = (38)— (39) =log. tan. {” -----__--. (41) =, for beginning of tho serics. (42) (20).. es J R(N2)=R/(Ny) 28M SNE! (QRS) atc el EON Sins | oe. 1/990 Pt S log. 0.133 +9.1239 log. R(N,)_| +9-420 ()=I*=..214.76._(2 dec.); (10)=P*=... 55.2032 dec.); (11)=/(K,)*... eB 9T...(3 dec.) log. R(2N) 8-5440 (00) 5a) a) (0 dec.); (13)=(h—p,)*=..--- 4h. _-(0 dee.) ||___ (14)=acceleration in S, due to K,=Table 23x f(K,)=.----Le4&____.°(1 dec.) R, +7.9031 (15)=acceleration in S, due to T,=Table 25 -...-.-.-- = 72.0 (1 dec.) +9-50 (6)=(2)) bse = 2006 __°(1 dec.) 7°4048 (17)=resultant amplitude, S,and K,=1+4[Table 24x /(K,)]= ____2e01____(2. dee.) |J-=== a= (OREM nico Rts, Coen Geer OE Terai £2 enone (19)=log. (17)-Hlog. (18,.. ... . = _ 929955 _(4 dee.) Mog. RSs) | 95017 (20)=(M2—Nj)=....2Le5_ (1 dec.); (21) =(S}—Mi)=__=4e2__(1 dec.) R(S8,) rap aera SL ee oe SAT RE EE ee Te | log. 0.059 8.7709 Exrplanation.—Obtain from Form 194 the amplitudes and epochs indicated by the accent (’); Ts ee R(S,) Hi 0 and from Form 244 or 244a the quantities indicated by the asterisk (*). = - If the series is less than 355 days, omit 2SM._ For all short series, the values in columns (4) 8 -2726 and (8) will be computed in accordance with the Equivalents and Factors in columns (3) and (7), respectively; but if the series is 192 days or more in le! , the x of K, for column (4) and the +7.8461 log. R of K, for column (8) may be taken directly from Form 194; and if the series is 365 days or S more in length, the « and log. R of all the components for which analysis has been made may +0 -09 be taken directly from the eame form. When a value is thus taken directly from the analyais, 7 -9404 the corresponding equivalent in column (3) and factors in column (7) should be crossed ont. The tabular values for (14) to (18) may be obtained from Tables 23 to 26 in Special Publi- caiion No. 98, or from plotted curves representing these tables. For a series of 355 days or more, + 8.3802 (14)=(15)=(16)=0; Qe: and {i920 ; +0 -0958 Obtain the « of 2 by applying (16) to (S;) from Form 194, and use this corrected « in com- x puting (2). If the two angles in either (20) or (21) differ by more than 180°, add 360° to the | R 8 °4755 smaller before taking the difference, which may be either positive or negative. (a4) _In computing column (8) it will be noted that the corrected log. R of S, is to be used in in- z ferring other components depending upon this one. +9 4201 | REMARES 22 Se ee eee Rv) os Oa | esate (28d) ines R (Sa) = R’ (2a) FIGURE 18. HARMONIC ANALYSIS AND PREDICTION OF TIDES ial constituents. The symbol A represents the constituent to be cleared, and the symbol B is the general designation for the disturbing con- stituents. The symbol applying to constituent A is to be crossed out in column (1) and entered in column (8). The values for items (9) and (19) are to be taken from columns (1) and (2) of form 452. 327. For obtaining column (2) it will be found convenient to copy the logarithms of the R’s of B from column (8) of form 452 on a hori- zontal strip of paper spaced the same as table 29. Applying this strip successively to the upper line of the tabular values for each con- Form 246 DEPARTMENT OF COMMERCE ‘COAST AND GEODETIC SURVEY TIDES: ELIMINATION OF COMPONENT EFFECTS m d. h Peper LOR ebm NO ease ne Ole zat RB)x | Nat. No. 2)| Table 29 (5) (arxcn.(sy | @yxees (5) (8) Table29 | Table2? | —3(B:) | “+09 | “Tableso | ‘Table 30 RESULTS Tog (dec) | ft. dec) | nodec) | (modec.) | (dec) | (dec) | Use 4dec. for logarithms, 3 dec. for amplitudes, 1 dec. for angles Component A,;=.-Kee. (9) =R'(A;) from analysis (19) =(9) — (7) (11) =log (6) (12) =log (10) (13) = (11) — (12) =log tan 6 ¢ *(14)=6 (15) =log cos 6 ¢ (16) = (12) — (15) =log R(A:) (16a) =log F(A2) (17) = (16) +log F(A2) =log H(A?) (18) = H(A:) (19) =¢'(A2) from analysis (20) = (14) + (19) =¢ (A2) (20a) =(V.+u) (21) = (20) + (Vo+u) =x(A2) ribbon Component A,=..12... (9) = R’(A;) from analysis (10) = (9) — (7) (11) =log (6) (12) =log (10) _{| (13) =(11) — (12) =log tan 6 ¢ | *(14)=6 ¢ (15) =log cos 6 (16) = (12) — (15) = log R(A2) (16a) =log F(A2) (17) = (16) +log F(A2) =log H(A?) (18) = H(A2) (19) =¢'(A)) from analysis (20) = (14) + (19) =¢ (A2) (20a) =(V.+u) (21) = (20) + (Vo+u) =«(A,) Vit tnd ee pon ot in Component A,;= (9) = R'(A,) from analysis (10) =(9) —(7) (11) =log (6) (12) =log (10) (13) = (11) — (12) =log tan 6 ¢ *(14) =6 £ (15) =log cos 6 (16) = (12) — (15) =log R(A2) (16a) =log F(A2) a (17) = (16) +log F(A2) =log H(A.) (18) = H(A2) (19) =¢'(.42) from analysis (20) = (14) + (19) =¢ (42) (20a) =(V.+u) (21) = (20) + (Vo+u) =x(A2) * 8¢ or (14) is in the Ist quadrant when (6) is + and (10) is +. Takako Feb. 28, 1921 $f or (14) is in the 2d quadrant when (6) is + and (10) is —. Computed by 8 or (14) is in the 3d quadrant when (6) is — and (10) is —. $f or (14) is in the 4th quadrant when (6) is — and (10) is +. When (6) is 0 and (10) is positive, (11) & (13) = — oo, and (14) & (15) = 0. BV Tne clio ase eee een ee FIGURE 19. 118 U. S. COAST AND GEODETIC SURVEY stituent the logarithms of the resulting products for column (2) may be readily obtained. Similarly, for column (4), the ¢’s of B from column (6) of form 452 may be copied on a strip of paper and applied to the bottom line of the tabular values for each constituent and the differences obtained. The natural numbers for column (3) correspond- ing to the logarithms in column (2) can usually be obtained most expeditiously from table 27, this table giving the critical logarithm for each change of 0.001 in the corresponding natural number. If the logarithm is less than 6.6990, the natural number will be too small to appear in the third decimal place, and the effects of the corresponding constituent may be considered as nil. The products for columns (6) and (7) may be conveniently obtained from table 30. In column (8) the references to (6) and (7) are to the sums of these columns. The values of log F(A) and (V)+ 4) for column (8) may be obtained from forms 244 and 244a. 328. In the use of this form it will be noted that the R’s and ¢’s referring to constituent B are to be the best known values whether derived from the analysis or by inference, but the R’ and ¢’ of con- stituent A, entered as items (9) and (19), respectively, must be the unmodified values as obtained directly by form 194. ANALYSIS OF TIDAL CURRENTS 329. Tidal currents are the periodic horizontal movements of the waters of the earth’s surface. As they are caused by the same periodic forces that produce the vertical rise and fall of the tide, it is possible to represent these currents by harmonic expressions similar to those used for the tides. Constituents with the same periods as those con- tained in the tides are involved, but the current velocities take the place of the tidal heights. There are two general types of tidal cur- rents, known as the reversing type and the rotary type. 330. In the reversing type the current flows alternately in opposite directions, the velocity inereasing from zero at the time of turning to a maximum about 3 hours later and then diminishes to zero again, when it begins to flow in the opposite direction. By considering the velocities as positive in one direction and negative in the opposite direction, such a current may be expressed by a single harmonic series, such as V=Acos (at+a)+B cos (6t+ 8)+C cos (ct+y)+ete. (450) in which V=velocity of the current in the positive direction at any time ¢. A, B, C, ete.=maximum velocities of current constituents. a, 6, c, ete.=speeds of constituents. a, B, y, etc.=initial phases of constituents. 331. In the rotary type the direction of the current changes through all pomts of the compass, and the velocity, although varying in strength, seldom becomes zero. In the analysis of this type of cur- rent it is necessary to resolve the observed velocities in two directions at right angles to each other. For convenience the north and east directions are selected for this purpose, velocities toward the south and west being considered as negatives of these. For the harmonic HARMONIC ANALYSIS AND PREDICTION OF TIDES 119 representation of such currents it is, therefore, necessary to have two series—one for the north and the other for the east component. 332. For the analysis of either type of current the original hourly velocities or the resolved hourly velocities are tabulated in the same form used for the hourly heights of the tide. To avoid the incon- venience of negative readings in this tabulation, a constant, such as 3 knots, is added to all velocities. These hourly velocities are then summed with the same stencils that are used for the tides, and the hourly mean velocities are analyzed in the same manner as the hourly heights of the tide. The same forms are used for the currents, with the necessary modifications in the headings. The rotary currents will be represented by a double set of constants, one for the north components and the other for the east components. 333. For a 29-day series of observations, it is recommended that the analysis be made for the M series, the S series, and for No, K,, and O,. For longer series additional constituents may be included. In the analysis of current velocities, the harmonics of the higher degrees such as M, and M, may be expected to be of relatively greater magnitude than they are in the tides. From theoretical considera- tions it may also be shown that the magnitude of the diurnal constit- uents as compared with the semidiurnal constituents in a simple tidal oscillation is only about one-half as great in the current as in the tide. However, because of the complexity of the tidal and current move- ment, the actual relation between the various constituents as deter- mined by the analysis is subject to wide variations. The constituent S,, which is usually negligible in the tides, may be found to be of ap- preciable magnitude in offshore currents because of the effect of daily periodic land and sea breezes. However, as this constituent has a speed very nearly the same as that of K, it can be separated from the latter only by a long series of observations, preferably a year or more. 334. Form 723 (fig. 20) provides for the determination of harmonic constants from a series of current observations by comparison with corresponding constants from a tidal series covering the same period of time. This comparison is to be used if the series of observations is less than 29 days and may be used for longer series if desired. For the purpose of this comparison the hourly predicted heights at the tide station are usually to be preferred to actual observations since meteor- ological irregularities appearing in observed tides do not necessarily appear in a similar manner in the observed currents. In this work both currents and tides for the simultaneous period are to be summed for constituents M, S, N, K, and O; and the analysis is then carried through form 194 (Tides: Harmonic Analysis) to obtain the values of R’ and ¢’ for each constituent. The harmonics M,, Mz, and Mg are to be obtained for the current series, but may be omitted in the tidal series. 335. Enter in Form 723 the accepted H and « of the principal tidal constituents for the reference station and also the values of R’ and ¢’ obtained from the analyses of the simultaneous series of tides and currents. The necessary calculations in the form are self-explana- tory. The corrected velocity amplitude of each current constituent is obtaied by a ratio on the assumption that for each constituent the relation of the corrected amplitude to the uncorrected amplitude is the same for both tide and current. The ratio derived for the con- ‘0G GUNDIA ———<—<——$—<—$— sss Mtn = ao Be Sa ee ee 2 eS o(Stx4) OTT (Tots Too Tes “GAT TUFOU) qe JO UOPFIeIT a ae a Eee Sa ae SN ee Ce ee oe SS ee o(Bnr3) oL8e (TSTFTOOTOA GAT FTsod) pools Jo UoTzosIT ‘SYIBULOY EP COCe Mfojrmn lls Gers |) bekts fe Magis IP ELA P Etetstoy i ekehs™ 898°O | Seo'O al =) aanee ‘oO zB HL 88o toy SP Ulex ees S°00g p nsephto) |i TSp°O | g99°T | 608°C |” COE Oa Se eras a r mH ie Sl SO Gss OlO meen ehh Ag allie! TLS ae (he) Apaw ee GeLvl Ones ‘aCeeOnmli a VARIO 1 Soo"O 990°0 Gl ee Sra. tN nN sep eke eee Rl a a eccee beer errs beeen ao -cnad Romp pence bese eee |E E-7| RO coved ,f| inberee eee és 6°OtTt 0°0 Seito= wrest TéT°O T22"0 298°O | OBE" AMO ts eH H MOSES: slime Coto "SS Sar CoLS all aos oe OSOLO S| = ee ee f ) fH cos (at+a) (458) for height of tide at any time, and >) af A sin (at+a)=0 (459) for times of high and low waters. Formula (458) may be easily solved for any single value of ¢, but for many values of ¢ as are neces- sary in the predictions of the tides for a year at any station the labor involved by an ordinary solution would be very great. Formula (459) can not, in general, be solved by rigorous methods. The in- vention of tide-predicting machines has rendered the solution of both formulas a comparatively simple matter. TIDE-PREDICTING MACHINE 353. The first tide-predicting machine was designed by Sir William Thomson (afterwards Lord Kelvin) and was made in 1873 under the auspices of the British Association for the Advancement of Science. This was an integrating machine designed to compute the height of the tide in accordance with formula (458). It provided for the sum- mation of 10 of the principal constituents, and the resulting pre- dicted heights were registered by a curve automatically traced by the machine. This machine is described in part I of Thomson and Tait’s Natural Philosophy, edition of 1879. Several other tide- predicting machines designed upon the same general principles but providing for an increased number of constituents were afterwards constructed. HARMONIC ANALYSIS AND PREDICTION OF TIDES 127 354. The first tide-predicting machine used in the United States was designed by William Ferrel, of the U. S. Coast and Geodetic Survey. This machine, which was completed in 1882, was based upon modified formulas and differed somewhat in design from any other machine that has ever been constructed. No curve was traced, but both the times and heights of the high and low waters were indicated directly by scales on the machine. The intermediate heights of the tide could be obtained only indirectly. A description of this machine is given in the report of the Coast and Geodetic Survey for the year 1883. 355. The first machine made to compute simultaneously the height of the tide and the times of high and low waters as represented by formulas (458) and (459), respectively, was designed and constructed in the office of the Coast and Geodetic Survey. It was completed in 1910 and is known as the United States Coast and Geodetic Survey tide-predicting machine No. 2. The machine sums simultaneously the terms of formulas (458) and (459) and registers successive heights of the tide by the movement of a pointer over a dial and also graphi- cally by a curve automatically traced on a moving strip of paper. The times of high and low waters determined by the values of ¢ which satisfy equation (459) are indicated both by an automatic stopping of the machine and also by check marks on the graphic record. 356. The general appearance of the machine is illustrated by figure 21. Itis about 11 feet long, 2 feet wide, and 6 feet high, and weighs approximately 2,500 pounds. The principal features are: First, the supporting framework; second, a system of gearing by means of which shafts representing the different constituents are made to rotate with angular speeds proportional to the actual speeds of the constituents; third, a system of cranks and sliding frames for obtaining harmonic ‘motion; fourth, summation chains connecting the individual constitu- ent elements, by means of which the sums of the harmonic terms of formulas (458) and (459) are transmitted to the recording, devices; fifth, a system of dials and pointers for indicating in a convenient man- ner the height of the tide for successive instants of time and also the time of the high and low waters; sixth, a tide curve or graphic represen- tation of the tide automatically constructed by the machine. The machine is designed to take account of the 37 constituents listed in table 38, including 32 short-period and 5 long-period constituents. 357. The heavy cast-iron base of the machine, which includes the operator’s desk, has an extreme length of 11 feet and is 2 feet wide. This forms a very substantial foundation for the superstructure, increasing its stability and thereby diminishing errors that might result from a lack of rigidity in the fixed parts. On the left side of the desk is located the hand crank for applying the power (J, fig. 24), and under the desk are the primary gears for setting in motion the various parts of the machine. The superstructure is in three sections, each consisting of parallel hard-rolled brass plates held from 6 to 7 inches apart by brass bolts. Between these plates are located the shafts and gears that govern the motion of the different parts of the machine. 358. The front section, or dial case, rests upon the desk facing the operator and contains the apparatus for indicating and registering the results obtained by the machine. The middle section rests upon @ depression in the base and contains the mechanism for the harmonic motions for the principal constituents Mo, S., Ki,O1, N2, and My. The 128 U. S. COAST AND GEODETIC SURVEY rear section contains the mechanism for the harmonic motions for the remaining 31 constituents for which the machine provides. 359. The angular motions of the individual constituents, as indicated by the quantity at in formulas (458) and (459), are represented in the machine by the rotation of short horizontal shafts having their bear- ings in the parallel plates. All of these constituent shafts are con- nected by a system of gearing with the hand crank at the left of the dial case and also with the time-registering dials, so that when the machine is in operation the motion of each of these shafts will be proportional to the speed a of the corresponding constituent, and for any interval of time or increment in ¢ as indicated by the time dials the amount of angular motion in any constituent shaft will equal the increment in the product at corresponding to that constituent. 360. Since the corresponding angles in formulas (458) and (459) are identical for all values of t, the motion provided by the gearing will be applicable alike to the solution of both formulas. The mechanism for the summation of the terms of formula (458) is situated on the side of the machine at the left of the operator, and for convenience this side of the machine is called the “height side” (fig. 21), and the mech- anism for the summation of the terms of formula (459) is on the right- ee side of the machine, which is designated as the “time side” (fig. 22). 361. In table 37 are given the details of the general gearing from the hand-operating crank to the main vertical shafts, together with the details of ai! the gearing in the front section or dial case. It will be noted that S—6 (fig. 25) is the main vertical shaft of the dial case and is connected through the releasable gears to the hour hand, the minute hand, and the day dial, respectively. The releasable gears permit the adjustment of these indicators to any time desired. After ‘an original adjustment is made so that the hour and mmute hand will each read 0 at the same instant that the day dial indicates the begin- ning of a day, further adjustment will, in general, be unnecessary, as the gearing itself will cause the indicators to maintain a consistent relation throughout the year, and by use of the hand-operating crank the entire system may be made to indicate any time desired. The period of the hour-hand shaft is 24 dial hours, and the hand moves over a dial graduated accordingly (3, fig. 23). The minute-hand shaft, with a period of 1 dial hour, moves over a dial graduated into 60 minutes (2, fig. 23). 362. The day dial, which is about 10 inches in diameter, is graduated into 366 parts to represent the 366 days in a leap year. The names of the months and numerals to indicate every fifth day of each month are inseribed on the face of the dial. This dial is located just back of the front plate or face of the machine, in which there is an are-shaped open- ing through which the graduations representing nearly two months are visible at any one time (J, fig. 23). The progress of the days as the machine is operated is indicated by the rotation of this dial past an index or pointer just below the opening (6, fig. 23). This pointer is secured to a short shaft which carries at its inner end a lever arm with a pin reaching under the lower edge of the day dial, against which it is pressed by a light spring. A portion of the edge of the dial equal to the angular distance from January 1 to February 28 is of a slightly larger radius, so that the pin pressing against it rises and throws the day pointer to the right one day when this portion has passed by. On Specia! Publication No. 98 FIGURE 21.—COAST AND GEODETIC SURVEY TIDE-PREDICTING MACHINE. ‘ACIS AWIL ‘ANIHOVW SONILOIGSYd-3AGII1—' 22 JYHNDIA we ee oe 96 “ON vorreot{gng [e!29dg Special Publication No. 98 ae RECORDING DEVICES PREDICTING MACHINE, .—TIDE FIGURE 23 Special Publication No. 98 & a Sern a eee FIGURE 24.—TIDE-PREDICTING MACHINE, DRIVING GEARS. Specia! Publication No. 98 ER aap one? pinhopentnee : i ise FIGURE 25.—TIDE-PREDICTING MACHINE, DIAL CASE FROM HEIGHT SIDE. No. 98 ication Special Publ —TIDE-PREDICTING MACHINE, DIAL CASE FROM TIME SIDE. FIGURE 26. Special Publication No. 98 FIGURE 27.—TIDE-PREDICTING MACHINE, VERTICAL DRIVING SHAFT OF MIDDLE SECTION. Special Publication No. 98 FORWARD DRIVING SHAFT OF REAR SECTION PREDICTING MACHINE, .—TIDE FIGURE 28 No. 98 ication Special Publ DTS) nO Y IR FIGURE 29.—TIDE-PREDICTING MACHINE, REAR END. Special Publication No. 98 FIGURE 30.—TIDE-PREDICTING MACHINE, DETAILS OF RELEASABLE GEAR. FIGURE 31.—TIDE-PREDICTING MACHINE, DETAILS OF CONSTITUENT CRANK. HARMONIC ANALYSIS AND PREDICTION OF TIDES 129 the last day of December this pointer will move back one day to its original position. 363. On the same center with the day pointer there is a smaller index (7, fig. 23) which may be turned either to the right toward a plate in- scribed ‘“‘Common year,” or to the left to a plate inscribed ‘Leap year ’’ When this smaller index is turned toward the right, the day pointer is free to move in accordance with the change in radius of the edge of the dial. If the smaller index is turned toward the left, the day pointer is locked and must hold a fixed position throughout the year. For the prediction of the tides for two or more common years in succession the day dial must be set forward one day at the close of the year in order that the days of the succeeding year may be cor- rectly registered. The day dial can be released for setting by the nut (5, fig. 23) immediately above the large dial rmg. | fl. ana o arora Sena aan 2 |). delQ..|. 8209 H. 1.30 54 4 2-10 | 214 |, 3255 | 255. oe Marigram scale .....-7°.7.~ ..-..... 2.2... E2U7a 228 ie P, || 1,10 | 242 eae [ee Glens 5 Cae |e 32. --|}---220.} 149 |. FIGURE 33. 146 U. S. COAST AND GEODETIC SURVEY The values for column C may then be obtained by applying 360° to the negative values in column D. A429. Form 445, settings for tide-predicting machine (fig. 33).—This form is designed for the computations of the settings for the predicting machine for the beginning of each year of predictions. The forms are bound in books, a separate book being used for each year of predictions. This form is used in connection with Form 444, and for convenience the order of arrangement of the constituents is identi- cal in the two forms. The name of the station, the time meridian, the height dial, marigram gear, marigram scale, and datum plane are copied directly from Form 444. 430. For the amplitude settings the amplitudes of column B of Form 444 are multiplied by the factors f from table 14 for the year for which the predictions are to be made. A convenient way to apply these factors is to prepare a strip of paper with the same vertical spacing as the lines on Form 444 and enter the factors f for the required year on this strip. The strip may then be placed alongside of column B of Form 444 and the multiplication be performed. The same strip will serve for every station for which predictions are to be made for the given year. It has been the recent practice to enter the amplitude settings to the nearest 0.05 foot as being sufficiently close for all practical purposes. 431. For the dial settings for January 1, 0 hour, the Greenwich equilibrium arguments of (V,+u)’s from table 15 are to be applied, ac- cording to the indicated sign, to the angles of column C or D of Form 444, using the angle in column D if it is less than the argument, otherwise using the angle in column C. For the application of the (V,+w)’s a strip similar to that used for the factors f should be pre- pared. The same strip will serve for all stations for the given year. _ For the dial settings it is customary to use whole degrees, except for constituent M,, for which the setting is carried to the first decimal of a degree. 432. The settings for February 1 and December 31 are used for checking purposes to ascertain whether there has been any slipping of the gears during the operation of the machine. To obtain the dial settings for February 1, 0°, and December 31, 24", prepare strips similar to those for the f’s and (V,+w4)’s. On one enter the angular motion of the constituents from January 1, 0", to February 1, 0"; on a second and a third strip, the angular motion for February 1, 0°, to December 31, 24°, for a common and leap year, respectively. For checking purposes a fourth and fifth strip may contain the angular - changes for a complete common and a complete leap year, respectively. The values for these strips may be obtained from table 36. These strips will be found more convenient if arranged with two columns each, one column containing the values in a positive form and the other column containing the equivalent negative value: which is obtained by subtracting the first from 360°. These strips are good for all years, distinction being made between the common and leap years. By applying the first strip to the dial settings for January 1 the values for February 1 are readily obtained, and by applying the second or third strip to the latter settings those for the end of the year are obtained. The values obtained by applying the fourth or fifth strips to the settings for January 1 should also give the correct setting for the end of the year, and thus serve as a check. The HARMONIC ANALYSIS! AND PREDICTION OF TIDES 147 angular changes for computing the settings for any day of the year may be obtained from tables 16 and 17. PREDICTION OF TIDAL CURRENTS 433. Since the tidal current velocities in any locality may be expressed by the sum of a series of harmonic terms involving the same periodic constituents that are found in the tides, the tide- predicting machine may be used for their prediction. For the cur- rents, however, consideration must be given to the direction of flow, and in the use of the machine some particular direction must be assumed. At present the machine is used for the prediction of reversing currents in which the direction of the flood current is taken as positive and the maximum velocity in this direction corre- sponds to the high water of the predicted tide. The ebb current is then considered as having a negative velocity with its maximum corresponding to the low water of the predicted tide. Rotary cur- rents may be predicted by taking the north and east components separately but the labor of obtaining the resultant velocities and directions from these components would be very great without a machine especially designed for the purpose. Predictions can, how- ever, be made along the main axis of a rotary movement without serious difficulties. Formulas for referring the harmonic constants of the north and east components to any desired axis are given in Coast and Geodetic Survey Special Publication No. 215, Manual of Current Observations. 434. The harmonic constants for the prediction of current velocities are derived from current observations by an analysis similar to that used in obtaining the harmonic constants from tide observations. In the current harmonic constants, however, the amplitudes are expressed in a unit of velocity, usually the knot, instead of the linear unit that is used for the tidal harmonic constants. Forms 444 and 445 for the computation of the settings for the tide-predicting machine are applicable for the current predictions and the procedure in filling out these forms is essentially the same as described in paragraphs 421-432 for the tide predictions. The node factors (f) and arguments (V,+ 1) are the same as for the tides. The height dial, marigram gear and scale suitable to the current velocity can be obtained from the table on page 138. Instead of a sea level elevation there should be entered in the column of ‘“‘Remarks’’ the velocity of any permanent current along the axis in which the predictions are tobemade. This velocity should be marked plus (+) or minus (—) according to whether the permanent current is in the flood or ebb direction. 435. The predicting machine is set with the current harmonic con- stants in the same manner as for the tidal harmonic constants. To take account of the permanent current the height summation wheel should be brought to its zero position and the height hand then set at a dial reading corresponding to the velocity of the permanent current, the hand being set to the right of the scale zero if the per- manent current is in the flood direction and to the left if in the ebb direction. The hand crank should be then turned to bring the height hand to its zero position and the curve-pen set at the medial line of the paper, this line now representing zero velocity or slack water. 148 U. Si: COAST AND GEODETIC SURVEY 436. The operation of the machine for the prediction of the cur- rents is similar to that for the prediction of the tides. The machine automatically stops at each maximum flood and ebb velocity and the corresponding times and velocities are then recorded, the flood veloci- ties being read to the right and the ebb velocities to the left of the scale zero. In the prediction of the currents the times of slack water are also desired. These are indicated by the zero position of the recording hand as well as by the intersections of the curve and medial line in the graphic record. The velocity of the current at any inter- mediate time can be read directly from the height dial when the machine has been turned to the time desired and it may be also scaled from the graphic record. 437. Predictions of hydraulic currents in a strait, based upon the difference in the tidal head at the two entrances, may be made by means. of harmonic constants derived from the tidal constants for the entrances. Differences in tidal range or in the times of the high and low waters at the two ends of a strait will cause the water surface at one end alternately to rise above and fall below that at the other end, thus creating a periodic reversing current in the strait. Theo- retically, disregarding friction or inertia, the velocity of the current would vary as the square root of the difference in head, being zero when the surface is at the same level at both ends and reaching a maximum when the difference is greatest. Actually there will gen- erally be a lag of some minutes in the response of the current movement to the difference in head which must be determined from observations. 438. Let the two ends of the strait be designated by A and B, with the flow from A toward B considered as flood or positive and the flow in the opposite direction as ebb or negative. With the waterway receiving the tide from two sources, the application of the terms “flood” and ‘‘ebb” will be somewhat arbitrary, and care must be taken to indicate clearly the direction assumed for the flood move- ment. In the following discussion tidal constants pertaining to entrances A and B will be distinguished by subscripts a and b, respec- tively, and those pertaining to the difference in tidal head by the subscript d. Since the usual constituent epochs known as ‘“‘kappas”’ refer to the local meridian, it will be necessary for the purpose of comparison between places on different meridians to use the Green- wich epochs ‘‘G”’ (par. 226), these being independent of local time and longitude. 439. For any one constituent let 7 represent time as expressed in degrees of the constituent reckoned from the phase zero of its Greenwich equilibrium argument. Also let Y, and Y, represent the height of the constituent tide for any time 7 as referred to the mean level at locations A and B, respectively; and let Y, equal the difference (Y,—Y,). Formulas for heights and difference may now be written Y,=H, cos (T—G,) for location ‘‘A’’ (467) Y,=H, cos (T—G,) for location ‘‘B”’ (468) Y.=H, cos (T—G,) —H, cos (T— G;) =(H, cos G,—H, cos G,) cos T+ (A, sin G,— Hy sin G,) sin T =H, cos (T—G;) (469) HARMONIC ANALYSIS! AND PREDICTION OF TIDES 149 in which H,=|H2+ H}?—2H,H, cos (G,—G,))? (470) _, H, sin G.—H, sin G, H, cos G,—H, cos G, The proper quadrant for G,z is determined by the signs of the numer- ator and denominator of the above fraction, these being the same, respectively, as for the sine and cosine of the angle. Formulas (470) and (471) may be solved graphically (fig. 34) by drawing from any point C a line CD to represent in length and direction H, and G,, respectively; from the point D a line DE to represent in length and direction H, and (G,+180°), respectively. The connecting line from Ga=tan (471) 90° 180 FIGURE 34. C to FE’ will represent by its length the amplitude H, and by its diree- tion the epoch G). 440. Formulas (470) and (471) may be modified to adapt them for use with tables 41 and 42. From (470) we may obtain Ala/Ha=([1 + (A/Ha)’+ 2(Hp/H_) cos (Gy—Ga+180°)]} (472) A1,/H,=[1 + (H./He)?+2(Ha/Hr) cos (Ga—Gr+180°)}! (478) and from (471) we have (H,/H,) sin (G,—G,+£180°) or 150 U. S. COAST AND GEODETIC SURVEY or (H,,/H,) sin (G.—G,+180°) 1+ (H,/H,) cos (G.— G,+180°) Formulas (472) and (474) are to be used when the ratio H,/H, does not exceed unity. In this case take argument’r of the tables=H,,/H,, and argument 2=(G),—G,+180°). If the ratio H,/H, exceeds unity use formulas (473) and (475) and take argument r=H,/H, and argument z=(G,—G,+180°). The tabular values will give the ratios and angular differences represented in the first terms of the formulas. Therefore, in order to obtain the amplitude H,, the tabular value from table 41 must be multiplied by H, if the ratio H,/H, does not exceed unity, or by H, if this ratio does exceed unity. Also to obtain the epoch Gj, the tabular value from table 42 must be increased by G, if the ratio does not exceed unity or by (G,+180°) if the ratio is greater than unity. 441. By the formulas given above separate computations are made for each of the tidal constituents. The values obtained for H, and Gz are the corresponding amplitudes and Greenwich epochs in an har- monic expression for the continually changing difference in elevation of the water surface at the two entrances to the strait. When only a single time zone is involved, the small g’s or modified kappas («’) per- taining to that zone may be substituted for the Greenwich epochs (G) in the formulas. For the prediction of the current, further modifica- tions are necessary in the amplitudes to reduce to velocity units and in the epochs to allow for the lag in the response of the current to the changing difference in water level at the two entrances to the strait. 442. Since the velocity of an hydraulic current is theoretically proportional to the square root of the difference in head, we may write Tan (Ga— Gy te 180°) — (475) (Velocity)?=constant (C) Xheight difference (476) If we now let V equal the average velocity of the current at time of strength as determined from actual observations and assume that the corresponding difference in water level is 1.02 times the difference resulting from the principal constituent M>, we may obtain an approxi- mate value for the constant (C) by the formula C=V?/(1.02Mp) (477) in which M,j is the amplitude of the constituent M, in the harmonic expression for the difference in head. The application of the factor (C) to all the constituent amplitudes in this expression has the effect of changing the height units into units representing the square of the velocity of the resulting current. 443. The lag in the current is usually determined by a comparison of the times of strengths and slacks from actual observations with preliminary predictions of the corresponding phases based upon the harmonic constants derived by the method just described. This lag expressed in hours is multiplied successively by the speed of each constituent and the result applied to the preliminary epoch for that constituent. 444. In order that the magnitude of the constituent amplitudes may be adapted for use with the predicting machine, a scale factor (S) is HARMONIC ANALYSIS AND PREDICTION OF TIDES 151 introduced. This factor, which depends upon the velocity of the cur- rent, is selected with the view of obtaining a reasonably large working scale without exceeding the limitations of the predicting machine. The following scale factors are suggested: ' Average velocity of current at time of strength: Scale factor WessithamnOcocknO tee Sie) ae) a a eee Mees ee Ess See 20 ERROTIAN Os nUON Oso UKeIn Olle atone as WS erat pee UE VOR ey ae ee yee 10 FROMMAOF OS FOw le OFM O beete ed eyes SB ke hee Seperate. 5 Brome Oktomle sok Ot se. esis on pigdh oa ely ee oR Ss 3 EOE LOWS. Os KT OGS ay eh peach ee ee ee ee cake ee 2 romeo OtoOrs Oy KO tse oo ee ee aes ae Fee We Oe emcee! ee 1 romeo OrtovaOsknotsies lee tis Oe Bye! Se ae, shee ae 0.5 BromeavOntor a OV Knots 205 ke ei in eh blondy Eee Bin ai 0.25 Brom o-OstowliQvOvknotssefi ss: s2ns8 see eke eee et eee ee 0.1 In practice, the scale factor is usually combined with the factor (C) and the product applied to each of the constituent amplitudes in the expression for difference in head. i 445. Using the harmonic constants, modified in the manner de- scribed above, in the predicting machine, the resulting dial readings will represent the square of the current velocity. In order to avoid the necessity of extracting the square root of each individual reading, a square-root scale may be improvised and substituted for the regu- lar height dial on the machine. From a consideration of the con- struction of this machine, it can be shown that with a scale factor of unity the angular position of a velocity graduation as measured in degrees from the zero point will be 9° (velocity)?.. Thus the 1-knot graduation will be spaced 9° from the zero, the 2-knot graduation at 36°, the 3-knot graduation at 81°, etc. For any scale factor (S), the formula for constructing the square-root scale becomes Angular distance from dial zero=9° XS (velocity)? (478) 446. To take account of any nontidal current not attributed to difference in head at the two entrances to the strait, a special -gradu- ation of the square-root scale is necessary. Let V, represent the nontidal current velocity, positive or negative according to whether it sets in the flood or ebb direction, and let V represent the resultant velocity as indicated by any scale graduation, positive or negative according to whether it is flood or ebb. The angular distance of any scale graduation as measured from an initial point, usually marked by an arrow, may then be expressed by the following formula: Angle in degrees=9 x SX (V —V,)? (479) The required angle is to be measured to the right or to the left of the initial point according to whether the angle (_V—YV,) is positive or negative. When setting the predicting machine the velocity pointer must be at the initial point marked by the arrow when the sum of the harmonic terms is zero. 447. In the graphic representation of the summation of the har- monic terms by the predicting machine, the scale of the marigram depends upon the marigram gear ratio as well as upon any scale factor which may have been introduced. With a gear ratio of unity, the scale of the marigram is 0.1 inch per unit of machine setting. In the summation for the hydraulic currents, the marigram read by a natural lea? U. S$. COAST AND GEODETIC SURVEY scale would indicate the square of the velocity. A special square-root reading scale for taking the velocities direct from the marigram may be prepared as follows: Let }’=distance of any velocity graduation from zero of scale. Then Y (an inches) =0.1 X (scale factor) < (gear ratio) X (velocity)? (480) With the scale factor and gear ratio each unity, 1 knot of velocity would be represented by 0.1 inch on the marigram, 2 knots by 0.4 inch, 3 knots by 0.9 inch, etc. With a scale thus constructed the velocity of the tidal current may be taken directly from the mari- gram. Any nontidal current which is to be included may afterward be applied. TABLES EXPLANATION OF TABLES Table 1. Fundamental astronomical data.—This table includes fundamental constants and formulas with references which form the basis for the computation of other tables contained in this volume. Because of the smallness of the solar and lunar parallax no distinction is made between the parallax and its sine. The eccentricity of the earth’s orbit and the obliquity of the ecliptic are given for epoch January 1, 1900. The former changes about 0.000042 per century and the latter about 0.013 of a degree in a century. The values given may therefore be considered as applicable to the present century. The formulas for longitude of both sun and moon are the same as used in the previous edition of this book and are from the work of Simon Newcomb. In a later work by Earnest W. Brown, slightly different values are obtained for the elements of the moon’s orbit but the differences may be considered negligible in so far as the tidal work of the present century is concerned. In these formulas it will be noted that J is the number of Julian centuries reckoned from Greenwich mean noon on December 31, 1899, of the Gregorian calendar which corresponds to December 19, 1899, by the Julian calendar. In the application of these formulas to early dates special care must be taken to make suitable allowances for the particular calendar in use at the time. See page 4 for information in regard to calendars. Table 1 includes the numerical values of the mean longitude of the solar and lunar elements for the beginning of the century years 1600 to 2000 and also the rate of change in these longitudes as of January 1, 1900. As the variations in these rates are very small, they are applicable without material error for all modern times. This table includes also the principal astronomical periods depending on the solar and lunar elements with formulas showing how they are derived. In these formulas the longitude symbol is used to represent its own rate of change according to the unit in which the period is expressed. Table 2. Harmonic constituents —This table includes the arguments, speeds, and coefficients of the constituent harmonic terms obtained in the development of the tide-producing forces of the moon and sun. They are grouped with reference to the formulas of the text from which they are derived, the long-period constituents first, followed by the diurnal, semidiurnal, and terdiurnal terms. The reference numbers in the first column correspond to the numbered terms of the formulas of the text, the letter A indicating a term in the lunar development and the letter B a term in the solar development. In the second column the usual symbols are given for the principal constituents, parentheses being used when the term only partially represents the constituent. For an explanation of the constituent argument (E) see page 22. The argument consists of two parts—the VY which contains the 153 154 U. S. COAST AND GEODETIC SURVEY uniformly changing elements and determines the speed and period of the constituent, and the w which is a function of the moon’s node with slow variations and which is treated as a constant for a limited series of observations. Because of the very small change in the element p, it may for practical purposes also be treated as a constant with a mean value of 282° for the present century. The constituent speeds are obtained by adding the hourly rates of change in the elements appearing in the V of the arguments. The hour angle (7’) of the mean sun changes at the rate of 15° per hour. The pourly rate of change for each of the other elements will be found in table 1. For an explanation of the constituent coefficients (C) see page 24. The coefficients of the solar terms include the solar factor S’ (para- graph 118), and coefficients of the lunar terms involving the 4th power of the moon’s parallax include the factor a/c (paragraph 108); in order that all terms may be comparable when used with the common basic factor U. It is to be noted that in the present system of coef- ficients for the terms of the principal tide-producing force there is. included a factor ‘‘2’’ which was formerly incorporated in the general coefficient. For the terms involving the 4th power of the parallax there is a corresponding factor of ‘3”’ in order that all terms may be comparable in respect to the vertical component force. In general the coefficients have been computed in accordance with the coefficient formulas of the text, but exceptions were made for the evectional and variational constituents p,, v2, 2, and po, the coeffi- cients of which are based upon computations by Professor J. C. Adams who was associated with Darwin in the investigation of har- monic analysis and who carried the development of the lunar theory to a higher order of precision than is provided in this work. (See pp. 60-61 of the Report of British Association for the Advancement of Science for year 1883.) The node factor (f) is explained on page 25. The last column of table 2 contains references to the formulas for the node factors of the various constituents. Table 2a. Shallow-water constituents.—In this table there are listed the overtides and compound tides which are described on page 47. Table 3. Latitude factors —This table includes numerical values of the latitude (Y) functions which appear in the text as factors in formulas representing component tidal forces and the equilibrium height of the tide. Tle combination symbol at the head of each column is taken to suggest the formula to which it applies. Thus, the letters v, s, and w refer respectively to the vertical, south, and west components of the force, the letter » being applicable also to the formulas for the equilibrium height of the tide which have the same latitude factors as the corresponding terms in the vertical component of the force. The first numeral ‘3’ or ‘4” indicates whether the formula is from the development of the principal force involving the cube of the parallax or from the development involving the 4th power of the parallax of the tide-producing body. The last digit “0,” “1,” “2.” or “3” refers to the species of the constituents and indicates whether they are long-period, diurnal, semidiurna]l, or terdiurnal. In several cases the same latitude factor is applicable to a number of different groups as indicated at the head of the column in the table. HARMONIC ANALYSIS AND PRHDICLION OF TIDES 155 The following formulas were used in computing the latitude factors. The maximum value (irrespective of sign) with corresponding latitude is also given for each function. Vp B32 Si) Oe eee maximum—1 when Y=+90°. WY sats) PP) Gee aes aR dati maximum-+1 when Y=+45°. 330, Vss2, and Yi, same as V4; WA COsn) ree Len ee maximum-+1 when Y=0. Y 4g SAME as Y 439 Ware cose Yee ee maximum+1 when Y=0 or+90°. cus ciia, 1 a ee eee inet maximum +1 when Y=+90°. Wa cos ee oe erty? maximum-+1 when Y=0. Yo=sin Y (cos?¥Y—2/5)_-- maximum +0.4 whenY= +90°. Y,u=cos Y (cos’Y—4/5)_-_ maximum—0.2754 when Y=-+58.91°. Yao Same as Y yay Vac Y cos*Y. 2 maximum +0.3849 when Y= -+35.26°. Y 543 same as ees AE Cos ye. eR AIS maximum- 1 when Y=0. Y.u=sin Y (cos?¥Y—4/15)__ maximum +0.2667 when Y=+90°. Yy2=cos Y (cos?¥—2/3)__. maximum—0.2095 when Y=-+61.87°. Ve (COs 4/5) aes maximum—0.8 when Y= +902. Table 4. Mean longitude of lunar and solar elements.—This table contains the mean longitude of the moon (s), of the lunar perigee (p), of the sun (h), of the solar perigee (p,), and of the moon’s ascending node (NV), for January 1, 0 hour, Greenwich mean civil time, for each year from 1800 to 2000, the dates referring to the Gregorian calendar. These values are readily derived from table 1, the rate of change in the mean longitude of the elements for the epoch January 1, 1900, being applicable without material error to any time within the two centuries 1800 to 2000 covered by table 4. The same rate of change may also be used, without introducing any errors of practical impor- tance, to extend table 4 to dates beyond these limits. In extending the table, care should be taken to distinguish between the common and leap years, and for the earlier dates due consideration should be given to the kind of calendar in use. (See p. 4 for discussion of calendars.) It will be noted that each Julian century contains 36,525 days, while the common Gregorian century contains only 36,524 days, with an additional day every fourth century. Table 5. Differences to adapt table 4 to any month, day, and hour.— These differences are derived from the daily and hourly rate of change of the elements as given in table 1, multiples of 360° being rejected when they occur. The table is prepared especially for common years, but is applicable to leap years by increasing the given date by one day if it is between March 1 and December 31, inclusive. The cor- rection for the hour of the day refers to the Greenwich hour, and if the hour for which the elements are desired is expressed in another ane of time the equivalent Greenwich hour must be used for the table. Table 6. Values of I, v, &, ’, and 2v"’ for each degree of N.—Referring to figure 1 (page 6), note that by construction are 87’ equals arc 8. Then in the spherical triangle 2 7A, the three sides are N, », and (N—£), and the opposite angles are respectively (180°—J), 7, and w. 156 U. S. GOAST AND GEODETIC SURVEY Therefore we have the following relations which may be used in com- puting the values of J, v, and é in the table: cos [=cos 7 cos w—sin 7 sin w cos N —(0,91370— 0.03569 cos N cos 4(w—12) tan 3(N—£-+») > Gis SE) tan 3N=1.01883 tan $N _ sin 3(w—1) sin 3(w +1) For the computation of v’ and 2»v’’, formulas (224) and (232) on pages 45-46 may be used. The tabular values themselves were taken from the preceding edition of this work where they were based upon formulas differing slightly from those given here but any differences arising from the use of the latter may be considered as negligible. Table 7. Values of log R, for amplitude of constituent Ly.—Values in this table are based upon formula (213) on page 44. Table 8. Values of R for argument of constituent L,—Values in this table are derived from formula (214) on page 44. Table 9. Values of log Q, for amplitude of constituent M,—Values in this table are based upon formula (197) on page 41. Table 10. Values of Q for argument of constituent M,—Values in this table are derived from formula (203) on page 42. Table 11. Values of wu for equilibrium arguments.—This table is based upon the u-formulas in table 2 and includes values for the principal lunar constituents for each degree of N. The w’s of L, and M,, which are functions of both N and P are given separately in table 13 for the years 1900 to 2000. Table 12. Log factor F for each degree of I—The factor F is the reciprocal of the node factor f to which references are given in table 2. The values in table 12 are based upon the formulas for these factors and are given for all the lunar constituents used in the tide-predicting machine, excepting values for L, and M, which are given separately in table 13. Table 13. Values of u and log F for Ly and M,.—From a com- parison of the u’s of constituents L,, M;, and Mz in table 2, it will be noted that the following relations exist: tan 4(N—é—p) tan 4N=0.64412 tan 3N u of Ly»=(u of M,)—R u of M,=i(u of M2) +Q Also, the following relations may be derived from formula (215) on page 44 and formula (207) on page 43 since the factor F' is the recip- rocal of the node factor f: log F(L,)=log F(M2)+log R. log F(M,) =log F(O;) +log Q, The values for table 13 were computed by the above formulas, the component parts being taken from tables 7 to 12, inclusive. The values for log F(M,) in this table are in accord with Darwin’s original HARMONIC ANALYSIS AND PREDICTION OF TIDES 157 formula from which a factor of approximately 1.5 was inadvertently omitted (see page 43). Table 14. Node factor f for middle of each year 1850 to 1999.—The factor f is the reciprocal of factor F. The values for the years 1850 to 1950 were taken directly from the Manual of Tides, by R. A. Harris, and the values for 1951 to 1999 were derived from tables 12 and 13. Table 15. Equilibrium argument (V,+u) for beginning of each year 1850 to 2000.—The equilibrium argument is discussed on page 22. The tabular values are computed by the formulas for the argument in table 2, the V, referring to the value of V on January 1, 0 hour Greenwich mean civil time, for each year, and the wu referring to the middle of the same calendar year; that is, Greenwich noon on July 2 in common years and the preceding midnight in leap years. The value of the 7 of the formulas is 180° for each midnight, and the values of the other elements for the V may be obtained from table 4. The wu of the argument may be obtained from tables 11 and 13 after the value of NV has been determined for the middle of each year from tables 4 and 5. In constructing table 15 the values for the years 1850 to 1950 were taken directly from the Manual of Tides, by R. A. Harris, and the values for the years 1951 to 2000 were computed as indicated above. Tables 16, 17, and 18.—These tables give the differences to adapt table 15 to any month, day, and hour, and are computed from the hourly speeds of the constituents as given in table 2. The differ- ences refer to the uniformly varying portion V of the argument, it being assumed that for practical purposes the portion wu is constant for the entire year. The approximate Greenwich (V,-+w) for any desired Greenwich hour may be obtained by applying the appropriate differences from tables 16, 17, and 18 to the value for the first of January of the required year, as given in table 15. To refer this Greenwich (V,+ 4) to any local meridian, it is necessary to apply a further correction equal to the product of the longitude in degrees by the subscript of the constituent, which represents the number of periods in a con- stituent day. West longitude is to be considered as positive and east longitude as negative, and the subscripts of the long-period constituents are to be taken as zero. This correction is to be subtracted. The (V,+wu) obtained as above will, in general, differ by a small amount from the value as computed by Form 244, because in the former case the wu refers to the middle of the calendar year and in the latter case to the middle of the series of observations. Table 19. Products for Form 194.—This is a multiplication table especially adapted for use with Form 194, the multipliers being the sines of multiples of 15°. Table 20. Augmenting factors —A discussion of augmenting factors is given on page 71. The tabular values for the short-period constit- uents are obtained by formulas (308) and (309) on page 72, and those for the long-period constituents by formulas (403) and (404) on page 92. For constituents S,, S:, etc. the augmenting factor is unity. Tables 21 to 26.—These tables represent perturbations in K, and S. due to other constituents of nearly equal speeds. They are based upon formulas (359) to (864), inclusive, on page 83. 158 U. S. COAST AND GEODETIC SURVEY Table 27. Critical logarithms for Form 245.—This table .us de- signed for quickly obtaining the natural numbers to three decimal places for column (3) of Form 245 from the logarithms of column (2). The logarithms are given for every change of 0.001 in the natural number. Each logarithm given in this table is derived from the natural number that is 0.0005 less than the tabular number to which it applies. Intermediate logarithms, therefore, apply to the same natural number as the preceding tabular logarithm. For example, logarithms less than 6.6990 apply to the natural number 0.000 and logarithms from 6.6990 to 7.1760 apply to the natural number 0.001, etc. Table 28. Constituent speed differences.—The constituent speeds as given in table 2 were used in the computation of this table. Table 29. Elimination factors—These tables provide for certain constant factors in formulas (389) and (390). Separate tables for each length of series and different values for each term of the formulas are required. The tabular values are arranged in groups of three, determined as follows: So St (fh First value=logarithm of oe sin 4 (6—a)r. 4(b—a)r & omnaid int Second value=natural number oe always taken as positive. Ne Rechte le hid alent pe ale ae eee “ay aOStinine: sin 4(b—a) or 4(6—a)r+180, if 7 is negative. 4(b—a)r Table 30. Products for Form 245.—This table is designed for ob- taining the products for columns (6) and (7) of Form 245. Table 31. For construction of primary stencils——This table gives the differences to be applied to the solar hours in order to obtain the constituent hours to which they most nearly coincide. Each differ- ence applies to several successive solar hours, but for brevity only the first solar hour of each group to which the difference applies is given in the table. An asterisk (*) indicates that the solar hour so marked is to be used twice or rejected according to whether the constituent speed is greater or less than 15p, when in the summation it is desired to assign a single solar hour to each successive constituent hour. For the usual summations in which each solar hour height is assigned to the nearest constituent hour no attention need be given to the asterisk. The table is computed by substituting successive integral values for d in formula (243) and reducing the resulting solar hour of series (shs) to the corresponding day and hour. The solar hour to be tabulated is the integral hour that immediately follows the value of (shs) of the formula. If the fractional part of (shs) exceeds 0.5, the tabular solar hour is marked by an asterisk (*). The successive values of d, although used positively in formula (243), are to be considered as negative in the application of the table when the speed of the con- stitutent is less than 15p. When the constituent speed is greater than 15p, the difference is to be taken as positive. All tabular differences are brought within the limits +24 hours and —24 hours by rejecting multiples of +24 hours when necessary, and for convenience in use all differences are given in both positive and negative forms. HARMONIC ANALYSIS AND PREDICTION OF TIDES 159 The following example will illustrate the use of the table: To find constituent 2Q hours corresponding to solar hours 12 to 23 on 16th day of series. By the table we see that solar hour 12 of the 16th day of series is within the group beginning on solar hour 8 of the same day with the tabular difference of +19 or —5 hours, and that the differ- ence changes by —1 hour on solar hours 15 and 21, the latter being marked by an asterisk. Applying the differences indicated, we have for these solar hours on the 16th day of series: Solar hour-_-_-_- aS el SA ore G aes Uli/pmel Senin Oma 2 On eZ iE 2S Difference__-_. —5 —5 —5 —6 —6 —6 —6 -—6 —6 —7 —-7 —7 Constituent 2Q hour! - _- ie 8, eo Ge VOL IL, 2. ES? ase il Aoki? EG In the results it will be noted that the constituent hours 9 and 14 are each represented by two solar hours. If it should be desired to limit the representation to a single solar hour each, the hours marked with the asterisk should be rejected. To find constituent OO hours corresponding to solar hours 0 to 18 on the 22d day of series. The 0 hour of the 22d day is in the group beginning on solar hour 14 of the preceding day with the tabular difference of +14 or —10 hours, and changes of +1 hour in the differences occur on solar hours 3 and 17 of the 22d day. It will be noted that the hour 3 is marked by an asterisk. Applying the differences from the table as indicated, we have for the 22d day of series: Solar hours O10) o12, 913) Taps 16ers i7 eas Derrek al fish 1s 418 448 Hie yas <9) <9) =o 9 oe, eo) 8 es Constituent Ollicues 149 °15,/ 16) “1s, 19) 20) 21, 92)) 23, 0 4, 2s a 5, 6, 7) (8 “40 In the results it will be noted that constituent hours 17 and 8 are missing. If it is desired to have each of these hours represented also, the solar hours marked by asterisks will be used again. In this table the constituents have been arranged in accordance with the length of the constituent day. Table 32. Divisors for primary stencil sums.—This table contains the number of solar hourly heights included in each constituent hour group for each of the standard length of series when all the hourly heights have been used in the summation. Table 33. For construction of secondary stencils —Constituent A is the constituent for which the original primary summations have been made, and constituent B is the constituent for which the sums are to be derived by the secondary stencils. The ‘‘Page’’ refers to the page of the original tabulations of the hourly heights in Form 362. The differences in this table were calculated by formula (252), and the corresponding ‘‘Constituent A hours” from formula (250), m being assigned successive values from 1 to 24 for each page of record. Special allowance was made for page 53 of the record to take account of the fact that in a 369-day series this page includes only 5 days of record. The sign of the difference is given at the top of the column. For K-P and R-T the positive sign is to be used for constituents K and R and the negative sign for constituents P and T. For brevity all the 24 constituent hours for every page of record are not directly represented in the table. The difference for the omitted hours for any page should be taken numerically one greater 160 U. S. COAST AND GEODETIC SURVEY than the difference for the given hours on that page. For an example, take the hours for page 2 for constituent OO as derived from con- stituent J. According to the table the difference for the constituent hours 10 to 3, inclusive, is 9 hours; therefore the difference for the omitted hours 4 to 9, inclusive, should be taken as 10 hours. For constituent 2Q as derived from constituent O the three differences usually required for each page are given in full. ee use of the table may be illustrated from the example above, as follows: Page 2— J-hours —----------- 0, il 2; 3, 4, 5, 6, 7, 8, 9, 10, 11 Difference__._---- +9, oF (OS OS 10S LO 10s LOS aOse tO} wae 9 OO-hours___--_-- 9 SAO Dd alse WA e155) iG, P73 eS SAO ae Jehourse= = 12,. 13, 7,14, 15, 16, 17, 718, . 19, — 205 Soe es Difference______-_- +9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9 OO-hours___---- Pl, “Fey WB (Oh Dee DM B3t pruy4ereeOpa | Cee LS The period 24 hours should be added or subtracted when necessary in order that the resulting constituent hours may be between 0 and 28. Table 34. For summation of long-period constituenis——This table is designed to show the assignment of the daily page sums of the hourly heights to the constituent divisions to which they most nearly correspond. The table is based upon formula (395). The constituent division to which each day of series is assigned is given in the left-hand column. For Mf, MSf, and Mm there will frequently occur two consecutive days which are to be assigned to the same constituent division. In such cases the day which most nearly corresponds to the constituent division is the only one given in the table, and this is marked by an asterisk (*). The missing day, whether it precedes or follows the one marked by the asterisk, is to be assigned to the same constituent division. For Sa a number of consecutive days of series are assigned to each constituent division. In the table there are given the first and last days of each group. Table 35. Products aS/15 for Form 444.—This table contains the products of constituent speeds and time meridian longitudes for formula (466) which is used in obtaining values of («’—«) for column A of Form 444. Table 36. Angle differences for Form 445.—This table gives the differences for obtaining and checking the dial settings for February 1 and December 31, as entered in Form 445. The differences are de- rived from tables 16 and 17. Table 37. Coast and Geodetic Survey tide-predicting machine No. 2— General gears.—This table gives the details of the general gearing from the hand-operating crank to the main vertical shafts, together with the details of the gearing in the front section or dial case. In this table the gears and shafts are each numbered consecutively for con- venience of reference, the gears being designated by the letter G and the shafts by the letter S. In the second column are given the face of each bevel or spur gear and the diameter of each shaft. The next two columns contain the number of teeth and pitch of each bevel and spur gear. The pitch is the number of teeth per inch of diameter of the gear. The worm screw is equivalent to a gear of one tooth, as it requires a complete revolution of the screw to move the engaged wheel HARMONIC ANIALYSIS AND PREDICTION OF TIDES 161 one tooth forward. The period of rotation of each shaft and gear is relative and refers to the time as indicated on the face of the machine, which for convenience is called dial time. Table 38. Coast and Geodetic Survey tide-predicting machine No. 2— Constituent gears.—This table contains the details of the gearing from the main vertical shafts to the individual constituent cranks. Column I gives the number of teeth in the bevel gear on the main vertical shaft; column II, the number of teeth in the gear on the intermediate shaft that meshes with the gear on the vertical shaft; column III, the number of teeth in the gear on the intermediate shaft that meshes with the gear on the constituent crank shaft; and column IV, the number of teeth in the gear on the crank shaft. For the long-period constituents the worm gear is taken as the equivalent of one tooth. For each of these constituents there is a short secondary shaft on which sliding gears are mounted, but the extra gears do not affect the speed of any of the crank shafts except that for constituent Sa in which case a ratio of 1:2 is introduced. The crank-shaft speed per dial hour for each constituent is equal to 3qos¢ column I << ea nua gag column II’*column IV both values appearing in each of the columns II and III is to be taken as the value for the column. The column of ‘Gear speed per dial hour” contains the speeds as computed by the above formula. For comparison the table contains also the theoretical speed of each of the constituents and the accumulated error per year due to the difference between the theoretical and the gear speeds. For convenience of reference the table includes also the maximum amplitude settings of the constituent cranks. Table 39. Synodic periods of constituents.—This table is derived from table 28, the period represented by 360° being divided by the speed difference and the results reduced to days. Table 40. Day of year corresponding to any date.—This table is convenient for obtaining the difference between any two dates and also in finding the middle of any series. Table 41. Values of h in formula h=(1+7r°+2r cos x)?.—This table may be used with formulas (472) and (473) on page 149 to obtain constituent amplitudes for the prediction of hydraulic currents. Table 42. Values of k in formula kta Tog ge rhs table may be used with formulas (474) and (475) on pages 149-150 to obtain constituent epochs for the prediction of hydraulic currents. For constituent Sa the product of 162 U. S. GOAST AND GEODETIC SURVEY Table 1.—Fundamental astronomical data Mean distance, earth to sum_—--_+___=--—-_=-=- ===) 92, 897, 416 miles « Mean distance} earth to moon _=-__=_-2-=-- =.= 2 eee 238, 857 miles « Equatorial radius of earth (Hayford’s Spheroid of 1909) ___ 3, 963. 34 miles « Polar radius of earth (Hayford’s Spheriod of 1909)_______ 3, 949. 99 miles « Mean radius of earth (a), (Intern. Ell.) 6,371,269 meters ® i 20,903,071 feet = 8, 958. 91 miles Solar parallax (Paris Conference) -___-----____- 8.80’ 2 = 0. 000, 042, 66 radian Lunar equatorial horizontal parallax (Brown) _-57’ 2.70’ ¢ = 0. 016, 59 radian Mean solar parallax in respect to mean radius (a/c;)________ 0. 000, 042, 61 radian Mean lunar parallax in respect to mean radius (a/c) ______-_- 0. 016, 57 radian Eccentricity of earth’s orbit (e;), epoch Jan. 1, 1900________ 0. 016, 75 ¢ E-ccentricitysofomoonysionpity(€) eae 0. 054, 90 4 Obliquity of the ecliptic (w), epoch Jan. 1, 1900 DO mEe ae OMMeo 23. 452° Inclination of moon’s orbit to plane of ecliptic (7) 5° 08’ 48.3546’ 4 = 5. 145° Ratio of mass of sun to combined mass of earth and moon (Sitter) S28 426 Vee a ee ee he to 327, 932 ° Ratio of mass of earth to mass of moon (Hinks) _________- 81. 53° Mass (of sun/mass) of earthy GS//i) en a ee ee eee 331, 954 Mass of moon/mass of earth (M/E)____---.--_-_-_-_--__-- 0. 012, 27 Solar coemicient Wi i1— 10S/E) (a/c 4) 2 35 — ee .2569 x 10-7 Basichtactor) Umm CVE) = (Gic)h sees ene ee 0S .5582 x 1077 Solarphactorusy 16 Wai xo See oer Foe dhe ee Se 0. 4602 In the following formulas for longitude, 7’ represents the number of Julian centuries (36525 days) reckoned from Greenwich mean noon, December 31, 1899 (Gregorian Calendar). Mean longitude of sun (h) \ = 279° 41’ 48.04’’ + 129,602,768.13’’ T+ 1.089’ T2«¢ | Longitude of solar perigee (p 1) et S280 137. 15.0/" 16, 189103" 1632" 2 —— Ol012” 7 ar J Mean longitude of moon (s) \ = 270° 26’ 14.72’’+ (1336 rev.+1,108,411.20’’) T+ 9.09’’ T 2+ 0.006,8/’ T3 « |Longitude of lunar perigee (p) |} == 334° 19’ 40.87" + (11 rev. + 392,515.94’’) T — 37.24’ T2 — 0.045’ T3 « | Longitude of moon’s node (N) | == 259° 10’ 57.12’’ — (5 rev. + 482,912.63’) T -++ 7.58’ T?2 + 0.008” Te \Ratio of mean motion of sun to that of moon (m)_---------- 0. 074, 804 a American Ephemeris and Nautical Almanac for year 1940, p. xx. > Table of astronomical constants by W. de Sitter, published in Bulletin of the Astronomical Institutes of the Netherlands, Vol. VIII, No. 307, July 8, 1938, pp. 230-231. Oe Astronomical Papers for the American Ephemeris, by Simon Newcomb; Vol. VI, pp. 9-10, and Vol. pt. 1, p. 224. @’The Solar Parallax and Related Constants, by William Harkness, p. 140. 163 HARMONIC ANALYSIS! AND PREDICTION OF TIDES Table 1.—Fundamental astronomical data—Continued MEAN LONGITUDE OF SOLAR AND LUNAR ELEMENTS FOR CENTURY YEARS F Solar Lunar Moon’s Epoch, Gregorian calendar Sun F Moon 7 Greenwich mean civil time perigee perigee node h Di 3 D N ° ° ° ° ° COO Mans 1) O}HOUT © eee Sere. eee 279. 857 276. 067 99. 725 7.417 301. 496 OO seems ea Oia OUT e see eee 280. 624 277. 784 47. 604 116. 501 167. 343 SOO Tan le On Our 2o2 Sess 2 AS ee 280. 407 279. 502 342. 313 225. 453 33. 248 LIOO ane OUROUR Me eae ee Sa ate 280. 190 281. 221 277. 026 334. 384 259. 156 | QUO} ante Oth oun 38 =e oe eee ake 279. 973 282. 940 211. 744 83. 294 125. 069 RATE OF CHANGE IN MEAN LONGITUDE OF SOLAR AND LUNAR ELEMENTS (EFOCH, JAN. 1, 1900) << __ Per Julian cea- Per common Elements tury (36525 days)| year (365 days) Per solar day | Per solar hour ° ° ° ° Sun; (QD) oye ee ee 1007+ 0.769 359. 761, 28 0. 985, 647, 3 0. 041, 068, 64 SOMALI CrISCel (Dy) = ame ee ae ee 1. 719 0. 017, 18 0. 000, 047, 1 0. 000, 001, 96 IVROOTIN(S) eee yee ene eu eee 8 1336r+307. 892 | 137-+-129. 384, 82 13. 176, 396. 8 0. 549, 016, 53 WiMar perigee) () aes se eee 117-109. 032 40. 662, 47 0. 111, 404, 0 0, 004, 641. 83 IV ROOTYS iT GG CCN») Bae se eee —5r—134. 142 —19. 328,19 | —0. 052, 953.9 | —O, 002, 206, 41 MEAN ASTRONOMICAL PERIODS (Symbols refer to rate of change in mean longitude) Solar days Siderealid ay so00e/ (S60 Satoh ss ee a a RMS ST SS Siar RI Le 0. 997, 270 MUG ay s60r/(S00c 1S) see e hee oe ont aoe he ke ais ea a ee ee ie ae 1. 035, 050 INodicalynron ths s602)/(SN)) sass] oo eR Nan a oo 27. 212, 220 PERG Picalea Onmthisg a0 Sass eee ss ee en ee ae ee Oe ee oe Se 27. 321, 582 ATIOMAlISticunon th ps60s/(S=—D) eneea =a ee ee ee ee ek a ee eee en eee 27. 554, 550 Synodicalemonth"o60e) (SN) ee wen he aa eee crete Ul ee Reeds OE oe 29. 530, 588 Moons evectional period, 8607/(s—2h--p) 252. 2- 22 ee ee ee 31, 811, 939 FE CLIPSOMY Calis tO OU) (IU IIND) ee aes cos Mere cent i aaah uf ye ae Sr Sea 346. 620, 0 PRO picale-y.Oay,: S60 >) ies ene Set ee eS ee ee ere _--- 365. 242, 2 FATIOMIALISEICHy CATA OU s/t 1) ee es eee ae ee ee ee Sa 365. 259, 6 COTTON RY CAT ae eee ee ean aN Re cient een re Sa Si acu L: Sat eee ee we ke ea See ae a 365. 000, 0 (NEGATE COL OF ATV CAT ae ee te as ae eee INES Ge ee ee a ee ae 365. 242, 5 IMIEA TTT A sy CAL = So eee aE Se es a Se So ha a meee oe Rr ee ee 365. 250, 0 CADDY CAT ee eee ote BRINE Bs EN E28 ss ya eh ei NE sl te ci NO Ne eee 366. 000, 0 Evectional period in moon’s parallax, 360°/(h—p) ----------------_----------------------------- 411. 784, 7 RevolutionroMlunar perizeewo00e/ Desens eae ee ee eee 8.85 Julian years VE VOLUELOMNOL IN OOMVSsMOG CoG Oo) eee ee eae ogee em ee ee 18.61 Julian years Revolution of solar perigee, 360°/p1 209 Julian centuries 164 U. S. COAST AND GEODETIC SURVEY Table 2,— Harmonic constituents Bie Argument (E) ie Ref. peed per No, | Symbol solar hour V u LUNAR LONG-PERIOD TERMS, FORMULA (62) Au eet ee ee Zero (permanent term)_--_. -|_____--------------- zero Aor 5 Mm_._-___ Cy eee eee De Siete ZRT ON seer tone ane 0. 544, 374, 7 Als Sash | ae 28— 2p ee ashe. 2. 3 SOS ee ZET OWA See a ee a 1. 088, 749, 4 Ye Sel SD st el ee OT Obes oh ye 0. 471, 521, 1 Algae MSf____- 28 Of aie LL ee Sa Ss ZETO We cs inte moe 1.015, 895, 8 Ao._.--- Wei gi Malte abe WS tae SREREES. AT fas Ti Wai eg 1. 098, 033, 1 TN Bees bate « SSD ae Rn a ee are Corea a DE eer meen peat 1. 642, 407, 8 Agente llc be Bk Spt 1 808s ae ek 7} Sa ee ee a 0. 558, 658, 4 PA Ras mal eee ee CN OY a i tr a ae ripe tet al OE neta tee eee 2. 186, 782, 5 Alpes |e ee Sh = hase ee Fee eee ita ee See 1. 569, 554, 3 Aljysteneet ioe o. 2. sel 8+2h—p+180°_____._____-__- OEE Seer ne ee 0. 626, 512, 0 VA otetatad roses eee AS — Dita oe Se eee were ee 01 Rien tha ie ee 2. 113, 928, 8 Al gMerea a Tae a DiS FEN F toh gee SSS Pps Jae Ss 0. 082, 137, 3 LUNAR LONG-PERIOD TERMS, FORMULA (137) 0. 549, 016, 5 1. 093, 391, 2 0. 004; 641, 8 1. 647, 049, 6 2.191, 424, 3 SOLAR LONG-PERIOD TERMS, FORMULA (185) ~ dtl ages ZeroON(PELMAneNE term) sae see | eae ae ee ee zero Tepe | eee eae | Sp The ee pa a alee aan ete EE ZL Ose eee aR 0. 041, 066, 7 7 es Ur ee DOD eee wk Sa Ey ZOLO nee saan 0. 082, 133, 4 Betas Ssa__.___ (HE Hip Se a UT ZOTO’ Sse es asa 0. 082, 137, 3 By late ae ee Shp ihe ee eae ZOLO seas a at ee 0. 123, 204, 0 Bees Aa SS hE Di- 180 ose en ee ZOLOe. =o sass 0. 041, 070, 6 UPB 9 eae |e 3 Aaa AyD ene Seer oe os ae ZOLQ te eee es aa 0. 164, 270, 6 SOLAR LONG-PERIOD TERM, PARAGRAPH 119 Bos_---- Saeki e. [ Sa ees eae [iizerosete see eee | 0.041, 068, 6 LUNAR DIURNAL TERMS, FORMULA (63) cA gee ee Ope sc T—2s+h+90°______- ee 13, 943, 035, 6 Algo tee Qi es T—3s+h+p+90°-_ 13, 398, 660, 9 Age (NT) aa pes — 90 14. 487, 410, 3 Aj7_-__- 20 eee T—4s+h+2p+90°___.._______ QE yy. 2 Se 12, 854, 286, 2 Ajg____- eee a T—38s+3h—p+90°___-_______- = Bey. Se ee 18, 471, 514, 5 C7. peta [eee a T-—s—h+p—90°__-...____-__- sO E aya a 14, 414, 556, 7 An_-__- Clea T—48s+3h+90°____-_---_ QE putes Sees 12, 927, 139, 8 Again Sue ney. ww Tra [=f Q0 leeds Ea LORS See ee Ss aad ee 14, 958, 931, 4 Alege eK) ae Eh — 9002 See ee — ph eit See eae 15. 041, 068, 6 Ag3____- (M})_.-_| T--s+h+p—90°_____________- Se Rte er amen a 14. 496, 693, 9 Age Vig T+s+h—p—90°___.________-- ae ee ee De 15. 585, 443, 3 Asp Se c|h sain aka 3 T—2s+h+2p—90°__....-___-- Ya we ee ee 13. 952, 319, 2 Aso T+2s+h—2p—90°____________ aa Ye ea re 16. 129, 818, 0 Ao7____- Ninowewne T—s+3h—p—90°___________-- ale ale Re a 14. 569, 547, 6 Agg_..-- Oe T+s—h+p—90°___...-------- SUS usec Se eee 15. 512, 589, 7 LUNAR DIURNAL TERMS, FORMULA (63) 14. 025, 172, 9 16. 056, 964, 4 16. 139, 101, 7 16. 683, 476, 4 15. 594, 727, 0 17. 227, 851, 1 lees leccoedeces T+3s—h-+p—90°_ ...-------- —2E—v__..-.------ 16. 610, 622, 8 Alsgeaen |e ocea en od Ts 3h—p4-900e ee eee 2 eee 15. 667, 580, 6 Fy ES Bees T-4s—h— 902222 see eee DEY eee aoe eee 17. 154, 997, 5 Aa eeese easooessen TSR O08 se ose eee ee omy oe noane eee 15. 128, 205, 9 Coeffi- Factor-f cient (C) | formula HARMONIC ANALYSIS! AND PREDICTION OF TIDES: 165 Table 2.—Harmonic constituents—Continued Argument (E) Ref. Ce ee Speed per Coeffi- | Factor-f No, | Symbol solar hour | cient (C)| formula V u LUNAR DIURNAL TERMS, FORMULA (138) : Alege a et. mene I RBS Rien emt oS MA ee cree et ah ne Sea Oe 13. 394, 019, 0 0. 0116 (143) Ang eee a A T —4s+h-+p._-----..-.-.-_.- +3£—v____-_---.-- 12. 849, 644, 4 0. 0032 (143) Al -2--" (Cass Gy) ees SY A se eae, Sa rac at rl po 14. 492, 052, 1 0. 0367 (144) 7a | Eee es T—23--h--p a 2 oe Epps beens 13. 947, 677, 4 0. 0060 (144) WAY ames lek sundite. 2 Oe Sy Ya SOT ea ES GS ee SS 15, 036, 426, 8 0. 0020 (144) PAT Ale = ee CS ee ele ee fps en ee 15. 590, 085, 2 0. 0134 (145) Agee | Pe oe eT 28i- the ae eee — 16. 134, 459, 9 0. 0022 (145) SOLAR DIURNAL TERMS, FORMULA (186) Big Aa ey ee ee NEY a AYU te feel tee eee ZOT OSS se ae Caen sen 14. 958, 931, 4 0. 1755 unity Bi----- Cee ee T—2h+91+90°._.-.-------.-- ZeTO san 4 eee ee 14, 917, 864, 7 0. 0103 unity Bigases | 252335 a. D—Dpig- 90ers See See AES ZOLO ne se ene 14. 999, 998, 0 0. 0015 unity Jet RE Se | ie ee T—3h+2pi+90°_____________- ZOTOS 2 ese 14, 876, 798, 0 0. 0004 unity IE (CE a Ph = 90 See aipia 2 Sine ZOLQS A awe Ae aS 15. 041, 068, 6 0. 1681 unity IBaguen | Cee PE Di 90k ens Pe CaS eae ee 15. 000, 002, 0 0. 0042 unity Boge (Zee ae T+ 2h—pi—900. ee ZOVOe sae eae anne 15. 082, 135, 3 0. 0042 unity AE oe Np ae | 2s Bday ea T—h+2p;—90°_____ ZOTO es 282 RL eee 14, 958, 935, 4 0. 0001 unity iBipesee || Ene ne T+3h—2pi—90°____________-- ZOTO Sane eee 15. 123, 202, 0 0. 0001 unity Base Olece aoe ol — 90! an nsee eee ee oe ZETON a2 oe aes 15. 123, 205, 9 0. 0076 unity 18 Fp ie ah ah T+4h—pi—90°____.__-_-_-_-- ZOTO Map core ue 15. 164, 272, 6 0. 0004 unity 18h ol le i eg ei T2h--pi+-900_- 2 eee ZOTO Hite eee ANON 15. 082, 139, 2 0. 0001 unity Eg atest | ad crn no ap 8 T+5h—2pi—90°__.___-------- ZOTO nose eee 15. 205, 339, 3 |.__..--..- unity SOLAR DIURNAL TERM, PARAGRAPH 119 Brees Sie ANA a Se ENE [iZerols 2 sae otis ees | 15. 000, 000, 0 |__.__.__._ unity COMBINATION DIURNAL TERMS, FORMULAS (194), (201), AND (222) Note 1_} My_____- T—s+h+p—90°____.-_------ —v—Qu-_---------- 14. 496, 693, 9 0. 0209* (206) T—s+h—90°____..--- +é—-7+Q__.------ Note 2_| Ky___-__- Eh G0ce- wea lees ole me eerie Ni eS NS 15. 041, 068, 6 0. 53805 (227) LUNAR SEMIDIURNAL TERMS, FORMULA (64) Ago. Mise ae (= esc eh ake to hee Ere Opt ety oe 28.984, 104,2 | 0. 9085 (78) Ago_-__- IN pee =8S-FOn-Fp___._..----2---- SBQE= 2p. ee 28. 439, 729, 5 0. 1759 (78) Ag____-| (Le)-_-_- 2T—s+2h—p+180°____-_____- = 2E—27_ ===. 29. 528, 478, 9 0. 0251 (78) Anomanee 2Noee ee 2T—48+2h+2p___..-.__-_---- +-2£—2p_....--_.-- 27. 895, 354, 8 0. 0235 (78) Ag3____- Paes 2T—8s+-4h—p_---.._--_------ 2E— 2p ees ese 28, 512, 583, 1 0, 0341 (78) Agg___ Ag_-----.| 2T—s+p+180°____- +2&--2p___ 29, 455, 625, 3 0, 0066 (78) Ag5____- 1 eee ate 2d AACS Sd RD ee eo te eee +2£—2y__ 27. 968, 208, 4 0, 0219 (78) A4g_-.__| (Se)_-_-- DY Ms i oat eel ae OE an Bre Nol ~-2E—Op_ 30. 000, 000, 0 0, 0006 (78) Aa7____- (Kg) ___-- SF RD (esi ee See ee CPD ee ent ae 30. 082, 137, 3 0, 0786 (79) Agg____- (to) pee 2T—s+2h+p.-..----------.- = Dy See eee 29. 537, 762, 6 0, 0064 (79) Aggies. KJo_____ OL Se Om Dna hen oe Dee — Dy tee ee ae 30. 626, 512, 0 0. 0064 (79) BAG eee es || See nee 2T—28s+2h+2p._.-...--------- Dy hei e ase ee 28. 993, 387, 9 0. 0005 (79) aA sete al SS 2T+28+29h—Op___...-.-------- a DD eae lo eae 31, 170, 886, 7 0. 0005 (79) PA pg Norse se oe 2T—s+4h—p__-..-.---_------ Se DE eee sae ara 29, 610, 616, 2 0, 0009 (79) ZU ga aS ae Co EEC 1 ea eeee pian Se mean se DY ne bet onto 30, 553, 658, 4 0, 0009 (79) Alp 4 vsaeteee |feien ya Debate leeten wee aide 2 Ae Se Dy eM ay oe 29. 066, 241, 5 0. 0007 (79) VAR ena (mae ke 5d RAS Sy oe a URS Sa OY ome aye 2 31. 098, 033, 1 0. 0007 (79) PAlgg tesa ene nse es OTe esa Chee oe ee ee a DE — apr iene ae 31, 180, 170, 3 0, 0017 (80) VA heh |hivs oe tema os ‘2T+38s+2h—p__....---------- | en a 31. 724, 545, 0 0. 0003 (80) AlRoeeeni ee ea 2T+s+2h+p+180°___-.-_-_ =—2&=—2p__ 8. 30. 635, 795, 6 |.------___ (39 7, (ye te eee 2T+48+2h—2p__-..---------- a ae — Dp aber Rene B25 268491 O57 Menno 80) PAGO eee tll eset OR Gee scene, Sis os Nae DE Dy So 31. 651, 691, 4 |_-_.-.____ (80) AG yea | ies aera ec 2T-+s+4h—p+180°_____--_._- se DE me yee anna 30. 708, 649, 3 |_...-.___- (80) Aigo ellos oa 8 DD IahG yarn aenstint tn wea setae es we me DY Hah ee ra 32. 196, 066, 1 |_-...-.-_- (80) Allg sea ne ei EAS DEAS) oR Ta ee ee asp ee SDE Dy ee ee eee 305164527456] Soe (80) *Adapted for use with tabular node factors, theoretical value is 0.0317. See p. 43. 166 _ U. §. COAST AND GEODETIC SURVEY Table 2.—_ Harmonic constituents—Continued Argument (E) Ref. Symbol Speed per Coeffi- | Factor-f No. | °* solar hour |cient (C)| formula V u LUNAR SEMIDIURNAL TERMS, FORMULA (139) FAG ict a Ram ald 2T—8s+2h+90°_____________- eS hy s Senile ave 28. 435, 087, 7 0. 0223 (146) Allg sie cs Be ey sens 2T—48+2h+p+90°__.-.--____ +8£=—2p os so2 ces 27. 890, 713, 0 0. 0062 (146) Al aie | vanes 2T—23+2h—p—90°___________ ALI ON oe ld 28.979, 462,4 | 0.0012} (146) Alrgeetar ats Blehn ago 9 R— S| oh) — 8) 2a ee +&—-2y_________ Le 29. 533, 120, 8 0. 0209 (147) SA spss | RES eee 2T—2s+2h+p—90°__--____- Pallic paar ae ee 28. 988, 746, 0 0. 0034 (147) 7 \y S| ee a 27-3 8h — 900258 Shes see Epa eee 30. 631, 153, 8 6. 0019 (148) SOLAR SEMIDIURNAL TERMS, FORMULA (187) B39____- Sone dst TPE ww Ss rahe pt ot eg pe VOTO aes ya ie 30. 000, 000, 0 0. 4227 unity I Qeecine Gites He EY DEO) (2 0 ee ae ieee li ZENO eae eee 29. 958, 933, 3 0, 0248 unity JEM eee Row ecen 2T+h—pi+180°____-_________- ZOLO 4 eee ee 30. 041, 066, 7 0, 0035 unity Byes ee eA TTL Y ee ge ea PERO ss S4 sla e emo 29.917, 866,6 | 0.0010 unity Bignes (LEE ce 08 IE Pe AE Te ta a ay ae ZELOS 2 Shan Bee 30. 082, 137, 3 0. 0365 unity IBigeas t= eeanneae OUND teas nae aaa noe ZOLO eee ee 30. 041, 070, 6 0. 0009 unity Big ites |Maeeen ORAS hppa s ae ee EE ZOTOS: ae ee en 30. 123, 204, 0 0. 0009 unity Bese e | sone OT Ae Dien ete we ER eee TOTO see Seay eR eee 30500050035 9) 22a unity Be see eee JERE UA ty eye ee eee ZENO Ne 305164527056); Dae unity © Beas | aetna = Be OH Re ioe Bae pe eee = |\Zerostmae 22a saeees 30. 164, 274, 6 0. 0008 unity Beye w alee a OR b = Dien sare eee ZT Os nen ek eee eee 08205 (S420 eee unity Jet aa ie eae 2T+3h-+pi1+180°__...-___--_- ZOLO Ne wee een SO SL 23920;710 yeaa unity 1B d SN Wives DOOD ian eee ae Seen ZETOU Se ae ee eal 28 [:305:246;5/407;'9) |£ 2a unity COMBINATION SEMIDIURNAL TERMS, FORMULAS (212) AND (230) INOte)(3=|) Lee2- =e == Air id Oe D--Lk0Ss2 eee +2&-2v—R______- 29. 528, 478, 9 0. 0251 (215) Notes Koes OT t = hteeaah i a es ee ee Opt unre SL A 30. 082, 137, 3 0, 1151 (235) LUNAR TERDIURNAL TERMS, FORMULA (140) VA'gon n= Mgueeee Phd Mami SCE) (ue teeta i cE Scat | ee 43. 476, 156, 3 0. 0178 (149) WAN gis eS Mie ee 8T—J48+-3h+-p__-..__.__------ +3£—3y__.____--_- 42. 931, 781, 6 0, 0050 (149) PA Aertel te ee 8T—2s+8h—p+180°___.______ =pab—oy ie See ke eee 44, 020, 531, 0 0. 0010 (149) AN is es ee ee 8T—Ss+-3h+2p___.-.------.---| 4+8&—3y. 22 ee 42. 387, 406, 9 0. 0009 (149) A igg tein tek 8 iets 8T—As+bh—p__..-_.-------- +3£—3y_ 43. 004, 635, 2 0. 0007 (149) v1) Re Se ity See eee ST SOPs ee eee Fe oe) 44. 574, 189, 4 0. 0024 (150) Alpe seta | meses op 8T—28s+3h+p___-____-_------ +E-8y_ ee 44, 029, 814, 7 0. 0004 (150) SOLAR TERDIURNAL TERM, PARAGRAPH 119 Bape. Sate teal Gus a see eee ee aes eR [izeroteceeeeten eee | 45.000, 000, 0 |__-----__- unity Note 1—Combines terms Ais and A23. Note 3—Combines terms Aq; and A4g Note 2—Combines terms Ax» and By. Note 4—Combines terms A47 and Bur. HARMONIC ANALYSIS AND PREDICTION OF TIDES 167 Table 2a.—Shallow-water constituents Argument ESR a OY Speed Factor-f Origin V u degrees per h. Semidiurnal MNS,.._-| M2-+-N2—S2------- 2T—5s+4h+p..__----- 44&-—4y__ ee 27. 423, 833, 7 | f2(Ma) 2SM2__--| 2S2—Me____-_____ 27-+-2s—2h_____-.__--- —2§+2p_.________- 31. 015, 895, 8 | f(M2) Terdiurnal MKs3.___.- Me+Ki-________-- 37—2s+3h—90°______- +2§—2y—p!______ 44. 025, 172,9 | f (Mes) x f(K1) 2MK3___- 2Mo—Ki__-.___--. 3T—4s+3h+90°______- +4&—4y-+y’______ 42. 927, 139, 8 | f2 va Xf(K1) Sie 0 Sas Kge sss. eS 3T+h—90°_________-_- SR ee SSC ea eee 45. 041, 068, 6 | f (K1) SO3__-:--| Se+O1-.-.-------- 3T—2s+h+90°_._____- <2e—p) bee ee 43. 943, 035, 6 | f (O01) Quarter diurnal 4T—4s+-4h___.______-- 57. 968, 208, 4 | f2 (Ma) 4T—2s+2h___________- 58. 984, 104, 2 | f2 (Mo) 4T—5s+4h+. Piet 57. 423, 833, 7 | {2 (Ma) 4T—2s+4h__ we 59. 066, 241, 5 | f (M2) Xf(Ks) AT Se. SA 6 aS. 60. 000, 000, 0 | uxity Sizth diurnal 6T—6s+6h___.__-_____ 6T—4s+4h____ 6T—7s+6h+p-_ 6T—2s+2h_-___ 3 67—5s+4h+p-____----- CY pea ek eee Eighth diurnal Mighell Mig 2 oe Senet 2: et 8T—8s+8h______-___-- +8§—8y_________-- 115. 936, 416, 9 | f4 (Ma) 3MSs_-_--| 3M2+So_________- 8T—6s+6h______.__--- +6&—6r_________-- 116. 952, 312, 7 | f3 (Ma) 2 (MS)s_-| 2M2+2Se_________ 8T—4s+4h______------ +4&—4y_______ le 117. 968, 208, 4 | f2 (Ms) 2MSNs3.---| 2Mo+S2+Np-_ _--_- 8T—7s+6h+p____.--_- +6&—6y_._______-- 116. 407, 938, 0 | {2 (Me) Seuss. = CASE ie 5 EMU haf fa CS PR a pe TOTO WS... 2. ae 120.000, 000,0 | unity 168 U. S. COAST AND GEODETIC SURVEY Table 3.—Latitude.factors Y!} Yvso ° 0 0. 500 1 . 500 2 - 498 3 - 496 4 . 493 5 . 489 6 . 484 7 . 478 8 471 9 . 463 10 . 455 11 . 445 12 ~ 435 13 . 424 14 412 15 . 400 16 . 386 17 . 372 18 Ont, 19 . 341 20 . 320 21 . 807 22 . 290 23 ol 24 . 252 25 E232 26 212 27 191 28 . 169 29 . 147 30 125 31 . 102 32 . 079 33 . 055 34 . 031 35 . 007 36 | —.018 37 | —. 043 38 | —. 069 39 | —.094 40 | —.120 41 | —.146 42 ples 43 | —.198 44 | —, 224 45 | —. 250 *In these columns reverse signs for south latitude. latitude. Yo31 035 “000 Y w31 Y ws2 ‘719 |707 Y vo Yui Yoo Y vae Yus3 . 312 . 322 . 331 . 339 . 347 . 354 . 360 . 366 . 371 . 375 . 378 . 381 . 383 . 384 . 385 . 385 . 384 . 382 . 380 377 . 374 . 370 . 365 . 359 . 354 . 688 . 671 . 650 . 630 . 610 - 590 . 570 . 550 . 530 . 509 . 489 - 469 . 450 - 430 . 410 . 391 372 . 354 . 206 199 - 191 . 183 .174 - 165 Yass 07 S18 coy Our WHF OO Other values are applicable to either north or south HARMONIC ANALYSIS AND PREDICTION OF TIDES' Table 3.—Latitude factors—Continued 169 Y! Yvs3o ° 45 |—0. 250 46 | —.276 47 | —.302 48 | —.328 49 | —.354 50 | —.380 51 | —. 406 62 | —. 431 53 | —.457 54 | —. 482 55 | —.507 56 | —. 531 57 | —. 555 58 | —.579 59 | —.602 60 | —.625 61 | —.647 62 | —. 669 63 | —.691 64 | —.712 65 | —.732 66 | —.752 67 | —.771 68 | —. 790 69 | —.807 7 —. 825 71 | —. 841 72 | —.857 73 | —.872 74 | —. 886 75 | —.900 76 | —.912 77 | —.924 78 | —.935 79 | —.945 80 | —. 955 81 | —. 963 82 | —.971 83 | —.978 84 | —. 984 85 | —. $89 86 | —.$$3 87 | —.996 88 | —. 998 89 |—1.000 90 '|—1.000 *In these columns reverse signs for south latitude. latitude. Yv31 Yeso | Yvae Yoes2 | Y wi3 Y wie * 1.000 |0. 500 0.999 | . 483 998 | .465 995 | .448 .990 | . 480 .985 | . 413 .978 | .396 .970 | .379 .961 | . 362 .951 | , 345 .940 | , 329 .927 | .313 .914 | . 297 899 | . 281 . 882 | . 265 . 866 | . 250 . 848 | . 235 . 829 | . 220 .809 | . 206 . 788 | .192 .7€6 | .179 . 743 | .165 .719.| . 153 .695 | .140 . 669 | . 128 .643 | .117 - 616 | .106 . 588 | . 095 . 559 | .085 - 530 | .076 . 500 | . 067 - 469 | .059 - 438 | .051 -407 | .043 .375 | .036 . 342 | .030 . 3809 | .024 . 276 019 . 242 | .015 208 | .011 -174 | .008 . 139 | .005 -105 | .003 . 070 | -001 . 035 ; .000 .000 | .000 Y wai | Y wa2 * 0. 707 |0. 707 .719 | .695 .73l | .682 . 743 | .669 . 755 | .656 . 766 | .643 .777 | .629 . 788 | .616 .799 | .602 . 809 | . 588 .819 | .574 . 829 | .559 839 | .545 . 848 | .530 .857 | .515 . 866 | .500 .875 | .485 . 883 | . 469 891 | . 454 -899 | .438 . £06 | .423 914 | .407 .921 | .391 927 | .375 . 984 | .358 . 940 | .342 . 946 | . 326 .951 | .309 - 956 | . 292 .961 | .276 . 966 | . 259 -970 | . 242 .974 | .225 -978 | .208 - 982 | .191 -985 | .174 - 988 | . 156 - 990 | .139 .993 | .122 295 | .105 .996 | .087 .998 | .070 .999 | .052 .999 | .035 1.000 | .017 1.000 ; .000 Yvso Yui Yosso —_——— —0. 212 —. 221 —. 228 —. 236 —. 242 —. 249 —. 254 —. 259 —. 263 —. 267 —. 270 —.272 —.274 —.275 —.275 —. 275 —.274 —.272 —.270 —. 266 . 263 . 258 —. 253 —. 247 —. 241 —. 234 —. 226 —. 218 —. 209 —. 200 = 190 —.179 —. 169 —.159 —. 146 —. 134 —.121 —.109 —. 096 —. 082 —. 069 —.055 —.042 —. 028 | —.014 - 000 Yvse Yous 0. 354 . 335 317 . 300 . 282 . 266 . 249 . 233 . 218 | . 203 . 189 175 . 162 . 149 . 137 - 125 114 . 103 094 . 084 075 . 067 . 060 053 . 046 . 040 . 035 . 030 . 025 021 017 .014 O11 . 009 . 007 . 005 . 004 - 003 . 002 001 . 001 . 000 . 000 . 000 . 000 . 000 —. 258 —. 261 —. 264 —. 265 —. 266 —. 267 Yos2 Yow —0. 118}—0. 300 —.128 | —.317 —.137 | —.335 —.146 | —.352 —.155 | —.370 —.163 | —. 387 —.170 | —. 404 —.177 | —.421 —.183 | —. 438 —.189 | —. 455 —.194 | —.471 —.198 | —. 487 —.202 | —.503 —.204 | —.519 —.207 | —. 535 —.208 | —.550 —.209 | —. 565 —.210 | —. 580 —.209 | —.594 —.208 | —.608 —.206 | —.621 —.204 | —. 635 —.201 | —. 647 —.197 | —. 660 —.193 | —.672 —.188 | —. 683 —.183 | —. 694 —.177 | —.705 —.170 | —.715 —.163 | —. 724 —.155 | —. 733 —.147 | —. 741 —.139 | —. 749 —.1380 | —. 757 —.120 | —.764 —.111 | —.770 —.100 | —.776 —.090 | —.781 —.079 | —. 785 —.069 | —. 789 —.057 | —. 792 —.046 | —. 795 —.035 | —. 797 —.023 | —. 799 —.012 | —.800 .000 | —.800 Other values are applicable to either north or south 170 U. §. COAST AND GEODETIC SURVEY Table 4.—Mean longitude of lunar and solar elements at Jan. 1, 0 hour, Greenwich mean civil time, of each year from 1800 to 2000 [s=mean longitude of moon; p=mean longitude lunar perigee; h=mean longitude of sun; pi=mean longitude Year. Ss ° 1800__... 342. 31 1801____- 111. 70 1802225 241. 08 180322222 10. 47 1804_____ 139. 85 1805___.- 282. 41 1806____- 51. 80 SO (eee 181.18 1808____- 310. 57 18092222 93. 13 LSi0===== 222. 51 1811_____] 351. 90 Sisson 121. 28 1813_____ 263. 84 1814_____ 33. 23 Tk 162. 61 1816_____ 292. 00 18t 7a 74. 56 _ 1818_____| 203. 94 1819 eee 333. 33 1820____- 102. 71 1821 245. 28 1822 eeeee 14. 66 S23 aoe 144. 04 1824_____ 2738 43 1S25eeeee 55. 99 1826____- 185. 38 TPA fe 314. 76 1828beens 84.15 1829_____ 226. 71 1830E=== 356. 09 1831_____ 125. 48 18322oee" 254. 86 IheBR YS 37. 42 1834ea 166. 81 1835_—- = 296. 19 1836_____ 65. 58 183 (eee 208. 14 1838_____ 337. 52 1839_____ 106. 91 1840_____ 236. 29 1841_____ 18.85 1842eeeee 148. 24 1843____- 277. 62 1844_____ 47.01 1846_-_ 189. 57 18465==3 318. 95 1847se 88. 34 1848_____ 217.72 1849_____ 0. 28 1850_____ 129. 67 1851ee 259. 05 sola perigee; N=longitude of moon’s node] Pi ° 279. 50 279. 52 279. 54 279. 55 279. 57 279. 59 279. 61 279. 62 279. 64 279. 66 279. 67 279. 69 279. 71 279. 73 279. 74 279. 76 279. 78 279. 79 279. 81 279. 83 279. 85 279. 86 279. 88 279. 90 279. 91 279. 93 279. 95 279.97 279. 98 280. 00 280. 02 280. 03 280. 05 280. 07 280. 09 280. 10 280. 12 280. 14 280. 16 280. 17 280. 19 280. 21 280. 22 280. 24 280. 26 280. 28 280. 29 280. 31 280. 33 280. 34 280. 36 280. 38 Year. 1852____ 1853 __ _- 1854____ 1855_-__- 1856___- 18a (eee 1858 ____ 1859 see 1860___- 1861____ 1862___- 1863 ___- 1864___- 1865____ 1866___- 1867 ___- 1868 ____ 18692e2= 1870 Ss== 18 7a 1872___- 1873 _-_- Uy 1875 22e= UY (Qs = LS ieee 1878____ 137.9 eae 1880____ 1881____ 1882___- 1883 ____ 1884____ 1885____ 1886 ___- MEY (a 1888____ 1889 see 1890____ 189 lease 1892____ 1893 ___- 1894____ 1895____ 1896 ____ TY 1898 ___- T899nee= HARMONIC ANALYSIS AND PREDICTION OF TIDES el Table 4.—Mean longitude of lunar and solar elements at Jan. 1, 0 hour, Greenwich mean civil time, of each year from 1800 to 2000—Continued ° 277. 03 46. 41 175. 80 305. 18 74. 57 217. 13 346. 51 115. 90 245. 28 27. 84 157. 23 286. 61 56. 00 198. 56 327. 94 97. 33 226. 71 9. 27 138. 66 268. 04 37. 48 179. 99 309. 37 78. 76 208. 14 350. 71 120. 09 249. 47 18. 86 161. 42 290. 81 60. 19 ° 280. 19 279. 95 | 279. 71 279. 47 279. 23 279. 98 279. 74 279. 50 279. 27 280. 01 279. 77 279. 53 279. 30 280. 04 279. 80 279.57 279. 33 280. 07 279. 84 279. 60 279. 36 280. 10 279. 87 279. 63 279. 39 280. 14 279. 90 279. 66 279. 42 280. 17 279. 93 279. 69 279. 45 280. 20 279. 96 279. 72 279. 48 280. 23 279. 99 279. 75 279, 51 280. 26 280. 02 279. 78 279. 54 280. 29 280. 05 279. 81 279. 57 280. 32 280. 08 279. 84 1955_-_- 1956____ 1957____ 1958___- 1059E==s 1960___- Ii 1962____ 1963____ 1964____ 1965____ 1966___- 1967____ 1968 ____ IS69re=s 1970=eas 1971S ss 1972222" 1973___- 1Oftaaee 19752225 1976___- Wome 1O7Saee= WWVOscse 1980___- 10Sizaas 1982____ 1983____ 19842. 1985____ 1986____ 1987____ 1988___- 1989____ 1990____ 1991 ____ 1992ee=s 1993____ 1994____ 1995____ 1996____ 1997____ 1998ee23 1999=e=S 2000___- 211. 74 172 U. $. COAST AND GEODETIC SURVEY Table 5.—Differences to adapt table 4 to any month, day, and hour of Greenwich mean civil time DIFFERENCES TO FIRST OF EACH CALENDAR MONTH OF COMMON YEARS! Month s Dp h P1 N Month s p h D1 N ° °o ° ° ° ° ° ° ° ° ° Jan. 1 0. 00 0. 00 0.00 0. 00 0.00 || July 1 | 224.93 | 20.16 | 178.40 0.01 —9. 58 Feb. 1 48. 47 3.45 | 30. 56 0. 00 —1.64 || Aug. 1 | 273.40 | 23.62 | 208.96 0.01 | —11. 23 Mar. 1] 57.41 6.57 | 58.15 0.00 | —3.12 || Sept. 1 | 321.86 | 27.07 | 239. 51 0.01 | —12.87 Apr. 1) 105.88} 10.03] 88.71 0. 00 —4.77 || Oct. 1 | 357.16 | 30.41 | 269.08 0.01 | —14. 46 May 1 | 141.17 13. 37 | 118. 28 0.01 —6.35 || Nov. 1| 45.62 | 33.87 | 299.64 0.01 | —16.10 June 1 | 189.64 16.82 | 148.88 0.01 —8.00 || Dec. 1 | 80.92 37.21 | 329.21 0.02 | —17. 69 Day 8 p h Pi N Day 8 p h P1 N °o ° ° ° ° °o ° ° ° ° 1 a eee 0. 00 0. 00 0. 00 0. 00 OXO0R ive 210. 82 1. 78 15.77 0. 00 —0.85 Oe 13. 18 0.11 0.99 0.00 |} —0.05 |} 18____-- 224. 00 1.89 16. 76 0.00 | —0.90 Ce eae 26. 35 0. 22 1.97 0.00 — O'S pL Ose=— a= 237. 18 2.01 17. 74 0. 00 —0. 95 39. 53 0. 33 2. 96 0. 00 —0. 16 2.12 18. 73 0.00 —1.01 52.71 0. 45 3.94 0. 00 —0,. 21 2. 23 19. 71 0. 00 —1. 06 65. 88 0. 56 4.93 0. 00 —0. 26 2.34 | 20.70 0. 00 —1.11 79. 06 0. 67 5.91 0. 00 —0. 32 2.45 | 21.68 0.00 —1.16 92. 23 0. 78 6.90 0. 00 —0. 37 2. 56 22. 67 0. 00 —1, 22 105. 41 0. 89 7. 89 0. 00 —0. 42 2.67 | 23.66 0. 00 —1,27 10___.-__| 118. 59 1.00 8. 87 0.00 —0. 48 2.79 24. 64 0. 00 —1.32 Ati eee ee 131. 76 1,11 9. 86 0. 00 —0. 53 2.90 25. 63 0. 00 —1.38 QB 144, 94 1, 23 10. 84 0. 00 —0. 58 3. 01 26. 61 0.00 —1. 43 1 eee 158. 12 1, 34 11, 83 0. 00 —0. 64 3.12 | 27.60 0. 00 —1. 48 4a 171. 29 1,45 12.81 0. 00 —0. 69 3. 23 28. 58 0.00 —1. 54 peers 184. 47 1. 56 13. 80 0.00 —0. 74 3. 34 29. 57 0. 00 —1. 59 V6 SoS 197. 65 1, 67 14, 78 0. 00 —0. 79 3. 45 30. 56 0. 00 —1. 64 Hour s p h Pi N Hour s p h Di N °o ° ° ° ° ° ° ° ° ° Qu aeee 0. 00 0. 00 0. 00 0. 00 OX00) ||P 2e=== 6. 59 0. 06 0. 49 0. 00 —0.03 ek os 0. 55 0. 00 0. 04 0. 00 0500713 2=22== 7. 14 0. 06 0. 53 0. 00 —0. 03 Dis eee = 1.10 0. 01 0. 08 0. 00 0.00 || 14._---- 7. 69 0. 06 0. 57 0. 00 —0. 03 Oe SS 1. 65 0.01 0.12 0. 00 —0.01 || 15__-___- 8. 24 0. 07 0. 62 0. 00 —0. 03 (ieee 2. 20 0. 02 0. 16 0. 00 —0.01 |} 16__--_- 8. 78 0. 07 0. 66 0. 00 —0. 04 O8seeott 2.75 0. 02 0. 21 0. 00 —ONO1 |} 07_----= 9. 33 0. 08 0. 70 0.00 | —0.04 62 ae se 3. 29 0. 03 0. 25 0. 00 —0.01 182202 9. 88 0. 08 0. 74 0. 00 —0. 04 (eae 3. 84 0. 03 0. 29 0. 00 —0. 02 || 19______ 10. 43 0. 09 0. 78 0. 00 —0. 04 (ae eee 4.39 0. 04 0. 33 0.00 | —0.02 |} 20__-._- 10. 98 0. 09 0. 82 0.00 | —0.04 9A sos 53 4.94 0. 04 0. 37 0. 00 —0.02 || 21_____- 11. 53 0. 10 0. 86 0. 00 —0.05 102222222 5.49 0. 05 0. 41 0. 00 —0.02 || 22____-. 12. 08 0. 10 0. 90 0. 00 —0.05 Ves 6. 04 0. 05 0. 45 0. 00 —0. 02 || 23___-__ 12. 63 0.11 0. 94 0. 00 —0.05 _.1 The table may also be used directly for dates between Jan. 1 and Feb. 29, inclusive, of leap years; but if the required date falls between Mar. 1 and Dec. 31, inclusive, of a leap year, the day of month should be increased by one before entering the table. HARMONIC ANALYSIS AND PREDICTION OF TIDES 173 Table 6.— Values of I, v, £, v', and 2y'' for each degree of N. Positive when NV is between 0 and 180°; negative when NV is 3; eee. between 180 and 360° 3 I v é vy! | 2p!’ Q a ‘ Diff. 2 Diff. iz Diff. 2 Diff. Diff o eh at EMD gl) OCD eas | OCD be sel) OGD | am || Oto Aa ee Mere ee 5 he 28. 60 0.19 0.17 0.13 0. 28 359 0 19 17 14 29 ee 28.60 | 4 0138) 5 eel ae 0.27| 43 0157 II) 5a |) 388 Cet Se Sa 28. 59 0 0. 56 19 0. 51 16 0. 40 14 0.85 29 357 (1 NS Sener eee 28. 59 0.75 0. 67 0. 54 1.14 356 1 19 17 13 28 eee ae 28. 58 0 0. 94 18 0. 84 17 0. 67 13 1.42 28 355 Geet = eee 28. 58 1 1.12 19 1.01 17 0. 80 14 1.70 29 354 56/1] ago] 3] xas) 1] Yor| ws] bor] 28] 38 28. 55 2 1.68 19 1.51 17 1.20 14 2.55 28 351 | se] 4) goo) 18) res] @| var] 3) Sh] | se 1 ae eee 28. 50 1 2. 24 18 2. 02 16 1.60 13 3.39 28 348 a} oka] 2] mer] | as; Z| reel | Sos] 38] Bas 1 ee eae 28. 45 2 2.79 19 2.51 17 1.99 13 4.23 28 345 | beat} 2] te] S| ee] 1S] Sos] | fae] Bas 1S asses shel 28. 39 3 3.34 18 3.01 16 2.38 13 5. 06 97 342 a] mae} 2 S| | Sa] 7) oer) B) ge) 27] Bo Des See 2 28. 31 2 3. 88 18 3. 50 16 2.77 13 5. 87 27 339 | bee) 3| fo] | se) 28) Sos] Bl) eal 2) Sy Qe 28. 23 3 4.42 18 3. 98 16 3.15 13 6. 68 26 336 MB] gee] 4] 2s] 38) £30] 1] sao] 2) not] | Bee Deter 28.13 4 4.95 18 4.46 16 3. 53 12 7.47 26 333 30] anos} 63} a0] | aes] 18) sma] | Teo] 28) gai SOS eke: J 28. 02 4 5. 48 17 4,94 16 3.90 12 8. 25 25 330 gas] mee] 4| sez] 2] sa) 38] Sig] 2] ge] 2) See Sowaes eee fe 27.90 4 5. 99 17 5. 41 15 4. 26 12 9.00 25 327 Bac) gee] #| 63s; 1 sn] i] f50| 2} sa] 25) dee SOea eset Slee 27.77 4 6. 50 16 5. 86 15 4.62 12 9.74 24 324 Oo Sea fie 27.73 6. 66 6.01 4.74 9.98 323 5 17 15 11 24 SOU eae sere 27. 68 5 6. 83 16 6.16 15 4.85 12 10. 22 24 322 gOS ees ee 27. 63 5 6.99 16 6. 31 15 4.97 11 10. 46 93 321 27.58 7.15 6. 46 5.08 10. 69 320 5 16 15 11 24 27. 53 5 7.31 16 6. 61 14 5.19 li 10.93 23 319 27.48 5 7.47 16 6. 75 15 5.30 ll 11.16 29 318 CB eee 27.43 7. 63 6.90 5.41 11.38 317 CV ernie ek |e mo: nee EOL Fp Buss ee ||| wantin) eras este Gwe: Sea 27.32 7.94 7.18 5. 63 11.82 315 174 U. S..COAST AND GEODETIC SURVEY. Table 6.—Values of I, v, &, v', and 2v'' for each degree of N—Continued Positive when N is between 0 and 180°; negative when N is Fositive between 180 and 360° N always Ian anne ata ores AbD an en RTD SR EMNE SSE woe ee N I py é yp! 2p!’ ° ° Diff. ° Diff. ° Diff. ° Diff ° Dif.| ° aeete & Se np) || Tao Gt a6 in G3 || ap 11,82) | eee AGW sd ioe 27.27 8.10 7.32 5.74 12. 04 314 6 6 15 14 10 22 rite. A ela) 272i er ee geil Gace AGN aes fe 12. 26/9 s 7am Wee A apes 27a (oil aeee 8.40] 42 7x60 is Babul sail al! 47am elmer doe 9 Sah 27200) a Su55 Nay 7.138 6.05;|" jqi|. i268) emaqt ammeml 0. ee POS.) 8 8.69) I5 Bol | ee 6:16 101 rose) oe eta Binet se ots 265974 ee SEA | aa 8: 003|me ts 6.25) hi 194] 213408) ears neo a 26.91 A Sac yr Su14 | aes 6.351] jo i| 2 13828) eer mses a. ee ee 26.85| ° O41) BE ag siondianss 645| 10 13,48 | 20) 307 7 OR 26:78)l| a O826i|| ea 8.40| 13 6.54] 40 13. 674) ore SOG ie eee B62 Eo 9.40] 44 8252180, 6.64] 9 13.86 | 49] 305 BoM ok aoe 26.65| 2 EE ee 8165 ||ep4s 6.7 3 14.05 | 49) 304 eee ae 26.59| 8 oy6%4| Bra Baza llme ts Gel) = 14) 23) oa ess ee ae 26.52] 7 DIC hs SYC0i ess 6.91 5 14.40| 4,3] 302 ae a1 7) Me Bi Se) Ble 8) tae 60. __-------- . 7 : 12 : 11 8 17 ia ee 26. 31 10. 19 9. 25 7.17 14. 92 299 Gale LEE |e Deke f 10532) er 45 eran 2a 7, 26 3 15.08 | j6| 298 Ge is say eae |p ee 10.44| 32 ont) | mae bee | 8 15.24 | 18) 297 Ge a ee 26. 10 10. 56 9.59 7.42 15. 39 296 Gott 18 ee 26. 03 d TONGA Nh oc 9.70) 31 7.49 g 15.54| 12 | 295 Goer twee | Pe 25 051 ee S 109704 Cems ae 7.57 : 15.69| 12 | 204 BU dee 25. 88 10.90 9.92 7.64 15. 83 293 BRR ketal, 25, 80i) ©) 8 ey MvOlG! p22.) SA 0.02 | crt) $y. 72.) 0, (8 |) Beate. Oo eae 69 wel nmsse72 nl ipas WET) |e 10512) ac 779) aa 16, 101] oreo eee 70 “|. 2868 11.23 10. 22 7. 86 16. 23 290 71 Tule one : 11.33 . 10.32 0 7.92 g 16. 35 lle 72a POS 4011 tee THES) | Sif OEE | 7.99| 16.47] 12) 98 jee Pee 25.41 11. 53 10. 50 8.05 16. 58 287 7 aenneiqantin | WNeO535 3 163) ers 10. 59 3 8.11 3 16.69 | jt | 286 i, ee 25.25| &§ mbes |) 1 10.68| Saiz 40 16.80 | 35 | 285 Tenia bee 25.17 11.81 10.77 8. 23 16.90 284 Tiga heat 2 25.09| 8 11589 |r 5 T0355 5 S28 gue 17.00| 10] . 983 eG Vie PR || ies || 8 10.93] &8 Bete) 2 172 09 || ee TOP ae 24.92 12. 06 11.01 ie 8.39 17.17 281 Sire tiga! a4g4| 8 121aiiae e iw’ || 8.44 3 17.25 3 280 gree ee 2) P2476.) en. 12.21 Oh iat [Pe Ba 17. ae Ret ae 24. 67 12. 28 11.22 8. 53 17.40 278 gana 1 a's) Bar snl ey 8 Diab. e 1129) |e Si57il lee, 17.46 3 277 ase: doe oe 241/50) eure gen 12542) aa ibe a SOUS 7.5 8 gia ees 24. 42 12. 48 11. 42 8. 64 17.58 275 apie § ask 0 Mioa 33" ere 12. 54 4 11.47 8. 68 : 17, 63) ea 274 e7ee ieee ep Mosca oe 8h 12.604) me ol Ort 5s | mble eure 17.67 || ag Rave ct ne 24.16 12. 65 11.58 8.74 17.71 272 gore ee 7 DEG | 5 do|| 2 11.63 S Si76llig & 7. 74.| <3: \e wom Oke neque tana) mm gugR 12.75 11.68 | 8.79 17.77 270 HARMONIC ANALYSIS AND PREDICTION OF TIDES 175 Table 6.—Values of I, v, — v', and 2v’’ for each degree of N—Continued oe Positive when WN is between 0 and 180°; negative when NV is between Positive 180 and 360° N always N I y & py Qv” ° ° Diff. ° Diff. ° Diff. ° Diff. ° «| Dig.| © ome. ner .7 23. 98 12.75 11.68 8.79 17.77 270 9 4 4 2 2 QI hee 23. 89 5 12.79 A 11.72 A 8. 81 5 17.79 9 | 269 Oy Nie ee 23, 80 : 12. 83 4 11.76 i 8, 83 z 17.81 Alf oie Ch i eae 23. 72 5 12. 87 5 11.80 5 8.85 j 17.82 tS Ba Oy eee 23. 63 5 12. 90 5 11.83 5 8. 86 ; 17. 83 o| 266 i aie fo ae 23. 54 5 12. 93 : 11. 86 3 8, 87 ] 17. 83 O | 265 demi 23. 45 3 12.95 - 11.89 3 8. 88 ; 17. 82 | 264 Gye ew 23. 36 5 12.97 5 11. 92 , 8.89 ; 17.81 a || 2S xm eae 93.27 5 12. 99 3 11. 94 5 8.90] 4 17.79 Si) eee Cae ae 23. 18 5 13.01 : 11. 95 , 8. 90 : 17.77 Bl val AGG ae yi.) 23.09 5 13007, 11. 96 ; SEDI" 17.74 Al 280 Try a a oa 23.00 3 130024 (meee 11. 97 : 8 89 p 7 3 | 259 1008. Pune 22. 91 13.02 11.98 67 258 - 2a 2 ===> 9 0 0 1 5 tLe eee 22, 82 A 13.02 1 1.98). 9 8. 87 : 17. 62 Bl 2a i048 cee 22.73 5 13.01 } 11.98 ; 8, 86 S 17.57 ens i050 | 0i 1 4 a) yo'64 13. 11.97 8. 84 17.51 255 9 1 1 2 6 TOE ee we 22. 55 ; 12. 99 5 11. 96 ; 8. 82 4 17.45 7 | 254 Vi 7s 22, 46 3 12.97 3 11. 95 : 8. 80 2 17.38 Z| 253 fone ee 22. 37 12.95 11.93 8. 17.30 252 - 9 3 2 3 8 GON eA 22. 28 ; 12. 92 5 11.91 5 8.75 . 17.22 g| 251 Tid ae 22. 20 5 12. 89 3 11.89 g 8, 72 3 17.14 S| 250 Ti ee 22.11 : 12. 85 7 11. 86 3 8. 69 A 740 yin 228 ie ee 22. 02 5 12. 81 A 11. 88 n 8.65 ; 16954) all 228 it Ee LI 21. 93 ; 12.77 11.79 A 8.61 A TOLS4 ay neta ise. | 21. 84 5 12.72 5 11.75 8.57 3 16.73| J, | 246 MIGh oR k.. lee pi, « 12. 67 ‘ 11.70 A 8. 52 i 16.62| j5| 245 DIG a ak | 21.67 5 12.61 3 11. 65 3 8. 48 - 16.50 | 32 | 244 Tee 21.58 3 12. 55 : 11.60 4 8.43 3 16 37a ar as ise eres 21. 50 5 12. 48 : 11.54 é 8. 37 16.24) 5, | 22 Osman. cae 21.41 3 12. 41 i 11.48 2 8.31 3 WO || |p 2A oie aoa 21.32 5 12. 33 3 11.41 8. 25 8 15.96] 5 | 240 ee eae 21.24 A 12, 25 3 11.34 j Se101| | a ERE || ea 21.15 3 12.17 11.26 5 8.13 y TAG oars | P36 see Ba aN 21.07 S 72108) 00 11.18 3 8,06 g 15950) Hee nas ie Sa 20. 99 5 TLCS | ap 11.10 5 7.99 5 123 || Gol 236 OSPR. | ile SOpOT B ie fa 11.01 5 7.91 5 15.16 | 42) 235 1 Re 20.82 5 1785 ere 10.92] 4¢ 7.83 8 14.99] 32 | 234 Cal. ea 20. 74 11. 67 10. 82 7.75 14. 81 233 iene ae 20.66| 8 55) ee Te a 7.67 B Te) 232 OOM Umi Pte 500. 58 ECD | a TONG ae 7.58 5 Chg) || I) Sox 180} 20.51 11.31 10. 50 7.49 14.23 230 ils Reet 20.43} 8 THES ae 10, 38)\e0 43 A || 14.03 | 30) 229 ioe eee 20. 35 : 11.05) 33 10.26 | 43 LED | 13.83 | 32 | 228 es aa ee 20. 28 10. 91 10.13 7.20 13. 62 297 fate At 20. 20 g 10.77 * 10.00 3 mone 2p 13.40 | 32 | 936 1350 Ree e 20. 13 10. 62 9. 87 7.00 13.18 295 176 U. S. COAST AND GEODETIC SURVEY Table 6.—Values of I, v, £, v’, and 2v'’ for each degree of N—Continued Positive when NN is between 0 and 180°; negative when N is Positive between 180 and 360° N always N I v é y! 2p"! ° o log. | 9 hpi | © olupig.| © 2 linig.| aan Fra he 0113) ig | enO%62) alta ONS Til m4 7300 | "5, |b: tal) eels _| 20.05 10. 47 9.73 6. 89 12.96 224 toy esl ae proves. Ry? | ego toe 9.50} 15 6.78 | | 3} |)--i2\73 eee ieee Aegina 19.91 g 10.15 | 47 9.44]. 35 6.66] 35 12 49) a 222 19. 84 9.98 9. 29 6. 55 12. 25 221 tpmeee cal MeiONTT i avetile 2 onisille ae 6/43 |) 12 || la erst 24) 990 i eee eee 19.71 7 9.63) 8 8.97] 47 6.31] 43 11.76) 55] 219 19. 64 9. 45 8. 80 6.18 11.51 218 aon tone || 81 solar) 18) 9 bees |) |e oon |) 221) onal eo vireo we eee ips) A OK08 1) barg Bideilie ia 5.98 ||, 12 |e) St:(001 eae mete 19. 45 8. 89 8. 28 5.80 10. 74 215 tau snus 51a cto 30)||) | ©))| We w8.60)|)) 20 |e us. Toll) 5) lm 2 5.66 8 rg ie te i pee eee 19.33] 6 8.49} 5 Coll || ses 5.52115 gilli O21 eres 19. 27 8. 28 7.72 5.38 9.94 212 Ma ate | aio 22 5 BiOmlliesss “Aw || Tt 5.24] te 0166 | cs [amet a ae 19.16} § 7uG5inas wee | et 5.09 | 19 9.38 lin 3 |e 19. 11 7.63 7.12 4.95 9.10 209 aie amamml |. CKO: g 7A 2 Biol 4.80] 3° 8.61 l|/m sou leeas i ae ame 19. 00 5 7.18 | 53 6.70 | 94 4.65] 45 8.52] 59 | 207 18.95 6.95 6. 49 4.50 8. 23 206 Cem pall cic ; eu72ile as Bazil ee asa |\eae 7. 94\|0 2 gees (56.2 ke al TS 86)|/g fe Ga8ile 5, 6.05| 3a ZG) 7: 64)|., 29)" 9208 | 18.82 6.24 5, 82 4.03 7.34 203 iy eee ag 78 |, $e soo |) 221) sco 3 balay |) oe a7 ole mee agua 18474 Bu7Eellmuae 5.36 | 53 a0 || oe 6:74 | 20) 20 16S 2 BSR 70 5. 49 5.13 | 6.43 | 3, | 200 SIGS ome m8 | 4) 5.24 |) 2) aco | 2h] cia a7 |i) of | tots eee ra lee 16200 wea 1 || 5 4.93] 36 4.65 | 34 S320 5.81 (i. oy antes 18. 59 4.72 4.41 3.03 5.50] 9,| 197 18.56] 3 4.46| 5 Au161 lee mens || a 5.19 at 196 1vo3) Mes ait) |) gl aro lense Bien) ee a Te6t- Stee = 18. 50 3.92 3. 66 25524 lee 4.55| 9 | 194 ley me 147] 3| 365| 3) 341) SP] 234) q7| 428] gp] 198 reads ae 18.45| 2 3438) tad ay16)|lMeise Bia 3.91) | ages ae Vie ey 18. 43 3.10 2.90 1.99 3.59| a9] 191 ied Sale fai la 2)| aes fp 22 | eel Se Sst tees 27 lie iyi: eee 1009) | Pe Bisa) lie es Dissill a 163) |e) Ae 2.94) |e eats Die Seed 18. 37 2.27 2.12 1.45 2.62| 9] 188 a ae 16136)| eg 1.99| 38 hei | oe 17d aes 2.29 35] 187 17a ie ie |G 17 |e 1,60) |her oe 109} aes 1.97| 32) 186 Bi geepisuss 1.42 1.33 0.91 1.64) ean) eis ep ease Maa a ISA) acs O74 ae oe Oe ae 131) 3>| 184 17. Raa 18432) Ols6r lie as OO || 0.55] 38 0,99) |) @<24.1(0ee8 Tyee ee ws 18, 31 0.57 0.54 0.37 0.66] 33] 182 tees ILE || 0,22) fs es aoa | 0518) eae 0.33 33] 181 Tso seals 18. 31 0.00 0.00 0.00 | 000 177 HARMONIC ANALYSIS AND PREDICTION OF TIDES 0LZ =| 06 cl G6 082 | OOT G8S cOT 062 «| OTT S6Z SIT 00g | O2I SOE Gor OIE | O&T STE cI 0cg | OFT GSE GFT O&€ OST cee GST Ore | O9T Crs GOT ose OLT cc CLT 09 O8T ° ° d a LE I GeS8 6 898 6 6868 6 L998 6 £918 6 L188 6 T&06 6 S126 6 0&6 6 9196 6 £966 6 2920 ‘0 4690 ‘0 2960 0 SI&T “0 8S9T 0 €S6r ‘0 gcTc 0 8222 0 £298 6 9£98 6 9298 6 CLs 6 VE88 6 $568 6 ZOT6 6 8226 6 E876 6 LIL6 6 6166 6 8920 0 8290 0 4060 0 O&ZT °0 GEST “O Z6LT ‘0 996I ‘0 1606 ‘0 086 OTL8 6 £228 6 T928 6 ¥C88 6 S168 6 . 6606 6 OLT6 6 8EE6 6 £&96 6 GS16 6 T000 ‘0 0220 ‘0 290 °0 £9800 LIT 0 LIFT 0 TP9T 0 O6ZT ‘0 &P8T 0 ot S618 °6 L088 °6 vP88 6 4068 °6 0668 °6 0016 6 GEZ6 6 S6E6 6 0896 6 6826 6 6100 0 6920 0 £890 0 2080 ‘0 S9OT ‘0 SOT ‘0 OOST ‘0 829T “0 PLOT ‘0 096 8188 6 6888 6 468 6 6868 6 4906 6 6916 6 8626 6 6746 6 £296 6 6186 6 ¥£00 0 $920 “0 9090 0 0920 ‘0 9860 0 L611 ‘0 L9ET ‘0 6ZFI ‘0 LIST 0 o8G 8968 6 6968 °6 £006 6 8506 6 9816 6 9€26 6 LS€6 6 0096 6 £996 6 9786 6 $400 ‘0 8920 0 8240 °0 8690 ‘0 8060 ‘0 S60T “0 PPT 0 OFET ‘0 €Z&T “0 ofS 206 6 L¥06 6 6206 6 T&T6 6 [026 °6 6626 ‘6 FIP6 6 896 6 0026 °6 0286 6 4900 0 6620 0 670 0 26900 480 0 8660 0 SZIT 0 II@T ‘0 OFZT “U0 086 €116 6 &C16 6 GST6 6 6066 6 TL26 6 09€6 6 L9¥6 6 6656 6 ¥EL6 6 1686 6 0900 °0 8&20 0 8170 0 9690 0 2920 0 9060 ‘0 610T ‘0 2601 0 LITT ‘0 0G 9816 6 9616 6 4626 6 0126 °6 GEE6 6 8IP6 6 8196 6 FE96 6 G96 6 O166 6 7900 0 $220 ‘0 8880 0 990 0 £690 °0 0280 0 8160 0 1860 ‘0 GOT “0 olG L926 6 9926 ‘6 6626 6 9&6 6 966 6 €L76 6 9996 6 £196 6 F626 6 9266 °6 9900 ‘0 6120 0 2g€0 0 8660 ‘0 8290 0 6&20 0 4280 0 8280 0 2680 0 00% 9286 6 PEE6 6 8986 6 8686 6 PSF6 6 9296 °6 T196 '€ O16 6 0286 °6 0666 6 9900 0 L610 0 2€0 ‘0 c940 0 ¢9¢0 ‘0 2990 0 9€20 0 £820 0 6620 '0 61 T686 6 66£6 6 16¥6 6 8976 6 O1S6 6 $196 6 £996 6 £426 6 £686 6 TS66 6 $900 ‘0 Z810 0 2620 0 2070 0 9090 0 0660 °0 4990 0 $690 0 8040 0 o81 012 06 ¢9Z G8 092 08 oct cL 0Sz 0L biG G9 OFZ 09 GES g¢ 6 og GZS cP 022 OF CIS ce o1z | og 906 © 002 06 G6. cT 06T Or CsI ¢ ost | 0 cr} ° os Sry 2U9N214SUOI JO apNzNAdWUD «104 “Y GoOT—"), FAI, 178 U. S. COAST AND GEODETIC SURVEY S29 96©OM O86SD 8WOD OND ®SN SO A, SHm FHM ANN Bas SHR DON BF \ 0 MOO OOO DOC COO MANN ANN AN N S2S 69H O00 WOM ODO wOowM 2 0 OER SCHOH HHH MOHAN ATH SOG ® iS \\ Se ee oe ann bo en oon Se Ie OO cl ane are COM FHID DODD AMM ARN WEA O SON KON BHO HSH Sod 51d 5 DAN ANN Aas sae — (a) ° Q SCOn HOO nROH MWOM NOH Otr O S6H 6HS FAH SOK BAS 51d i ) ° ce mon AANA Ana Seas eS © oO ° a a me) q 3 3 | SND BHD AND HON MNO Ono O =) S56GB 86SH BSG SGKS HANS KSA SC oa ° eel AAS aoe Soni eel ts] ~~ ° a aN iol i=} S) — =) eS) CORD WAH NOt nWON HOt HOt Oo p=! Gel Std ASK BHD NSS HHS NHN S q ° rene Son oe Eee Sn I oon Eh ol ere ~ fo} co ° SS iN » 4 m=} {S| = ~~ . d Kosee eae SHAD ANDO BMHO HHH MOO RNMM OS om & Swit dois SkKX SdH aidwH Sta S e q ° A on ih oe | 5 oe Dh oe on | ane rere o 1 © 3° q N So ee o Se oly = o~r ontnr le No ron ON o 2 Sao~ SNH i6idtd BH HSH S¥N CS o °o De Ih oe Boe | eee 5 oe oon onl nae eo & H ° gg e n we) = 4a 3 Sp oH 5 1S OM CNH AMM ANH- DMMO DRO O SS SHS GBHN Hit HHA SOK WGN S rs o ° lon thom mod poe ie eee] hod Se a i Se | Et oe aq Se + wu — ° oe SMD COM MOH AHN CHH tHhRO CO 2 SNS ASH Ads ANH Sor wWHH CS wa [ves] @ ne A oe oe Bee | 5 oe he oe | re Vg N ° mS 45) N T= 4a S 8 q oy SDS HOM MOM OHM AMM COHN CO 2D oS S056 Nos HHH HHS SNS wWSH SG ioe) B a Se wae weed o ~ ° SE x na ! H é |l- a © COOH HON HOO ONt WHO ONO © S SNH SAG SSS SOG OKs HGH So nm 6 an ne 3 > ° = A fo] > u 3 = & COm SCHN ONY WHM OOW NOt oO 3 SNS HSKH BBB GSO ~KsH HANG CS St 2 7) ° Sal COt ANM OCOHK- ONK ASCO DMHM OO SHwe TOK AHH AOHrKY SSIS BAH S ° ee) ° An S /)| (882 S82 225 RSS S82 Sse = J o FSS SRR ARAN AAA AAN AAA A ff OSneo MOH SHO HON ODS WOH 2 Vi, Q, ° a KAN MOH THO SSK ~MO FS o Oor Dorp wWnmr HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 9.—Log Q, for amplitude of constituent M, Log Qa 9. 71383 9. 7133 9. 7135 9. 7137 9. 7141 9.7145 9. 7151 9. 7158 9. 7165 9. 7174 9. 7184 9. 7194 9. 7206 9. 7219 9. 7232 9. 7247 9. 7263 9. 7280 9. 7298 9. 7317 9. 7337 9. 7358 7380 7403 7427 7452 7479 7506 JODO ss 7534 7564 7595 7626 7659 7693 DI OOH wo woo Diff ISOF BPNNH CO 135 P Log Q, 9. 8182 9. 8229 9. 8278 9. 8328 9. 8379 9. 8430 9, 8482 9. 8536 9. 8590 9. 8645 9. 8701 9. 8757 9. 8814 9. 8872 9. 8931 9. 8990 9. 9049 9. 9109 9. 9169 9. 9229 9. 9289 9. 9349 9. 9408 9. 9468 9. 9527 9. 9585 9. 9642 9. 9698 9. 9753 9. 9807 9. 9859 9. 9909 9. 9957 0. 0002 0. 0045 0. 0085 0. 0122 0. 0156 0. 0186 0. 0213 0. 0236 0. 0255 0. 0271 0. 0282 0. 0288 0. 0290 Diff. 179 180 U. S. COAST AND GEODETIC SURVEY Table 10.—Values of Q for argument of constituent M, woo Oort WHF CO Hee Nur oO ee One Ww Q | Diff. P Q Diff. P Q Diff. P Q ° ° ° ° ° ° ° 00) |e 4591 FHSS eaiats G0)1/MRGSONO! 2S 135 | 154.2 0.5 46| 26.6 gi} 92.1 136 | 155.0 THOM ee Cee 47 97.4 ee 92] 94.1 20 137 | 155.8 BES (ha-aaee 48) 8 p28)2 | ae Golligeoowal 27 138] 156.5 1.9 49| 29.1 94} 98.2 139 | 157.2 rg a Oil) | 2050) |liaees 95] 1003} 2111 140] 157.9 : 0 Bodie) 0? Bi 9 62053) ake oi oii penoze | 2/4 141] 158.6 3.4 52) |b ez 97| 104.3 142 | 159.3 aioqee 0.8 bat aepsereinga 6 98 | 106.2 Aa 143 | 160.0 AD oe Be G36) | gee eo | Parosiay| 28 144] 160.7 4.9 55| 34.6 100 | 110.1 145 | 161.3 ci) |e 56) RaEEG 10 101} 111.9 18 146| 162.0 Ee) (a il eet) os 192) aettsigae) 8 147| 162.6 GNA 58 | 37.7 103 | 115.5 148 | 163.2 e.oqlen oe 59| 38.8 id 104| 117.3 18 149 | 163.8 es 60’| § e39%9) 1) fers 105'| ep1ies0)|) | 12 150 | 164.4 28 | ae Gui || # yn 1 - 108)| WI207 |. | 1 g 151 | 165.0 geal 0s 60s) 9 e423 | gal? 1074 amet nee 152 | 165.6 j ; 5 2) [eee Gils EZenis| feet 108 |jegpia3:eu| | 28 153 | 166.2 9.4 64| 44.7 109 | 125.5 154 | 166.7 1050) ee oe 65 | | 460 | jee {I Genito | onm27.0e 2) ihene es eters 10.55] ue oe 66 | 47.3 ze I | @aleSsEh| ease 156-| 167.9 LC | aay Ga | ran) mare TIP | MLeDTONI Sg 157| 168.4 11.6] 8 O36 68 | 50.1 a 3 || emis Sie 8 ag 158 | 169.0 Taiji coe Got | Rete ||) Were Ti I Sees 159 | 169.5 DONT Ma Yair 70)|' } e52:0/|) OF. 115 | BEBO) 4 160 | 170.0 a313) 70 OG mi |) 4454. 5 He 116 | OBS S|e | es 161 | 170.6 18,84) eee? 72| 56.1 1.6 17-0036. § | 2 162| 171.1 14:4 | aie Pn | 4 wseste| | aise 18 (ORS lL 1S 163 | 171.6 iE) | ee 7-4 153 | | ye 8 19)|) puaBia eee 164 | 172.1 15.6 0.6 75 61.0 7 120 140. 1 1] 165 172.6 1682) rane 76) $1627 |\hune 21 |) MIE | Te 166 | 173.1 16:8)|) ae cae ml) Dull wee i292 |euria |) tt 167| 173.6 74s ee dae 78) | 1G6:24l | weed 493 || masa ie oat 168 | 174.1 1810 0%, 79| 681 a 24] 1444] 4 9 169 | 174.6 8.7) || oe Oe g0)| $1609] | pe-8 125), 145.4) LD 170 | 175.1 19,3: 08 81 mes || wee 126] 64} 28 171 | 175.6 20K | ae gale} tase) | oe, 127 (aut 172| 176.1 20:7) Us Ot S38) wae74| | wate 108 | gaocae 210 173 | 176.6 pide Oe ge] Fame! ae? 129] 149.2) 08 174| 177.1 BIW aie 85] 1 gOk7 ||| me 130 | 150.1] 175 | 177.6 Bois me coe SOE FSC a 31) eAlBOsO ee ee 176 | 178.1 36 2. 1 p35: 1.2 87 garaged 1321)) wxow eee 1028 177| 178.5 24.2 ss | 85.9 133 | 152.6 178 | 179.0 25.04 h 28 go 79] | oe ll) weaae | gunsa 21) FF neeaizo | panes 25.8 90] 90.0) *1\! 135) 1542 . 180 | 180.0 if Ss 925 299 SSS SSS SSS SSS S295 Ss=9S SSS SSS S25 Ss=9S SSS SSS Ss oon Our Oo crc on Cran a ooo oro on cron cron [+ To- Ker} [oi erkop) [ornor nor) Onn Ona Nuc “I~71 00 > +} HARMONIC ANALYSIS AND PREDICTION OF TIDES 181 Table 10.— Values of Q for argument of constituent M,—Continued Diff. es S92 999 S85 S99 SSS S95 S99 SSS SSS SSS SSS SSS SSS SSS = G0 00 bo ae fies f av SION DDO AAC Dorm Oo orc oom Oren Or o1cor cr croc orci cr crorc Oro o P 270 bo ~~ SONG GA Suys > CIR Ses co oo~y loron or} lone ee f oo to to is aU PAS SIRO: Oe WO awr Diff. Rai Soa Sco | HOS ONO ©8M OMH w Hee Hb Oe ROC wrb bre Pee DH He eee RONIN) GN INS F HO HOF COD COM COND IND Aon NN 315 NNN NNN WN 5 Fhe ORG NWO WOH OOD COOH ee ee a a a ee ro pnww Cee Oro woo 22 2922 Sel Sik — oreo C@co MOM SHO COG OF FIOb 360 eS sss ess sss sS9 SSS SSS S99 SSS SSS SSS SSS SSS SSS ssf Ss or or or OF oor or Crore Crore Crt crorc c1orc [Stor or} [3 Torkor} DAD Onrn~ forbs is f avn ~I~1 00 co 182 U. S. COAST AND GEODETIC SURVEY Table 11.— Values of u for equilibrium arguments [Use sign at head of column when N is between 0 and 180°, reverse sign when N is between 180 and 360°} M2, No O Q N Ji Ki Ke |2N, MS} M3 |Ms4MN/] Me | Mg 20 . OO | MK |2MK]| Mf |N X, BY ; ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° — = = — — — — — + = — + = 0} 0.00} 0.00) 0.00 0.00 | 0.00 0.00 | 0.00 | 0.00 | 0.00} 0.00} 0.00} 0.00} 0.00 | 360 1} 0.19 | 0.13 | 0.28 0.04 | 0.05 0.08 | 0.11! 0.15 | 0.15 | 0.53} 0.17} 0.06] 0.34 | 359 2} 0.38 | 0.27] 0.57 0.08 | 0.11 0.15 | 0.23 | 0.30} 0.30 1.05 | 0.34] 0.12} 0.67 | 358 3 | 0.56 | 0.40 | 0.85 0.11 | 0.17 0.23 | 0.34 | 0.45 | 0.45 1.57 | 0.52} 0.17] 1.01 | 357 4] 0.75 | 0.54 1.14 0.15 | 0.23 0.30 | 0.45 | 0.60 | 0.60] 2.10] 0.69} 0.28 1.35 | 356 5 | 0.94 | 0.67] 1.42 0.19 | 0. 28 0.38 | 0.56 | 0.75 | 0.75; 2.62] 0.86] 0.29) 1.68] 355 6 | 1.12] 0.80] 1.70 0. 23 | 0.34 0.45 | 0.68 | 0.90 | 0.90] 3.14] 1.03] 0.35; 2.02 | 354 7 1.31 | 0.94 1.99 0.26 | 0.40 0.53 | 0.79 | 1.05 1.05 | 3.67] 1.20} 0.41 2.36 | 353 8 1.50 | 1.07 | 2.27 0.30 | 0.45 0.60 | 0.90 | 1.20 1.20} 4.19 1.37 | 0.47 | 2.69 | 352 9 1.68 | 1.20 | 2.55 0.34 | 0.51 0.68 | 1.01 | 1.35 1.35} 4.71 1.54] 0.53 | 3.03 | 351 10 1.87 | 1.34 | 2.83 0. 37 | 0.56 0.75 | 1.12 | 1.49 1.49 | 5.23 1.71 | 0.59] 3.36 | 350 li 2.05 | 1.47] 3.11 0.41 | 0.62 0.82 | 1.24 | 1.64 1.64} 5.75 1.88 | 0.64] 3.70 | 549 12} 2.24] 1.60} 3.39 0.45 | 0.67 0.90 | 1.34 | 1.79 1.79 | 6.27) 2.05] 0.70] 4.03 | 348 13 | 2.42] 1.73 | 3.67 0.48 | 0.73 0.97 | 1.45 | 1.94] 1.94] 6.79 | 2.21) 0.76] 4.36 | 347 14/ 2.61 | 1.86] 3.95 0.52 | 0.78 1.04 | 1.56] 2.09 | 2.09] 7.31 2.38 | 0.82] 4.70 | 346 15 | 2.79 | 1.99 | 4.28 0.56 | 0.84 12/4 1.67 | 2.23 | 2:23 7.82 | 2.55 | 0.88} 5.03 | 345 16 2.98 | 2.12 | 4.51 0.60 | 0.89 1.19 | 1.79 | 2.38) 2.38 | 8.34] 2.721 0.93] 5.36 | 344 17} 3.16 | 2.25 | 4.78 0.63 | 0.95 1. 26 | 1.90 | 2.53 2.53 | 8.85 | 2.89 | 0.99] 5.69 | 343 18 | 3.34] 2.38 5. 06 0.67 | 1.00 1.34 | 2.00 | 2.67 | 2.68 | 9.36] 3.05] 1.05] 6.02) 342 TQ) 93552) 2951 || MS: 33 0.70 | 1.06 ae Ps 2581 2.82 | 9.87 | 3.22] 1.11] 6.35] 341 20 | 3.71 | 2.64] 5.60 0.74 | 1.11 1.48 | 2.21 | 2.95 | 2.97 | 10.38 | 3.38 1.17] 6.67 | 340 21 | 3.89 | 2.77 | 5.87 Onn, ele G 1.55 | 2.32 | 3.09 | 3.11] 10.89] 3.54] 1.23} 7.00 | 339 22 | 4.07 | 2.90 | 6.14 0.81 | 1.21 1.62 | 2.42 |-3.23 | 3.26 | 11.29 | 3.71 1.28 | 7.33 | 338 23 | 4.25 | 3.03 | 6.41 0.84 | 1.26 1.69 | 2.53 | 3.37 | 3.40 | 11.89 | 3.87] 1.384 7.65 | 337 24) 4:42) 3.15 | 6.68 0.88 | 1.31 1.75 | 2.63 | 3.51 | 3.55 | 12.39} 403] 1.40] 7.97 | 336 25 | 4.60 | 3.28! 6.94 0.91 | 1.37 1.82 | 2.73 | 3.64 | 3.69 | 12.89) 419 1.46 | 8.29 | 335 26 | 4.78 | 3.40 | 7.21 0.94 | 1.42 1.89 | 2.83 | 3.78 | 3.83 | 18.39 | 4.35 1.52 | 8.61 | 334 27 | 4.96 | 3.53 | 7.47 0.98 | 1.47 1.96 | 2.94 | 3.92 | 3.98 | 13.89} 4.51 1.57 | 8.93 | 333 28 | 5.13 | 3.65 | 7.73 1.01 | 1.52 2.02 | 3.04 | 4.05 | 4.12] 14.38] 4.67] 1.43] 9.25 | 332 29] 5.30} 3.78 | 7.99 1.04 | 1.57 2.09 | 3.13 | 4.18] 4.26 | 14.87] 4.82] 1.69] 9.57 | 331 30 | 5.48] 3.90 | 8.24 1.08 | 1.62 2.16 | 3.23 | 4.31 4.40 | 15.36] 4.98 1.75 | 9.88 | 330 31 5.65 | 4.02 | 8.50 abl |) ab Gy 2.22 | 3.33 | 4.45 | 4.54] 15.84] 5.13 1.80 | 10.19 | 329 32} 5.82] 4.14] 8.75 1.14 | 1.72 2.29 | 3.43 | 4.58 | 4.68 | 16.32] 5.29 1.86 | 10.50 | 328 23} 5.99 | 4.26; 9.00 LZ |alero 2.35 | 3.52 | 4.70 | 4.82 | 16.80) 5.44] 1.92] 10.81 | 327 34] 6.16 | 4.38 | 9.25 1.20 | 1.81 2.41 | 3.61 | 4.82 | 4.96 | 17.28 | 5.59] 1.97 | 11.12 | 326 35 | 6.33 | 4.50 | 9.50 1. 24 | 1.85 2.47 | 3.71 | 4.94] 5.10 | 17.76] 5.74] 2.03 | 11.43 | 325 36 | 6.50 | 4.62] 9.74 1.27 | 1.90 2.53.) 3.80 | 5.06 | 5.23 | 18.23 | 5.89] 2.09 | 11.73 | 324 37 | 6.66] 4.74 | 9.98 1.30 | 1.94 2.59 | 3.89 | 5.18 | 5.37 | 18.69] 6.03 |) 2.15 | 12.03 | 323 38 | 6.83 | 4.85 | 10. 22 1.33 | 1.99 2.65 | 3.98 | 5.30] 5.50] 19.16) 6.18 | 2.20] 12.33 | 322 39 6.99 | 4.97 | 10.46 1.36 | 2.03 2.71 | 4.07 | 5.42 | 5.64] 19.62] 6.32} 2.26} 12.63 | 321 40 | 7.15 | 5.08 | 10.69 1.38 | 2.08 2.77 | 4.15 | 5.54 | 5.77 | 20.08 | 6.46] 2.31 | 12.92 | 320 41 7.31 | 5.19 | 10.93 1.41 | 2.12 2.82 | 4.24 | 5.65 | 5.90! 20.53] 6.60 | 2.37 | 13.22 | 319 42] 7.47] 5.30 | 11.16 1.44 | 2.16 2.88 | 4.32 | 5.76 | 6.03 | 20.98 | 6.74] 2.42 | 13.51 | 318 43 | 7.63 | 5.41 | 11.38 1.47 | 2.20 2.94 | 4.40 | 5.87 | 6.16 | 23.48 | 6.88 | 2.48 | 13.80 | 317 44) 7.79 | 5.52 | 11.60 1.50 | 2.24 2.99 | 4.49 | 5.98 | 6.29 | 21.87 | 7.02) 2.53 | 14.08 | 316 45 | 7.94] 5.63 | 11.82 1.52 | 2.28 3.04 | 4.57 | 6.09 | 6.42 | 22.31 7.15 | 2.59 | 14.37 | 315 Note.—For L2 and M; see Table 13; for 285M and MSf, take u of Me with sign reversed; for P1, Ro, S1, So, S3, Ss, T2, Mm, Sa, and Ssa, take u=0. HARMONIC ANALYSIS AND PREDICTION OF TIDES 183 Table 11.—Values of u for equilibrium arguments—Continued [Use sign at head of column when N is between 0 and 180°, reverse sign when NN is between 180 and 360°] M2, No O Q N Ji K, K, |2N,MS/} M3 |M4MN| Me Ms 20 i 0O MK |2MK| Mf N A, By 2 ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° 45 7.94 | 5.63 | 11.82 1, 52) || 2.28 3.04 | 4.57 | 6.09 | 6.42 | 22.31 7.15 | 2.59 | 14.37 | 315 46 8.10 | 5.74 | 12.04 1.55 || 2.32 3.10 | 4.64 | 6.19 | 6.55 | 22.75 7. 28 2.64 | 14.65 | 314 47 8.25 | 5.84 | 12.26 1.57 | 2.36 3.15 | 4.72 | 6.30] 6.68 | 23.18 7.41 2.69 | 14.93 | 313 48 8.40 | 5.95 | 12.47 1.60 | 2.40 3.20 | 4.80 | 6.40 6.80 | 23.60 7. 54 2.75 | 15.20 | 312 49 8.55 | 6.05 | 12.68 1.62 | 2.44 3.25 | 4.87 | 6.50 6.92 | 24.02 7. 67 2.80 | 15.47 | 311 50 8.70 | 6.15 | 12.88 1.65 | 2.47 3.30 | 4.94 | 6.59 7.05 | 24.44 | 7.80 | 2.85 | 15.74 | 310 51 8.84 | 6.25 | 13.08 1.67 | 2.51 3.34 | 5.01 | 6.68 7.17 | 24.85 7.92 | 2.91 | 16.01 | 309 52 8.99 | 6.35 | 13.28 1.69 | 2.54 3.39 | 5.08 | 6.78 7.29 | 25.26 | 8.04 2.96 | 16.28 | 308 53 9.13 | 6.45 | 13.48 1.72 | 2.58 3.44 | 5.15 | 6.87 | 7.41 | 25.66] 8.17] 3.01 | 16.54 | 307 54 9.27 | 6.54 | 18.67 1.74 | 2.61 3.48 | 5.22 | 6.96 7.53 | 26.06 | 8.28] 3.06 | 16.80 | 306 55 | 9.41 | 6.64 | 13.86 1.76 | 2.64 3.52 | 5.28 | 7.04 7.65 | 26.46 | 8.40 | 3.12 | 17.05 | 305 56 | 9.54 | 6.73 | 14.05 1.7 2.67 3.56 | 5.34 | 7.12 7.76 | 26.85 8.51 3.17 | 17.30 | 304 57 9.68 | 6.82 | 14. 23 1.80 | 2.70 | 3.60 | 5.40 | 7.20 7.88 | 27.23 | 8.62 | 3.22 | 17.55 | 303 58 9.81 | 6.91 | 14.40 1.82 | 2.73 3.64 | 5.46 | 7.28 7.99 | 27.61 8.73 3.27 | 17.80 | 302 59 | 9.94 | 7.00 | 14.58 1.84 | 2.76 3.68 | 5.52 | 7.36 | 8.10 | 27.98 | 8.84] 3.32 | 18.04 | 301 60 | 10.07 | 7.09 | 14.75 1.86 | 2.79 3.72 | 5.58 | 7.44 8.21 | 28.34 8.95 | 3.37 | 18.28 | 300 61 | 10.19 | 7.17 | 14.92 1.88 | 2.82 3.76 | 5.63 | 7.51 8.32 | 28.70 9.05 | 3.42 | 18.51 | 299 62 | 10.32 | 7.26 | 15.08 1.90 | 2.84 3.79 | 5.69 | 7.58 | 8.42 | 29.06 | 9.15 | 3.46 | 18.74 | 298 63 | 10.44 | 7.34 | 15.24 1.91 | 2.87 3.82 | 5.74 | 7.65 | 8.53 | 29.41 9.25 | 3.51 | 18.97 | 297 64 | 10.56 | 7.42 | 15.39 1.93 | 2.89 3.86 | 5.78 | 7.71 8.63 | 29.75 | 9.35 3.56 | 19.19 | 296 65 | 10.68 | 7.49 | 15.54 1.94 | 2.92 3.89 | 5.83 | 7.78 | 8.73 | 30.09 | 9.44] 3.61 | 19.41 | 295 66 | 10.79 | 7.57 | 15.69 1.96 | 2.94 3.92 | 5.88 | 7.84 | 8.83 | 30.42 | 9.53 | 3.65 | 19.63 | 294 67 | 10.91 | 7.64 | 15.88 1.98 | 2.96 3.95 | 5.93 | 7.90 8.93 | 30.74 | 9.62 3.69 | 19.84 | 293 68 | 11.02 | 7.72 | 15.96 1.99 | 2.98 3.98 | 5.97 | 7.96 | 9.03 | 31.06) 9.71 3.74 | 20.04 | 292 69 | 11.12 | 7.79 | 16.10 2.00 | 3.00 4.00 | 6.01 | 8.01 9.12 | 31.37 9.79 3.78 | 20.25 | 291 70 | 11.23 | 7.86 | 16.23 2.02 | 3.02 4.03 | 6.05 | 8.06 | 9.22 |} 31.68] 9.87 3.83 | 20.45 | 290 71 | 11.33 | 7.92 | 16.35 2.03 | 3.04 4.06 | 6.08 | 8.11 9.31 | 31.98 | 9.95 3.87 | 20.64 | 289 72 | 11.43 | 7.99 | 16.47 2.04 | 3.06 4.08 | 6.11 | 8.15 | 9.40 | 32.27 | 10.03 | 3.91 | 20.83 | 288 73 | 11.53 | 8.05 | 16.58 2.05 | 3.08 4.10 | 6.15 | 8.20 9.48 | 32.55 | 10.10 3.95 | 21.01 | 287 74 | 11.63 | 8.11 | 16.69 2.06 | 3.09 4.12 | 6.18 | 8.24 9.57 | 32.82 | 10.17 3.99 | 21.20 | 286 75 | 11.72 | 8.17 | 16.80 2.07 | 3.10 4.14 | 6.21 | 8.28 | 9.65 | 33.09 | 10.24 | 4.03 | 21.37 | 285 76 | 11.81 | 8.23 | 16.90 2.08 | 3.12 4.16 | 6.24 | 8.32 | 9.73 | 33.35 | 10.31 4.07 | 21.54 | 284 77 | 11.90 | 8.28 | 17.00 2.09 | 3.13 4.18 | 6.26 | 8.35 | 9.81 | 33.60 | 10.37 4.11 | 21.71 | 283 78 | 11.98 | 8.34 | 17.09 2.10 | 3.14 4.19 | 6.29 | 8.38 | 9.88 | 33.85 | 10.43 4.15 | 21.87 | 282 79 | 12.06 | 8.39 | 17.17 2.10 | 3.15 4.20 | 6.31 | 8.41 9.96 | 34.09 | 10.49 4.18 | 22.02 | 281 80 | 12.14 | 8.44 | 17.25 2.11 | 3.16 4.22 | 6.32 | 8.48 | 10.08 | 34.31 | 10.54 4.22 | 22.17 | 280 81 | 12.22 | 8.48 | 17.33 Peak Bs we/ 4.23 | 6.34 | 8.46 | 10.10 | 34.53 | 10.60 | 4.25 | 22.32 | 279 82 | 12.29 | 8.53 | 17.40 2°12) 3218 4.24 | 6.36 | 8.48 | 10.17 | 34.74 | 10.65 4,29 | 22.46 | 278 83 | 12.36 | 8.57 | 17.46 2.12 | 3.19 4.25 | 6.37 | 8.50 | 10.23 | 34.95 | 10.69 | 4.32 | 22.59 | 277 84 | 12.42 | 8.61 | 17.52 2.13 | 3.19 4,26 | 6.38 | 8.51 | 10.30 | 35.14 | 10.73 | 4.35 | 22.72 | 276 85 | 12.49 | 8.64 | 17.58 2.13 | 3.20 4.26 | 6.39 | 8.52 | 10.36 | 35.33 | 10.77 | 4.38 | 22.84 | 275 86 | 12.55 | 8.68 | 17.63 2.13 | 3.20 4.27 | 6.40 | 8.53 | 10.41 | 35.50 | 10.81 4.41 | 22.96 | 274 87 | 12.60 | 8.71 | 17.67 2.14 | 3.20 4.27 | 6.41 | 8.54 | 10.47 | 35.67 | 10.84 | 4.44 | 23.07 | 2738 88 | 12.65 | 8.74 | 17.71 2.14 | 3.20 4.27 | 6.41 | 8.54 | 10.52 | 35.88 | 10.87 | 4.47 | 23.17 | 272 89 | 12.70 | 8.76 | 17.74 2.14 | 3.20 4.27 | 6.41 | 8.54 | 10.57 | 35.98 | 10.90 4.49 | 23.27 | 271 90 | 12.75 | 8.79 | 17.77 2.14 | 3.20 4.27 | 6.41 | 8.64 | 10.62 | 36.12 | 10.93 | 4.52 | 23.37 | 270 Note.—For La and M; see Table 13; for 25M and MSI, take u of M2 with sign reversed; for Pi, Ra, Si, Sa, S3, Sa, T'2, Mm, Sa, and Ssa, take w=0. 184 Wo S. COAST AND GEODETIC SURVEY Table 11.— Values of u for equilibrium arguments—Continued [Use sign at head of column when N is between 0 and 180°, reverse sign when N is between 180 and 360°] Mo, Ne Ji K, K, |2N,MS A, HY ° ° ° ° 12.75 | 8.79 | 17.77 2.14 12.79 | 8.81 | 17.79 2.14 12. 83 | 8.83 | 17.81 94; 1183 12.87 | 8.85 | 17.82 2.13 12.90 | 8.86 | 17.83 2013 12.93 | 8.87 | 17.88 Pala} 12.96 | 8.88 | 17.82 PAP) 12.98 | 8.89 | 17.81 PAP) 13.00 | 8.90 | 17.79 Py val 13.01 | 8.90 | 17.77 oA Ab 13.02 | 8.89 | 17.74 2.10 13.03 | 8.89 | 17.71 2.09 13.03 | 8.88 | 17.67 2.09 13.02 | 8.87 | 17.62 2.08 13.02 | 8.86 | 17.57 2.07 13.01 | 8.84 | 17.51 2.06 12.99 | 8.82 | 17.45 2.05 12.97 |} 8.80 | 17.38 2. 04 12.95 | 8.78 | 17.30 2.03 2ROSmSaomlplierae 2. 02 12.90 | 8.72 | 17.14 2.00 12.86 | 8.69 | 17.05 1.99 12.82 | 8.65 | 16.95 1.98 12.77 | 8.61 | 16.84 1.96 12.72 | 8.57 | 16.73 1. 94 12. 67 | 8.52 | 16.62 1.93 12.61 | 8.48 | 16.5 1.91 12.55 | 8.43 | 16.37 1.90 12.48 | 8.37 | 16.24 1. 88 12.41 | 8.31 | 16.10 1. 86 12.34 | 8.25 | 15.96 1. 84 12.26 | 8.19 | 15.81 1. 82 12.17 | 8.13 | 15.66 1. 80 12.08 | 8.06 | 15.50 1.78 11.98 | 7.99 | 15.33 1.76 11.88 | 7.91 | 15.16 1.74 11.78 | 7.83 | 14.99 1 7} 11.67 | 7.75 | 14.81 1.70 11.56 | 7.67 | 14. 62 1. 67 11.44 | 7.58 | 14.43 1.65 11.31 | 7.49 | 14.23 1. 63 11.18 | 7.40 | 14.03 1. 60 11.05 | 7.30 | 18.83 1. 58 10.91 | 7.20 | 13.62 OD 10.77 | 7.10 | 13.40 153 10.62 | 7.00 | 13.18 1. 50 E2G2 wKDI Gees sorts wood worK me = ~J = 00 00 G0 CO wooo oO Nee oo NNN NNN Nw “10 bo ES ace] tie ww M;,MN| Me ° ° 4.27 | 6.41 4.27 | 6.41 4.27 | 6.40 4.26 | 6.40 4.26 | 6.39 4.26 | 6.38 4.25 | 6.37 4.24 | 6.35 4.22 | 6.34 4.21 | 6.32 4.20 | 6.30 4.19 | 6.28 4.17 | 6.26 4.15 | 6.23 4.14 | 6.20 4.12 | 6.17 4.10 | 6.14 4.08 | 6.11 4.06 | 6.08 4.03 | 6.05 4.00 | 6.01 3.98 | 5.97 3.95 | 5.93 3.92 | 5.88 3.89 | 5.83 3.86 | 5.78 3.82 | 5.74 3.79 | 5.69 3.76 | 5. 64 3.72 | 5.59 3. 69 | 5.53 3.65 | 5.47 3.61 | 5.41 Shu |) Ghaw Bae) | ay e49) 3.48 | 5.22 3.44 | 5.15 3.39 | 5.09 3. 34 | 5.02 3.30 | 4.95 3.26 | 4.88 3.21 | 4.81 3.16 | 4.74 3.11 | 4.66 3.06 | 4.58 8.00 | 4.51 Ms Nnww wwe mow ono He CO r+ tt et et mero w wRo oon “100 oom em bDoO Work Corr aro [or or) ano POP BOM APH SON NNN NNN NNN NNN Nom m9 wo 00 Ooo WH Orn C2) He On Oon [lla So) Nowe O1, 2 SOKO KON COKOKON SOO Qi Q, Pl oo ror ot On OO > tb SeS2 S22 ARR RAN ABB wes BSS ese Wad 26S HNN NWW WNNHN HOOD BOING BW octes BPP He Or Or ood No OoOwg one NOR GIBBS BOERS UBSESSS ESEABAT PSbSh> LSE ES BERS SS ESESES ES EStS ESCSIS ESEAT AS SESS CALAIS IESESES SESESES aA meebo ooo ORO BO Mf 23.37 23. 46 23. 54 23. 61 23. 67 23. 73 23.79 23. 84 23. 88 23. 91 23. 93 23. 95 23. 96 23. 97 23. 97 23. 96 23, 94 23. 91 23. 88 23. 84 23. 79 23. 73 23. 66 23. 59 23. 50 23. 41 23. 31 23, 21 23. 09 22. 96 22. 83 22. 69 22. 54 22. 37 22, 20 22. 03 21, 84 21. 64 21. 44 21. 23 21.00 20. 76 20. 52 20. 27 20. 01 19. 75 Notre.—For Lz and Mi see table 13; for 28M and MSf, take wu of M2 with sign reversed; for Pi, R2, S:1, S2, Ss, Sa, T2, Mm, Sa and Ssa, take w=0. HARMONIC ANALYSIS AND PREDICTION OF TIDES 185 Table 11.—Values of u for equilibrium arguments—Continued [Use sign at head of column when N is between 0 and 180°, reverse sign when WN is between 180 and 360°] Mo, Na N Ji Ki Ke |2N, MS| M3 A, My ° ° ° ° ° ° 135 10.62 7,00 | 13.18 1.50 | 2.25 136 | 10.47 | 6.89 | 12.96 1.48 } 2.21 137 | 10.31 | 6.78 | 12.73 1.45 | 2.17 138 | 10.15 | 6.66 | 12.50 1.42 | 2.13 139 9.98 | 6.55 | 12. 26 1.39 | 2.09 140 9.81 | 6.43 | 12.02 1.36 | 2.05 141 9.64 | 6.31 | 11.77 1,34 | 2.00 142 9.46 | 6.18 | 11.52 1.31 | 1.96 143 9.27 | 6.06 } 11.27 1.28 | 1.91 144 9.08 | 5.93 | 11.01 1.25 | 1.87 145 8.89 | 5.80 | 10.74 1.22 | 1.82 146 8.69 | 5.66 | 10.48 1.19 | 1.78 147 8.49 | 5.52 | 10.21 1.16 | 1.738 148 8.28 | 5.38 9. 94 1.12 | 1.69" 149 8.07 | 5.24 9. 66 1.09 | 1.64 150 7.85 | 5.10 9. 38 1.06 | 1.59 151 7.63 | 4.95 9.10 1.03 | 1.54 152 7.41 | 4.80 8. 81 1.00 | 1.49 153 7.18 | 4.65 8. 52 0.96 | 1.44 154 6.95 | 4.50 8. 23 0.93 | 1.39 155 6.72 | 4.34 7. 94 0.90 | 1.34 156 6.48 | 4.19 7. 64 0.86 | 1.29 157 6.24 | 4.03 7. 34 0.83 | 1.24 158 5.99 | 3.87 7.04 0.79 | 1.19 159 5.74 | 3.70 6. 74 0.76 | 1.14 160 5.49 | 3.54 6. 43 0.72 | 1.09 161 5. 24 | 3.37 6.12 0.69 | 1.04 162 4.98 | 3.20 5.81 0.66 | 0.98 163 4.72 | 3.03 5. 50 0.62 | 0.93 164 4.46 | 2.86 5.19 0.58 | 0.88 165 4.19 | 2.69 4. 87 0.55 | 0.82 166 3.92 | 2.52 4.55 0.51 | 0.77 167 3.65 | 2.34 4. 23 0.48 | 0.71 168 3.38 | 2.17 3.91 0. 44 | 0.66 169 3.10 | 1.99 3. 59 0.40 | 0.61 170 2.83 | 1.81 3. 27 0.37 | 0.55 171 2.55 | 1. 63 2.94 0.33 | 0.50 172 2.27 | 1.45 2. 62 0.30 | 0. 44 173 1.99 | 1.27 2.29 0. 26 | 0.39 174 1.71 | 1.09 1.97 0.22 | 0.33 175 1.42 | 0.91 1, 64 0.18 | 0.28 176 1.14 | 0.73 1.31 0.15 | 0.22 177 0.86 | 0.55 0. 99 0.11 | 0.17 178 0.57 | 0.37 0. 66 0.07 | 0.11 179 0.29 | 0.18 0. 33 0.04 | 0.05 180 0.00 | 0.00 } 0.00 0.00 | 0.00 MiMN| Me ° ° 3.00 | 4.51 2.95 | 4.43 2.90 | 4 34 2.84 | 4.26 2.78 | 4.18 2.73 | 4.09 2.67 | 4.01 2.61 | 3.92 2.55 | 3.83 2.49 | 3.74 2.43 | 3.65 2.37 | 3.56 2.31 | 3.47 2520) ||| dod 2.18 | 3.28 2.12 | 3.18 2.06 | 3.08 1.99 | 2.99 1.92 | 2.89 1.86 | 3.7 1.79 | 2.69 1.72 | 2.59 1.66 | 2.48 1.59 | 2.38 1.52 | 2:28 1.45 | 2.17 1.38 | 2.07 as ale ed 1.24 | 1.86 gal? |) ale as} 1.10 | 1.64 1.02 | 1.54 0.95 | 1.43 0.88 | 1.32 0.81 | 1.21 0.74 | 1.10 0.66 | 1.00 0. 59 | 0.89 0.51 | 0.77 0. 44 | 0.66 0.37 | 0.55 0.30 | 0.44 0. 22 | 0.33 0.14 | 0.22 0.07 | 0.11 0.00 | 0.00 Ms Oe WRo DD ne ss eon oO o> OO 62 Pe) POSES MOSS GOPOCS CICS COCO Shab bebe hme ba salbab Sas Salt oo or Oo w 88 ON ee tet teeta O1, Q: 20, p 0O MK |2MK]| Mf N ° ° ° ° ° ° a — = + - 9.12 | 30.37 8. £0 3.99 | 19.75 | 225 9.00 | 29.94 8. 36 3.94 | 19.47 | 224 8.87 | 29.49 8. 22 3.88 | 19.18 | 223 8.73 | 29.04 8.08 3.82 | 18.88 | 222 8.59 | 28. 56 7.94 3.76 | 18.58 | 221 8.45 | 28.08 7.79 3.70 | 18.26 | 220 8.30 | 27.58 7.64 3.64 | 17.94 | 219 8.15 | 27.07 | 7.49 | 3.57 | 17.61 | 218 8.00 | 26. 54 7.33 3.50 | 17.27 | 217 7.84 | 26.00 7.17 3.43 | 16.92 | 216 7.67 | 25.45 7.01 3.36 | 16.56 | 215 7.50 | 24.89 6. 84 3.29 | 16.20 | 214 7.38 | 24.31 6. 68 3.21 | 15.82 | 213 7.16 | 23.72 6. 51 3.14 | 15.44 | 212 6.98 | 238.12 6. 33 3.06 | 15.05 | 211 6.80 | 22.51 6. 16 2.98 | 14.65 | 210 6.61 | 21.88 5. 98 2.90 | 14.24 | 209 6.42 | 21.24 5. 80 2.81 | 13.83 | 208 6. 22 | 20. 59 5. 61 2.73 | 13.41 | 207 6.03 | 19.93 5. 43 2.64 | 12.98 | 206 5. 82 | 19. 26 5. 24 2.55 | 12.54 | 205 5.62 | 18.58 5.05 2.46 | 12.10 | 204 5.41 | 17.89 4.85 2.37 | 11.65 | 203 5.20 | 17.19 4, 66 2.28 | 11.19 | 202 4.99 | 16. 48 4. 46 2.18 | 10.73 | 201 4.77 | 15.75 4.26 2.09 | 10.26 | 200 4.55 | 15.02 4.06 1.99 9.79 | 199 4.33 | 14.29 3. 86 1.89 9.31 | 198 4.10 | 13. 54 3. 66 1.79 8.82 | 197 3.87 | 12.78 3.45 1.70 8.33 | 196 3.64 | 12.02 3. 24 1. 60 7.83 | 195 3.41 | 11.25 3.03 1. 49 7.33 | 194 3.18 | 10.48 2. 82 1.39 6.83 | 193 2.94 9.7 2.61 1. 29 6.32 | 192 2.70 8.91 2. 39 1.18 5.80 | 191 2. 46 8. 12 2.18 1.08 5.29 | 190 2.22 7.32 1.97 0.97 4.77 | 189 1,97 6. 51 1.75 0. 86 4.24 | 188 1, 783 i 7Al 1. 53 0. 76 3.72 | 187 1. 49 4.90 1.31 0.65 3.19 | 186 1. 24 4.09 1.10 0. 54 2.66 | 185 0.99 hy 20 0.88 0. 43 2.13 | 184 0.75 2. 46 0. 66 0.33 1.60 | 183 0. 50 1. 64 0. 44 0. 22 1.07 | 182 0. 25 0.82 0. 22 0.11 0.53 | 181 0.00 0.00 0.00 0.00 0.00 | 180 Note.—For Lz and M; sce Table 13; for 25M and MSf, take wu of Ma with sign reversed; for Pi, Rs, S:, Ss, 83, S4, T2, Mm, Sa, and Ssa, take w=0. 186 U. S. COAST AND GEODETIC SURVEY Table 12.—Log factor F corresponding to every tenth of a degree of I SS Led 18.3° Constituent isk eae le 5 wa 0. 0827 oe ncee ene eo oo 0. 0547 Rea a Poetic f 0. 1263 M2*, No, 2N____| 9. 9839 Ys ct Brees 9. 9758 Miu, MN_-_-__-_- 9. 9678 Vig Sie Seve he 9. 9516 1 os Be ee 9. 9355 ae Qi 2Q, pi_--| 0.0939 ae aajome Vio Nae 0. 31389 VK eee eae 0. 0386 QIMKas 23) boas 0. 0224 1 De ee 0. 2039 Vin S32 ier. baie 9. 9465 HHS T | 189° Ghadinnenee Tite teeter! eed 0. 0707 eee eee Se 0. 0477 1 (Cea ee 0. 1134 M2", No, 2N___-| 9.9854 Mig ios = ses ee 9 9780 My, MN_--___-- 9. 9707 Mig 28h Sa 9. 9561 WY (eee a 9. 9415 or Qi, 2Q, Pi--- ~ 0811 be eee it a 2726 IMU Se Soh 0. 0330 DIMER Ss eee oe ae 0. 0184 1. 5 ees ee 0. 1769 Mime 2 Ee 9. 9514 T | 19.5° Constituent Unseen aa ae ee 0. 0592 Key au 2 ee 2 BR 0. 0408 Kops 2 0. 1001 M2*, No, 2N__-_| 9. 9869 14 Ce te See 9. 9804 Mi, MN_______ 9. 9738 Migs oe tee ena 9. 9607 Mig i: Saat k aes 9. 9476 aR Qi, 2Q, pi_--| 0. 0688 [ee oe ee . 2327 MiKicee te © See 0. 0277 QM Ke ae eee 0. 0146 Mifieien 12 See 0. 1508 Mim ss eee 9. 9564 Diff. 18.4° Diff. 18.5° Diff. 18.6° Diff. 18.7° 0. 0786 0. 0523 0. 1220 9. 9844 9. 9766 9. 9687 9. 9531 9. 9375 0. 0896 0. 3000 0. 0367 0.0211 0. 1948 9. 9481 *Log F of \2, 2, »2, MS, 28M, and MS&¢f are each equal to log F' of Mz. Log F of P1, Ro, Si, S2, Ss, Ss, T2, Sa, and Ssa are each zero. For log F of L2 and M, see Table 13. HARMONIC ANALYSIS AND PREDICTION OF TIDES 187 Table 12.—Log factor F corresponding to every tenth of a degree of I—Con. 20.2° | Diff. | 20.3° | Diff. | 20.4° | Diff. | 20.5° | Diff.| 20.6° | Diff. ALT sh Sh er ee oe 0.0482 | —18 | 0.0464 | —17 | 0.0447 | —18 | 0.0429 | —18 | 0.0411 | —17 | 0.0394 | —17 Reper ae eet. 0.0340 | —11 } 0.0329 | —11 | 0.0318 | —11 | 0.0307 | —11 | 0.0296 | —11 | 0.0285 | —11 1 Cagle SR ah ae 0.0864 | —23 | 0.0841 | —23 | 0.0818 | —23 | 0.0795 | —24 | 0.0771 | —23 | 0.0748 | —23 Ae » No, 2N__-| 9.9885 +3 | 9.9888 +2 | 9.9890 +3 | 9.9893 +3 | 9.9896 +3 | 9.9899 +2 op we 2a a. 9.9827 | +4 | 9.9831 +4 | 9.9835 +5 | 9.9840 +4 | 9.9844 +4 | 9.9848 +4 Me VIN eee 9. 9770 +5 | 9.9775 +6 | 9.9781 +5 | 9.9786 +6 | 9.9792 +5 | 9.9797 +6 1 Ce ae 9.9655 } +8 | 9.9663 +8 | 9.9671 +8 | 9.9679 +8 | 9.9687 +9 | 9.9696 +8 WY pe Reever 9.9540 | +10 | 9.9550 | +11 | 9.9561 | +11 | 9.9572 | +11 | 9.9583 | +11 | 9.9594 | +11 an Qi, 2Q, pi--- a 0570 | —19 | 0.0551 | —20 | 0.0531 | —19 | 0.0512 | —19 | 0.0493 | —18 | 0.0475 | —19 AR He de tea 1940 | —63 | 0.1877 | —63 | 0.1814 | —63 | 0.1751 | —62 | 0.1689 | —62 | 0.1627 | —62 IVEKes oes See 0.0225 | —8 | 0.0217 —9 | 0.0208 —8 | 0.0200 —9 | 0.0191 —8 | 0.0183 —8 OIVEKE Beene 0.0109 | —5 | 0.0104 —6 | 0.0098 —5 | 0.0093 —6 | 0.0087 —5 | 0.0082 —6 Mise aso er 0.1255 | —41 | 0.1214 | —41 | 0.1173 | —41 | 0.1132 | —41 | 0.1091 | —40 | 0.1051 | —41 Terma eee 9.9617 | +9 | 9.9626 +9 | 9.9635 +9 | 9.9644 +9 | 9.9653 +9 | 9. 9662 +9 i 20.7° | Diff.| 20.8° | Diff. |} 20.9° | Diff. | 21.0° | Diff.| 21.1° | Diff. | 21.2° | Diff. Constituent Vite ee UE 0.0377 | —17 | 0.0360 | —17 | 0.0343 | —17 | 0.0326 | —17 | 0.0309 | —17 | 0.0292 | —16 KG os ee eee 0.0274 | —11 | 0.0263 | —11 | 0.0252 | —11 | 0.0241 | —11 | 0.0230 | —11 | 0.0219 | —10 KKge sa eee e ne 0.0725 | —24 | 0.0701 | —23 | 0.0678 | —24 | 0.0654 | —24 | 0.0630 | —23 | 0.0607 | —24 M2*, No, 2N__.-| 9.9901 | +3 | 9.9904 +3 | 9.9907 +3 | 9.9910 +2 | 9.9912 +3 | 9.9915 +3 Mg aes ek 9.9852 | +4 | 9.9856 +4 | 9.9860 +4 | 9.9864 +5 | 9.9869 +4 | 9.9873 +4 Ma, MN_-_-__-- 9.9803 | +5 | 9.9808 +6 | 9.9814 +5 | 9.9819 +6 | 9.9825 +6 | 9.9831 +5 1 [ee ae 9.9704 | +8 | 9.9712 +8 | 9.9720 +9 | 9.9729 +8 | 9.9737 +9 | 9.9746 +8 Might een 9.9605 | +11 | 9.9616 | +11 | 9.9627 | +12 | 9.9639 | +11 | 9.9650 | +11 | 9.9661 | +12 On Q1, 2Q, pi--. 0. 0456 | —19 | 0.0437 | —18 | 0.0419 | —19 | 0.0400 | —18 | 0.0382 | —1i8 | 0.0364 | —18 epee ae Sh EO 0.1565 | —61 | 0.1504 | —61 | 0.1443 | —61 | 0.1382 | —61 | 0.1321 | —60 | 0.1261 | —60 0.0175 | —8 | 0.0167 —8 | 0.0159 —8 | 0.0151 —8 | 0.0143 —8 | 0.0135 -—8 0.0076 | —5 | 0.0071 —6 | 0.0065 —5 | 0.0060 —5 | 0.0055 —5 | 0.0050 —5 0.1010 | —40 | 0.0970 | —39 | 0.0931 | —40 | 0.0891 | —39 | 0.0852 | —40 | 0.0812 | —39 9.9671 | +9 | 9.9680 | +10 | 9.9690 +9 | 9.9699 | +10 | 9.9709 +9 | 9.9718 | +10 ~ | ob 21.3° | Diff.) 21.4° | Diff. | 21.5° | Diff. | 21.6° | Diff. | 21.7° | Diff. | 21.8° | Diff. 0. 0346 0. 1201 -| 0.0127 0. 0045 0.0259 | —16 | 0.0243 | —16 0.0198 | —11 | 0.0187 | —10 0.0559 | —25 | 0.0534 | —24 9.9921 | +3 | 9.9924} +3 9.9882 | +4 | 9.9886 | +4 9.9842 | +6 | 9.9848 | +6 9.9763 | +9 | 9.9772 | +8 9.9684 | +12 | 9.9696 | +11 —18 | 0.0328 | —18 | 0.0310 | —18 —60 | 0.1141 | —59 | 0.1082 | —59 —8 | 0.0119 | —8/ 0.0111} —8 —5 | 0.0040 | —5 | 0.0035 | —5 —38 | 0.0735 | —39 | 0.0696 | —38 +9 | 9.9737 | +10 | 9.9747 | +10 0. 0227 0.0177 0. 0510 9. 9927 9. 9890 9. 9854 9. 9780 9. 9707 0. 0292 0. 1023 0. 0103 0. 0030 0. 0658 9. 9757 —16 —11 —24 0. 0211 0. 0166 0. 0486 9. 9930 9. 9894 9. 9859 9. 9789 9. 9719 0. 0275 0. 0964 0 0096 0. 0025 0. 0619 9. 9767 —16 —10 —4 0. 0195 0. 0156 0. 0462 9. 9933 9. 9899 9. 9865 9. 9798 9. 9730 0. 0257 0. 0906 0 0088 0. 0021 0. 0581 9. 9776 *Log F of do, ua, v2, aM 28M, and: MSf are each equal to log F of Me. Log F of Pi, Ra, Si, 2, 84, Se, T2,S a, and Ssa are each zero. For log F of Le and mM, see Table 13. 188 U. S. COAST AND GEODETIC SURVEY Table 12.—Log factor F corresponding to every tenth of a degree of I—Con. Se T | 21,9° Constituent ~._ Vico ee 0.0179 Ko eee ee 0.0145 KGa bee ee a 0. 0438 M2*, Ne, 2N_--| 9.9936 Viger oe 9. 9903 IMIZINUN 2s ese5 9. 9871 IMige he eee eae 9. 9806 1 pe ae 9. 9742 O1, Qi, 2Q, p1--| 0.0240 (ONO) Bi ee iepeeai S 0. 0848 IV Kanatiecer ere Se 0. 0081 QININKel: Seen oe 0.0016 VEL sone ae 0. 0544 I Bag ee ee ars 9. 9786 IS I | 22,5° Constituent A yc pa cease 0.0086 TG (ep opel he pels 2 0. 0083 EG rae aaa a ae 0.0298 M2*, No, 2N_-_-| 9.9953 CNR bakin iad pi he ce 9. 9930 Mi MNee ae 9. 9907 Td (a See Se 9. 9860 Nig ke SS 9, 9813 O1, Qu, 2Q, pi--| 0.0137 O= 0 ees ae 0. 0504 Mike = Peel 0.0036 DINKes Saki 2 9. 9990 TY Ee pe eta 0. 0321 IN tingle Ges oe 9. 9847 SS LY | ae Cee Diff. PPAI2 | ON, | eA | IOyKie|| Pape 0.0163 | —15 | 0.0148 | —16 | 0.0132 0.0135 | —11 | 0.0124 | —10 | 0.0114 0.0414 | —24 | 0.0390 | —25 | 0.0365 9.9938 | +3 | 9.9941 | +3 | 9.9944 9.9908 | +4 | 9.9912] +5 | 9.9917 9.9877 | +6 | 9.9883 | +6 | 9.9889 9.9815 | +9 | 9.9824] +9 | 9.9833 9.9754 | +12 | 9.9766 | +11 | 9.9777 0.0222 | —17 | 0.0205 | —17 | 0.0188 0.0790 | —58 | 0.0732 | —57 | 0.0675 0.0073 | —8 | 0.0065 | —7 | 0.0058 0.0011 | —4 | 0.0007] —5 | 0.0002 0.0506 | —37 | 0.0469 | —38 | 0.0431 9.9796 | +10 | 9.9806 | +10 | 9. 9816 22.6° | Diff.| 22.7° | Diff.} 22.8° 0.0071 | —15 | 0.0056 | —15 | 0.0041 0.0073 | —10 | 0.0063 | —11 | 0.0052 0.0268 | —24 | 0.0244 | —25 | 0.0219 9.9956 | +3 | 9.9959 | +8 | 9.9962 9.9935 | +4 | 9.9939 | +5 | 9.9944 9.9913 | +6 | 9.9919} +6 | 9.9925 9.9869 | +9 | 9.9878 | +9 | 9.9887 9.9825 | +13 | 9.9838 | +12 | 9.9850 0.0120 | —16 | 0.0104 | —17 | 0.0087 0.0448 | —56 | 0.0392 | —56 | 0.0336 0.0029 | —7 | 0.0022 | —7 | 0.0015 9.9985 | —4 | 9.9981 | —4 | 9.9977 0.0284 | —36 | 0.0248 | —36 | 0.0212 9.9857 | +11 | 9.9868 | +10 | 9.9878 23.2° |Diff. | 23.3° | Diff.| 23.4° 9.9982 | —14 | 9.9968 | —14 | 9.9954 0.0012 }| —10 | 0.0002 | —10 | 9.9992 0.0121 | —25 | 0.0096 | —24 | 0.0072 9.9975 | +3 | 9.9978] +3 | 9.9981 9.9963 | +4 | 9.9967 | +5 | 9.9972 9.9950 | +6 | 9.9956 | +6 | 9.9962 9.9924 | +10 | 9.9934 | +9 | 9.9943 9.9899 | +13 | 9.9912 | +12 | 9.9924 0.0022 | —16 | 0.0006 | —16 | 9.9990 0.0116 | —54 | 0.0062 | —55 | 0.0007 9.9987 | —7 | 9.9980 | —6 | 9.9974 9.9962 | —4 | 9.9958 | —3 | 9.9955 0.0069 | —35 | 0.0034 | —35 | 9.9999 9.9921 | +10 | 9.9931 | +11 | 9. 9942 *Log F of \2 u2, v2, MS, 28M and MSf are each equal to log F of Me. Log F of Pi, Re Si, Se, 81, Ss, T2 Sa, and Ss2 are each zero, For log F of Ly and M; see Table 13. Diff. 22,4° Diff. 0. 0103 0. 0341 9. $247 9. 9921 9. 9395 9. 9842 9. 9789 0.0171 0.0618 0. 0051 9. 9998 0. 0394 9. 9827 22.9° 0. 0026 0. 0042 0.0194 9. 9966 9. 9948 9. 9931 9. 9897 9. 9862 0. 0071 0. 0281 0. 0008 9. 9973 0. 0176 9. 9889 23.5° 9. 9940 9. 9982 0. 0047 9, 9984 9. 9976 9. 9968 9. 9953 9. 9937 9. 9974 9. 9953 9. 9967 9. 9951 9. 9964 9. 9953 HARMONIC ANALYSIS AND PREDICTION OF TIDES 189 Table 12.—Log factor F corresponding to every tenth of a degree of I—Con. Pa 23.7° | Diff.| 23.8° | Diff. | 23.9° | Diff.| 24.0° | Diff. | 24.1° | Dift.| 24.2° | Diff, Constituent Livks ie Binet 9.9912 | —14 | 9.9898 | —14 | 9.9884 | —14 | 9.9870 | —13 | 9.9857 | —14 | 9.9843 | —13 ieee ed 9.9963 | —9 | 9.9954 | —10 | 9.9944 | —10 | 9.9934 | —10 | 9.9924 | —9 | 9.9915 | —10 Kgs ASRS a 9.9998 | —25 | 9.9973 | —25 | 9.9948 | —24 | 9.9924 | —25 | 9.9899 | —25 | 9.9874 | —24 M>*, No, 2N____| 9.9991 | +3 | 9.9994 | +3 | 9.9997] +3 | 0.0000} +3] 0.0003 | +4 | 0.0007] +3 1 Ife ee Oe 9.9986 | +5 | 9.9991 +4 | 9.9995 +5 | 0.0000 +5 | 0.0005 +5 | 0.0010 +5 M;, MN.-_.--- 9.9981 | +6 | 9.9987} +7 | 9.9994 | +6 | ¢.c000) +7] 0.0007] +6 | 0.0013 | +7 ie oe 9.9972 | +9 | 9.9981 | +10 | 9.9991 | +10 | 0.0001 | +9 | 0.0010 | +10 | 0.0020 | +10 Maye) RMON k 9.9962 | +13 | 9.9975 | +13 | 9.9988 | +13 | 0.0001 | +12 | 0.0013 | +13 | 0.0026 | +13 01, Q1, 2Q,p:_--| 9.9942 | —15 | 9.9927 | —16 | 0.9911 | —15 | 9.9896 | —16 | 9.9880 | —15 | 9.9865 | —15 On Tawa 9.9846 | —53 | 9.9793 | —53 | 9.9740 | —53 | 9.9687 | —53 | 9.9634 | —52 | 9.9582 | —52 NK) SORE Ba 9.9954 | —7 | 9.9947 | —6 | 9.9941 | —7 | 9.9934 | —7 | 9.9927] —6 | 9.9921 | —6 Div Konaee 17 9.9944 | —3 | 9.9941 | —4 | 9.9937 | —3]| 9.9934 | —3 | 9.9931 | —3 | 9.9928] —3 cee ee 9.9894 | —34 | 9.9860 | —35 | 9.9825 | —34 | 9.9791 | —34 | 9.9757 | —33 | 9.9724 | —34 NG ee 9.9975 | +11 | 9.9986 | +11 | 9.9997 | +12 | 0.0009 | +11 | 0.0020 | +11 | 0.0031 | +12 | Se I 24.3° | Diff.| 24.4° | Diff. | 24.5° | Dift.| 24.6° | Diff.! 24.7° | Dift.| 24.8° | Diff. alee 9.9816 | —13 | 9.9803 | —13 | 9.9790 | —13 | 9.9777 | —13 | 9.9764 | —13 9.9896 | —9 | 9.9887 | —10 | 9.9877 | — 9 | 9.9868 | —10 | 9.9858 | —9 9.9825 | —25 | 9.9800 | —24 | 9.9776 | —25 | 9.9751 | —25 | 9.9726 | —25 0.0013 | +3 | 0.0016 | +4 | 0.0020] +3 | 0.0023} +3 | 0.0028 | +4 0.0020 | +5 | 0.0025 | +5 | 0.0030] +5 | 0.0035 | +5 | 0.0040 | +5 0.0026 | +7 | 0.0033 | +6 | 0.0039] +7 0.0046 | +7 | 0.0053 | +6 0.0039 | +10 | 0.0049 | +10 | 0.0059 | +10 | 0.0069 | +10 | 0.0079 | +10 0.0053 | +13 | 0.0066 | +13 | 0.0079 | +13 | 0.0092 | +13 | 0.0105 | +14 9.9835 | —15 | 9.9820 | —15 | 9.9805 | —15 | 9.9790 | —15 | 9.9775 | —15 9.9478 | —52 | 9.9426 | —51 | 9.9375 | —51 | 9.9324 | —51 | 9.9273 | —51 IVE SON A 9.9915 | —6 | 9.9909 | —6 | 9.9903 | —6 | 9.9897} —6 | 9.9891 | —6 | 9.9885 | —6 Ne eS 9.9925 | —3 | 9.9922 | —3 | 9.9919 | —3| 9.9916 | —2| 9.9914) —3 | 9.9911) —2 Bp Uae oh 9.9690 | —34 | 9.9656 | —33 | 9.9623 | —33 | 9.9590 | —33 | 9.9557 | —33 | 9.9524 | —33 Mine Seni a 0.0043 | +11 | 0.0054 | +12 | 0.0066 | +11 | 0.0077 | +12 | 0.0089 | +12 | 0.0101 | +11 “ar if 24.9° | Diff.| 25.0° | Diff. 25.1° | Dift.| 25.2° | Diff. | 25.3° | Diff.| 25.4° | Diff. even eas teas eer anatio tt 9.9751 | —13 | 9.9738 | —12 | 9.9726 | —13 | 9.9713 | —12 | 9.9701 | —13 | 9.9688 | —12 Ke kris 2 9.9849 | —9 | 9.9840 | —9 | 9.9831 | —9 | 9.9822 | —10 | 9.9812 | —9 | 9.9803 | —9 oe oe wae 9.9701 | —24 | 9.9677 | —25 | 9.9652 | —24 | 9.9628 | —25 | 9.9603 | —24 | 9.9579 | —25 M>*, No, 2N___| 0.0030 | +3 | 0.0033 | +3 | 0.0036 | +4 | 0.0040 | +3 | 0.0043 | +4 | 0.0047] +3 NM; OOS Be 0.0045 | +5 | 0.0050 | +5 | 0.0055 | +5 | 0.0060] +5 | 0.0065} +5 | 0.0070 | +5 Mi, MN______- 0.0059 | +7 | 0.0066 | +7 | 0.0073 | +7 | 0.0080] +6 | 0.0086 | +7 | 0.0093 | +7 Mie b RERO 0d. 0.0089 | +10 | 0.0099 | +10 | 0.0109 | +10 | 0.0119 | +11 | 0.0130 | +10 | 0.0140 | +10 Mig: DRA eb A 0.0119 | +13 | 0.0132 | +14 | 0.0146 | +13 | 0.0159 | +14 | 0.0173 | +13 | 0.0186 | +14 01, Qi, 2Q, pi---| 9.9760 | —14 | 9.9746 | —15 | 9.9731 | —14 | 9.9717 | —15 | 9.9702 | —14 | 9.9688 | —14 OXON age ee 9.9222 | —51 | 9.9171 | —50 | 9.9121 | —50 | 9.9071 | —50 | 9.9021 | —50 | 9.8971 | —49 EKER 2 9.9879 | —6 | 9.9873 | —6 | 9.9867] —6 | 9.9861 | —5 | 9.9856} —6 | 9.9850| —6 ONE KMAeE 8 1S. 9.9909 | —3 | 9.9906 | —2| 9.9904 | —3 | 9.9901) —2| 9.9899} —2 | 9.9897| —3 Mif 2 ee ey 9.9491 | —32 | 9.9459 | —33 | 9.9426 | —32 | 9.9394 | —32 | 9.9362 | —32 | 9.9330 | —32 Vir Seeeeh 2b 38! 0.0112 | +12 | 0.0124 | +12 | 0.0136 | +12 | 0.0148 | +12 | 0.0160 | +12 | 0.0172 | +13 *Log F of do, u2, v2, MS, 28M, and MSf are each equal to log F of Mz. Log F of Pi, Ro, S:, S2, S4, Ss, T2, Sa, and Ssa are each zero. For log F of L2 and M; see Table 13. 190 U. S. COAST AND GEODETIC SURVEY Table 12.—Log factor F corresponding to every tenth of a degree of I—Con. TS I 25.5° | Diff.| 25.6° | Diff. | 25.7° | Diff. | 25.8° | Diff. | 25.9° | Diff. | 26.0° | Diff. Constinent Ae ie Bua. | phe RO ae 8. 9.9676 | —12 | 9.9664 | —12 | 9.9652 | —13 | 9.9639 | —12 | 9.9627 | —11 | 9.9616 | —12 Kye sae 8 9.9794 | —9 | 9.9785 | —9 | 9.9776 | —8 | 9.9768 | —9 | 9.9759] —9 | 9.9750] —9 1s a: 9.9554 | —25 | 9.9529 | —25 | 9.9504 | —24 | 9.9480 | —25 | 9.9455 | —24 | 9.9431 | —25 M2*, No, 2N-_--| 0.0050 | +3 | 0.0053 +4 | 0.0057 +3 | 0.0060 +4 | 0.0064 | +3 | 0.0067 | +4 ig. meee tt 0.0075 | +5 | 0.0080 +5 | 0.0085 +6 | 0.0091 +5 | 0.0096 | +5 | 0.0101 +5 M4, MN_------ 0.0100 | +7 | 0.0107 +7 | 0.0114 +7 | 0.0121 +7 | 0.0128 | +7 | 0.01385 sei IMig= tees 0.0150 | +10 | 0.0160 | +11 | 0.0171 | +10 | 0.0181 | +11 | 0.0192 | +10 | 0.0202 | +11 Is Eyes ds es 0.0200 | +14 | 0.0214 | +14 | 0.0228 | +13 | 0.0241 | +14 | 0.0255 | +14 | 0.0269 | +14 O1, Qu, 2Q, pi_---| 9.9674 | —14 | 9.9660 | —14 | 9.9646 | —14 | 9.9632 | —14 | 9.9618 | —14 | 9.9604 | —14 ____________.] 9.8922 | —49 | 9.8873 | —49 | 9.8824 | —49 | 9.8775 | —49 | 9.8726 | —49 | 9.8677 | —48 MEK Sei es 9.9844 | —5 | 9.9839 —6 | 9.9833 —5 | 9.9828 —5 | 9.9823 —6 | 9.9817 —5 QIK Poot £2 9.9894 | —2 | 9.9892 —2 | 9.9890 —2 | 9.9888 —2 | 9.9886 —1 | 9.9885 —2 Mf ae ees 9.9298 | —32 | 9.9266 | —31 | 9.9235 | —32 | 9.9203 | —31 | 9.9172 | —31 | 9.9141 | —31 Mim 2 es eee 0.0185 | +12 | 0.0197 | +12 | 0.0209 | +13 | 0.0222 | +12 | 0.0234 | +13 | 0.0247 | +12 I 26.1° | Diff.| 26.2° | Diff. | 26.3° | Diff.| 26.4° | Diff.| 26.5° | Diff.| 26.6° | Diff. Constituent eee a 9.9604 | —12 | 9.9592 | —12 | 9.9580 | —11 | 9.9569 | —12 | 9.9557 | —11 | 9.9546 | —11 KG) 3 SSeS. 4 9. 9741 —9 | 9.9732 —8 | 9.9724 —9 | 9.9715 —9 | 9.9706 —8 | 9.9698 -9 Keg oo ae ee 9.9406 | —24 | 9.9382 | —25 | 9.9357 | —24 | 9.9333 | —25 | 9.9308 | —24 | 9.9284 | —24 M2*, No, 2N-__--| 0.0071 | +3 | 0.0074 +4 | 0.0078 +3 | 0.0081 +4 | 0.0085 +4 | 0.0089 +3 aood0onm oOcen1o [Sor ho cos 62. to 1) DOr COMI OREO DOAIrHD Ww or hoe SOHO i SA wNown a =m ams. Owoorn-7 ISO HK» CO bo NPR Oo bo =I (5 Tory e.2) CoOO-, Nor ; sa boc oor O10 © CORrmo NAD ob ~100 Or Orr to . No . nace Bases Seren PAD moana oor NOOm Tor -] Log F (Le) 0. 0964 0. 1125 0. 1052 0. 0793 0. 0445 0. 0092 9. 9779 9. 9529 9. 9347 9. 9233 9. 9183 9. 9193 9. 9254 9. 9369 9. 9523 9. 9713 9. 9926 0. 0148 0. 0357 0. 0523 CORB KOIDE OS) 1S) FOS IO) OKO CO KOKORO CORDON ma S =) o BOSS SSO 0 osoKoe atte 2 S ie) o Log F(M,)| Diff. Diff aie 9. 7295 9.7260 ae 9. 7338 ae 9. 7527 : 9, 7824 353 9. 8228 a0 9, 8734 ee 9, 9330 9.9977 1 0, 0586 HH 0.0998 a 0. 1056 0.0780 ; 5 0. 0283 rep 9. 9760 ae 9. 9275 9. 8861 au 9, 8525 a 9, 8269 a 9, 8091 5 9, 7991 62 9.7970 109 9, 8033 ae 9, 8185 9, 8433 Ze 9. 8782 Be 9, 9228 ie 9, 9744 0.0238 Pt 0.0521 3 0. 0397 fe 9, 9883 9.9195 282 9. 8512 390 eae 487 Bate 536 | 8: t03 463 : 191 9. 6883 209 9. 6789 9. 6822 a) 9. 6986 602 9. 7285 556 9. 7722 449 : 9, 8286 ee 9. 8921 9.9461 77 9. 9626 9. 9283 219 9. 8640 ous 9. 7959 aZ0 RQs pee 9.7374 E 9. 6924 et 9. 6616 Bi 9. 6447 7S 9. 6414 i 9.6515 a 9. 6755 os 9.7133 9.7651 HARM 7 ONIC ANALYSIS AND PREDICTION OF TIDES 193 a e ae g f 2 1 y s 1900 t 2 —Con. Year : N | wofle | Diff. | wof M i eee ene 1 Diff. Log v7 (L») Diff ; =| Xiat iff. | Log #(M;)| Diff. N ° — 1914 | 330] — a ~ rd TSR7AL 9 sgl a CORN a x 1915 | 325| —14.3 10.9 POWBON| 1 gam 9. 7651 390 3.8 Ban 219. 4 if 9, 9283 643 330 315 2.9 5.9 De @ | oat 9.8 295 9. 8294 35 | 29) go, BEF) gos] 8 sus | 725 o.suen | | P| ap 1916 | 305| +86 a |e Re 9.8815 | 198 foe | ag 315 oalmeelacen |) } Oma lll sanec 9. 8923 s16 | 20 995 | +174 3.8 314.9 | 17.0 oe 230 9. 9425 Beeele | Gondal «(22-2 SNe as 9.3344 | 581 308 1817 Be ee ; 9.4 9. 9499 449 9. 8244 600 300 285 419.4 0.2 336. 8 9. 9948 506 295 280| +lei| 2&2 344, 3 7.5 Ghow 510 9.7738 a5; 496; ©5| 3502 56; Gece 1200s | pages oe 285 AOTSMY |i 270)\|)) Shetty a 3 0.1305 | %g3 eral e Pinnte|) eeeY 265 aa 7.1 1.4 0. 1368 15 275 Seo eemonaiin koe Ghouls | Se 0.11 297 9. 7033 O55) ot ele | Seale GG BLE | Goa |, |. 00 ORR CNH O POWDMD OMWD me Rone KOO ONwo OWwWWwa DOP Ne woo wT Reo CORD “IQ bo 0 WwWoawrn uw of Mi ~ Roe ese worn ATH =" wo a SIRS RII Gae SOUS TOO CORO Ic Sisus mre CO ADWW PNDMD UAMKHD ANNI ONS DOWN Diff — = REED. POS EIED CUES Colca fen eae aor) RPNIONMN CrWWwW Omn WROD [ NHnNne oa SO SOT SUS Coie 1 CONT = S Gare BSUS ON IHN aR APSR C aes tC CNMPO HOMNMN ONT AWN BARAWD WIND BERS PRO — S 9. 8923 9. 9576 0. 0134 0. 0840 0. 1614 0. 2202 0. 2218 0. 1646 0. 0865 0.0145 9.9569 9.9157 9.8895 9.8776 9.8794 9.8945 9.9227 9. 9637 0.0160 0.0752 0. 1302 0. 1618 0. 1548 0.1155 0. 0625 0.0129 9.9711 9.9398 9.9190 9.9087 9. 9063 9.9126 9.9262 9.9458 9. 9698 9.9960 0.0216 0. 0432 0. 0573 0.0617 0.0561 0.0422 0.0228 0.0010 9. 9791 9.9591 9.9420 9.9290 9.9207 9. 9176 9.9204 9.9295 9.9455 9. 9685 9.9982 0. 0331 0. 0696 0. 1002 Log F (L2) 9.9175 | Diff Log F (M1) 9. 9184 9. 8498 9. 7828 9.7270 9. 6850 9. 6569 9. 6428 9. 6421 9. 6550 9. 6817 9.7225 9.7771 9. 8433 9.9110 9.9578 9.9572 9.9123 9. 8490 9.7896 9.7418 9.7078 9. 6874 9. 6803 9. 6858 9.7039 9.7349 Chad 9. 8331 9.8989 9.9690 0. 0285 0. 0555 0. 0395 9.9949 9. 9427 9.8952 9. 8569 9.8285 9. 8099 9. 8002 9. 7989 9.8057 9. 8202 9. 8425 9.8726 9.9107 9.9564 0.0077 0. 0586 0. 0966 0. 1044 0. 0745 0. 0182 9.9530 9.8906 9.8364 9.7923 9.7589 9. 7364 Diff. HARMONIC ANALY NALYSIS AND PREDICTION OF TID D ES 195 o Table 13.—V 3 alues of u and log F of L, and M, for years 1900 to 2000—Con Ye ar N ota le Dit . | wof Mi | Diff. | L ° ° ares . og. F (Le) Diff. Log EF (M 7 1944 uo; 43.2 ° »)| Dif. | NW ’ 6. 354. 1945 105 14, 2 6 bi Areas 100 —9.6 6.6 359.7 147 9. 7364 95 | —14.9 5i3 4.8 5.1 0. 1149 115 110 90 | —17.7 2.8 10.3 5.5 0. 1061 88 9.7249 1946 0.2 16.5 6.2 0.0757 304 9. 7246 3 105 85 | —17.9 7.5 0. 0331 426 9. 7363 117 100 80 | —16.1 1.8 24.0 446 9. 7603 240 95 75 | 12.7 3.4 33.5 9.5 9. 9885 370 90 1947 4.5 46.1 12.6 9. 9488 397 9. 7973 70 8.2 17.0 9. 9163 325 9. 8467 494 85 65 ity | pice 63.1 200 9.9050 | 588 80 60 42.6 5.6 gag | 21-5 9. 8963 553 75 55 +8.1 5.5 107.2 | 22:6 9. 8864 99 9. 9603 1948 5.1 126.2| 19.0 9. 8884 20 9. 9882 279 70 50 | +13.2 14.5 9. 9030 146 9. 9677 205 65 45 | +17.4 4.2 140.7 282 9. 9088 589 60 40 | +19.8 2.4 151.4 | 10.7 9. 9312 706 55 35 | +19.0 0.8 159.8 8.4 9. 9740 428 9, 8382 1949 5.6 166.7 6.9 0. 0319 579 9. 7734 648 50 Es 30} +13.4 6.0 0. 1025 706 9.7208 526 45 25 499 | 1.4 172.7 724 9. 6820 388 40 20} 11.2] 132 178. 4 5.7 0. 1749 250 35 0) meade | Logto| | aaed 5.6 0.2196 | 447 9. 6570 1950 3.0 189.8 5.8 0. 2046 150 9. 6455 115 30 10| 23.1 6.6 0. 1407 639 9. 6475 20 25 5| 22.0 11 196. 4 764 9. 6629 154 20 ON) 51852 3.8 204. 1 Tei 0. 0643 295 15 355 | —12.9 5.3 213.9 9.8 9. 9963 680 9. 6924 1951 ; 6.1 2079 | 13.1 9.9431 532 9. 7361 437 10 350 =6.8 17.7 9.9054 | 322 9.7933 | 572 5 Se anal Gall sae SET Gace |) OO 0 340 +6.1 6.4 267.0 22.3 9. 8827 634 355 335 +12. 1 6.0 289. 9 22.9 9. 8743 84 9. 9237 1952 5.2 309. 0 19.1 9. 8797 54 9. 9572 335 350 330 | +17.3 14.2 9. 8990 193 9. 9398 174 345 325 491.1 3.8 323. 2 332 9. 8843 555 340 320 | +22.6 as 333.9 | 10.6 9, 9322 652 335 315| +20.5| 22 a9 | oh 2 9.9793 471 9. 8191 a 1953 6.7 348.7 6.7 0. 0390 597 9. 7612 579 330 310 | +13.8 6.0 0.1056 666 9.7162 450 325 305 43.9 | 10.6 354.7 583 9. 6852 310 320 300 Si 4 0.2 5.5 0, 1639 172 315 295| —13.6| &4 57 | 55 0. 1881 242 9. 6680 1954 3 11.4 5.7 0. 1643 238 9. 6642 38 310 290 | —15.9 6.3 0. 1085 558 9. 6733 91 305 285 | —15.0 0.9 17.7 600 9. 6958 225 300 230 | 12.2 2.8 25.1 7.4 0. 0485 349 295 1955 3.8 34.4 9.3 9. 9948 537 9. 7207 275 —8.4 12.2 9. 9528 420 9.7791 484 290 270 4.0 44 46.6 298 9. 8399 608 285 2e6 +0.5 4.5 62.7| 16-1 9. 9230 703 280 260 44.8 pie | Peso eau 9. 9055 175 9.9102 1956 aay lose 21.8 9. 8987 68 9. 9776 674 275 255 48.5 19.0 9. 9020 33 0.0225 449 270 250} +11.5 3.0 123.7 124 0. 0237 12 265 245 | +13.4 1.9 138.3 | 146 9.9144 379 260 240 | +14.0 0.6 149.9 | 10.9 9. 9348 204 9. 9858 1957 0.9 157.6 8.4 9. 9615 267 9. 9320 538 255 235 | +13.1 6.7 9, 9923 308 9. 8804 516 250 230 | +10.6 2.5 164.3 315 9. 8380 494 245 225 46.7; 39 169.9 5.6 0.0238 312 240 220 jing | | 44 175.0 Bal 0.0511 273 9. 8068 1958 4.3 179.7 4.7 0. 0693 182 9. 7866 202 235 215 ~2.0 4.6 0.0746 53 9. 7769 97 230 210 —5.4 3.4 184.3 77 9.7770 1 225 205 Dee |p eed 1390| 47 0.0669 92 220 Sen? 4 Babe aie e Gee 5.1 0.0492 | 177 9. 7862 1959 0.0 199. 7 5.6 0. 0260 232 9. 8042 180 215 195 8.5 6.6 0.0014 246 9. 8308 266 210 190 “7.6 0.9 206. 3 233 9. 8658 350 205 185 8,8) 1.3 214.4 8.1 9. 9781 432 200 180 —4.4 1.9 224. 6 10.2 9. 9579 202 9. 9090 237.7 gp 9.9417 162 9. 9596 506 195 9. 9299 118 0.0151 555 190 0. 0692 541 185 180 196 U.S. COAST COAST AND GEODETIC SURVEY © g 2 y Year | N u of : jase Rees, Rusa Diff. | wofM: | Diff ee | ee een . | Log F (12) P ke ° Sesare acetone’ Diff. | Log F(Mi)| Di 1959 180 An ° Peel coetssotel ey Bi sponilines oven iff. N : 2.1 237.7 ——_|———— 1960| 175| —23 16.4 | 9.9299 ; 170 0.0 2.3 254. 1 71 0.0692 165 42.3 2.3 972.7 | 18-6 9. 9228 404 180 160 44.5 2.2 291.0 | 183 9. 9207 21 0. 1096 1961 i 2.0 306. 6 15.6 9. 9237 30 0. 1203 107 175 155 G3 12.3 9. 9320 83 0.0951 252 170 150] +81 1.6| 318.9 137 0.0444 | 907 165 145 48.9 0.8 328. 5 9.6 9. 9457 603 160 1962 0.2 336. 1 7.6 9. 9648 191 9. 9841 140 | +8.7 6.4 9.9890 | 242 9.9252 | 589 155 135 47,2 1.5 342. 5 280 9. 8729 523 150 130 44.1 3.1 348.1 5.6 0. 0170 435 145 125 0.6 4.7 353.2 5.1 0.0465 295 9. 8294 1963 1 5.4 358.0 4.8 0. 0728 263 9. 7950 344 140 120 —6.0 4.9 0. 0894 166 | - 9: 7708 247 135 115 FeO 5.0 2.9 8 9. 7554 149 130 110 VR 3.5 8.0 5.1 0. 0902 47 125 105 | —16.0 1.5 13.6 5.6 0.0733 169 9. 7507 1964 0.5 20. 2 6.6 0. 0428 305 9. 7565 58 120 100 ia B 8.1 0. 0058 370 9. 7735 170 115 95| —13.4 2.1 28.3 368 9. 8025 290 110 90 | —10.1 3.3 38.3 | 10.5 9. 9690 412 105 85 59 4.2 52.9 14.1 9. 9370 320 9. 8437 1965 : 4.8 71.6| 187 9. 9125 245 9. 8956 510 100 80 54 22.1 9, 8969 156 9. 9523 567 95 75 +4.0 5.1 93.7 48 9. 9988 465 90 70 +9.0 5.0 114.9 | 21-2 9. 8921 61 85 65| +13.3 4.3 131.9 | 17-0 9. 8981 60 0.0049 1966 Su) 144.7 12.8 9. 9161 180 9. 9657 392 80 60} +16.5 9.6 9. 9470 309 9. 8991 666 75 55 | +17.9 1.4 154.3 447 9. 8289 702 70 50 | +15.9 2.0 162.0 7.7 9.9917 615 65 45 +8. 9 7.0 168. 5 6.5 0.0498 581 9. 7674 1967 ; 11.5 174.3 5.8 0.1173 675 9.7185 489 60 40 —2.6 5.5 0.1795 622 9. 6832 353 55 35 | —142| 11.6 179.8 274 9. 6614 218 50 30 —91.2 7.0 185.4 5.6 0. 2069 85 45 25 —93.1 1.9 191.4 6.0 0. 1786 283 9. 6529 1968 ‘ 1.8 198. 2 6.8 0. 1135 651 9.6577 48 40 200) e283 8.2 0.0418 717 9. 6759 182 35 15 17.2 4,1 206. 4 624 9. 7081 322 30 10} 1108 5.4 917.0 | 10-6 9. 9794 464 25 5 5 6.2 231.3 14.3 9. 9310 484 9. 7545 1969 6.5 250. 5 19.2 9. 8975 335 9. 8137 592 20 0 40.9 23.0 9. 8785 190 9. 8803 666 15 355 47.4 6.5 273.5 50 9. 9370 567 10 350 || tras | | toe pom 2242 9. 8735 185 5 1970 5.1 313.4 17.7 9. 8826 91 9. 9555 345 | +18.6 13.0 9. 9058 232 9. 9227 328 0 340 | +22.2 3.6 326. 4 379 9. 8603 624 355 335 | +23.1 0.9 336.2 | 28 9. 9437 657 350 330 | +19.8 3.3 343. 9 ri 9. 9965 528 9. 7946 1971 : 8.7 350. 4 6.5 0. 0628 663 9. 7388 558 345 : B25 gecttle 5.9 0.1355 | 727 9e06s | 428 340 320 ee wee 356.3 583 9. 6683 282 335 S16) |) ait | | S0x8 1.8 5.5 0. 1938 143 330 B10) 170 5.4 7.4 5.6 0.2059 | 121 9. 6540 1972 ; 0.9 13.4 6.0 0. 1643 416 9. 6531 9 325 305 —18.0 6.8 0.0974 669 9. 6656 125 320 300 —15.9 Oli 20. 2 661 9. 6912 256 315 295 | 12.2 3.7 28, 4 8.2 0.0313 393 310 290 =7.B 4.7 38.3 | 10.4 9. 9767 546 9.7305 1973 5.1 52.8 14.0 9. 9359 408 9. 7833 528 305 285 2.4 18.5 9. 9088 271 9. 8482 649 300 280 42.7 5.1 71.3 143 9.9191 709 295 275 47.3 4.6 93.3 | 22-0 9. 8945 | 611 290 270 SSNS 4.0 114.9 21.6 9. 8919 26 9. 9802 1974 : 3.1 132.0 | 17-1 9. 9005 86 0.0058 256 285 265 | +14.4 12.8 9. 9188 183 9. 9834 294 280 260 | +16.0 1.6 144.8 274 9, 9326 508 275 on || is.9 | ie 154.5 | 9% 9. 9462 566 270 | ume | @ell aaa Sarit Vosga05)| | 1a 9. 8760 f : 168.3 6.2 0.0185 380 9. 8269 491 265 0. 0549 364 9. 7895 374 260 9, 7642 253 Be HAR MONIC AN'ALYSIS AND PREDICTION OF TIDES 197 n Year N u . of L alg Saga i ia 2 Diff. u of My, Diff. | Log F(L2)| Dit. | L : 4 a =| oe) Ce oak og F(M))| Diff. N 1974 | 250 ° = e TEESE ey 168. 3 0.0. E 1975 245 +9.6 5.5 ee 274 9. 7642 240 44.2 5.4 173, 8 0. 0823 134 250 235 ii 5.3 178.8 28 0.0 116 9. 7508 Sidley ee 193.6 | 48 0939 ue 9. 7486 22 zee : ae jae 49 0. 0871 a 9. 7568 82 240 1976} 295/ —8.0 5.3 0.0660 | 3a 9.7751 | 88 Zp 220 —9.0 1.0 1g8 Be 0.0374 282 230 21 _3 0.2 6 : 9. 80: 1977| 205| —5.9 ae AO |) 10ks 9.9573 | iT a oagg | 554 215 200 a7 2.2 Bete OR 9. 9399 606 210 05 |) geesicds nen 208 Ber | ace gyezei || (U8 0.0631 | 589 205 1978} 190] +41.0 18.7 9. 9218 10 0.1070 | 439 200 185 43.2 2.2 274.2) 4 9. 9208 129 195 1390| +59| 20 poniah|) 181 olga 40 0.1199 Was || eater) cL ae || Hee pe By Gee aes a0 : 12 319.5 11.9 9. 9337 Be 0, 0502 469 185 1979 170 48.1 : 9.3 9. 9472 ie 9.9949 553 180 BS) |" pese | 1 ONS oe 9. 9649 536] 77 160 48.3 0.3 336. 2 7.4 9.9 213 9, 9413 i5| +69| 14 342.4 Bop 862) Toa | | | 2 8086 at 170 1980 | 150] +43 26 : mg] 0.0340) 3 Oe | al? 160 145 40.6 3.7 352.7 0. 0553 232 155 140 3.7 4,3 357.3 4.6 0.0 143 9. 7989 135 lee 4.3 1.9 4.6 aes 3 9. 7842 147 150 8.0 eo 6.7 4.8 0. 0729 aa 9.7783 59 145 igen | ER OE ee a 2a i) osees0)| 4/0 ozsie |e) as 125) |) 134 2.0 11.9 0. 0416 131 135 100| —138| 04 iyo.) PERO 0.01 287 9. 7947 il Veoie be eed 0) i 0129) 319 g.gis4 | 237 130 : 2.3 34.0 9.0 9. 9817 Ba 9. 8533 349 125 1982 110} —10.5 : 12.0 9. 9524 546 9. 8990 457 120 105 —73 3.2 46.0 9.9 533 115 aM Ad, Me 62.9 | 16.0 9278 | 77 9, 9523 os | 400) 14:3 gai | 20.1 9.9101 ot 0.0031 | 508 ay - Aaa |) Dik 9. 9006 , 0. 0307 276 105 1983 90 45,3 : 19.1 9. 9004 100 0. 0146 161 100 85°) bctoys | “12 oe ea 9. 9104 9 ve i 73 | 44183 23 Ms. 7 eo SEE 330 0. 8804 703 80 1984 70 | 15.4 c : mo 0.0099 | 560 0, 7646 572 80 By so | ee 164.3 Gteccs 445 % 60 +42 7.8 170.4 6.1 0. 604 9. 7201 ‘ 176.0 5.6 . 1263 312 70 10.8 . 0. 17. 478 9. 6889 1985 55 =RR 5.4 . 1741 98 9, 6706 183 65 60| —162| 2% te 0. 1839 af a 45] —a15| 58 iazagn ||" 1888 0.14 365 9. 6652 40 29 4 0.9 193.2 6. 2 - 1474 62, 9. 6729 77 55 : 2.3 200. 3 7,1 0. 0849 be 9. 6941 212 50 1986 35 ~20.1 ; 8.7 0.0199 a 9. 7291 350 45 SN Ta 4,2 209. 0 9.963 490 40 | Gent Be gga} 4 Ree alee azo) 9. 781 30) eahon wane? 2361 wy RE woo 9.8395 | 614 a 1987 Wg oS ee bal 23e4 prs7es | 122 Rage 71 25 10| 486| 6&3 280.0 9. 87 18 20 Bil elt |) Be 301.9 | 21-9 Bees | 126 9. 9524 a 4.9 317.3 | 16.3 8880 | on 9.9068 | 456 15 .4 39 309.9] 119 9.9150 7 ae 9. 8400 668 10 1988} 355| +22.6 9.1 9.9572 | 579 9.7750 | £50 5 350 | +22.7 0.1 338. 3 0.015 535 0 2 | eben | ew 345.6 | 8 Rae ale 9. 7215 340 47.4] 10.7 ae 9 e é 0. ee 761 9. 6820 a ae : 5 1989 | 335| —6.0 Li Za crea okerGs: ge oee P6805 i | aay | gees 330 15.8 9.8 3.3 0. 212 18 340 325| —198| 40 9.0 5.7 OAS e573 9. 6469 320 | —19.4 0.4 15.2 6.2 1553 m4 9, 6623 154 335 22.5 7.3 0. 0809 aoe 9 289 330 0.0132 | 97 poe 395 9. 7341 429 320 198 ue Tc S. COAST AND GEODETIC SURVEY Table 13.—Val : ues of u and log F of L, and M, for years 1900 to 2 o 2000—Con. Year N , ty of Le Diff. | wof M,; | Diff . | Log F(L2) | Di Be iff. | Log F i g F(Mi) | Diff N 1989 320| —19.4 : } : 22.5 1990 311511 aee16 | a bea 4 ae op Rap 9. 7341 an ‘ , Kes 565 320 305 EiGi5 | | 2,3 0.0 58 oo us| 5 || tae | dine 209 | 388 | 9.8582 ONE is. 7915 | 72,2 9.8967 | 19g cone | 8 A 1991 205 | +4 + | a ee ale s : a, ss a og 9.9776 | °00 oon 285 | +14.0 4.3 12216) ig 0.9011 7 : : mol seize | @2°| saaae 15 cout} 2 = : 1.3 149.7] 21-5 9. 9257 a 0. 888 i 1992 275 | +18.3 8.9 sooo | $88 | 9 aaas zs a 270 | +17.3 1.0 158. 6 on = : ' A 0. 002 i e 265| +136| 22 ae 0 : ee] 3 0.0029 | 459 9.7819 27 ‘ = a | 0 0.0879 | 40g 9.7493 | 326 a : 4 a 250 9. 7290 cus cite se fia] se 16.9) 5.4 0. 1137 9 1 245 8.6 316 187.0 Ped 0.0933 i ee i S imo] 0.1147 | 914 9. 7249 8 2 1994 240 | —10 te |e bednes : 7 a on 9.7662 | 262 oe : iH i 371 245 a a gi Seee| jee 0.0218 9 225 GA 21 213.5] i ¢ eS eo zs j 2.6 904.1 | 10.6 Blue 209 see sn 195 | 220| 28] 9 138 | 99889) 337 cel |e ee 4 e ale ue e273 ||| eas oe BG emeeG | | a2 2 | eos 2) gies 0 : : 205 eT 2.5 DIANS)|| Taig 9.9177 ‘ a 3 2] 293,1| 18-6 eon 1 0 1h 1996 200| +6.2 si wer ip a i 200 i 308. 5 i ne 190 if a jee | 370-4) 743 0.0477 8 . oe | ees | ORS mara 9.3 9177 | 84 se | is oS | Lmataoy 153 a ee 200 ako : i i ae) oH : 13 9.9661 | 369 9.9382 | 494 ~~ alae |e “ 20 9.8055 | 340 iB ie| 488) 4s 43.0 5.5 0.0091 9 i to | 4ne| 23| gers ie wor) 12 cae: i‘ | 33] g5r5| 45) 0.056 | dee |\ ae 1998 160 25 . 4.4 0. 0586 ae 9.8085 u13 in : 4 fs 9. 8035 1s 165 ‘8 at 3.6 1.9 ie 0.0605 9 i 145) | te 1 2.0 13] 56 0.0355 f i i 0.7 16.9| 56 0.0855 | 331 4 i 1999 1440| —11.8 . 6.5 0.0124 oa 9.8305 za 195 | seeste4 0.4 23. 4 a on i : i Ha] 4 4) gy 9. 9866 9 ‘| 7 | _ ad] a 8.9865 | 95g 9. 8714 140 = “ts 1.5 9. 9613 a 9.9130 | 416 1B : Bee ss a 940627, || Maen 30 i “|g 55.9] 1.0 9.9215 0 ‘i 110 +2.8 3.7 94.8] 99 9: 9060 ‘a et : ip : a) ae 9. 9103 ae 0.0516 75 12 9.9111 45 bord 08 0.0142 | 398 110 HARMONIC ANIALYSIS AND PREDICTION OF TIDES Table 14.—Node factor f for middle of each year, 1850 to 1999 199 Constituent 1850 | 1851 | 1852 | 1853 | 1854 | 1855 | 1856 | 1857 | 1858 | 1859 STs ese ea 0.892 | 0.948 | 1.007] 1.061 | 1.105] 1.138] 1.158| 1.165 | 1.160] 1.141 Igy oe Be il aa 0.922 | 0.959 | 0.999 | 1.037 | 1.069] 1.092] 1.107] 1.113 | 1.108] 1.095 Thy ap aie deel ele 0.816 | 0.887 | 0.977 | 1.075 | 1.168] 1.246| 1.298] 1.317] 1.302] 1.254 Tips spre 1.163 | 0.905 | 0.725 | 1.055] 1.263 | 0.944] 0.469] 0.962 | 1.283 | 1.001 NEE E Ewan | Fat 1.023 | 1.675] 1.974| 1.559| 1.118] 1.860] 2.348] 1.872] 1.177| 1.776 Mz*, No, 2N, dz, u2,v2-| 1.027 | 1.017] 1.005 | 0.993 | 0.981 | 0.972] 0.966] 0.963 | 0.965 | 0.971 Pee eros 1.042 | 1.026] 1.008| 0.989) 0.972 | 0.958 | 0.949 | 0.945 | 0.948 | 0.957 IUCR ANDN tc So RSE 8 1.056 | 1.035 | 1.011 | 0.986] 0.963) 0.944 | 0.932] 0.928 | 0.931 | 0.942 Tig Secs LONG Sas 1.085 | 1.053 | 1.016] 0.978 | 0.944 | 0.918 | 0.900 | 0.894 | 0.899 | 0.915 Vee Balle i 1.114] 1.071 | 1.021] 0.971 | 0.927] 0.892] 0.869 | 0.861 | 0.867 | 0.888 OmOn ZO, pee 0.874 | 0.933 | 0.998 | 1.059} 1.110] 1.150] 1.174] 1.183] 1.176] 1.153 (OKO Se 0.631 | 0.786 | 0.983 1.204] 1.422] 1.608] 1.735 | 1.783 | 1.745] 1.627 INTIS BM 0 crite ss 0.948 | 0.976 | 1.004 | 1.029} 1.048] 1.062] 1.069] 1.072] 1.070] 1.063 NK eM 7 SHES 0.974 | 0.993 | 1.010] 1.022] 1.029] 1.032] 1.032] 1.032 | 1.032] 1.032 ENT fame eh Peo ne 0.743 | 0.856 | 0.990 | 1.129] 1.257] 1.360] 1.427] 1.452 | 1.432] 1.370 Nive bi 1.094 | 1.059 | 1.016 | 0.973 | 0.933 | 0.900 | 0.879 0.871 | 0.878 | 0.897 Constituent 1860 | 1861 | 1862 | 1863 | 1864 | 1865 | 1866 | 1867 | 1868 | 1869 Lee 1.110] 1.066} 1.013 | 0.955] 0.898 | 0.852] 0.829] 0.832 | 0.863] 0.912 1 a a 1.072 | 1.041 | 1.004] 0.964] 0.926] 0.898] 0.883 | 0.885 | 0.904] 0.936 Reln) Beh 1.179 | 1.086 | 0.988 | 0.897 | 0.823] 0.773 | 0.749 | 0.753 | 0.784] 0.840 fn ae On ee 0.568 | 0.924| 1.225] 1.117] 0.865 | 0.879 | 1.082] 1.190 | 1.091 | 0.840 10 ee 2.227 | 1.792] 1.046 | 1.260] 1.680] 1.609) 1.164] 0.812 | 1.189] 1.731 M2*, No, 2N, dz, u2, v2-| 0.980} 0.991 | 1.004 | 1.016] 1.026 | 1.034] 1.038 | 1.037 | 1.032] 1.024 Ak SE EAL ‘970 | 0.987 | 1.006 | 1.024] 1.040] 1.051] 1.057| 1.056 | 1.049 | 1.036 SVL. joe 0.960 | 0.983 | 1.008 | 1.032] 1.054] 1.069] 1.076 | 1.075 | 1.066] 1.048 iis eee Ree 0.941 | 0.974 | 1.011 | 1.049 | 1.081 | 1.105] 1.117] 1.115 | 1.100] 1.073 Rig 2 LABME DT BSE 0.922 | 0.966 | 1.015] 1.065] 1.110] 1.143] 1.159] 1.156] 1.135] 1.099 OQ@y, 2Q}/pib-s2e = 2 1.116 | 1.065] 1.005 | 0.941 | 0.880] 0.832] 0.808 | 0.812 | 0.843 | 0.896 ek, Miied OORT 1.447 | 1.230] 1.008! 0.807) 0.647] 0.540 | 0.489 | 0.497 | 0.563 | 0.685 za! aren 1.050 | 1.032} 1.007] 0.979] 0.951 | 0.928 | 0.916 | 0.918 | 0.933 | 0.958 PNK D Be FT | 1.029 | 1.023 | 1.011] 0.995] 0.976 | 0.960 | 0.950 | 0.952 | 0.964 | 0.981 i a 1.270 | 1.145 | 1.007] 0.872] 0.755 | 0.670 | 0.629 0.635 | 0.689 | 0.783 Nimik Gee t's: BELT | 0.928 | 0.968) 1.011 | 1.054] 1.091 | 1.117] 1.130] 1.128 | 1.112 | 1.082 Constituent 1870 | 1871 | 1872 | 1873 | 1874 | 1875 | 1876 | 1877 | 1878 | 1879 ce! eR aie 0.971 | 1.028 | 1.079] 1.120] 1.147] 1.162] 1.164] 1.154] 1.130] 1.094 AG Bie FO AS 0.974 | 1.014 | 1.050] 1.079] 1.099] 1.111] 1.112] 1.104 | 1.087] 1.061 KGa ean cE EO Ae 0.920 | 1.014] 1.112] 1.201 | 1.269] 1.309] 1.315 | 1.287] 1.227] 1.144 Try Oeil Spee en 0.828 | 1.148 | 1.224] 0.816] 0.545] 1.087] 1.270] 0.858 | 0.543 | 1.036 INC SMe TAS 1.811 | 1.300] 1.185 | 2.004] 2.286] 1.656 | 1.227] 1.998 | 2.269] 1.645 Mo*, No, 2N, dz, uz, »2-| 1.013 | 1.000] 0.988 | 0.977] 0.969} 0.964] 0.963 | 0.967 | 0.974 | 0.984 ak SE | aS 1.019 | 1.001 | 0.982] 0.966 | 0.954 | 0.947 | 0.946 | 0.951 | 0.961} 0.976 Ng NENTS Be es 1.026 | 1.001 | 0.976 | 0.955 | 0.939 | 0.930} 0.928 | 0.935 | 0.949 | 0.968 IGN nN wn 1.039 | 1.001] 0.965 0.933] 0.910 | 0.896 | 0.894 | 0.904 | 0.924] 0.953 Nie OT | EGS 1.052 | 1.002 | 0.953] 0.912] 0.881 | 0.864 | 0.862 | 0.874 | 0.900} 0.938 OmiQw2Q)) prec ce 0.958 | 1.022] 1.080] 1.127] 1.161] 1.179] 1.182] 1.169 | 1.140 | 1.098 COOL ae CT MT 0.858 | 1.067 | 1.290] 1.500| 1.666/ 1.764] 1.779 | 1.708 | 1.563 | 1.366 Kare uA nc 0.987 | 1.014] 1.037] 1.054] 1.065] 1.071 | 1.072 | 1.068 | 1.059 | 1.044 Mik. SNE 5 RE 1.000 | 1.015} 1.025] 1.030 | 1.032] 1.032 | 1.032] 1.032] 1.031 | 1.027 IVa Mees OMT Se 0.907 | 1.044 | 1.181] 1.300] 1.390] 1.442] 1.450] 1.413 | 1.335 | 1.224 Min AU OF PRR 1.043 | 1.000 | 0.957| 0.919 | 0.891 | 0.874 | 0.872] 0.884 | 0.908 | 0.943 *Factor f of MS, 25M, and MSf are each equal to factor f of Mo. Factor f of Pi, Re, 81, S2, S4, Se, T2, Sa, and Ssa are each unity. 200 U. S. COAST AND GEODETIC SURVEY Table 14.—Node factor f for middle of each year, 1850 to 1999—Continued Constituent 1880 1881 1882 1883 1884 1885 1886 1887 1888 | 1889 1,047 | 0,991 | 0.932) 0.878 | 0.840 | 0.827} 0.841} 0.880 | 0.934] 0.994 1.027 | 0.988 | 0.949 | 0.914} 0.890} 0.882] 0.891 | 0.915 | 0.950} 0.990 1.048 | 0.951 | 0.866 | 0.800 | 0.760 | 0.748 | 0.762} 0.803 | 0.869} 0.955 | a ae Oe 1.246 | 1.020] 0.786] 0.944] 1.152] 1.171 | 1.006 | 0.824 | 0.945 | 1.205 ING pee Oras ok Cuan 1.046 | 1.528 | 1.824] 1.529] 0.970 | 0.877) 1.364} 1.721] 1.593} 1.075 Mo2*, No, 2N, Azo, u2,v2-| 0.996 | 1.009 | 1.020} 1.030} 1.086] 1.088 | 1.0386 | 1.029 | 1.020] 1.008 Gi} pecan mee 2 0.994 | 1.013 | 1.0381 | 1.045 | 1.054] 1.057] 1.054] 1.044] 1.030] 1.012 VIG IV IIN eee 0.992 | 1.017} 1,041] 1.060] 1.073] 1.077] 1.072] 1.060 | 1.040] 1.016 Migs oe ote oe a sve! 0.988 | 1.026 | 1.062} 1.092] 1.111} 1.118} 1.111] 1.091 | 1.061] 1.024 UN [ies Se eas ees 0.984 | 1.035} 1.084] 1.124] 1.151] 1.160] 1.150] 1.123 | 1.082] 1.033 O1, Qi, 2Q, p1_-------- 1.043 | 0.980} 0.916 | 0.860] 0.820] 0.806] 0.821 | 0.862 | 0.919} 0.983 OLS ae 1,144 | 0.926} 0.739] 0,599] 0.513 | 0.486) 0.516 | 0.604 | 0.746 | 0.936 MEK hewn oye 1.023 | 0.997 | 0.968 | 0.941 | 0.922} 0.915 | 0.922] 0.942 | 0.969] 0.998 PANG He SE LL ee 1.019 | 1.005} 0.988 | 0.969] 0.955 | 0.950 | 0.955 | 0.970 | 0.988 | 1.006 IVE fos. 31 eee acm 2 1.092 | 0.953 | 0.828 | 0.717 | 0.649 | 0.626 | 0.651 | 0.721 | 6.828 |) 0.959 IVETE A oes ee ee 0.984 | 1.028} 1.069} 1.102} 1.124] 1.1381] 1.123] 1.101 | 1.067] 1.026 Constituent 1890 1891 1892 1893 1894 1895 1896 1897 1898 | 1899 Jie 3224 eel ya. 3 1.049 | 1.096 | 1.1382} 1.155} 1.165] 1.162) 1.146} 1.118 | 1 077} 1.026 Gnas Sires 1.028 | 1.062} 1.088] 1.105) 1.112) 1.110} 1.099} 1.078 | 1.054 | 1.012 Kip oot A ee 1,052 | 1.148) 1.230 | 1.289] 1.316] 1.308] 1.267] 1.197 | 1.108} 1.010 Tigsz 2 eke ee nk ee 1,153 | 0.709 | 0.683 | 1.185) 1.219 | 0.704 |) 0.607 | 1.141 | 1.229} 0.897 Min oh oe eee aA 1.323 | 2.091 | 2.158) 1.434 | 1.369 | 2.176} 2.240] 1.471 |) 1.166) 1.781 M:2*, No, 2N, Az, u2, v2-| 0.996 | 0.984 | 0.974 | 0.967 | 0.963 | 0.964} 0.969] 0.978 | 0.989 | 1.001 gis eee pe 0.993 | 0.976} 0.961 | 0.950 | 0.946) 0.947 | 0.954 | 0.967 | 0.983 | 1.002 IM, IMO 0.991 | 0.968 | 0.948 | 0.934 | 0,928} 0.930} 0.939} 0.956 | 0.977 | 1.002 Migr. - Plea Sark ing els 0.987 | 0.952] 0.923] 0.903 | 0.894 | 0.896 | 0.910} 0.934 | 0.966} 1.003 IM gee EE Sea 0.982 | 0.936} 0.898} 0.873 | 0,861 | 0.864] 0.882} 0.913 | 0.955 | 1.004 01, Qu, 2Q, p1.-------- 1.046 | 1.100} 1.142] 1.170} 1.182] 1.179] 1.160 | 1.125} 1.078 | 1.020 One See 1,153 | 1.375 | 1.571] 1.713) 1.780 | 1.761 | 1.660} 1.491 | 1.281 | 1.058 IM Kt ie ae bag oe 1,024 |} 1.045} 1.059] 1.068) 1.072) 1.071 | 1.065] 1.054 | 1.036] 1.013 2 Mike antes 1.019 | 1.028) 1.031 | 1.032} 1.082] 1.032) 1.032} 1.030} 1.025] 1.014 IN Bae epee oper oe ta oe Oe 1,098 | 1.230) 1.339) 1.416 | 1.451 |} 1.441] 1.387) 1.296 | 1.175 | 1.088 Mime sae aie Eee 0.983 | 0.941 | 0.907 |} 0.883 | 0.872 | 0.875 | 0.892] 0.921 | 0.958 | 1.001 Constituent 1900 1901 1902 1903 1904 1905 1906 1907 1908 | 1909 Wit 22S ORS EF Re 5 0.968 | 0.910} 0.861 | 0.832 | 0.829 | 0.854] 0.900] 0.957 | 1.016 | 1.069 Ke Soe 2 ae 0.973 | 0.934) 0.903 | 0.885 | 0.883 | 0.899 | 0.928 | 0.965 | 1.005 | 1.042 UG ee ae Se aes 0.916 | 0.838 | 0.782 | 0.752} 0.750 | 0.774] 0.825 0.900 | 0.992 | 1.090 Dp eee ee oe a 0.753 | 1.080 | 1.193 | 1.117 | 0.925] 0.858) 1.051 | 1.221 | 1.062] 0.653 Me fk ees SE Ee 1,902 | 1.399 | 0.858 | 1.069} 1.507] 1.643 | 1.340 | 0.946 | 1.479 | 2.112 M2*, No, 2N, do, w2, v2-| 1.013 | 1.024} 1.032) 1.037] 1.038} 1.034] 1.026] 1.016 | 1.003 | 0,991 32 bo Sig oy 1.020 | 1.036 | 1.049 | 1.056] 1.057) 1.051 | 1.039 | 1.023 | 1.005] 0.986 Wi IMO Se 1.027 | 1.049) 1.066 | 1.076} 1.076; 1.068} 1.053 | 1.031 | 1.007 | 0.982 IVE GO. 2 ee Ea ae 1,040 | 1.074) 1.101 | 1.115 | 1.117] 1.104) 1.080} 1.047 | 1.010] 0.9738 Migse arcane een 1.054 | 1.100] 1.186] 1.157 | 1.159} 1.142] 1.108] 1.063 | 1.013 | 0.964 01,7Qi, 2Q, pi. -------- 0.956 | 0.893 | 0.842] 0.811 | 0.808 | 0.834] 0.882] 0.944 | 1.008 | 1,068 KO ie eee ees Eee eS 0.850 | 0.679 | 0.559 | 0.496 | 0.490 | 0.543 |} 0.652) 0.814 | 1.017] 1.240 IVER. 2 tee CeO ge 0.986 | 0.957 | 0.933 | 0.918 | 0.916 |} 0.929) 0.952} 0.980 | 1.008 | 1.033 ZIVEKE ee set om eae 0.999 | 0.980} 0.963 | 0.952 | 0.951} 0.960 | 0.977 | 0.996 | 1.012 | 1.623 VES SS Sm 0.901 | 0.779 | 0.686 | 0.634 | 0.680 | 0.673 | 0.75 | 0.877 | 1.012 | 1.151 NST ea SO 1.045 | 1.083 1.128 | 1.130) 1.116] 1.089 |) 1.052] 1.010 | 0.966 *Factor f of MS, 28M, and MSf are each equal to factor f of M2. 1.112 Factor f of Pi, Re, 81, Se, Ss, Ss, T2, Sa, and Ssa are each unity. bs HARMONIC ANALYSIS AND PREDICTION OF TIDES 201 Table 14.—Node factor f for middle of each year, 1850 to 1999—Continued Constituent 1910 | 1911 | 1912 | 1913 | 1914 | 1915 | 1916 | 1917 | 1918 | 1919 TP Be doer GN oe 1.111] 1.142} 1.160] 1.165] 1.157] 1.137] 1.104] 1.059] 1.004] 0.945 ee 1.073 | 1.096 | 1.109] 1.113 1.107 | 1.092] 1.067| 1.035] 0.997 | 0.958 Kaen fee 1.182| 1.256 | 1.303] 1.317| 1.296| 1.243| 1.165| 1.071 | 0.973| 0.884 ipl ae eee 0.834 | 1.246) 1.135] 0.561| 0.729] 1.221| 1.172] 0.761 | 0.780] 1.118 iW rj eine Sipe 1.972 | 1.248 | 1.557] 2.297| 2.146] 1.310] 1.371] 1.993 | 1.909] 1.243 M2*, Nz, 2N, dz, 12, »2-| 0.980 | 0.970] 0.965 | 0.963} 0.966] 0.972] 0.982] 0.993] 1.006] 1.018 My | 9969 | 0.956 | 0.948 | 0.945 | 0.949 | 0.958 | 0.972] 0.990 | 1.009 | 1.027 NTRNEN Pete ee 0.959 | 0.942] 0.931 | 0.928] 0.933 | 0.945] 0.964] 0.987 | 1.012] 1.036 Mths hace eae 0.940} 0.914] 0.898] 0.804] 0.901| 0.918] 0.946] 0.980 | 1.017] 1.054 IViamenebers et ca 0.920} 0.887 0.867] 0.861 | 0.870] 0.893 0.928] 0.973 | 1.023] 1.073 O1; @), 20 sp ee 1.118 | 1.154] 1.176] 1.183] 1.173] 1.148] 1.109] 1.056] 0.995 | 0.931 ONO) = Raoul Matar 1.455 | 1.633 | 1.748 | 1.783 | 1.732] 1.602| 1.414] 1.195 | 0.974| 0.779 1.063 | 1.070] 1.072| 1.069] 1.061] 1.048] 1.028 | 1.003 | 0.975 1.032 | 1.032] 1.032] 1.032] 1.032] 1.028] 1.021 | 1.009| 0.992 1.373 | 1.434 | 1.452 | 1.425] 1.356) 1.252) 1.124 | 0.985] 0.851 0.896 | 0.877 | 0.871] 0.880} 90.902] 0.934} 0.975 | 1.018 | 1.060 Constituent 1920 1921 1922 1923 1924 1925 1926 1927 1928 | 1929 ee ee Sen ek OE aries! 2 0.890 | 0.847 | 0.827] 0.836} 0.870} 0.921 | 0.980} 1.037 | 1.086} 1.125 ee ee ak a ipl ae 0.921 | 0.894] 0.882] 0.887 | 0.909} 0.942] 0.981 | 1.020] 1.055 | 1.083 KG ME renee ce 0.813 | 0.767 | 0.748) 0.756 | 0.791 | 0.852] 0.934 | 1.030} 1.127] 1.214 Tigk soe eee eo eee 1.198 | 1.034} 0.870} 0.932 | 1.183 | 1.199 | 0.963 | 0.669 | 0.975 | 1.270 Mies abe 0.896 | 1.308 | 1.597 | 1.503) 1.082] 0.954) 1.619] 2.063 | 1.739} 1.138 Mo*, No, 2N, Az, we, v2-| 1.028 | 1.035 | 1.038 | 1.036] 1.031 | 1.022] 1.011] 0.998 | 0.986] 0.976 IVER Seis, SE tee 1.042) 1.052) 1.057} 1.055} 1.047} 1.034] 1.016 | 0.998 | 0.979 | 0.964 Wig, VUNG c= oboe oe 1.056 | 1.071 | 1.077) 1.074] 1.063 | 1.045 | 1.022) 0.997 | 0.973 | 0.952 DY ast oe ae 9 RRR Ree ae 1.086 | 1.108 | 1.118] 1.114] 1.096 | 1.068} 1.033] 0.995 | 0.959 | 0.929 I ya Sees ees 1116) 1.146] 1.160} 1.154] 1.180} 1.092] 1 044) 0 994 | 0946] 0 906 01, Qi, 2Q, p1--------- 0.871 | 0.827] 0.806] 0.815 | 0.850] 0.905] 0.968] 1.032] 1.088] 1.134 OO Rarer Ta Clee 0.626 | 0.528 | 0.487 | 0.504] 0.579) 0.710] 0.889] 1.102] 1.325) 1.530 Mikes) © eka ee shee ue 9.947 | 0.925) 0.915 | 0.920} 0.937 | 0.963] 0.992] 1.018] 1.040] 1.056 Mikes fet 0.973 | 0.958 | 0.950} 0.953} 0.966] 0.984] 1.002} 1.017 | 1.026] 1.031 IVE ees be ek 0.739 | 0.660] 0.626] 0.641 | 0.701) 0.801} 0.928 | 1.066 | 1.201] 1.317 Mime oe eae 1,096 | 1.120} 1.131 | 1.126] 1.107 | 1.076 | 1.036] 0.993 | 0.950] 0.914 1930 1931 1932 1933 1934 1935 1936 1937 1938 | 1939 1.150 | 1.163 | 1.164) 1.151 | 1.126] 1.088} 1.038] 0.982 | 0.923] 0.871 1.102} 1.112 | 1.112] 1.102] 1.083] 1.056] 1.021 | 0.982] 0.943 | 0.909 1.278 | 1.312) 1.313] 1.279] 1.216] 1.130] 1.033] 0.987 | 0.8541! 0.792 1.022); 0.471 | 0.873 | 1.270) 1.078] 0.636 | 0.859] 1.190] 1.162] 0.935 1.312 | 2.353 | 1.992) 1.197] 1.614] 2.148 | 1.850] 1.098 | 1.079 | 1.534 0.968 | 0.964 | 0.964 | 0.968} 0.975 | 0.986 0.998 | 1.011 | 1.022] 1.031 0.952 | 0.946) 0.946 | 0.952} 0.963 | 0.979 | 0.997) 1.016 | 1.033 | 1.047 0.937 | 0.929) 0.929} 0.936 | 0.951 | 0.972] 0.996] 1.021 | 1.044] 1.063 0.907 | 0.895 0.895 | 9.906 | 0.928) 0.958} 0.994 | 1.032 | 1.067] 1.096 0.878 | 0.863 | 0.862 | 0.877] 0.905 | 0.945} 0.992] 1.043 | 1.091] 1.130 1.165 | 1.181 | 1.181} 1.165 | 1.184] 1.090] 1.034] 0.970 | 0.907] 0.852 1.686 | 1.772 | 1.773 | 1.690] 1.535] 1.332} 1.108 | 0.894] 0.714] 0.581 1.066 | 1.071 | 1.071} 1.067 | 1.057] 1.041] 1.019} 0.992 | 0.964] 0.938 1.032 | 1.032) 1.032] 1.032] 1.031] 1.026} 1.017] 1.003 | 0.985 | 0.966 1.402 | 1.446 | 1.447 | 1.403 | 1.320] 1.205 | 1.070] 0.931 | 0.804 | 0.704 0.887 | 0.873 | 0.873 | 0.887] 0.913] 0.949 | 0.991 | 1.085 | 1.075] 1.107 ee ee ee eee *Factor f of MS, 25M, and MSf are each equal to factor f of Mo. Factor f of Pi, Re, Si, S2, S4, Se, T2, Sa, and Ssa are each unity. 202 U. 8S. COAST AND GEODETIC SURVEY Table 14.—Node factor f for middle of each year, 1850 to 1999—Continued Constituent 1940 | 1941 1942 | 1943 1944 | 1945 | 1946 | 1947 | 1948 | 1949 ST ghee SCA Aah ta! 0.836 | 0.827 0.846 | 0.888] 0.944] 1.003] 1.057] 1.103] 1.136 | 1.157 Tee GD eae 0.888 | 0.882} 0.894] 0.920] 0.956] 0.996 | 1.034] 1.067] 1.091 | 1.107 epee od, Se RT OF 0.757 | 0.748 | 0.766 | 0.812] 0.882] 0.970] 1.068] 1.162] 1.242] 1.295 Tee ala heel 0.860 | 1.021] 1.180} 1.144] 0.876 | 0.748] 1.091] 1.255 | 0.894! 0.482 iY ee i, a 1.623 | 1.313] 0.879 | 1.076 | 1.714] 1.944] 1.480] 1.138 | 1.927] 2.339 Me*, No, 2N, Ao, #2, ¥2-| 1.036 | 1.038} 1.035] 1.028] 1.018] 1.006] 0.994] 0.982 | 0.972 | 0.966 Nik I 1.055 | 1.057] 1.053 | 1.042] 1.027] 1.009} 0.990 | 0.973 | 0.959 | 0.949 IND aS TN ao 1.074 | 1.077] 1.071 | 1.057] 1.036] 1.012] 0.987] 0.964 | 0.945 | 0.933 11M (Poe see ee Se 1.113 | 1.118] 1.108] 1.086] 1.055] 1.018] 0.981 | 0.947] 0.919 | 0.901 Nghe. ee 1.154} 1.160] 1.147] 1.117] 1.074] 1.025] 0.975 0.929 | 0.804! 0.870 O1, Qi, 2Q, p1--------- 0.816 | 0.806 | 0.826 | 0.870] 0.929] 0.994] 1.055] 1.107] 1.147] 1.173 Of 2 ee Le 0.505 | 0.486 | 0.526} 0.623] 0.774] 0.969} 1.189] 1.408 | 1.598 | 1.729 IMIR cel eddie de 0.920 | 0.915] 0.925] 0.946] 0.974] 1.002] 1.028] 1.047] 1.061 | 1.069 INE: aE Se ed 0.953 | 0.950 | 0.957} 0.973] 0.991] 1.008] 1.021 1.028 | 1.032] 1.032 Nfs eer ee Se 0.642 | 0.626 | 0.659} 0.736] 0.848] 0.981} 1.120] 1.249] 1.354] 1.494 Mints Oe 1 1.126 | 1.131] 1.121] 1.096 | 1.061] 1.019 | 0.976 C.935 | 0.902 0.880 Constituent 1950 1951 1952 | 1953 1954 1955 1956 | 1957 | 1958 | 1959 1.165 | 1.160] 1.143] 1.112] 1.070] 1.002] 0.959] 0.901 | 0.855 | 0.829 1.113 | 1.109} 1.096] 1.074] 1.043] 1.001 | 0.966) 0.929 0.900 | 0.883 1.317] 1.303] 1.257] 1.184] 1.092] 0.995] 0.903] 0.827 | 0.776 | -0. 749 Tigi. ip Soa pitied ee 1.074 | 1.330] 1.014] 0.653 1.001] 1.260] 1.112.] 0.867] 0.915] 1.115 Nip ok = CRU 7 ARS MD 1.717 | 1.120] 1.778] 2.161] 1.664] 0.964] 1.276| 1.656 | 1.527] 1.053 Mz*, No, 2N, dz, 2, »2-| 0.963 | 0.965] 0.970] 0.979} 0.990 | 1.003] 1.015] 1.026 | 1.033] 1.038 Bete ROE AR NER? A .945 | 0.948 | 0.956 | 0.969] 0.986} 1.004] 1.023] 1.039] 1.051] 1.057 Wily NON sooo eee 0.928 | 0.931 | 0.941.] 0.959} 0.981] 1.006] 1.031 | 1.052 | 1.068| 1.076 Mig coral eco sckialane 0.894 | 0.898] 0.914] 0.939] 0.972] 1.009] 1.646] 1.079 | 1.104] 1.116 Regt ee Os 0.861 | 0.867 0.887] 0.920} 0.962] 1.012] 1.062] 1.107] 1.141] 1.158 01, Qu, 2Q, pi--------- 1.183 | 1.177] 1.155] 1.119] 1.069] 1.010] 0.945] 0.884 | 0.835] 0.808 OOS. see URIS 1.784] 1.750! 1.637] 1.459] 1.246] 1.023] 0.819] 0.656 | 0.546 | 0.491 IVa Bh ii a le 1.072} 1.070] 1.063] 1.051] 1.033} 1.009] 0.981] 0.953 | 0.930] 0.916 SND: OR BRE 1.032 | 1.032] 1.032] 1.029] 1.023} 1.012] 0.996] 0.977] 0.960] 0.951 NA fia ese ee 1.452 | 1.435] 1.375] 1.278] 1.154] 1.016] 0.880] 0.761 | 0.675 | 0.630 IN Gite me aes gaa 0.872 | 0.877 | 0.896] 0.926] 0.965} 1.008] 1.051] 1.088 | 1.116] 1.130 Constituent 1960 | 1961 1962 1963 1964 1965 1966 | 1967 | 1968 | 1969 nj iste Spe eae eee ney 0.831 | 0.860] 0.909 0.766] 1.025] 1.076] 1.117] 1.146] 1.161] 1.165 qe ab ee EN ST 0.885 | 0.903] 0.934] 0.972] 1.011] 1.048] 1.077] 1.098] 1.110] 1.113 } 0.781 | 0.836] 0.914] 1.008] 1.106] 1.195] 1.265] 1.307] 1.316 1.081 | 0.849 | 0.893] 1.200] 1.237] 0.838] 0.690 | 1.185} 1.310 1.197] 1.690] 1.699] 1.166] 1.175 | 1.976] 2.175 | 1.503 | 1.197 1.033 | 1.024] 1.014] 1.001 | 0.989] 0.978] 0.969 | 0.964] 0.963 1.049 | 1.037] 1.020] 1.002] 0.983 0.967] 0.954 | 0.947} 0.945 1.066 | 1.050 | 1.027] 1.003] 0.978] 0.956 0.940 | 0.930 | 0.928 1.111.| 1.075 | 1.041] 1.004] 0.967] 0.935.) 0.911 | 0.897 | 0.894 1.137 | 1.102] 1.055] 1.005] 0.956] 0.914] 0.883 | 0.865] 0.861 O1, Qi, 2Q, p1--------- 0.810 | 0.840 | 0.891 | 0.954] 1.018 | 1.076] 1.124] 1.159] 1.178] 1.182 OULU. ose ae 0.495 | 0.557 | 0.675 | 0.845] 1.053 1.276] 1.487] 1.655 | 1.758 | 1.782 NER = ee eee 0.917 | 0.982] 0.956} 0.985] 1.013] 1.036] 1.053] 1.064] 1.071] 1.072 SMe 2. S80 Ve 0.952 | 0.962) 0.980} 0.998] 1.014] 1.024] 1.030] 1.032] 1.032] 1.032 Mifpeccere ct 2) eens 0.633 | 0.684 | 0.776] 0.898] 1.035] 1.172] 1.293] 1.385 | 1.439 | 1.451 Mima Sta er ea 1.128 | 1.113 | 1.084] 1.046 | 1.003] 0.959 | 0.922] 0.893 | 0.876 | 0.872 *Factor f of MS, 28M, and MS&f are each equal to factor f of Mz. Factor f of Pi, Re, 81, S2, Sa, Se, T2, Sa, and Ssa are each unity. HARMONIC ANIALYSIS AND PREDICTION OF TIDES 203 Table 14.—Node factor f for middle of each year, 1850 to 1999—Continued Constituent 1970 1971 1972 1973 1974 1975 1976 1977 | 1978 | 1979 1.1382 | 1.097] 1.051 | 0.995 | 0.936 | 0.881] 0.842 | 0.827] 0.839 1.088 | 1.063] 1.020] 0.991] 0.951] 0.916] 0.891 | 0.882] 0.890 1,232 | 1.150] 1.055] 0.957] 0.871 | 0.804] 0.763 | 0.748 | 0.760 Mg? oes oe ee ele te 0.882 | 0.668 1,118 1. 270 1.014] 0.808 | 0.988 | 1.179 | 1.169] 0.994 IN ee on ee 1.987 | 2.176 | 1.503] 1.012} 1.535] 1.777] 1.428] 0.870] 0.874] 1.361 ae » No, 2N, Az, #2, 72-| 0.966 | 0.973] 0.983 | 0.995] 1.008) 1.020] 1.029] 1.035 | 1.038 | 1.036 ie a ee .950 | 0.960} 0.975 | 0.993 | 1.012] 1.029) 1.044] 1.054 | 1.057 | 1.054 Mz, INN] 222 Wes ee 0.934 } 0.948 | 0.967} 0.991 1.016 1.039 1. 059 1.072 | 1.077 1. 073 INigheneeter ere ee 0.903 | 0.922] 0.951] 0.986] 1.024] 1.060] 1.090] 1.110] 1.118] 1.112 1 a aie a 82 toe a 0.873 | 0.898 | 0.935} 0.981 | 1.032} 1.081 | 1.122] 1.149 | 1.160] 1.151 ae Qu, 2Q, pi--------- 1.170 | 1.143 | 1.101 | 1.047] 0.984] 0.920} 0.863} 0.822 | 0.806] 0.819 Peete nye cee 1.716 | 1.575 | 1.380] 1.159 | 0.940) 0.750] 0.607] 0.517 | 0.485 | 0.512 1 ee Sa hs 1.068 | 1.059 | 1.045 | 1.024] 0.998] 0.970] 0.943] 0.923] 0.915 | 0.922 DNs Se Ae 1.032 | 1.031 | 1.028 | 1.020] 1.006] 0.989} 0.970] 0.956 | 0.950} 0.955 IND PAE oe ee 1.417 | 1.341 | 1.233 | 1.102] 0.962] 0.831 | 0.723 | 0.652 | 0.625] 0.647 Mime ei Bee 0.882 | 0.906] 0.940] 0.982] 1.025] 1.067] 1.100 1.123 | 1.131] 1.124 Constituent 1980 1981 1982 1983 1984 1985 1986 1987 | 1988 | 1989 Tie tes Saperecom 0.877 | 0.930 | 0.989 | 1.045 1.093 | 1.130) 1.153] 1.164 | 1.163) 1.148 LGA ae Mes eee ee 0.913 | 0.948 | 0.987 | 1.026 | 1.060} 1.086 | 1.104] 1.112] 1.111 | 1.100 omenmnc ect eet 0. 799 0. 864 | 0.949 1.045 1,142 1, 226 1, 285 1.315 | 1.310 1. 270 Meowbaee ee 0.848 | 1.001 | 1.238) 1.157 | 0.745 | 0.811) 1.263] 1.244 | 0.749] 0.746 Vipame ee SUES ere 1.656 | 1.468 | 0.974 | 1.323 | 2.050 | 2.032 | 1.292] 1.367 | 2.142] 2.122 M>*, No, 2N, Az, w2, v2-| 1.030 | 1.021 | 1.009 | 0.997] 0.984] 0.974] 0.967] 0.964 | 0.964] 0.969 UE it is. ye. Bs es 1.045 | 1.031 | 1.013 | 0.994] 0.977] 0.962 0.951 | 0.946 | 0.947 | 0.954 Mas IMEN 222 1.061 | 1.042} 1.018] 0.993] 0.969] 0.949] 0.935] 0.928 | 0.930] 0.939 Ig eee ee Gt in. Sh as 1.092 | 1.063 | 1.027] 0.989 | 0.954] 0.924] 0.904] 0.894 | 0.896 | 0.910 Wigs Se. Seeks ae Pes 1.125 | 1.085 | 1.036 | 0.986 | 0.939 | 0.901 | 0.874] 0.862 | 0.864] 0.881 te Qi, 2Q ,p1_-------- 0.858 | 0.915 | 0.979 | 1.041] 1.096] 1.140] 1.168] 1.182] 1.180] 1.161 ee fh A ae ae ae 0.596 | 0.785 | 0.921 | 1.187] 1.361 | 1.560 | 1.706 | 1.778 | 1.766] 1.668 TA cot AR RE) A 0.941 | 0.967 | 0.996 | 1.022] 1.043 | 1.058] 1.068) 1.072] 1.071 | 1.065 2 Ke ea ee 0.969 | 0.987 | 1.005} 1.019} 1.027] 1.031 | 1.032] 1.032 | 1.032] 1.032 IN, Bake A 2 aac 0.715 | 0.820} 0.949} 1.088 | 1.221 | 1.383) 1.412) 1.450 | 1.443] 1.392 IVE Sewer se 1. 103 1.070 1.029 | 0.986 | 0.944 0.909 |} 0.884} 0.872] 0.874] 0.891 Constituent 1990 1991 1992 1993 1994 1995 |- 1996 1997 1998 | 1999 Tig ee Se. er Se 1.120 | 1.080] 1.030] 0.972] 0.914] 0.864] 0.833] 0.829] 0.852 0.896 Ke ee ie are tee BAe 1.079 | 1.051 | 1.015 | 0.976] 0.937] 0.905] 0.886] 0.883 | 0.897] 0.926 IK Seeds alee ee 1.203 | 1.115} 1.016] 0.922] 0.842] 0.785] 0.754! 0.750 | 0.772] 0.821 Ngee ots dea SG he See 1.216 | 1.248] 0.898] 0.801] 1.077] 1.208] 1.107] 0.921 | 0.893] 1.096 IN ee LE ieee trae a 1.334 | 1.156] 1.778 | 1.829] 1.282] 0.800] 1.083 | 1.487] 1.560] 1.214 ee , Na, 2N, Aa, v2, v2-| 0.977 | 0.988} 1.000] 1.013] 1.024] 1.032] 1.037] 1.038 | 1.034] 1.027 oe Se Me Phe epee 0.966 | 0.982; 1.000} 1.019 | 1.036} 1.048] 1.056] 1.057] 1.051 | 1.040 uM VINEE SE 22 es 0.955 | 0.976 | 1.000} 1.025) 1.048} 1.065] 1.075 1.076} 1.069] 1.054 pe ee a ee 0.932 | 0.964] 1.000] 1.038] 1.072] 1.099} 1.115} 1.117] 1.105 | 1.082 Mig oes Se ees 0.911 | 0.52] 1.000] 1.051) 1.€98 | 1.134] 1.156] 1.159] 1.143} 1.111 oe Q1, 2Q, p1--------- 1.128 | 1.081] 1.024] 0.960] 0.897] 0.844] 0.812] 0.808 | 0.832] 0.879 ea pee eo ANS 1.505 | 1.296 | 1.072] 0.863] 0.688] 0,565] 0.498 | 0.489 | 0.538 | 0.643 SIV IR Be Bg EE 1.054 | 1.038] 1.015 | 0.988] 0.959] 0.934] 0.918] 0.916 | 0.928] 0.950 INKS aE aC es 1.030 | 1.025] 1.015] 1.000] 0.982] 0.964] 0.952 0.951 | 0.959 | 0.976 AU AEs. Se oes Se eee 1.303 | 1.184] 1.048 | 0.910] 0.786] 0.691 | 0.636 | 0.629 | 0.669] 0.752 I ap ees See a ee ee 0.918 | 0.956} 0.998} 1.042] 1.081 | 1.110} 1.128] 1.130] 1.117] 1.091 * Factor f of MS, 2SM, and MSf are each equal to factor f of Moe. Factor f of P1, Ry, 81, Se, S4, Ss, T2, Sa, and Ssa are each unity. COAST AND GEODETIC SURVEY Ss. U. 204 EE eee b'10¢ | 6661 | F 00% | 800% | €10Z | 8 66T | £00 | 8002 | E102 | 8°66T | Z 002 | 2002 | 2 10z | 4 ‘66T | 2002 | 2002 | 110% | 9'G6T | 100% | 900% |~~-~~~ 77777777 esg 4086 | 6612 | % 08% | F082 | 408% | 6 622 | Z 08% | F ORS | 9 082 | 6 62Z | 108% | F 08 | 9 08% | 8 6LZ | 1 08Z | 082 | 9 08z | 86Lz | T 082 | E08 [~~~ Bg 8 OE | 0 GES | COST | 919 | 8 BEE | T TEs | EI | 9'ES | 6 HE | Tess | PEL | Loh | BOT | Z'oIz | P:9ZT | 2°28 | O'GOE | Z'20z | BIT | 8°6Z | TT TTT WAL 0'89 | S {rl | Gers | OVE | O'F8 | BEST | 2.09% | 4:0 | GTOT | Z BAT | F 64% | BOG | ECZI | F66I | L008 | OZ | O'SFI | FIZ | B'6IE | 209 [~~ TTT ISIN SOP | LCE | HLL | 8'SBl | F662 | TTS | BS2I | F'2ee | POZE | ELE | S'eel | Eee | Tse | ste | 2-921 | 272s | TOze | Oke | B FET | O'OFZ |--7777 JIN 0°89 | S'ErI | 6'ehs | O'FPE | O'F8 | BEST | 209% | 4:0 | GTOL | BLT | F642 | 8'0Z | Sel | bOI | L008 | Oh | OPI | PIS | BIE | 2:09 |~-~7777TT TTT SZ 0 6S | Z91e | TOIL | O'9T | 0920 | Z 002 | 8°66 | EGE | G'Bc% | 8 181 | 9°08 | Zee | LL | 9091 | EOS | O'RIE | O'LZIZ | 9 °OFL | TOF | 8°66Z |~~~~~777 7 SIN Gers | E861 | 618 | f OSE | 1612 | E°69T | ELS | 6 FOE | O'6I | SOFT | 8'9G | LZ | G'8cT | T°90T | T'zGe | F'8e% | T-GZt | Tbe | 8'tze | 860% |--~-7 me NW 60cG | L'L9 | G¥ee | FOG | L'OLT | €'€% | OSL | J 6EE | T'BET | G'OFE | Z‘9PT | 9°GOE | O'GOT | T'FIs | SEIT | 872% | BTL | P:O8% | 9'BL | L9G [TT MING T°96G | 8°02 | O'FZT | 4°22 | 2162 | 2412 | G'BIL | SBE | F'L2% | G'86T | S'S6 | O'ZGE | Z'8hs | BLOT | 89 | F'1ZE | 261% | 9 IHL | SIF | 808 |-~-7-- MI @ L0G | 2 O1€ | Z'9T | 608 | L'9PT | 26h | GLTE | L9G | E26 | FSO | E222 | 9'GFE | O'%9 | TILT | F'8kZ | ESTE | 9°9Z | Beer | T*E0% | 6020 |~~7 7 id F906 | PCIe | 63S | FG | 8 COT | 6692 | TOPE | OOS | BGIT | 6 Fes) FHA | P'e | HCL | SLATE | G9 | 9'GTE | 1°92 | 9OEL | 9-002 | BOL |~-~ ~~ oa 91900 | PL | 9'ES | STE | OTT | BBE | 4461 | FOS | GST | ST | 86ST | LTE | GSTT | OE | BOIL | 6'LLz | 2°9L | F'e8% | TB | OTH | on 9°L61 | 6666 | £°6G | L'8IT | 1802 | G OTE | 9°6E | G'8ZI | SLI | L'8TE | 6 OF | O'GET | O'€ss | L'H2E | 2:7 | F OFT | 6'Be2 | LOGE | 96S | B°BhL | n eX OO (20 NEO TO OW VAD WO CMe OER TAD iA EO) GES OW AW GW SE OW SW ey ese esessec case aL OO DOD FOO. 100 1 OO OW shOW TOO OW 2 1 ON OO. 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OF ‘eS ST 19 L9 012 02 “81 ° tI 10 'T €9 0 80°24 1@ “81 20 FL TG &F 6L OF TS 93 $0 “861 €& GIS &T SLT 99 ‘OL 69 G6 GG LV 66 00 ‘06 00 ‘09 00 ‘0& 00 “SST €9 ‘0€ IT “291 81 PLT LY F61 T8 602 92 “I8T 49% ol 6 L119 8& ‘0S 69 '8E 61,902 62 OT OF ‘88T 9F “881 18 “& 20 T& £9 “S6T T9 20% ° &T 66 0 6h 0 £99 61 GL SI ST 61 GI 18 “LP 60 “628 &1 “SST 0€ “89T 99 “I9T GT GPE 69 “SEE LY €S§ TS “69¢ 00 0 00 0 00 0 00 ‘08T 6h 0 GZ “PST 84 ‘09T TS “62T L9 “861 GE “LOT bL ‘PEE 86 ThE ve ILE SP “8S 69 SEE TZ T9T 18 “LPE 06 “EZT 96 “ELT ve PSE 66 0 6F O8T €0 “Z81 ° ra yuenjrsu0g penuyu0j—fivp so anoy yava so Gu1uuibaq 07 ST 31902 Jdvpv O}7 SadUaLaYyIG— 8 2B, 218 U. S. COAST AND GEODETIC SURVEY Table 19.—Products for Form 194 [Multiplier=sin 15°=0, 259] 04 ee 010 269 . 528 787 1.046 1. 305 1. 564 1. 823 2. 082 [2.341 O5:<: eee ee 013 272 . d31 790 1.049 1. 308 1. 567 1. 826 2. 085 f2. 344 O63: Se ae 016 275 . 534 793 1. 052 1.311 1. 570 1. 829 2. 088 | 2. 347 Ova ee ee 018 277 . 536 795 | 1.054] 1.313] 1.572] 1.831 | 2.090 2.349 (a eee 021 280 . 539 798 1.057 1.316 1. 575 1. 834 2. 093 2. 352 (\ (i) Sages ie ope aed 023 282 541 800 | 1.059} 1.318] 1.577 | 1.836 | 2.095 2. 354 102 a a ees 026 285 544 803 | 1.062} 1.321 | 1.580} 1.839] 2.098 2 357 Liles eee ee 028 287 . 546 805 1. 064 1. 323 1. 582 1. 841 2.100 2. 359 DI Senta ee apres et ap 031 290 . 549 808 1. 067 1. 326 1.585 |- 1.844 2. 103 2. 362 LS Sees I ae 034 293 . 552 811 1.070 1. 329 1. 588 1. 847 2. 106 2. 365 1: BRS ANB apie 8 036 295 . 554 813 | 1.072} 1.331 1.590 | 1.849 | 2.108 2. 367 Ui Tee eats eet, 039 298 . 557 816 | 1.075] 1.334] 1.593 | 1.852] 2.111 2. 370 1; See nee eee 041 300 . 559 818 | 1.077] 1.336] 1.595] 1.854] 2.113 2. 372 Spe Taree dpe bs = 044 303 562 | .821 | 1.080] 1.339 | 1.598] 1.857] 2.116] 2.375 i Be he eae eS 047 306 565 | .824| 1.083 1.342] 1.601 | 1.860] 2.119] 2.378 LOR ie ee ee 049 308 567 | .826| 1.085] 1.344] 1.603] 1.862] 2.121 2. 380 O11 Seeeenee eee Sr 052 311 570 829 1. 088 1.347 1.606 1. 865 2.124 2. 383 Oe 29 ee a 1. 090 1.349 1. 608 1. 867 2.126 2.385 OD oe ee 1.093 1.352 1.611 1. 870 2.129 2. 388 SOK See eae 1.096 | 1.355 | 1.614 | 1.873 | 2.132 2.391 NOAA 9 be Does 5m) eee 1. 098 1.357 1.616 1.875 2.134 2. 393 8 Bb Be Se She 1.101 1. 360 1.619 | 1.878 | 2.137 2. 396 $06 Sa ee a 1.103 | 1.362 | 1.621 | 1.880] 2.139 2. 398 OT 2 eres arene 1. 106 1. 365 1.624 1. 883 2. 142 2.401 6 As RN care Bie 1.109 | 1.368} 1.627] 1.886] 2.145 2. 404 1! ee ee, ee! 1.111 1.370 1. 629 1. 888 2. 147 2. 406 MOQ Bence Ae aes 1.114 1. 373 1. 632 1.891 2.150 2.409 AO) Oe Seno neers 1.116 1.375 1. 634 1. 893 2. 152 2.411 NOD te ee eee ae 1.119 1. 378 1. 637 1. 896 2.155 2.414 Lb ee eee ee 1.121 1. 380 1. 639 1. 898 2. 157 2.416 AGE es ee =k eS SESS 1.124 1. 383 1.642 1.901 2. 160 2.419 Sh eee eee 1. 127 1. 386 1. 645 1. 904 2. 163 2. 422 RSG Ss 2 sae ee yee ee 1.129 1. 388 1. 647 1. 906 2.165 2.424 AST ae) a ee 1.132 1.391 1. 650 1. 909 2.168 2.427 OG ee ee Be eae 1.134 1.393 1.652 1.911 2.170 2.429 TiC Le peepee ae ey 1. 137 1.396 1. 655 1.914 2.173 2. 432 a4 0A 2 os ee 1.140 } 1.399 | 1.658 | 1.917 | 2.176 2. 435 oy Ue ee eee 1.142 | 1.401 | 1.660 | 1.919} 2.178 2. 437 HAO Ne Ee OO Sas one 1.145 1.404 1. 663 1, 922 2.181 2. 440 Dac |e SOME earn Ce 1.147 1. 406 1. 665 1. 924 2. 183 2. 442 wags. 528 oe 8 1.150 | 1.409 | 1.668 | 1.927 | 2.186 2.445 tho ee Suk SE eee 1.153 | 1.412 | 1.671 |} 1.930 | 2.189: 2.448 AAO Soe Ps Fe 1.155 1.414 1.673 1. 932 2.191 2. 450 Be (ire eae ee See ee es 1.158 1.417 1. 676 1.935 2.194 2.453 Bc Wee ae es See 1. 160 1.419 1. 678 1. 937 2. 196 2.455 pt eee ee oe 1. 163 1. 422 1. 681 1. 940 2.199 2.458 26023. ee 1. 166 1. 424 1. 684 1. 942 2. 202 2. 460 HARMONIC ANALYSIS AND PREDICTION OF TIDES P19 Table 19.—Products for Form 194—Continued {[Multiplier=sin 15°=0. 259] 220 U. S. COAST AND GEODETIC SURVEY Table 19.—Products for Form 194—Continued [Multiplier=sin 30°=0.500] 1 6 7 8 9 (0:00 3). 22 ee 6 ee 0. 000 0. 500 3. 000 3. 500 4. 000 4.500 A) ee he ape ee . 005 . 505 3. 005 | 3.505 | 4.005 4. 505 100 eee hie oF . 010 . 510 3.0:2 , 3.510] 4.010 4.510 109 S52.) Bees es .015 .515 3.015 3. 515 4.015 4.515 A ROR Se are en | . 020 . 520 3.020 | 3.520] 4.020 4. 520 ROD. 2 . 025 . 525 3.025 | 3.525 | 4.025 4. 525 B06 ss. Pe OE . 030 . 530 3.030 | 3.530 | 4.030 4, 530 LOT aa ogee pee gd - 035 - 535 3.035 | 3.535 | 4.035 4. 535 08S Se ae ad . 040 - 640 3.040 | 3.640 | 4.040 4. 540 109 Stee ee .045 . 546 3. 045 3. 545 4.045 4.545 21 Oe 2k ee . 050 - 550 3.050 | 3.550 | 4.050 4, 550 Pilents menaced 055 | 555 3.055 | 3.555| 4.055 | 4.555 OSS ck Magee ae . 060 . 560 3.060 | 3.560] 4.060 4. 560 BS |b nee oe 2) hae - 065 . 565 3. 065 3.565 | 4.065 4.565 Lg SS alias oA eh . 070 - 570 3.070 3.570 4.070 4. 57 MPH Ace Wiis ae al 075 . 575 3.075 | 3.575 | 4.075 4.575 Oy. oo teeeee ee A Oe . 080 . 580 3.080 | 3.580 | 4.080 4.580 A es eee ee eee ey - 085 . 085 3.085 | 3.585 |] 4.085 4. 585 sien Center ge Paes | . 090 . 590 3. 080 3. 590 4.090 4.590 ps erie) Sea eS ee a - 095 . 595 3.095 | 3.598 | 4.095 4. 595 20 ne 2 here . 100 - 600 3. 100 3.600 | 4.100 4.600 Hp) WS er eee . 105 - 605 3.105 3.605 4.105 4.605 DOD seg ye ot .110 . 610 3.110 3.610 4.110 4.610 OSs ene oe ok .115 . 615 3.115 3.615 4.115 4.615 Ae RY ee Ae . 120 - 620 3.120 | 3.620] 4.120 4. 620 BO Dias <= eae at ae - 125 - 625 2, 125 3.625 4.125 4.625 ROG So iE ge ee ae . 1380 . 630 3. 130 3. 630 4.130 4. 630 OA facanetyea eee es oie . 185 - 635 3.135 3.635 | 4.135 4.635 op). pian oe ce eee ey . 140 . 640 3. 140 3. 640 4.140 4.640 2p: een oe elope eee a - 145 - 645 3.145 3.645 4.145 4.645 Pf |) eee a ae . 150 . 650 3. 150 3. 68 4.150 4. 650 es} Uren eee Oe aes . 155 . 655 3. 155 3. 655 4, 155 4.655 PoP Ae fea, see SM Ee Bee . 160 . 660 3. 160 3. 660 4. 160 4. 660 ps ree Re eee ae» . 165 . 665 3. 165 3. 665 4. 165 4. 665 any le tee ee ee . 170 . 670 3.170 3. 670 4.170 4.670 i i) eels ae eee EE Dy .175 . 675 3.175 3.675 4.175 4.675 BOG Se eee ee . 180 . 680 3. 180 3. 680 4.180 4. 680 AS (es eae ae oe ee ee . 185 . 685 3. 185 3. 685 4.185 4. 685 OS re ete pa Ss Be . 190 . 690 3. 180 3. 690 4.190 4. 690 309 Sec ee . 195 . 695 3. 195 3. 695 4.195 4. 695 HV Ra eae ee . 200 . 700 3. 200. 3. 700 4. 200 4. 700 RG heen ee pear i iiss Be . 205 . 705 3. 205 3. 705 4. 205 4.705 Ue LE ae . 210 710 2.210 | 3.710 | 4.210 4.710 F(T a> abe Secret apr odies Bed) .215 . 15 3. 215 3.715 4.215 4.715 AGS = in Ba ya ie aed . 220 . 720 3. 220 3, 720 4, 220 4.720 PAD he Soe eee plea 6225 . 725 3. 225 3. 725 4. 225 4.725 (4G oe 22 aoe See . 230 . 730 3. 230 3. 730 4. 230 4. 730 eAVa5 Sa eee ee ek B20 . 130 3. 235 3. 735 4. 235 4.735 248 5 = Se ae . 740 3. 240 3. 740 4. 240 4.740 iy. 0 ey Pier ee 2 . 745 3. 245 3. 745 4. 245 4.745 Be) Ye ape PP . 750 3. 250 3. 750 4. 250 4.750 ere sh —SEE Sana>>Des-— —=E——E. CSS SST SSS ee 1 6 7 8 9 HARMONIC ANIALYSIS AND PREDICTION OF TIDES 221 Table 19.—Products for Form 194—Continued [Multiplier =sin 30°=0.500} 9 [= aaa i 4,750 1. 4. 755 ik 4.760 il 4. 765 18 4.770 iL 4.775 iL 4. 780 1. 4,785 if 4.790 1. 4.795 ft 4. 800 if 4. 805 1. 4.810 1. 4.815 1. 4. 820 il 4, 825 ik 4. 830 wh 4, 835 ik 4. 840 1. 4.845 ie 4, 850 A 4, 855 i 4. 860 1. 4. 865 ik 4.870 ih 4. 875 il 4. 880 ih 4. 885 1. 4. 890 iL 4,895 ih 4. 900 1. 4. 905 a 4,910 1. 4,915 i 4.920 i 4. 925 ib 4. 930 1. 4.935 1.940 4.940 1.945 4. 945 1.950 4. 950 1. 955 4. 955 1. 960 4, 960 1. 965 4. 965 1.970 4.970 1. 975 4.975 1. 980 4. 980 1. 985 4, 985 1. 990 4. 990 1. 995 4.995 2.000 5. 222 U. S. COAST AND GEODETIC SURVEY Table 19.—Products for Form 194—Continued [Multiplier =sin 45°=0.707] o oO oO a (i=) a _ » — > i) = bo i _—$<$<$<$—$—$— | | ee | ff 007 | .714| 1.421 | 2.128 4.249 | 4.956 | 5.663} 6.370 014} .721 | 1.428] 2.135 4.256 | 4.963 | 5.670 | 6.377 021 728 | 1.435 | 2.142 4,263 | 4.970] 5.677] 6.384 028 | .735 | 1.442] 2.149 4,270 | 4.977 | 5.684 | 6.391 035 | .7421 1.449] 2.156 4.277 | 4.984 | 5.691] 6.398 042| .749!| 1.456] 2.163 4.284 | 4.991] 5.698 | 6.405 Sb Bee N Sue 049 | .756| 1.463] 2.170 4.291 | 4.998] 5.705 | 6.412 it oe foe 057 | .764| 1.471] 2.178 4.299] 5.006| 5.713 | 6.420 ie 2 ee 064 | .771| 1.478 | 2.185 4.306 | 6.013 | 5.720| 6.427 o 1 1 1 ' 1 ' 1 ' 1 i 1‘ 1 1 H oO a = NI I foe) an ~ (ora) ao i} an oO bo 51S agai senieenciey per 078 785 | 1,492 | 2.199 4.320 | 5.027] 5.734 6.441 {OES She ee 085 792 | 1.499 | 2.206 4.327 | 5.0384 | 5.741 6. 448 ns kg 092 799 | 1.506 | 2.218 4.334 | 5.041 | 5.748 6.455 eee eee Se 099 . 806 513 | 2.220 341 | 5.048 |] 5.755 6. 462 Mbt. A eee Te 106 . 813 520 | 2.227 348 | 5.055 | 5. 762 6. 469 5) ae 113 . 820 527 | 2.234 355 | 5.062] 5.769 6.476 »_ is on ~ SSP PPR RA ~ oo Oo LAL APR PRB] Pp on On oo on S) S ro) on Re) a xq > ca RS HARMONIC ANALYSIS AND PREDICTION OF TIDES Depa Table 19.—Products for Form 194—Continued [Multiplier=sin 45°=0.707] a (ee) on re) to on a oOo oo to D> a oo co J oo is a> oo oO ou uO) [o-3} Ni D> oS Ce) -~ for) foe) So — a oe) co or for) S bdo eo (JS) Oo Oo ~ oO = for) a v=) (JC) “I bo D> a tse oo (Jv) or ar we Oo or oo 6i8 | 3.365 | 4.672 665 | 3.372] 4.079 672 | 3.379} 4.086 680 | 3.387] 4.094 687 | 3.394 i c or HS a for) ve) r= oo) os Oo — ee ee ee See _ —_—_— |———<$<— |e“ je je —_ | __ 4 5 6 7 8 8 i) _ bw — ie) lee) nS (e-e) (Je) or Oo or pe bw rs bw ~ Jo) = =) on for) for) w for) ao nf o a i=) 224 U. S. COAST AND GEODETIC SURVEY Table 19.—Products for Form 194—Continued {Multiplier =sin 60°=0.866] m| B85 oo 02 9 Ros 7. Ue 7. 7. 7. 283 7. 292 7. 300 7. 7. 7. 7. suey e200 BRE HARMONIC ANIALYSIS AND PREDICTION OF TIDES 225 Table 19.—Products for Form 194—Continued [Multiplier=sin 60°=0.866] Hee alten i load ell oat ile 226 U. S. COAST AND GEODETIC SURVEY Table 19.—Products for Form 194—Continued {Multiplier=sin 75°=0.966] woes woods g9o9 se _ ~ oO 4.106 | 5.072 | 6.038 | 7.004] 7.970] 8.936 149 | 4.115] 5.081 | 6.047] 7.013| 7.979| 8.945 159 | 4.125] 5.091] 6.057] 7.023] 7.989] 8.955 168 | 4.134} 5.100] 6.066] 7.032] 7.998| 8.964 178 | 4.144] 5.110 | 6.076 | 7.042] 8.008] 98.974 1.256 | 2.222 | 3.188) 4. 6.086 | 7.052 | 8.018 8. 984 1.265 | 2.231) 3.197 | 4. 6.095 | 7.061 | 8.027 8. 993 1.275 | 2.241; 3.207 | 4. 6.105 | 7.071 | 8.037 9. 003 1.285 | 2.251 | 3.217 | 4. 6.115 | 7.081 | 8.047 9.013 1,294 | 2.260 | 3.226] 4. 6.124 | 7.090 | 8.056 9.022 1.304 | 2.270 | 3.236 : 6.184 | 7.100 | 8.066 9. 032 1.314 | 2.280 | 3.246 6.144 | 7.110} 8.076 9, 042 323 | 2.289 | 3.255 6.153 | 7.119 | 8.085 9.051 333 | 2.299 | 3.265 6.163 | 7.129 | 8.095 9. 061 343 | 2.309 | 3.275 6.173 | 7.139 | 8.105 9.071 352 | 2.318 | 3.284 6.182 | 7.148 | 8.114 9. 080 HARMONIC ANALYSIS AND PREDICTION OF TIDES 227 Table 19.—Products for Form 194—Continued [Multiplier=sin 75°=0.966) 0.483 | 1.449 -493 | 1.459 .502 | 1.468 .512 | 1.478 .522 | 1.488 -531 | 1.497 -541 | 1.507 .o51 | 1,517 -560 | 1.526 .570 | 1.536 . 546 . 555 - 565 . 575 . 584 . 594 - 604 - 613 . 623 - 633 a ate 1. 1. 1. 1. 1. 1. 1. 1. etl sell ae sel sell oe lh oo = et ee 228 U. S. COAST AND GEODETIC SURVEY Table 20.—Augmenting factors SHORT-PERIOD CONSTITUENTS,* FORMULA (308) Augment- ing factor Logarithm Remarks Diurnal J;, Ki, M:, 01, OO, Pi, Qi, 2Q, p1 1. 0029 0. 001241 ; Semidurnal Ke, L2, M2, Neo, 2N, Re, Te, 1. 0115 0. 004972 ||Each tabulated solar hourly height do, w2, v2, 28M. used once and once only in summa- Terdiurnal M3, MK, 2MK_______.______-- 1. 0262 0. 011220 tion; group covers one constituent Quarter-diurnal Ms, MN, MS_-_____---_-- 1. 0472 0. 020029 hour; constituent day represented by Sixth-diurnal Mg_...._._____-._---_-_----- 1, 1107 0. 045605 24 means. Eighth-diurnal Ms__....___-..----_-.----- 1. 2092 0. 082498 SHORT-PERIOD CONSTITUENTS,* FORMULA (309) Augment- : Augment- . ing factor Logarithm ing factor Logarithm Remarks 1. 0031 0.00134 || Py._-_-_- 1. 0028 0. 00123 1. 0029 0.00125 |} Qi__-___ 1. 0023 0. 00099 1. 0116 0. 00500 |} 2Q_____- 1. 0021 0. 00091 1. 0112 0. 00482 || Re_____.. 1.0115 0. 00499 1. 0027 0.00116 |} T2_____. 1.0115 0. 00496 1. 0107 0. 00464 |} A2_.-___- 1. 0111 0. 00479 || Each constituent hour of observa- tion period receives one and 1. 0244 0.01047 || po.-_____ 1. 0100 0. 00432 only one of solar hourly heights 1. 0440 0. 01868 || v2g___- -__ 1. 0104 0. 00449 in the summation; group covers 1, 1028 0. 04251 || p1_------- 1. 0028 0. 00100 one solar hour; each constituent day represented by 24 means. 1. 1934 0.07680 || MK_____ 1. 0250 0. 01074 1. 0103 0. 00447 || 2MK____ 1. 0238 0. 01021 1. 0099 0.00430 || MN____ 1. 0431 0. 01833 Ope 1. 0025 0.00107 ||} MS_____ 1. 0456 0. 01935 OOs22 1. 0033 0.00144 || 2SM____ 1. 0123 0. 00532 LONG-PERIOD CONSTITUENTS, FORMULA (403) aie Logarithm Remarks Mm..___._- 1.0050 | 0. 00218 1 ROE Pies 1. 0205 0.00880 ||Daily sums used as units in the summation for the divisional MSf2s.2-<-- 1. 0192 0. 00825 means, and all daily sums used; constituent month for Mm, Mf, Be eta e amen A: quae a eee MSf, and constituent year for Sa and Ssa represented by 24 means. SAsesscsse= . 0115 . 00497 ANNUAL AND SEMIANNUAL CONSTITUENTS, FORMULA (404) Augment- ing factor | Logarithm Remarks aearrs I tae a oe \For analysis of 12 monthly means. *For constituents S;, So, S3, etc., augmenting factor is unity. HARMONIC ANALYSIS AND PREDICTION OF TIDES 229 Table 21.—Acceleration in epoch of K, due to P, [Argument h—}’ refers to beginning of series] 58 days 87 days Ul cats 105 days or) we DOV NOR NNw Neo sat Sh5 bas 86 4Ad +++ 44+ +11 tet +44 134 days | oo Romo NED ON& COO Oro +++ +++ ++ Ps pt) AR MAN HOD WaT aH 163 days It] iS) | Oe Sik tes ONO FOOD KOO AO Orr OPW Miele =f 192 221 250 279 | 297 | 326 days | days | days | days | days | days ° ° ° ° oO ° +0.2 |+2.4 |+3.9 |+3.9 |+3.3 | +1.6 +0.7 |+3.0 |-+4.1 |+3.6 |+2.7 | +0.9 +0.9 |+3.2 |43.7 |+2.8/+1.7| 0.0 +1,0 |+2.9 |+3.0 |4+1.7 |+0.5 | —1.0 +1.0 |+2.4 |+2.0 |+0.3 |-0.8 | —1.7 +0.8 |+1.7 |+0.8 |—-1.0 |-2.0 | —2.0 +0.6 |+0.9 |—0.4 |—2.2 |-2.9 | —-2.1 10.4 |4+0.1 |—1.5 |—3.2 |—3.4 | —1.9 +0.1 |—0.8 |—2.6 |-3.8 |-3.3 | —1.5 =), 1 ISG |e S IES. | =9.0 || iit —0.4 |-2.3 |—4.0 |-3.5 |-2.2 | —0.5 —0.6 |-2.8 |—4.0 |-2.7 |-1.3| 0.0 —0.8 |—3.1 |-3.5 |—1.6 |—0.4 | +0.6 —1.0 |-3.1 |—2.5 |—0.3 |+0.7 | 41.1 1,0 |—2.5 |—1.1 |40.9 |41.6 | 41.6 —0.9 |—1.5 |+0.5 |+2.1 |+2.5 | +1.9 =(),7 |Get JS, Jeb it Ee) |) —0.3 |41.2 |+3.1 |4+3.8 |+3.4 | +2.0 0.2 |+2.4 |+3.9 |+3.9 |+3.3 | +1.6 Table 22.—Ratio of increase in amplitude of K, due to P; ™._ Series 14 29 Hue we days | days ° ° 0] 180 | +6.5 |+11.4 10 | 190 |+13.9 |+16. 4 20 | 200 |+17.9 |+18.3 30 | 210 |+19.0 |+17.6 40 | 220 |+17.6 |+15. 2 50 | 280 |+14.7 |+11.7 60 | 240 )+10.8 | +7.4 70 | 250 | +6.4 | +2.7 80 |} 260 | +1.5 | —2.2 90 | 270 | —3.5 | —6.9 100 | 280 | —8.2 |—11.3 110 | 290 |—12.5 |—14.9 120 | 300 |—16.0 |—17.5 130 | 310 |—18.4 |—18.3 140 | 320 |—18.9 |—16.7 150 | 380 |—16.8 |—12.1 160 | 340 |—11.3 | —4.7 170 | 350 | —2.8 | +3.9 180 | 360 | +6.5 |+11.4 WS stcs| 14 29 ep days | days 0} 180 | —0.31} —0. 26 10 | 190 | —0. 25} —0.17 20 | 200 | —0.15) —0.06 30 | 210 | —0.04| +0. 04 40 | 220 | +0.07) +0. 14 50 | 2380 | +0.17| +0. 23 60 } 240 | +0. 25) +0. 28 70 | 250 | +0.30) +0. 31 80 | 260} +0.33} +0. 32 90} 270 | +0.32} +0. 29 100 | 280 | +0. 28} +0. 23 110 | 290 | +0. 22} +0.15 120 | 300 | +0.13) +0.05 130 | 310 | +0.03] —0.06 140 | 320 | —0.08} —0.16 150 | 330 | —0.19] —0. 25 160 | 340 | —0. 28} —0.30 170: | 350 | —0. 32} —0.31 180 | 360 | —0.31] —0. 26 [Argument h—3y’ refers to beginning of series] 58 days —0. 12 —0.02 +0. 07 +0. 16 +0. 23 +0. 27 +0. 29 +0. 28 +0. 24 +0. 18 +0. 10 +0. 01 —0. 09 —0.17 —0. 23 —0. 26 —0. 25 —0. 19 —0. 12 87 days +0. 01 +0. 09 +0. 16 +0. 20 +0. 23 +0, 24 +40. 22 +0. 18 +0. 12 +0. 04 —0.03 —0. 10 —0. 15 —0. 18 —0. 19 —0.17 —0. 12 —0. 06 +0. 01 105 days +0. 06 +0. 12 +0. 17 +0. 20 +0. 21 +0. 19 +0. 15 +0. 10 +0. 05 —0.02 —0.07 —0. 11 —0. 14 134 days +0. 09 +0. 12 +0. 13 +0. 13 +0. 12 +0. 09 +0. 05 +0. 02 —0.01 —0. 04 —0. 06 —0.07 —0. 07 —0. 06 —0. 04 —0.01 +0. 03 +0. 06 +0. 09 163 days +0. 06 +0. 07 +0. 06 +0. 05 +0. 04 +0. 03 +0. 02 192 221 250 279 297 326 days | days | days} days | days | days +0. 01)—0. 02) 0. 00)/-++0. 03)-++0. 05)-+€. 05 +0.01| 0.00}-+-0. 02/+-0. 06)-+-0. 07)-++0. 06 +0. 02|-++0. 02/--0. 05|-+-0. 08|-+-0. 08]-+0. 07 +0. 02/-++0. 04/-++0. 07)++ 0 +0. 03/-++0. 05|-++0. 08]-+0 -++0. 03} -+-0. 06)-+-0. 09/--0. +0. 04)-++0. 07/-++0. 09)-++0. +0. 04|+-0. 08)-+-0. 09|--0. +0. 04/-++0. 08/-++0. 08)-+-0. +0. 04/-++0. 07/-++0. 06/-+-0. 02 0 +0. 04|-+0. 06/-+-0. 03) 0.00/—0.01! 0. +0. 04/-++0. 04/-++0. 01/—0. 02} —0. 02)—0. +0. 03/-++0. 02) —0. 01/—0. 03}—0. 03} 0. 0. +0.03} 0. 00)—0. 03)—0. 04/—0. 03) 0. +0. 02}—0. 01)—0. 04) —0. 04) —0. 02/-+0. +0. 02] —0. 02) —0. 04) —0. 03) —0. 01)-40. +0. 01]—0. 03) —0. 03) —0. 01/-+-0. 01/-0. +0. 01) —0. 03) —0. 02/ +-0. 01/-+0. 03/-++0 +0. 01)—0. 02) 0. 00/++0. 03/+-0. 05/-++0. S28 ses ses se 230 U. S. COAST AND GEODETIC SURVEY Table 23.—Acceleration in epoch of S, due to K, [Argument h—o”’ refers to beginning of series] Series — 5 29 58 87 105 134 163 192 | 221 | 250 | 279 | 297 | 326 heen PE days | days | days | days | days | days | days | days | days| days | days | days | days —9 ° ° ° ° ° ° ° ° °o ° ° ° ° 0} 180) +3.2 | +65.9 |+10.1 |+10.4 | +8.0] +3.2 | +0.4 | +0.1 |41.3 |+2.9 |4+3.2 |4+2.4/] +0.9 10| 190] +7.2| +9.6 |+12.3 |+10.0 |.+6.7/ +2.0] 0.0] +0.3 |+1.9 /+3.3 |+2.9 |+1.9 | +0.5 20 200 |+10.8 |+12.6 |+13.2 | +8.4 |) +4.7|) —0.6 | —0.5 | +0.5 |+2.4 |+3.3 |+2.2 |41.1 0.0 30 | 210 |+13.7 |+14.6 |+12.5 | +5.7| +2.3) —0.9 | —0.9 | +0.7 |+2.6 |+2.9 |+1.3 |40.3| —0.5 40 220 |+15.4 |+15.0 | +9.9 | +2.5 | —0.4 | —2.2]) —1.3 | 40.8 |42.5 |+2.0 |40.3 |—0.6 | —1.9 50 230 |+15.4 |+13.5 | +5.8 | —1.1] —3.0 | —3.4 | —1.6]| +0.8 }+2.0 |40.9 |—0.8 |—1.4 | —1.3 60 240 |+13.2) +9.6 | +0.8| —4.5| —5.4) —4.4|) —1.7 | +0.7 |+1.2 |—0.4 |—1.8 |—2.1 |] —16 70 250 | +8.6 | +3.7 | —4.4| —7.4| —7.2|) —4.9 | —1.7]} +0.5 |40.1 |—1.6 |—2.6 |—2.6 | —1.7 80 260 } +1.9} —3.0}] —8.8] —9.5 |] —8.3 | —4.9 | —1.3 | +0.2 |—1.0 |—2.6 |}—3.1 |—2.8 | —1.6 90 270 | —5.5 | —9.1 |}—11.9 |—10.4 | —8.3 | —4.2] —0.7 | —0.2 |—1.9 |—3.2 |—3.3 |—2.7 | —1.3 100 | 280 |—11.2 |—13.2 |—13.2 | —9.9 | —7.3 | —2.7 0.0 | —0.5 |—2.4 |—3.4 |—3.0 |—2.2 | —0.7 110 290 |—14.6 |—15.0 |—12.7 | —8.2 | —5.2| —0.8 | +0.8 | —0.7 |—2.6 |—3.1 |—2.3 |—1.4 0.0 120 300 |—15.6 |—14.7 |—10.9 | —5.6 | —2.6 1.2} +1.4 | —0.8 |—2.4 |—2.5 |—1.4 |—-0.4 | 40.8 130 310 |—14.7 |—12.9 | —8.0 | —2.4| 40.5 | +3.1] +1.7 | —0.8 |—2.0 |—1.7 | —0.3)/+0.7 | +1.3 140 320 }—12.4 |—10.0 | —4.4 | +1.0 |} +3.4 | +4.4 | 41.7 | —0.7 |—1.5 |—9.7 | +0.8/41.6 | +1.7 160 330 | —9.1 | —6.4] —0.6 | +4.4| +5.9 | +4.9 | +1.6|) —0.5 |-0.8 0.3 | +1.9)4+2.4 | +1.7 160 340 | —5.3 | —2.3 | +3.3 | +7.3 | +7.7 | +4.8 |] 41.3 | —0.3 |—0.1 |+1.3 |+2.7 |42.8 | +1.6 170 850 | —1.1 |} +1.9} +7.0] +9.4) +8.4] +4.2] +0.9 | —0.1 |+0.7 |+2.2 |4+3.2 |42.8 | +1.3 180 360 | +3.2 | +5.9 |+10.1 |+10.4 | +8.0 | +3.2 |) +0.4 0.1 |-+1.3 |4+2.9 |+3.2 |4+2.4 | +0.9 Table 24.—Ratio of increase in amplitude of S, due to K, [Argument h—v” refers to beginning of series] “Series 29 58 87 105 134 163 192 221 250 279 297 326 days | days | days | days | days | days | days | days | days} days | days } days | days ha” —~N i 0 180) +0. 26) +0. 24) +0. 15) +-0.03) —0.02) —0.04) —0. 01) ++0. 03/++-0. 05}++0. 04/++0.01) 0.00) 0.90 10 190} +0. 23) +0.19} +0.08) —0.03) —0.06) —0.05) —0.01} -++0. 03)-+-0. 04)+-0.02) 0.00)—0. 01)—0. 01 20 200} +0.18} +0.12} 0.00) —0.09) —0. 10) —0.06; —0.01} +-0.03)+-0.03) 0.00) —0. 02|—0. 02|—0. 01 30 210} +0.10} +0.04) —0.08) —0.13) —0.12) —0. 06 0. 00; +0. 02/-+0. 01} —0. 01) —0. 03) —0. 03) —0. 01 40 220) +0.01} —0. 05) —0. 15} —0. 15) —0.13) —0.05 0.00} +0.02) 0.00)/—0. 03/—0.04/—0.03) 0.00 50 230} —0.08) —0. 14) —0. 19) —0. 16) —0.11) —0. 03) +0.01) +0. 02}—0. 01|—0. 04|/—0. 03|—0.02) 0.00 60 240} —0.17| —0. 21) —0. 21) —0. 14) —0.09) —0.01) +0. 02) +0. 01) —0. 02)—0. 04) —0. 02|—0. 01/++0. 01 70 250} —0. 23} —0. 25} —0. 20) —0. 10) —0.05) +0. 02) +0. 03} +0. 01/—0. 03)—0.03|/—0.01) 0.00)-+0. 02 80 260) —0. 27; —0. 25) —0. 16) —0.05 0.00} +-0.05| +0. 04 0. 00) —0. 02} —0. 02) 0. 00}-++0. 02)-++0. 03 90 270| —0. 25) —0. 21} —0. 10} +0.01) +0.05) +0. 08) +-0. 05 0. 00}—0. 01} 0. 00/-++0. 02/--0. 04/-++0. 04 100 280) —0. 20) —0. 15) —0.02) +0.07) +0. 10) +0. 10) +0.05) +0.01] 0. 00/-++0. 02/-+-0. 04/-+-0. 05} 4-0. 05 110 290) —0. 13) —0. 06) +0.05) -+-0.12) +0.14) +0.11) +0.05} +-0. 01]-++0. 01|-++-0. 04/-+-0. 06/-+0. 06/-+-0. 05 120 300} —0. 03) +0.03) +0. 13) +0.17) +0.16) +0.11) +0. 04) ++0. 01)-+0. 03)-+-0. 05}-+0. 07|-++0. 07) -++0. 05 130 310} +0. 06) +0. 11) +0. 18] +0. 19) +0.17} +0. 09) +0. 03) +0. 02)/+0. 04/-+-0. 07|+-0. 08/-+-0. 07/-+0. 04 140 320) +0.14) +0.18) +0. 22) +0.19]) +0. 15! +0.07| +0. 02} +0. 02/-++0. 05}-++-0. 07)-++0. 07| +-0. 06] +0. 03 150 330} +0. 21] +0. 23] +0. 24] +0. 18) +0.13) +0.04/ +0. 01} ++0. 03}-+-0. 06) +-0. 07|-+0. 06) -F0. 05) +0. 02 160} 340] +0. 25} +0. 26) +0. 23) +0. 14) +0. 08) ++0. 01 0. 00} +0. 03}-+0. 06) -++0. 07|-+0. 05/-++-0. 03) -+-0. 01 170} 350) +0.27) +0. 26] +0. 20] +0. 09] +0.03} —0.02) 0.00) +0. 03/-+0. 06/-++-0. 06/-++0. 03)+0. 02) 0.00 180} 360] +0. 26} +0. 24) +0.15] -+0. 03} —0. 02} —0.04) —0. 01] -++0. 03/-++0. 05/+-0. 04)/+0.01) 0.00} 0.00 231. HARMONIC ANALYSIS AND PREDICTION OF TIDES h of S, due to T, ton in epoc Table 25.—Accelerat [Argument h— 7, reters to beginning of series] be es Hen HAN MSO SHH BHD ONT HOR BAM SOHN WHtst HMO AN of S oSS SSS SSS GGee SSS SSS SSS SSS SSS SSS SSS SSS 3,8 OT Tsp aesraP Sear arse Gar aaa a 0 Tit fit God tol yu ~ 2 st ano mNANM hun o~r~ yeyer 6 2 & Goo radi. o>) Host morn oon ole a=) DOr 19 400 SHO SH moO DAN SR | od cidiai Adidd Aa HHS SSH HAG Cidisd Adal AA HSS SSG riria is) | or Pe ttt 001 sae see osraspe srr crim arian DOU ul n oS at Sos) one fo voksn) fo or) SHANDY ANrKN 19 00 © hh) cO1d NI 00 0 SHND roo Ee oN ANA Sd AACN HHS SSS FHM Anes oe ANN Fis SSS FIN o 1 O00 Cb ott 000. USSR seer ser sre serie arime: sel te i tt n us Ont onn Ano co N00 Awe Id AO mid co AN NOOO Dri O mor 1 © OR | ort Halal coses aaa Aad HSS Grint Aelia cseded oA AA ASS Grint ss) Pout ae wd Wo 0a See SSP oer srasse sare saci Ol n fo a} CD00 NI CAIN OD OD OD 100 10 SCwm DOD OH OO rH oD 0 9 NI oro & sH 00 oD (00 ah 0S BHA ANS mod BAN AHS SSS Had Need ced GAM HIS SSS S r fin giv thet wit Pha WSS so seer cere ora sear sel 0 mie s one 1 Sk) Aw cA 400 Hos 00 NAD mor OAN OD HOD ADO AN hos a3 0S SH aidies weeded maa Addn SSS AHA Aaics ossded GAA Adri SSS fo fi Ie I et se gistae Sara Sees sei seers seal 5/8 SRS SBS RAS SSR #3 SEZ ASR RRR RES RRS SHR SEE iB | @ 8S BSB SSF S88 88s sds secs sds SBS SBS SBS Bz n ae S 666 SGna aoe saa See aaa aaa sae sae ads aa SAS bs] r& & 88S SBF SBS SSS SSS S55 SSS SSS S55 S88 S88 888 as SO S6ScSo SSS SSO HAA BAA HAA AAR FAA FAR FRA FHF OSS i} of BQ AVe SVQ SBS SSS SSS sss SSS SSS SSS SSS SSS BBS ag 42 663 666 68S GSS 484 464 Ba&S 3S4 46S 438 G58. 68S a & S) 2h SBRVBV SVS SVVZ SVS SSS sss SSS SSS SSS SSS SSS SES = as S Scddédc SSS SSS SSO SSH HHH BSA BAH FAA BAH FHF FHSS A = ie i= a = (2 SSe~ S55 SES SSS SSS Sss S838 SSS SSS SSS SSS SSS oO 2200 CO be OEE, Be omits Menno TO o>! PERO “SEOSONCy “SO = 3208 Qo ad Smeal Gono no IS) A % As So 60O0O SOO CSO COO COn AAA AN NAR NN ORR RAR RHO mM na _ iw oO Oo. 8 as S$ $32 255 555 SES SSS SSS SSS SSS SSS SES SBS SSE iz e J ag HH 6Sdd6 SSO SSO SSO SOS SHH HAA BAB BAA BAA BHA AAA 3 m § g B » P 2 & ao SSS 255 S88 SSS S55 28S SSS SBS SSS SSS SSS SSS a 2 2 a3 wei HOS SOS SSO SSO SSO SSO HAH HAH AAH AHA AAA AAG Oo £ £ 0 n Nn a 3 Ss xe S 38S S25 588 S88 SSS SERS SSS SSS SSS SSS SSS SSS y, we ge a8 maven SSS SoS SSO SSS SSS SCH HAH BAR BHA AAA BHA q & & = = = a = i 2 S$ 2885 SSR 558 SSS SSS SSR SSS SSS SSS SES SES SSS mM 3 3 a8 aoAade HOS SSO SSO SSO SSS SSO HHH FHA FAH FAH HAA < » § (e) i) q g 3 888 sS8S S58 SSS BSS SSE SBS SSS SS 33 885 8 8 aodadtdn AHS SSO Soo SSO SSS SSO FAH FHA FAA FAH BAA a a ed wD oath Se) we Oo ~ ~ s A Pa 3 BMS ASS SVS HSK SSS SSS SSS SSS SSS SSS SEES SSS 1a) @ g mowada ae SSO SSS SSS SSS SSS SSoH HHH AAA AAA FAA Q oo} © 10 i= Re $6 98s SSN sSRF BSE SSS SSS SSE FBS SSS SSS SSE EES g aodaxiA Ann HHO SSSo SSS SSS SSS SSO BHA FAH FAA FH ive) 10 S$ $83 $88 S85 S85 S88 S33 SSS SES BSS SSS SES SSS alc Rox AAR HAH SSS SSS SSS SSS SSO SHH AA AeA BAe om DOD Jl © 28R SRB BF BBR SSE SE SEL RE REL REZ ESE SEE 232 \ Series \ HARMONIC ANALYSIS AND PREDICTION OF TIDES 233 Table 27.—Critical logarithms for Form 245 Natural| Loga- Natural Loga- Natural Loga- || Natural Loga- Natural | Loga- number | rithm number | rithm number | rithm number | rithm number | rithm 0.000 |_.-------- 0.050 8. 6947 0. 100 8. 9979 0. 150 9, 1747 0. 200 9. 3000 001 6. 6990 051 8. 7033 .101 9. 0022 | 151 9.1776 . 201 9. 3022 . 002 7.1761 . 052 8.7119 . 102 9. 0065 . 152 9. 1805 . 202 9. 3043 . 003 7. 3980 . 053 8. 7202 . 103 9.0108 . 153 9. 1833 . 203 9. 3065 . 004 7, 5441 . 054 8. 7284 . 104 9. 0150 . 154 9. 1862 . 204 9. 3086 .005 7. 6533 . 055 8. 7365 - 105 9. 0192 . 155 9. 1890 . 205 9. 3107 . 006 7. 7404 . 056 8. 7443 . 106 9. 0233 . 156 9.1918 . 206 9.3129 . 007 7. 8130 . 057 8. 7521 . 107 9. 0274 . 157 9, 1946 . 207 9. 3150 . 008 7. 8751 . 058 8. 7597 . 108 9.0315 | 158 9. 1973 . 208 9, 3171 . 009 7.9295 . 059 8. 7672 . 109 9.0355 | . 159 9, 2001 . 209 9. 3192 .010 7.9778 . 060 8.7746 . 110 9. 0395 . 160 9, 2028 . 210 9. 3212 O11 8.0212 . 061 8.7818 iil 9.0434 . 161 9. 2055 , 211 9. 3233 .012 8.0607 . 062 8.7889 .112 9.0473 . 162 9, 2082 . 212 9. 3254 .013 8.0970 . 063 8.7959 .113 9.0512 . 163 9. 2109 . 213 9. 3274 . 014 8. 1304 . 064 8. 8028 .114 9.0551 . 164 9, 2136 . 214 9, 3295 .015 8. 1614 . 065 8. 8096 .115 9. 0589 . 165 9. 2162 .215 9. 3315 .016 8. 1904 . 066 8. 8163 .116 9. 0626 . 166 9, 2189 . 216 9. 3335 017 8.2175 . 067 8. 8229 5 Jst7/ 9. 0664 | . 167 9. 2215 . 217 9. 3355 .018 8. 2431 . 068 8. 8294 .118 9.0701 . 168 9. 2241 . 218 9. 3375 019 8. 2672 . 069 8. 8357 119 9.0738 | . 169 9, 2267 .219 9. 3395 . 020 8. 2901 070 8. 8420 . 120 9.077 .170 9, 2292 . 220 9.3415 .021 8.3118 .071 8. 8482 .121 9.0810 171 9, 2318 . 221 9. 3435 . 022 8.3325 .072 8. 8544 . 122 9. 0846 .172 9, 2343 . 222 9. 3454 . 023 8.3522 .073 8. 8604 . 123 9. 0882 . 173 9. 2368 . 223 9. 3474 . 024 8.3711 . 074 8. 8663 . 124 9. 0917 . 174 9. 2394 . 224 9. 3493 .025 8. 3892 075 8. 8722 . 125 9. 0952 .175 9. 2419 R220) 9. 3513 . 026 8. 4066 .076 8. 8780 . 126 9. 0987 : . 176 9. 2443 . 226 9, 3532 027 8. 4233 077 8. 8837 5 el 9.1021 | 177 9. 2468 APPL 9, 3551 . 028 8. 4394 .078 8. 8894 . 128 9. 1056 .178 9, 2493, . 228 9. 3570 . 029 8. 4549 O79 8. 8949 .129 9. 1090 .179 9, 2517 . 229 9. 3589 030. 8. 4699 . 080 8. 9004 . 130 9. 1123 . 180 9, 2541 . 230 9. 3608 . 031 8. 4843 . 081 8. 9059 . 131 9.1157 . 181 9. 2565 . 231 9. 3627 . 032 8. 4984 . 082 8.9112 . 132 9.1190 . 182 9. 2589 . 232 9. 3646 . 033 8. 5119 . 083 8.9165 . 183 9. 1223 . 183 9. 2613 . 233 9. 3665 - 034 8. 5251 . 084 8.9217 . 134 9. 1255 . 184 9. 2637 . 234 9. 3683 . 035 8.5379 .085 8. 9269 . 135 9. 1288 . 185 9. 2661 . 235 9. 3702 . 036 8. 5503 . 086 8. 9320 . 136 9. 1320 . 186 9. 2684 . 236 9.3720 . 037 8. 5623 . 087 8.9371 . 137 9. 1352 . 187 9. 2707 . 237 9. 3739 . 038 8. 5741 . 088 8.9421 . 138 9. 1384 . 188 9. 2731 . 238 9. 3757 . 039 8. 5855 . 089 8. 9470 . 139 9.1415 189 9, 2754 . 239 9. 3775 . 040 8. 5967 . 090 8.9519 . 140 9. 1446 . 190 9. 2777 . 240 9. 3794 041 8.6075 . 091 8. 9567 . 141 9.1477 . 191 9. 2799 . 241 9. 3812 . 042 8.6181 . 092 8. 9615 . 142 9. 1508 . 192 9. 2822 . 242 9. 3830 . 043 8. 6284 . 093 8. 9662 . 143 9. 1539 . 193 9, 2845 . 243 9. 3848 . 044 8. 6385 . 094 8.9709 . 144 9. 1569 . 194 9. 2867 . 244 9. 3866 045 8. 6484 . 095 8.9755 . 145 9. 1599 . 195 9. 2890 . 245 9. 3883 . 046 8. 6581 . 096 8: 9801 . 146 9. 1629 . 196 9, 2912 . 246 9. 3901 047 8. 6675 . 097 8. 9846 . 147 9. 1659 . 197 9, 2934 . 247 9. 3919 . 048 8. 6767 . 098 8. 9891 . 148 9. 1688 . 198 9. 2956 . 248 9. 3936 049 8. 6858 . 099 8. 9935 . 149 9.1718 . 199 9. 2978 . 249 9. 3954 . 050 8. 6947 . 100 8. 9979 . 150 9.1747 . 200 9. 3000 . 250 9. 3971 > eS 0 Tozsl'0 | 9FF06L°6 ©=—s | eayeog's —«| eesezt'0 —| @r9zb'0. —|tosez9°6 —‘| 0080100 OLLS6I 0 Cages OF Sot eres ae: eS 0 OSFECs T+ | BezLI9';0— | F9ecL0;0— | LTHSP I+ | zBgZ99%+ | IeaIZFO+ | BxIScoI+ | Feeeog T+ | @zeEIE et |---- pce P r9cbat 0 0 eueree 0 | eernoz'0 =| PIseT9's «| g99990°0 | o90Kz0'0 —_—*|,zesTOL's FISEI9'8 GSTLGL GN |, eee ce OS 829 “T— 0 FIGHT G— | 6EETOT— | 690TFO°0— | ZOTGEL I+ | F96980T— | 9OEEDG';0— | G90THOO+ | ErGBGO | -- Ig = 940626 ZLOTEE 0 0 geacez'6 | ezieze’o =| 1199190 | szeoe0'0 ~—|T¥ST2 ‘0 908688 ‘0 ED, ee ee ES seeut9o+ | PUSH c+ 0 cuenes 0+ | ororor'c+ | o1sFez'et+ | rzE80'T+ | Soro T+ | zezosT zt | zetIeL'et | : bz 2 ecizos's esrr0c 0 | B6RCEL'G 0 oozest'o | ozezer'0 =| sesgez6 —| eT 9050°0 IS}912 0 OOS8GE Of ll) ee on Sete aa S yoecl0"0+ — | 6eet09T+ | SzePHe 0- 0 oxzo9s T+ | THrOPL T+ | GzEFPS 0+ | eos T+ | sOReHO T+ | celoBT-et | is) O seyeut-0 viselo’s | eztece'o —‘|oze6r 0 0 e¥61L0'0 | 6F8900°0 —|¢98F99"6 6ESF16'8 CATR ieee eas eee ee Q Libs T— — | eoors00+ | e¥oK0r'z— | oxz099 T— 0 OLtOsT'T+ | g6ssTo'T— | Leczoro— | zeTeso"ot — | zig9¢D OF fo 'd A ertsze-0 eygogo'0 | 1199190 | ozezer0 —_—|sr6tZ0‘0 0 gros 0 | T8¥92 "0 9190400 CUCU: Ore |e sek eee Seer wgc99°2— | zolest'I— | orsrez-s— | teroFLz— | OLT08T‘I— 0 g90get’2— | soreeaT— | eeos6o'T— | sege9g0— fro 00 zi T0989 6 090200 | ezeggo0 §~—- | sesceze —| ors900°0 —‘| erate‘ 0 CHCEPL 6 9190400 TSPSICUOR St aoc on Semen ere Sweeter 0- | voesg0T+ | erusgoT— | szere-0— | gesgt0-T+ | 990061 e+ 0 segess ot | eeoscot+ | gorzea T+ 9 | 10 © 0080100 zeston'6 | tep91z'0 —s| st90%0'0 ~—s| sostog’s =| tetctzo —| cere ‘e 0 BOREL 6 sceoso’0, | Se ee oe ,, _ Se18@0°T— | Soeeos "0+ | Soret | seos6oT— | Leccor'0+ | goFeEOT+ | geese -0— 0 oxen O+ — | 662880 T+ "1N q . 9LLS6T 0 visetg’s | gogses'o ©=—s | isrere'0 —| eecete's —| st90%0°0 —| t90%0'0 —|_gaacen's 0 BORG! Cpe =| ees ae aXe +5 pog69g'T— | @90TFO'0— | zelosT'z— | sorzra'T— | zeIz80°0— | ee0s60'1+ | o860T— | ozer¥s 0- 0 oLenes 0+ 4 nsosze 0 esrloL'6 | Leegeh'0 =| goneee’0 =| eze9ez'6 —s| cere’ —s| isisiz‘0 —*|-sze9e0"0 BOREL °6 Osa [ge ornare = ee eceelL — | eFFSES0— | LeTTezz— | zei9st'z— | cis9zg'0— | seoecoot+ | soreraT— | eps80T- ‘| SLenFg-0- 0 tf == i Ig dz ie) ta 00 10 "NW If i ee ee = 234 SINHOALILSNOD IVNANIG (v—q) 60) pun (v—Q) saduasayip paads quanz1z8sUu0j— gz 29108 L HARMONIC ANALYSIS AND PREDICTION 239 OF TIDES 0 0 STS86E 0 eTgeoe o+ OL6E8F ‘0 289240 €+ OUZE6T “0 022098 “I+ 6S0F20 0 296990 “I+ 678900 0 968S10 I+ 826886 6 628746 0+ O&Zr6Pr 0 TPSOZT “8+ PL601F ‘0 99I9Z9 S+ 628208 ‘0 Z6L1E0 3+ EEFZLT 0 LIPL8Sb I+ SEZ0L6 6 692886 ‘0+ WS GTS86§ 0 STEE0S c— 0 0 868982 °6 GLEFES ‘O+ TEShL6 6 GEOEFE ‘O— €LZ09T ‘0 OSE9FP I— EEFSLT 0 LIPL8h T— TOGFBT “0 P8P8sS “T— OFFOBL 6 8ZCL19 ‘0+ €94698 °8 ¥S8620 ‘0+ TOSEL9 6 IST LP O— 678900 ‘0 968910 T— 9LLS6I “0 $SS69S T— ta OL6E8F 0 289260 ‘€— 86898 6 SLEEPS ‘O— 0 0 EEPSLI 0 LIPL8h T— T10662 0 $¢L066 IT— 648208 0 G6LTEO G— OLSOTE 0 898620 — £94298 '8 498240 0+ TOSELS *6 T@stZp O— 678900 ‘0 968910 T— O0ZEET “0 02¢099 “IT— 060SZE ‘0 626EIT C— orl O0ZE6T 0 022098 T— TESP26 6 ZFOEh6 O+ EEFCLT 0 LIPL8h T+ 0 0 PESIOL 6 80€E0S ‘O— 868982 6 SLEPPS O— €8PL9L 6 Trrsss 0— O0ZEGT 0 0242098 "I++ 6%8900 0 968910 T-+ TOSEL9 6 T2S12b ‘O+ E9498 °8 ¥98620 0— 666962 6 619929 O— KN 6S0%20 0 296990 T— €ZZ09T ‘0 ose9rr I+ T10662 0 $22066 ‘I+ PESIOL 6 80ge09 0+ 0 0 S6FET9 '8 L90T£0 ‘O— 8ISFI6 8 €€1Z80 O— IG9FTE 0 629890 ‘3+ 9T9T8I ‘0 vOZ6IS “I+ 826886 °6 62826 0+ LO6EE9 6 PSFOEPr O+ G2%9060 6 POSES “O— *L 678900 ‘0 826886 6 O&ZF6F ‘0 FLE0IF 0 628208 ‘0 SEFSLT ‘0 £2026 ‘0 968S10 I— | 6Z8F46'0O— | TFSOZI'E— | 99T9LG‘Z— | ZELIEO'Z— | LIFL8PI— | 6GZE86'0— |----~7TT TTT Sz E8FZLE 0 T9ZEST “0 YFF06L 6 E9498 °8 T0S829 6 6F8900 ‘0 9LLS61 ‘0 LIPL8P T+ | P8P8zS I-+ | 8zZZ19'‘O— | FS8cZ0'0O— | I2STZP'‘O+ | 968STO I+ | F99699'I-+ |~~-~-7TTT “" "ta 628208 ‘0 OLS9TE ‘0 €9F798 8 TOSEL9 6 6*8900 ‘0 00ZE6T 0 060828 ‘0 ZELTEO'Z+ | 8S8cZ0'Z+ | FS8cZ0'O— | TZSTZv'O-+ | 968STO I+ | OLZ099 T+ | 6Z68IL' St [~- onl S68SEL 6 €8P292'6 | OOZE6I 0 6#8900 ‘0 T0SE29 6 €9F98 '8 626962 ‘6 GLEFPS ‘O+ | TFFS8S ‘0+ | 022099 'I— | 968SIO'i— | IZSTZP‘O— | PS8cZ0'O+ | ZIS9Z9'O+ | ~ TTT oy E6PE19 °8 SISFI6 ‘8 1Z9PTe 0 QT9T8T ‘0 826886 ‘6 LZ6EE9 °6 GZ9060 ‘6 L90IF0 ‘0+ | 81280 '0+ | 649890 'Z— | FOZBIS I— | 6c8PL6'0— | FSPOSh'O— | FOcECL 0+ |--~~-- TTT smack; 0 E6FE19 “8 6LIEZE ‘0 00ZE6T ‘0 678900 ‘0 TOSEZ9 °6 GESFIG ‘8 0 L90IFO ‘O-+ | SE9FOT'‘Z— | OLZO9S'T— | 968STIO'I— | TzstzhO— | 2ZeI@sO'O-+ |--7 ag L6FE19 8 0 TLSTEee ‘0 €8F40Z 0 620420 ‘0 89L60L 6 GESeI9 ‘8 190140 ‘0— 0 ZILSPI G— | LESTO9"I— | 296990 I— | BBczIg‘O— | TZOIFO';O+ [7 oy 6LIEZE ‘0 TASTES 0 0 S68SEL 6 826980 ‘0 6I0ETZ 0 S0s6eeO = =| ee ChOFOT S+ | SILShI ‘3+ 0 Glens O+ | 6RL880I-+ | PZTSE9 T+ | ZBLZ9BT'St [OT NZ O0ZE6T ‘0 E8FF0Z “0 S68SEL ‘6 0 S68SEL ‘6 826980 ‘0 T8PS1Z ‘0 } 042099 ‘I+ | Zgg109 I+ | SZerrs 0— 0 GLEPPS O+ | GPL880 T+ | 80PZP9 T+ | oN 678900 ‘0 680¥Z0 ‘0 826980 ‘0 868982 °6 0 8689EL ‘6 s190p0'0, | 968910 ‘I+ | 296990 ‘I+ | 6FZ880°I— | SZerPs 0— 0 GLERES ‘O- | G80860 T+ | IN TOSEL9 6 89L60L ‘6 6I0ETZ ‘0 826980 ‘0 8682 °6 0 CICEVLIG fl eo oli IZSTLb O+ | 88SZIS 0+ | FZIEE9'I— | 6PL880'I— | SLerPS ‘0— 0 CHUNSH aye ces see ey ory 6ESF16 8 GECEl9 ‘8 90868 ‘0 I8FS1Z 0 ST90F0 ‘0 ZPCEPL 6 ON eg ee ey ee 1€1Z80 0— | TZ0I#0 ‘0 Z8L98I Z— | 8OFZE9T— | E0860 T— | scggcs o— (sealer ger sees cD | a a NY ea a Pa | a cam [oc ee = zg ro NZ tn aN ay ty “7 ee Ya =e ponunu0)—(v—q) Ho) pun (n—@Q) saduasayip paads 7uan21j8uU0)—'8Z 248 J, SLINGOALILSNOO IVNYOIGINGS 236 U. §. COAST AND GEODETIC SURVEY Table 29.—Elimination factors [ Upper line for each constituent gives the logarithms of the factors; middle line, corresponding natural numbers; lower line, angles in degrees] SERIES 14 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Ji Ki Mi O1 oo Pi Qi 2Q 81 pl Tele swan eee oes 9.7968 .| 8.2015 | 9.3150 | 9.7890 | 9.7203 | 8.3017 | 9.0913 | 9.7607 | 8.1357 sonose . 626 016 . 207 - 615 - 525 . 020 - 123 - 576 014 Con 269 357. 264 93 255 353 261 262 185 9. 7968 |..------ 9. 7968 | 8.3839 | 8.3839 | 9.9958 | 9.3150 | 8.3017 | 9.9990 | 9, 3344 oOPD > Seece8 - 626 . 024 . 024 . 990 . 207 . 020 . 998 . 216 91 eae 269 356 4 346 264 353 353 276 8.2015 | 9.7968 |-.------ 9. 7890 | 9.3150 | 9.8578 | 8.3839 | 9.3150 | 9.8290 | 8.6530 016 8620) enone 615 . 207 721 . 024 . 207 675 045 3 91 =o 267 96 78 356 264 85 188 9. 3150 | 8.3839 | 9.7890 }._------ 8.3826 | 8.7358 | 9.7968 | 8.2015 | 8.1361 | 9.8516 . 207 . 024 oO |) esence . 024 . 054 . 626 -016 | .014 711 96 4 93 cous 9 171 269 357 178: 281 9. 7890 | 8.3839 | 9.3150 | 8.3826 |-.------ 8.9571 | 9.0878 | 8.3320 | 8.7710 | 9.1065 . 615 . 024 . 207 0242 4]\- 22 52-- . 091 . 122 021 059 . . 128 267 356 264 351 Sooe 342 260 348 349 272 9. 7203 | 9.9958 | 9.8578 | 8.7358 | 8.9571 |__------ 9. 3355 | 8.2581 | 9.9990 | 9.3331 . 525 . 990 721 . 054 ROOTS ee === 217 018 . 998 . 215 105 14 282 189 18 santa 278 186 7 290 8.3017 | 9.3150 | 8.3839 | 9.7968 | 9.0878 | 9.3355 |_------- 9. 7968 | 9.3283 | 9.9967 . 020 . 207 . 024 . 626 , 122 ole ese - 626 . 213 . 992 7 96 4 91 100 82 sore 269 89 12 9.0913 | 8.3017 | 9.3150 | 8.2015 | 8.3320 | 8.2581 | 9.7968 |-------- 7.1244 | 9.7298 . 123 . 020 . 207 . 016 .021 . 018 0200) | teen ne 001 . 037 99 7 96 3 12 174 91 Bec 0 104 9.7607 | 9.9990 | 9.8290 | 8.1361 | 8.7710 | 9.9990 | 9.3283 | 7. 1244 |__-____- 9. 3369 . 576 . 998 . 675 - 014 . 059 . 998 . 213 OO Ls | eens . 217 98 7 275 182 11 353 271 0 mee 283 8. 1357 | 9.3344 | 8.6530 | 9.8516 | 9.1065 | 9.3331 | 9.9967 | 9.7298 | 9.3369 |-------- 014 . 216 - 045 711 - 128 215 . 992 . 537 Bea We Seeas 175 84 172 79 88 70 348 256 77 ft: HARMONIC ANALYSIS AND PREDICTION OF TIDES 237 Table 29.—Elimination factors—Continued SERIES 15 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constituent ed sought (A) L2 M2 N2 2N Ro S2 Ts r2 ya v2 28M. KG eee 8 ie |b oe: 9. 7534 |8. 9437 |9. 2424 |8.9063 |9.9986 |9.9950 |9. 9892 |9. 6707 |8. 7223 |9. 2966 |8. 8476 . 567 088 |..175 . 081 . 997 . 989 . 975 . 468 . 053 . 198 . 070 260 342 244 326 353 345 338 247 339 257 168 get ON 004 ls ce 9.7627 |8.9055 |9. 2507 |9. 7927 |9.8276 |9. 8585 |9. 9961 |9. 3018 |8. 1941 |9. 3301 pees . 579 . 080 .178 .620 | .672 . 722 . 991 . 200 016 . 214 as 262 344 246 92 85 77 347 259 357 88 Mie ee eee 9.7627 |______ 9. 7627 |8.9055 |8. 7291 |8. 1941 |8. 4114 |9. 8276 |8. 1941 |9. 8276 |8. 1935 S019 | 22 sce . 579 . 080 . 054 . 016 . 026 . 672 . 016 672 016 98 ae 262 344 10 3 175 85 357 275 6 IN ee od 8.9055 |9. 7627 |______ 9. 7627 |9. 2760 |9.3018 |9, 3204 |8. 1941 |9.8276 |9. 9961 |9. 0793 . 080 NOUS Mm bees = . 579 . 189 . 200 . 209 . 016 . 672 991 . 120 16 98 ae 262 108 101 93 3 275 13 104 DINE eae oh een 9. 2507 |8.9055 |9. 7627 |______ 8.8157 |8. 6888 |8. 4856 19.3018 |9.9961 |9. 6823 18. 5765 178 . 080 DIG |) eke . 066 . 049 . 031 . 200 . 991 481 - 038 114 16 98 sa 26 19 11 101 13 111 22 Ronen Pete 9. 7927 |8. 7291 |9. 2760 .|8. 8167 |______ 9. 9987 19.9950 |9. 7195 |8. 5420 |9.3168 |8. 4114 . 620 . 054 . 189 . 066 | _-_.- . 997 . 989 . 624 . 035 207 . 026 268 350 252 334 228 353 345 255 347 265 175 Soseeoc oe eee ce b 9. 8276 |8. 1941 |9. 3018 |8. 6888 |9.9987 |______ 9.9987 |9. 7627 |8. 1935 |9. 3301 |8. 1941 F . 672 . 016 . 200 . 049 907 | ee = . 997 . 579 . 016 214 . 016 275 357 259 341 7 ae 353 262 354 272 3 ee Sees 9.8585 |8. 4114 9. 3204 |8. 4856 |9.9950 |9.9987 |______ 9.8010 |7. 6684 |9. 3364 |8. 7291 722 | .026 | . 209 - 031 .989 | .997 | _____ 632 | .005 217 . 054 283 185 267 349 15 ai ae 269 182 280 10 tN Pee eee 9. 9961 |9.8276 |8. 1941 19.3018 |9. 7195 |9. 7627 19.8010 |_-____ 9. 3301 |8.7786 |9. 3018 991 672 016 . 200 524 579 632). 214 . 060 200 13 275 357 259 105 98 91 Ls 272 190 101 PSecaccsse 9. 3018 |8.9141 |9.8276 |9. 9961 |8. 5420 |8.1935 |7. 6684 |9.3301 |__.__- 7627 |8. 1926 .200 | .016 | .672 | .991 .0385 | .016 | .005 | .214 | ____. 579 | .016 101 3 85 347 13 6 178 88 mere 98 9 Vissse ess eeess 8. 1941 19.8276 19.9961 |9. 6823 |9. 3168 |9.3301 [9.3364 |8.7786 |9. 7627 |___-__ 9. 1043 .016 | .672 | .991 - 481 -207 | .214 | .217 -060 | .579 | -_.-- . 127 3 85 347 249 95 88 80 170 262 oe 91 2S Maes Aes 9. 3301 |8. 1935 |9. 0793 |8. 5765 |8. 4114 |8. 1941 |8. 7291 |9. 3018 |8. 1926 |9. 1043 | ____. . 214 . 016 . 120 - 038 . 026 . 016 . 054 . 200 016 127 Secs 272 | 354 | 256| 338| 185| 357] 350] 259| 361 238 U. §. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 29 DAYS. DIURNAL CONSTITUENTS CoOOOoOOOOOOOOoooeeeooooeel_eeeae*q®qoqoqooqqqqqq«q=— ae Disturbing constituents (B, C, etc.) eee sought; |e ai Ji Ki Mi O1 oo Pi Qi 2Q 81 pi | Roa ecmsSeeeceoHoos | esaect 8.6955 | 8.6896 | 8.7199 | 8.8144 | 9.2092 | 8.6937 | 8.6672 | 9.0538 | 8.3224 tees . 050 . 049 - 052 . 065 . 162 . 049 - 046 113. . 021 Mi 351 341 328 13 322 319 310 336 344 Kees eae oon ce 836950") eese== 8.6955 | 8.7517 | 8.7517 | 9.9818 | 8.7199 | 8.6937 | 9.9954 | 8.0542 3050), |) S2e22 . 050 . 056 . 056 -959 . 052 . 049 . 990 O11 9 ee 351 338 22 331 328 319 346 354 Mite sat sito Ser Sees 8.6896 | 8.6955 | ------ 8.8144 | 8.7199 | 9.0674 | 8.7517 | 8.7199 | 8.4418 7.9579 . 049 O50) | eeesee . 065 . 052 .117 . 056 . 052 . 028 .009 19 9 jae 347 32 161 338 328 175 183 Opt Ses, 2s 8.7199 | 8.7517 | 8.8144 | ____-. 8.7185 | 8.2616 | 8.6955 | 8.6896 | 8.3262 | 8.9810 . 052 . 056 BOGOR | meeeee . 052 .018 - 050 . 049 - 021 . 096 32 22 13 uss 44 174 351 341 8 196 OOL2-2 aR AD, SIS 8.8144 | 8.7517 | 8.7199 | 8.7185 | _-__-- 9.0332 | 8.6848 | 8.6504 | 8.9334 | 8.4666 . 065 . 056 . 052 A052 9e\> a 2e - 108 . 048 .045 . 086 -029 347 338 328 316 =i=5 309 306 297 324 332 Pieeeee eee ee 9.2092 | 9.9818 | 9.0674 | 8.2616 | 9.03382 | ---_-_- 7.7378 | 8.2260 | 9.9954 8. 6248 . 162 - 959 -117 -018 LOSS) ». S2ee2 -005 -017 . 990 . 042 38 29 199 186 51 pt 357 348 14 202 Qe: Ces 8.6937 | 8.7199 | 8.7517 | 8.6955 | 8.6848 | 7.7378 | _----- 8.6955 | 8.4846 | 9.9857 . 049 . 052 . 056 . 050 . 048 SO05E5)) > Sees . 050 - 031 . 968 41 32 22 9 54 3 zee 351 17 25 2QY NEL Aaee 0k ate 8. 6672 | 8.6937 | 8.7199 | 8.6896 | 8.6504 | 8.2260 | 8.6955 | _._._. 8.5377 | 9.1825 . 046 . 049 - 052 . 049 - 045 . 017 O50R |" esas . 084 . 152 50 41 32 19 63 12 9 bape 27 35 Sprse ses eset Be 9.0538 | 9.9954 | 8.4418 | 8.3262 | 8.9334 | 9.9954 | 8.4846 | 8.5377 | _----- 8. 1807 Si . 990 . 028 .021 . 086 .990 . 031 OBE) cee O15 24 14 185 352 36 346 343 333 pees 188 Ose cso2ese ek --| 8.3224 | 8.0542 | 7.9579 | 8.9810 | 8.4666 | 8.6248 | 9.9857 | 9.1825 | 8.1807 eccces) . 021 - O11 . 009 . 096 . 029 . 042 . 968 . 152 .015 ecoce 16 6 177 164 28 158 335 325 172 wees HARMONIC ANALYSIS AND PREDICTION OF TIDES 239 Table 29.—Elimination factors—Continued SERIES 29 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constituent sought (A) Ke Le M2 Na 2N Re 82 To re pe ve | 28M sia a Seach en 8.8144 |8.7517 [8.7199 |8. 6937 |9.9954 |9.9818 19.9587 |9. 2092 |8.3224 |8.0542 19. 0054 ase . 065 - 056 - 052 - 049 . 990 . 959 . 909 . 162 - 021 O11 - 101 ee: 347 338 328 319 346 331 317 322 344 354 145 0 pe tt Weed 8.8144 |______ 8.6955 |8. 6896 |8.6798 |7.9581 |8.9810 |9. 2842 /9. 9857 17.7378 |8. 2616 |8. 6248 oli) ose 050 | .049 | .048 | .009 | .096 | .192 | .968. | .005 | .018 | .042 TO Petes 351 341 332 178 164 150 335 357 186 158 Ninkess soe ee 8.7517 |8.6955 |------ 8.6955 |8. 6896 |8. 3262 |8. 2616 S fee 8.9810 |8. 2616 |8. 9810 |8. 2588 E0560 eu Cb0mn |ieesa= -050 | .049 | .021 | .018 096 | .018 | .096 | .018 22 9,0 ee 351 341 8 174 PED 164 186 196 167 Ngee Cs ewe ce 8.7199 |8. 6896 |8. 6955 |-..--- 8.6955 |8. 4846 |7. 7378 |8.3278 |8. 2616 {8.9810 |9. 9857 |7 5900 O52 049 050 tenans -050 | .031 | .005 | .021 | .018 | .096 | .968 | .004 32 19 9 aie 351 17 3 169 174 196 25 177 DNs eee» 8. 6937 |8. 6798 |8. 6896 |8.6955 |______ 8. 5377 |8. 2260 |7.4179 |7. 7378 |9.9857 |9. 1825 |7. 7379 .049 | .048 | .049 | .050 | _____ 034 | .017 | .003 | .005 | .968 | .152 | .005 41 28 19 9 Soa 27 12 178 3 25 35 6 Flom wee ones re 9.9954 17.9581 |8.3262 |8. 4846 {8.5377 |______ 9. 9954 |9.9818 9.0538 |7. 2754 |8. 1807 |8. 7772 NOOO | PSO0ONT i O21 03 0345. |eoeut OOO ee |e GOO ee rail sient OO2ee e015 an O60 14 182 352 343 B83 |) anos 346 331 336 359 188 159 Soe shee a 9.9818 |8.9810 |8. 2616 |7. 7378 18. 2260 |9. 9954 |______ 9.9954 |8. 6955 |8. 2588 |8. 6248 |8. 2616 ~ 959) |.096) | .018 | .005) 017 12990" | 22222 -990 | .050 | .018 | .042 |-.018 29 196 186 357 348 14 ee 346 351 193 202 174 Ut pA eta 9.9587 |9. 2842 |8.7772 |8.3278 |7,4179 19.9818 |9. 9954 |______ 8.4418 |8.5780 |8. 8324 18. 3262 .909 | .192 | .060 | .021 | .003 | .959 | .990 | _____ 028 | .088 | .068 | .021 43 210 201 191 182 29 14 Same 185 207 217 8 Ng ean ae ae 2 9. 2092 |9.9857 |8. 9810 |8. 2616 |7. 7378 |9.0538 |8. 6955 |8. 4418 |_.____ 8. 6248 18. 9640 17. 7378 516205968) 3) S096) 1} 2018) 12005) 118% 0502. 0282 | .042 | .092 | .005 38 25 196 186 357 24 9 175 woe 202 212 3 (TF eee ph aac 8. 3224 |7. 7378 |8. 2616 |8.9810 |9.9857 |7. 2754 18. 2588 |8.5780 /8.6248 |_____. 8. 6955 |8. 2539 021 . 005 . 018 . 096 . 968 . 002 - 018 . 038 S042 EG ee o! . 050 018 16 3 174 164 335 1 167 153 158 ein 9 161 Wate oe ee 8.0542 |8. 2616 |8. 9810 |9.9857 |9. 1825 |8. 1807 |8. 6248 |8.8324 18.9640 18.6955 |______ 8. 5015 O11 | .018 | .096 | .968 | 152 | .015 |..042 | .068 |..092 | .050 |._____ - 032 6 174 164 335 325 172 158 143 148 351 ee 151 25 Metin ns22)- 9.0054 |8. 6248 |8. 2588 |7.5900 |7.7379 |8. 7772 |8. 2616 |8.3262 |7. 7378 |8. 2589 |8. 5015 |_____. -101_ | .042 | .018 | .004 | .005 | .060 | .018 | .021 | .005 | .018 | .032 | _____ 215 202 193 183 354 201 186 352 357 199 209 eee 240 U. S. COAST AND GEODETIC SURVHx Table 29.—Elimination factors—Continued SERIES 68 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C. etc.) Soe cay sought qi Ki Mi O1 Ooo Pi Qi 2Q 81 pl BFE ee ee ee eee 8.6896 | 8.6657 "| 8.6504 | 8.8039 | 9.1056 | 8.5715 | 8.4713 | 9.0154 | 8.3059 soaes . 049 . 046 . 045 . 064 . 128 . 037 . 030 . 104 . 020 eae 341 322 297 25 284 278 259 313 329 Kegpiae sed eee 8.6896 | -..--- 8.6896 | 8.7185 | 8.7185 | 9.9254 | 8.6504 | 8.5715 | 9.9818 | 8.0520 6049 |b Tess= = . 049 . 052 . 052 . 842 . 045 . 037 . 959 O11 19 a 341 316 44 303 297 278 331 348 1 eee Se ecsese eos 8.6657 | 8.6896 | ------ 8.8039 | 8.6504 | 9.0427 | 8.7185 | 8.6504 | 8. 4403 7. 9572 . 046 #04900 |) Beste=e . 064 045 .110 . 052 . 045 . 028 . 009 38 19 aoe 335 63 142 316 297 170 186 QOysioce eee et ee 8.6504 | 8.7185 | 8.8039 | ------ 8.5737 | 8.2588 | 8.6896 | 8.6657 | 8.3224 | 8.9640 . 045 . 052 AO0G435) | gee = 2 . 037 .018 . 049 . 046 .021 . 092 63 44 25 ae 88 167 341 322 16 212 OOe ee ae ee 8.8039 | 8.7185 | 8.6504 | 8.5737 | --_--- 8.8349 | 8.4575 | 8.3057 | 8.8391 8. 4112 . 064 . 052 . 045 SUB You ee . 068 . 029 . 020 . 069 . 026 335 316 297 272 ase 259 253 234 287 303 Pyesnst sees es Sk 9.1056 | 9.9254 | 9.0427 | 8.2588 | 8.8349 | ----_- 7.7379 | 8.2155 | 9.9818 | 8.5907 . 128 . 842 .110 .018 O68 wa ls 2-2" . 005 . 016 - 959 . 039 76 57 218 193 101 ws 354 335 29 225 Qype sk ee A 8.5715 | 8.6504 | 8.7185 | 8.6896 | 8.4575 | 7.7379 | ------ 8.6896 | 8.4645 | 9.9418 . 037 . 045 . 052 . 049 . 029 SOOO aoe . 049 . 029 . 875 82 63 44 19 107 6 a4 341 35 61 QQ ees oeoee ee Sent 8.4713 | 8.5715 | 8.6504 | 8.6657 | 8.3057 | 8.2155 | 8.6896 | ------ 8.4887 | 9.0969 . 030 . 037 - 045 . 046 . 020 . 016 i049ee | fs eee - 031 - 125 101 82 63 38 126 25 19 ae 53 70 Sietext Ae es see 9.0154 | 9.9818 | 8.4403 | 8.3224 | 8.8391 | 9.9818 | 8.4645 | 8.4887 | ------ 8.1761 . 104 . 959 . 028 . 021 . 069 .959 . 029 O38t |i eee .015 47 29 190 344 73 331 325 307 pei 196 PIS eee 8.3059 | 8.0520 | 7.9572 | 8.9640 | 8.4112 | 8.5907 | 9.9418 | 9.0969 | 8.1761 | ------ . 020 .O11 . 009 .092 . 026 . 039 . 875 . 125 sOLG ne eees 34 12 174 148 57 135 309 290 164 Zoe a HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 29.—Elimination factors—Continued SERIES 58 DAYS. SEMIDIURNAL CONSTITUENTS 241 Constituent sought (A) 8. 7185 . 052 9. 9818 . 959 29 Sole eo dues 8 9, 9254 25 Vinee Saas 8. 9185 . 083 250 - 875 Disturbing constituents (B, C, etc.) M2 No 2N Re 8.7185 |8. 6504 |8. 5715 |9. 9818 . 052 - 045 . 037 - 959 316 297 27 331 8. 6896 |8. 6657 |8. 6244 |7. 9579 ,, 049 . 046 . 042 . 009 341 322 303 177 2 ae 8. 6896 |8. 6657 |8. 3224 ee . 049 . 046 . 021 one 341 322 16 8.6896 |_..._- 8. 6896 |8. 4645 049 | ---_- . 049 . 029 19 Les 341 35 8. 6657 |8.6896 |_-__-- 8. 4887 046 .049 | _____ . 031 38 19 vee 53 8.3224 |8. 4645 |8. 4887 |___-__ . 021 . 029. SOS eee 344 325 307 eee 8. 2588 17.7379 |8. 2155 |9. 9818 .018 005 ‘ - 95 193 354 305 8.7480 |8.3193 |7.4165 |9. 9254 . 056 . 021 . 003 . 842 222 203 184 57 8.9640 |8. 2588 |7. 7379 |9. 0154 . 092 .018 . 005 - 104 212 193 354 47 8. 2588 |8. 9640 |9. 9418 |7. 2736 . 018 . 092 . 875 . 002 167 148 309 3 8.9640 |9.9418 |9.0969 |8. 1761 . 092 . 875 - 125 . 015 148 309 290 164 8. 2475 |7. 5898 |7. 7356 |8. 7480 -018 . 004 . 005 . 056 206 187 348 222 S2 T. 2 pa v2 28M. 9, 9254 |9. 8237 |9.1056 |8. 3059 |8.0520 |8. 9185 .842 | .666. | .128 | .020 . O11 . 083 303 274 284 329 348 110 8.9640 |9. 2209 |9.9418 |7. 7379 |8. 2588 |8. 5907 . 092 . 166 . 875 . 005 . 018 . 039 148 120 308 354 193 135 8. 2588 |8. 7480 |8.9640 |8. 2588 |8.9640 |8. 2475 .018 | .056 092 | .018 .092 | .018 167 138 148 193 212 154 7. 7379 |8. 3193 |8. 2588 |8.9640 |9.9418 |7. 5898 . 005 . 021 . 018 . 092 - 875 . 004 6 157 167 212 61 173 8. 2155 |7.4165 |7. 7379 |9.9418 |9. 0969 |7. 7356 . 016 . 003 . 005 . 875 . 125 - 005 25 176 6 61 70 12 9.9818 |9.9254 |9.0154 |7. 2736 |8.1761 |8. 7480 . 959 . 842 - 104 . 002 . 015 . 056 331 303 313 357 196 138 pace 9.9818 |8.6896 |8. 2475 |8. 5907 |8. 2588 pase . 959 . 049 . 018 . 039 - 018 We 331 341 206 225 167 9.9818 |_.-.__ 8. 4402 |8. 5270 |8. 7366 |8. 3224 R959) {I eee .028 | .0384 | .055 . 021 29 Be 190 234 253 16 8.6896 |8.4402 |_____- 8. 5907 |8. 8933 |7. 7379 -049 | .028 | ___-_ 039 | .078 005 19 170 ae 225 244 6 8. 2475 18.5270 |8. 5907 |___--- 8. 6896 |8. 2286 . 018 . 034 SO0d9Re ete . 049 017 154 126 135 ae 19 141 8. 5907 |8. 7366 |8. 8933 |8. 6896 |_...__ 8, 4439 . 039 055 - 078 049 | .LL . 028 135 107 116 341 oa 122 8. 2588 |8.3224 |7. 7379 |8. 2286 |8. 4439 |_____- .018 . 021 . 005 017 O28) ieee 2 193 344 354 219 238 Soe 242 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 87 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constiuent sought ee at ea Vi Ky Mi O1 00 Pi Q1 2Q $i pl Jee ee Se Bh AAS | ee 8.6798 | 8.6244 | 8.5225 | 8.7857 | 8.9030 | 8.3232 | 7.9841 | 8.9481 8. 2780 ee . 048 . 042 - 033 - 061 . 080 . 021 . 010 . 089 . 019 Be. 2 332 303 265 38 246 237 209 289 313 DI ae ene ene eam 8567980] 8.6798 | 8.6607 | 8.6607 | 9.8237 | 8.5225 | 8.3232 | 9.9587 8. 0476 “0485 lie eas . 048 . 046 . 046 . 666 . 033 . 021 - 909 O11 28 ieee 332 294 66 274 265 237 317 341 Mee ee AE 8.6244 | 8.6798 | __._- 8.7857 | 8.5225 | 9.0002 | 8.6607 | 8.5225 | 8.4376 | 7.9556 . 042 #048. {isha . O61 . 033 . 100 . 046 . 033 . 027 - 009 57 28 ae 322 95 123 294 265 165 190 Opts Et Beek 4 ee: 8.5225 | 8.6607 | 8.7857 | _.___- 8. 2641 | 8.2539 | 8.6798 | 8.6244 | 8.3155 8.9351 . 033 . 046 BUGIS mee a .018 .018 . 048 . 042 . 021 - 086 95 66 38 Dake 133 161 332 303 23 228 OO) 2. 2282 22 - 2252.|) 8.7857 || 8.6607 |\8.'5225 | 8.2641 | -._-_- 8.3377 | 7.8138 | 7.4337 | 8.6579 8.3116 - 061 . 046 - 033 OLS Loa eseee . 022 . 007 . 003 . 045 . 020 322 294 265 227 aaek 208 199 351 251 275 JR teee: pute ot ee 8.9030 | 9.8237 | 9.0002 | 8.2539 | 8.3377 | _____- 7.7367 | 8,1982 | 9.9587 8. 5315 . 080 . 666 . 100 018 O22 ee oa . 005 . 016 . 909 . 034 114 86 237 199 152 233 351 323 43 247 Oi ae ee 8.3232 | 8.5225 | 8.6607 | 8.6798 | 7.8138 | 7.7367 | _____- 78. 6798 | 8.4303 9. 8640 . 021 . 033 . 046 . 048 . 007 O05 e|) + 22 == . 048 . 027 . 731 123 95 66 28 161 9 atse 332 62 76 2Q ee ae Se, OS. 7.9841 | 8.3232 | 8.5225 | 8.6244 | 7.4337 | 8.1982 | 8.6798 | _____- 8.4014 8.9351 . 010 . 021 . 033 . 042 . 003 . 016 0480"; 1-282 . 025 - 086 151 123 95 57 9 37 28 mee 80 104 Sie cee ah ere 8.9481 | 9.9587 | 8.4876 | 8.3155 | 8.6579 | 9.9587 | 8.4303 | 8.4014 | ______ 8. 1689 . 089 . 909 . 003 . 021 . 045 . 909 . 027 SOZD eet eeae= 015 71 43 195 337 109 317 308 280 asae 204 P1ek 5 ee Se 8.2780 | 8.0476 | 7.9556 | 8.9351 | 8.3116 | 8.5315 | 9.8640 | 8.9351 | 8.1689 | ___-_- . 019 O11 - 009 - 086 - 020 . 034 731 - 086 JOS) J}) seees 47 19 170 132 85 113 284 256 156 SSen HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 29.—Elimination factors—Continued SERIES 87 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) 243 Constituent sought (A) 2 Lo M2 Na 2N R2 Se T. 2 Tighe le Sear RB oo] te 8.7857 |8.6607 |8.5225 |8. 3232 |9.9587 |9.8237 |9. 5416 |8. 9030 ates . 061 . 046 . 033 . 021 . 909 . 666 . 348 . 080 Yas. 322 294 265 237 317 274 231 246 ge eee 8.7857 |_.-.-- 8.6798 |8. 6244 |8.5247 |7.9576 |8.9351 |9. 1055 |9. 8640 LOGIT, | es ee . 048 . 042 . 033 . 009 . 086 . 127 -73l 38 seek 332 303 275 175 132 89 284 Mghoe.. eee 8 8.6607 |8.6798 |_----- 8.6798 |8.6244 |8.3155 |8. 2539 |8.6976 |8. 9351 | .046 | .048 | LL. .048 | .042 | .021 -018 | .050 | .086 66 28 out 332 303 23 161 118 132 Nee es 8.5225 |8. 6244 {8.6798 |_.....- 8.6798 |8. 4803 |7. 7367 |8. 3068 |8. 2539 033 | .042 | .048 | __... -048 | .027 | .005 | .020 | .018 95 57 28 ss 332 52 9 146 161 QING Seek 8.3232 |8. 5247 |8.6244 |8.6798 |_..... 8.4014 |8. 1982 |7. 4165 |7. 7367 021 033 | .042 | .048 | __... -025 | .016 | .003 | .005 123 85 57 <=} aaa 80 37 174 Re ce seen 8 9. 9587 17.9576 |8.3155 |8. 4303 |8. 4014 |______ 9.9587 |9.8237 |8. 9481 -909 | .009 | .021 SO270s O25 0 een -909 | .666 | .089 43 185 337 308 280216 S235 317 274 289 tS ee eee 9. 8237 |8.9351 |8. 2539 17.7367 |8. 1982 |9.9587 | .___- 9. 9587 is 6798 666 | .086 | .018 | .005 | .016 | .909 | ___.. - 909 048 86 228 199 351 323 43 See 317 * "332 4 he eee 9. 5416 |9.1055 |8.6976 |8. 3068 |7. 4165 |9.8237 |9.9587 |.....- 8. 4376 048 | .127 | .050 | .020 | .003 | .666 | .909 | -.-_. 027 129 271 242 214 186 86 43 oe 195 Nero eee cence 8.9030 |9. 8640 8.9351 |8. 2539 |7. 7367 {8.9481 8.6798 |8. 4376 |_----- -080 | .731 | .086 | .018 | .005 | .089 | .048 | .027 | -.... 114 76 228 199 351 71 28 165 seve paecssce tS 8. 2780 |7. 7367 |8. 2539 18.9351 |9.8640 |7. 2740 |8. 2286 /8. 4358 |8. 5315 -019 | .005 | .018 | .086 | .731 002 | .017 | .027 | .034 47 9 161 132 284 4 141 98 113 Pons caasccoceue 8.0476 |8. 2539 |8.9351 |9. 8640 |8.9351 |8. 1689 |8.5315 |8. 6521 |8. 7629 -O11 | .018 | .086 | .731 | .086 | .015 | .034 | .036 | .058 19 161 132 284 256 156 113 70 85 IS Mere sconce 8.7538 |8. 5315 |8. 2286 |7. 5883 |7. 7314 [8.6976 |8. 2539 |8.3155 |7. 7367 057 | .034 | .017 | .004 | .005 | .050 | .018 | .021 | .005 285 247 219 190 342 242 199 337 351 7) v2 28M 8. 2780 |8.0476 |8. 7538 -019 -011 . 057 313 341 75 7. 7867 |8. 2539 |8. 5315 005 | .018 | .034 351 199 113 8. 2539 |8.9351 |8. 2286 -018 | .086 | .017 199 228 141 8.9351 |9. 8640 |7. 5883 - 986 731 | .004 228 76 170 9. 8640 8.9351 |7. 7314 ial 086 | .005 76 104 18 7. 2740 |8. 1689 |8. 6976 002 | .015 050 356 204 118 8. 2286 |8. 5315 |8. 2539 017 | .034 | .018 219 247 161 8. 4358 |8. 5521 |8.3155 027 | .0386 | .021 262 290 23 8.5315 |8. 7629 |7. 7367 034 | .058 | .005 247 275 9 Epes 8. 6798 |8. 1849 Sssce 048 | .015! _——s 28 122 8.6798 |_.--.. 8. 3401 $0480 ees - 022 S32 Meuees 93 8. 1849 |8.3401 | --... 015 | .022 aoe 238 267, | === 244 U. §. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 105 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C, ete.) Consticnent sought |. I EG its pine ee ee = A Ji K; M, O; oOo Pi Qi 2Q 8; p1 Bs ee ei NG 8.6704 | 8.5885 } 8.4422 | 8.4953 / 8.8322 | 8, 2332 | 7.7808 | 8.3722 | 8.1065 ada. . 047 . 039 - 028 . 031 . 068 .017 - 006 . 024 . 013 i oe 214 248 271 158 291 308 339 342 216 Ke 5 ee Se ee 8.6704 | _----- 8.6704 | 8.5381 | 8.5381 | 9.7311 | 8.4422 | 8.2332 | 9.9393 | 7.0766 OST Sees . 047 . 035 . 035 - 538 . 028 . 017 . 870 001 146 ae 214 236 124 257 271 305 308 182 Mii s8: 2288 BSS 8. 5885 | 8.6704 | .___-- 8.4953 | 8.4422 | 8.8219 | 8.5381 | 8.4422 | 8.9548 | 8.3679 . 039 ROA A eeeee= . 031 . 028 . 066 - 035 . 028 090 . 023 112 146 aa 202 89 42 236 271 94 328 OPS. 52 Dee Oso 8.4422 | 8.5381 | 8.4953 } _____- 8. 2803 | 8. 1856 | 8.6704 | 8.5885 | 8.6113 | 8.8929 028 035 O315 | Wee 019 . 015 047 039 . 041 07: 89 124 158 Ee 67 20 214 248 72 306 OOS ee = See 8.4953 | 8.5381 | 8.4422 | 8.2803 | _____- 8.4500 | 7.9556 | 6.4362 | 7.5174 | 8, 1640 - . 031 . 035 . 028 MOO) |y eee . 028 . 009 . 000 . 003 015 202 236 271 293 2 313 327 181 185 239 IP, Se ee eh SE 8.8322 | 9.7311 | 8.8219 | 8.1856 | 8.4500 | _____- 7.8500 | 8.2067 | 9.9393 | 8. 4685 . 068 . 538 . 066 -015 (028: Ci) 2 = . 007 - 016 - 870 . 029 69 103 318 340 47 -e 194 228 52 286 Qe S: Bete Fi Sa 8. 2332 | 8.4422 | 8.5381 | 8.6704 | 7.9556 | 7.8500 | _____- 8.6704 | 8.2396.| 9.7951 017 . 028 - 035 - 047 . 009 A007, eee . 047 .017 . 624 55 89 124 146 33 166 234 214 38 92 7] AE as RE ee ae 7. 7808 | 8.2332 | 8.4422 | 8. 5885 | 6.4362 | 8. 2067 | 8.6704 | .__.-- 7.1241 | 8.7943 006 017 028 039 ‘ . 016 O4Ti 2) see 001 . 062 21 55 89 112 179 132 146 ae 4 68 Spee eR FS ve eo 8.3722 | 9.9393 | 8.9548 | 8.6113 |] 7.5174 | 9.9393 | 8.2396 | 7.1241 | _._.-- 8. 3820 . 024 . 870 . 090 . 041 - 003 . 870 .017 O01. ses . 024 18 52 266 288 175 308 322 356 aoe 234 pies Rae A 8, 1065 | 7.0766 | 8.3679 | 8.8929 | 8. 1640 | 8.4685 | 9.7951 | 8.7943 | 8.3820 | __.__. . 013 . 001 . 023 . 078 015 029 624 - 062 - 024 Se 144 178 32 54 121} ° 74| 268 302 126 oe HARMONIC ANALYSIS AND PREDICTION OF TIDES 245 Table 29.—Elimination factors—Continued SERIES 105 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constituent sought (A) K2 L2 Me No 2N R2 Se Te 2 pe v2 28M. PRC errss Wee Be 8. 4953 |8. 5381 |8. 4422 |8. 2332 |9. 9392 |9. 7311 |9. 1892 |8. 8322 |8. 1065 |7.0766 |8. 6847 eh . 031 . 035 . 028 . 017 . 869 . 538 . 155 . 068 . 013 . 001 . 048 es 202 236 271 305 308 257 205 291 216 182 97 1 Ey ee 8.4953 |_..--- 8. 6704 |8. 5885 |8. 4347 |8. 9311 |8.8929 |7. 6403 |9. 7951 |7. 8500 |8. 1856 |8. 4685 OST eee .047 | .039 | .027 | .085 |.078 | .004 |.624 | .007 | .015 | .029 158 See 214 248 282 106 54 2 268 194 340 74 M3z.-.-.. Bore es 8.5381 (8.6704 |_..-_- 8. 6704 18.5885 {8.6113 |8. 1856 |8. 3896 |8. 8929 |8. 1856 |8. 8929 |8. 1585 085 | .047 | --..- 047 | .039 | .041 | .015 | .025 | .078 | .015 | .078 | .014 124 146 ok 214 248 72 20 148 54 340 306 40 INg itera ce 8.4422 |8. 5885 18.6704 |___.._ 8. 6704 |8. 2395 |7. 8500 |8. 4362 |8. 1856 |8. 8929 |9. 7951 |7. 2638 028 | .0389 | .047 |} _--.. .047 | .017 | .007 |.027 | .015 | .078 | .624 | .002 89 112 146 es, 214 38 166 114 20 306 92 6 ON Se Sees 8. 2332 |8. 4347 |8. 5885 |8.6704 |__.-.. 7, 1241 |8. 2067 |8.3366 |7. 8500 |9. 7951 |8. 7943 |7. 8368 .017 | .027 | .039 | .047 | -.--- .001 | .016 | .022 | .007 | .624 | .062 | .007 55 78 112 146 223 4 132 80 166 92 58 152 1 Aan Ae ep 9, 9392 18.9311 |8.6113 |8. 2395 |7. 1241 |_____- 9. 9392 |9. 7311 |8.3722 |8. 3410 |8.3820 |8. 3896 -869 | .085 | .041 | .017 | .001 | ----- .869 | .5388 | .024 | .022 | .024 | .025 52 254 288 322 356 ame 308 257 342 268 234 148 Some sees 9.7311 |8.8929 18.1856 |7. 8500 |8. 2067 |9.9392 |..__-- 9. 9392 |8. 6704 |8. 1585 |8. 4685 |8. 1856 .5388 |.078 |.015 | .007 | .016 | .869 | __-.. 869 | .047 | .014 | .029 | .015 103 306 340 194 228 52 eee 308 214 320 286 20 fa NR tear 9. 1892 |7. 6403 8. 3896 |8. 4362 |8. 3366 |9. 7311 |9.9392 |_--__- 8.9548 |7. 6654 |8.0785 |8. 6113 155 | .004 | .025 | .027 | .022 | .588 | .869 | _--.. .090 | .005 | .012 | .041 155 358 212 246 280 108 52 2 266 192 338 72 Agee sss soe 8, 8322 |9. 7951 |8. 8929 |8. 1856 |7. 8500 8.3722 |8.6704 |8. 9548 |-.---- 8. 4685 |8. 6609 |7. 8500 068 | .624 | .078 | .015 | .007 | .024 047 | .090 | ----. 029 | .046 | .007 69 92 306 340 194 18 146 94 mee 286 252 166 FTF a etre 8.1065 |7. 8500 |8. 1856 |8.8929 |9. 7951 |8.3410 18.1585 |7. 6654 |8. 4685 |_----_ 8.6704 |8. 1117 013 | .007 | .015 | .078 | .624 | .022 | .014 | .005 | .029 | -_-_- .047 | .013 144 166 20 64 268 92 40 168 74 ae 146 60 Cee 7.0766 |8. 1856 |8.8929 |9. 7951 |8. 7943 |8. 3820 |8. 4685 |8.0785 |8. 6609 |8.6704 |_....- 8. 2581 001 | .015 | .078 | .624 | .062 | .024 | .029 | .012 | .046 | .047 | ----- .018 178 20 54 268 302 126 74 22 108 214 ae 94 QSIMesssse2ecc 8.6847 |8. 4685 18.1585 |7. 2638 17.8368 |8. 3896 |8. 1856 |8.6113 |7.8500 {8.1117 |8. 2581 |_----- -048 | .029 | .014 | .002 | .007 | .025 | .015 | .041 | .007 | .013 | .018 | ----- 263 286 320 354 208 212 340 288 194 300 266 ioe 246 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 134 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) 8. 3946 025 131 8. 2695 . 019 121 8. 0361 011 190 8. 7345 . 054 107 8. 2094 . 016 96 8. 0930 012 72 8. 6047 . 040 41 7.7771 - 006 159 Mi 8. 3946 025 229 8. 2695 019 239 8. 4838 8. 2695 019 121 8. 8500 071 271 8. 2196 017 28 -O11 190 8. 0361 O11 8. 2628 018 146 8. 2695 . 019 121 8. 1796 015 111 8. 1133 013 87 7, 9812 . 010 62 8. 2156 - 016 32 7. 8315 . 007 149 7.9951 .010 144 9. 8992 - 793 294 8. 2746 -019 52 Disturbing constituents (B, C, ete.) Qi 2Q 8. 2094 | 8.0930 016 -012 264 288 8. 2695 | 8. 2094 . 019 . 016 239 264 8. 2628 | 8. 2695 018 -019 214 239 8. 4360 | 8.3946 . 027 . 025 205 229 8. 1133 | 7.9812 .013 . 010 273 298 7. 6424 | 7.9951 . 004 - 010 191 216 Bees 8. 4360 2-828 . 027 SS 205 8. 4360 | ------ O20 eee 155 =a 8. 2605 | 7.9233 - 018 . 008 305 330 9. 6387 | 8.7610 - 435 - 058 243 268 HARMONIC ANALYSIS AND PREDICTION OF TIDES. 247 Table 29.—Elimination factors—Continued SERIES 134 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constituent sought (A) Ko La Ma; Na 2N Ra Se TT. a pa vm | 283M IGA ee se RE ae 8.0361 |8 2628 |8. 2695 |8. 2094 |9,8992 |9.5078 |8. 9538 |8.7345 17.7771 |7.1819 |8. 5254 ee . 011 - 018 -019 | .016 | .793 -322 | .090 | .054 .006 | .002 | .034 hate 190 214 239 264 294 228 342 253 201 356 61 I Up ees See 8.0361 |------ 8.4360 |8.3946 |8.3215 |8.8285 |8. 6697 |8. 5871 |9. 6387 |7. 6424 |7.9151 |8. 2746 AOE | eee . 027 . 025 . 021 . 067 . 047 . 039 435 . 004 . 008 .019 IVA) |) Ses 205 229 254 104 38 152 243 191 346 52 IN Migwhe tis Seen 8. 2628 |8. 4360 |___-__ 8. 4360 |8.3946 |8.5206 |7.9151 |8. 4622 |8. 6697 |7.9151 |8. 6697 |7.9208 . 018 OLE Nh te - 027 . 025 - 033 . 008 . 029 . 047 . 008 . 047 . 008 146 155 Bee 205 229 80 14 128 38 346 322 27 IN greece emer 8. 2695 |8. 3946 |8. 4360 |______ 8.4360 |8. 2605 |7. 6424 |8. 3592 |7.9151 |8. 6697 |9. 6387 |6. 7753 019 |: .025 OZ vgn | eee . 027 . 018 . 004 . 023 . 008 . 047 - 435 . 001 121 ©, 131 155 ae 205 55 169 103 14 322 117 2 Niue as sal 8. 2094 |8. 3215 8.3946 |8.4360 |_.___- 7.9233 |7.9951 |8. 2280 |7. 6424 |9. 6387 |8. 7610 |7. 6344 . 016 . 021 . 025 NO2 (Ae eee .008 | .010 . 017 . 004 ~ 435 -058 | .004 96 106 131 155 HEE 30 144 78 169 117 92 158 DR gee ees nee 9. 8992 |8.8285 |8.5206 |8. 2605 |7.9233 |___._- 9.8992 |9.5079 |8. 6047 |8. 2346 |8.3143 |8. 4622 . 793 . 067 - 033 . 018 SO08% {i s2ee2 . 793 . 322 . 040 .017 - 021 . 029 66 256 280 305 330 ae 294 228 319 267 242 128 Lm Soren eee 9. 5078 |8.6697 |7.9151 |7.6424 |7.9951 |9. 8992 |__.___ 9. 8992 |8. 4360 |7.9028 |8. 2746 |7.9151 322 . 047 008 . 004 010 oS) |) eese5 . 793 027 008 019 008 132 322 346 191 216 66 soe 294 205 333 308 14 eens eee eee 8.9538 |8. 5871 |8. 4622 |8. 3592 |8. 2280 |9. 5079 |9. 8992 |_____. 8.8500 |8.0509 |7. 7834 |8. 5206 090 | .039 | .029 | .023 .017 5Orr) Wectes Wee O71 . O11 - 006 . 033 18 208 232 257 282 132 66 =a 271 219 194 80 ON BSR SE CE ee 8.7345 |9.6387 |8.6697 |7.9151 |7.6424 18.6047 |8.4360 |8.8500 |__-___ 8. 2746 |8. 5650 |7. 6424 435 047 008 004 . 040 027 AO TEA es 019 037 . 004 107 117 322 346 191 41 155 89 aoe 308 284 169 ITC ea eye 7.7771 |7. 6424 |7.9151 |8. 6697 |9.6387 |8. 2346 |7.9028 |8. 0509 |8. 2746 |_____- 8. 4360 |7. 8820 -006 | .004 . 008 .047 | .485 | .017 | .008 | .011 NOIQ Reese -027 | .008 159 169 14 38 243 93 27 141 52 weg 155 41 [Pp pee ne eee 7.1819 |7.9151 |8. 6697 |9. 6387 |8. 7610 |8. 3143 |8. 2746 |7. 7834 |8. 5650 |8.4360 |_.___- 8.1118 002 008 047 =| .435 058 | .021 .019 006 | .037 | .027 | -_--- 013 4 14 38 243 268 118 52 166 76 205 Per 65 25 Mena sae 8. 5254 |8. 2746 |7.9028 |6. 7753 |7. 6344 |8. 4622 |7.9151 |8. 5206 |7. 6424 |7.8820 /8.1118 | _...- . 034 . 019 . 008 . 001 . 004 . 029 . 008 . 033 - 004 . 008 033 goon 299 308 333 358 202 232 346 280 191 319 295 coo 248 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 163 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) ens sought = Ji Ki M, O1 OO Pi Qi 2Q 81 pi Cys a tes SR NE ie he 8.1495 | 8.1341 | 7.9150 | 7.43865 | 8.4234 | 7.9579 | 7.9582 | 8.6570 | 7.0948 eae . 014 . 014 . 008 . 003 027 | .009 . 009 . 045 . 001 meee 195 210 207 3 215 223 238 295 185 Kae see ase eos 8.1495 | ___-_- 8.1495 | 7.7528 | 7.7528 | 9.0723 | 7.9150 | 7.9579 | 9.8470 | 7.5128 AOL4 eee 014 . 006 . 006 . 118 . 008 - 009 . 703 . 003 165 see 195 192 168 199 207 223 280 350 Viens ae ee ere e 8.1341 | 8.1495 | _____- 7. 4365 | 7.9150 | 7.6604 | 7.7528 | 7.9150 | 8.7625 | 8.0859 . 014 OL ree nae . 003 . 008 . 005 . 006 . 008 . 058 - 012 150 165 os 357 153 4 192 207 84 335 OR ee 7.9150 | 7.7528 | 7.4365 | ______ 7. 7427 | 7.5513 | 8.1495 | 8.1341 | 8.4422 | 8.3724 . 008 . 006 10,15) aa ees . 006 . 004 . 014 . 014 . 028 . 024 153 168 Oo | eeee 156 u 195 210 87 338 OOS 2aF hr ss 7.43865 | 7.7528 | 7.9150 | 7.7427 | ____-- 8.1140 | 7.8343 | 7.8631 | 8.3776 | 6.6248 . 003 . 006 . 008 MOOG | Sees . 013 007 . 007 . 024 . 000 357 192 207 204 aces 212 220 235 292 182 de) ees Sees Ne i ite 8. 4234 | 9.0723 | 7.6604 | 7.5513 | 8.1140 | _____- 7.4230 | 7.7409 | 9.8470 | 7.9852 . 027 .118 . 005 . 004 SOLS ia || Pee . 003 . 006 . 703 . 010 145 161 356 353 148 sike 188 203 80 331 (Oc kate a Be eens 7.9579 | 7.9150 | 7.7528 | 8.1495 | 7.8343 | 7.4230 | ______ 8.1495 | 8.2410 | 9.3888 . 009 . 008 . 006 . 014 . 007 AO008S) |. see . 014 . 017 . 245 137 153 168 165 140 172 ae 195 72 142 ZORA Stee 7.9582 | 7.9579 | 7.9150 | 8.1341 | 7.8631 | 7.7409 | 8.1495 | ______ 8.0589 | 8.5769 . 009 . 009 . 008 . 014 . 007 . 006 201455\) See O11 . 038 122 137 153 150 125 157 165 Lane 57 127 [3 a eels 8.6570 | 9.8470 | 8.7629 | 8.4422 | 8.3776 | 9.8470 | 8. 2410 | 8.0589 | ______ 8. 2562 - 045 . 703 058 . 028 . 024 . 703 017 ‘O11. 5|, S-s525 . 018 65 80 276 273 68 280 288 57 seus 250 1 Ness Si eo ages Be 7.0948 | 7.5128 | 8.0859 | 8.3724 | 6.6248 | 7.9852 | 9.3888 | 8.5769 | 8.2562 | _..... 001 003 012 024 . 000 010 245 038 O1Se ih eeseeee 175 10 25 22 178 29 218 233 110 ae HARMONIC ANALYSIS AND PREDICTION OF TIDES 249 Table 29.—Elimination factors—Continued SERIES 163 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constituent sought (A) Ke Le Me No 2N Ro So TT. a po va 28M. 5) CC ils to | 7.4865 |7. 7528 |7.9150 |7.9579 |9. 8470 |9.0723 9.3179 |8. 4234 |7.0948 |7. 5128 |8. 1450 Lee . 003 . 006 . 008 . 009 . 703 .118 . 208 . 027 . 001 . 003 . 014 ue. 357 192 207 223 280 199 299 215 185 350 26 ot Sod ee Tes4365)) po eee 8. 1495 |8.1341 |8. 1078 |8. 7464 |8. 3724 |8. 7614 |9. 3888 |7. 4230 |7. 5513 |7. 9852 003 | ----- 014 - 014 . 013 . 056 . 024 . 058 . 245 . 003 . 004 . 010 3 LP 195 210 226 103 22 122 218 188 353 29 CW Ip jes Sem Sg 7, 7528 |8. 1495 |______ 8.1495 |8. 1341 |8. 4422 17.5513 |8. 4590 |8. 3724 |7. 5513 |8.3724 17. 5483 . 006 OVER ok , 014 . 014 . 028 . 004 . 029 . 024 . 004 . 024 . 004 168 165 se 195 210 87 ds 107 22 353 338 14 INos 5.) 6 SEB 7.9150 |8. 1341 /8. 1495 |______ 8.1495 |8. 2410 |7. 4230 |8. 2850 |7. 5513 |8.3724 |9. 3888 |6. 3062 . 008 . 014 OLE S|) 222 . 014 . 017 . 003 .019 . 004 . 024 ~ 245 . 000 153 150 165 Lee 195 72 172 92 7 338 142 179 JOIN ese oe eee ae 7.9579 |8. 1078 |8. 1341 /8. 1495 |______ 8.0588 |7. 7409 |8. 1397 |7. 4230 |9. 3888 |8.5769 17. 4186 . 009 . 013 . 014 2014 | ____- . 011 . 006 . 014 . 003 . 245 . 038 . 003 137 134 150 165 Lae 57 157 76 172 142 127 164 TED A ee AF 9.8470 |8. 7464 8. 4422 |8. 2410 |8. 0588 |______ 9. 8470 |9.0725 |8.6570 |8. 1488 |8. 2563 |8. 4590 .703 | .056 | .028 | .017 |.011 |} ____- .703 | .118 | .045 | .014 | .018 | .029 80 257 273 288 303 oe 280 199 295 265 250 107 hy ee ee 9.0723 |8.3724 |7.5513 |7.4230 |7. 7409 |9.8470 |______ 9. 8470 |8. 1495 |7. 5483 |7. 9852 |7. 5513 . 118 . 024 . 004 . 003 . 006 CALS Jan || = opera - 103 . 014 . 004 . 010 . 004 161 338 353 188 203 80 te 280 195 346 331 a RAM SLE ee eh ae 9.3179 |8. 7614 |8. 4590 |8. 2850 |8.1397 19.0725 |9.8470 |______ 8. 7629 |8. 1668 |8.1966 |8. 4422 . 208 . 058 . 029 019 . 014 . 118 5 (A |) ease . 058 015 016 . 028 61 238 253 268 284 161 80 ae 276 246 231 87 2 ee ee 8. 4234 |9. 3888 18.3724 |7.5513 |7. 4230 |8.6570 |8. 1495 |8. 7629 |______ 7. 9852 |8. 3386 |7. 4230 . 027 . 245 . 024 . 004 . 003 . 045 . 014 O58: 1 Sees - 010 . 022 . 003 145 142 338 353 188 65 165 84 aaes 331 315 172 pesca. ee 7. 0948 |7. 4230 |7. 5513 |8. 3724 19.8888 |8. 1488 |7. 5483 |8. 1668 |7.9852 | ____- 8. 1495 |7. 5426 - . 001 . 003 . 004 . 024 . 245 . 014 . 004 015 eM) |) Sees . 014 , 003 175 172 7 22 218 95 14 114 29 aa 165 21 W0seansseces cca 7.5128 |7. 5513 |8.3724 |9. 3888 |8. 5769 |8. 2563 |7. 9852 |8. 1966 |8. 3386 |8. 1495 |______ 7. 8424 . 003 . 004 . 024 . 245 . 038 . 018 .010 . 016 . 022 jOT4) S|) S22 2— - 007 10 7 22 218 233 110 29 129 45 195 as 36 2S Miers see 8.1450 |7.9852 |7. 5483 |6. 3062 |7.4186 |8. 4590 |7. 5513 |8. 4422 |7. 4230 |7. 5426 |7.8424 | _____ . 014 .010 . 004 . 000 . 003 . 029 . 004 . 028 . 003 . 003 . 007 re 334 331 346 181 196 253 353 273 188 339 324 es 250 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 192 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Soe sought Ni Ki Mi O1 0oO Pi Q1 2Q 81 pl qt et ees ee |e Ee 7.6613 | 7.6591 | 7.0855 | 8.0828 | 7.3819 | 6.5151 | 7.0698 | 8.6281 | 7.3308 2a . 005 . 005 - 001 .012 . 002 . 000 . 001 42° 002 oats 186 192 356 16 357 182 187 271 350 Rep ot Rs oh ee GH} |! k= 7.6613 | 7.5891 | 7.5891 | 8.6868 | 7.0355 | 6.5151 | 9.7807 | 7.6468 005), || Sates - 005 . 004 . 004 . 049 . 001 . 000 . 604 . 004 174 Le 186 350 10 351 356 182 265 344 MAS 35) SR See 7.6591 | 7.6613 | _-__-- 8.0828 | 7.0355 | 8.1441 | 7.5891 | 7.0355 | 8.6866 | 7.9586 . 005 O05 55 fees . 012 . 001 014 . 004 . 001 . 049 . 009 168 174 wSce 344 4 165 350 356 80 338 Ope te See NE Bee 7.0355 | 7.6891 | 8.0828 | ____-- 7. 5826 | 6.4230 | 7.6613 | 7.6591 | 8.3698 | 7.7679 . 001 . 004 O12)! | ReSte . 004 . 000 . 005 . 005 023 . 006 4 10 16 Sau 20 1 186 192 95 354 OOnr. A 2s wee 8.0828 | 7.5891] 7.0355 | 7.5826 | ..___- 7. 8388 | 7.3409 | 7.0344 | 8.3250 | 7.6132 . 012 . 004 . 001 MOOS si) oet os . 007 . 002 . 001 . 021 . 004 344 350 356 340 Ee 341 346 352 256 334 12 ay ee 7. 3819 | 8.6868 | 8.1441 | 6.4230 | 7.8388 | ____.- 7.1547 | 7.3491 | 9.7807 | 7.3097 . 002 . 049 . 014 . 000 HOOT --£ 33 . 001 - 002 - 604 - 002 3 9 195 359 19 aes 185 191 95 353 Qpta tse: Ree 6.5151 | 7.0355 | 7.5891 | 7.6613 | 7.3409 | 7.1547 | -_-_-- 7.6613 | 8.1911 | 8.8560 . 000 . 001 . 004 . 005 . 002 JOO1S | 228 . 005 . 016 .072 178 4 10 174 14 175 cas 186 89 168 2Q) fe eee ee 7.0698 | 6.5151 | 7.0855 | 7.6591 | 7.0344 | 7.3491 | 7.6613 | _-_-. 8.0615 | 8.0931 . 001 . 000 . 001 . 005 . 001 . 002 OOSNe |) pases -012 012 173 178 4 168 8 169 174 eS 84 162 Bye Sees RENE 8. 6281 | 9.7807 | 8.6866 | 8.3698 | 8.3250 | 9.7807 | 8.1911 | 8.0615 | ____.. 8. 2024 . 042 . 604 . 049 . 023 . 021 . G04 . 016 O12) Ses8 . 016 89 95 280 265 104 265 271 276 aa 258 (ee ee 7. 3308 | 7.6468 | 7.9586 | 7.7679 | 7.6132 | 7.3097 | 8.8560 | 8.0931 | 8.2024 | -___.. 002 . 004 . 009 . 006 . 004 . 002 . 072 -012 2016; ||) FS 10 16 22 6 26 7 192 198 102 soe HARMONIC ANALYSIS AND PREDICTION OF TIDES 251 SERIES 192 DAYS. Table 29.—Elimination factors—Continued SEMIDIURNAL CONSTITUENTS Constituent sought (A) IMig-e2. tebe. 7. 5891 . 004 10 INg2 2. 2.2222-5- 7. 0355 - 001 4 QN nes Seance 6. 5151 - 000 178 Roo seed 9. 7807 . 604 95 (STR ae eee 8. 6868 Agente eee cece 7. 3819. . 002 3 Poo coea doeks cee 7. 3308 10 28Merr sent ase 7. 6012 - 004 189 7. 6591 . 005 168 7. 6554 . 005 163 8. 6778 . 048 259 7. 7679 - 006 354 8. 7615 . 058 268 8. 8560 "186 7. 6591 . 005 8. 3698 "359 8. 4057 . 025 274 7. 7679 . 006 354 6. 4230 - 000 1 7. 7679 . 006 6 6. 4265 - 000 359 Disturbing constituents (B, C, etc.) "186 8. 2077 - 016 7. 3491 . 002 191 8. 0649 7. 1547 . 001 185 8. 8560 072 7. 1525 001 190 2 Beals 8. ass 9. 2922 |7.3819 |7. 3308 |7. 6468 |7. 6012 65 8. 6778 . 048 101 8. 3698 0 8. 6863 . 04 8. 0768 012 8. 4057 - 025 274 351 7. 7679 . 006 6 6. 4230 . 000 1 7. 1547 . 001 7. 3491 9. 7807 . 604 95 7. 6613 . 005 174 6. 4265 -196 | .002 | .002 | .004 | .004 256 357 350 344 171 8.7615 |8. 8560 |7. 1547 |6. 4230 |7. 3097 .058 | .072 | .001 | .000 | .002 92 192 185 359 7 8.4057 |7. 7679 |6. 4230 |7.'7679 |6. 4265 -025 | .006 | .000 | .006 | .000 86 6 359 354 1 8. 2077 |6. 4230 |7. 7679 |8. 8560 |6. 8803 -016 | .000 | .006 | .072 | .001 80 1 354 168 175 8.0649 |7.1547 |8. 8560 |8.0931 |7. 1525 012 | .001 | .072 | .012 | .001 8. ae 8. 6281 |8.0768 |8. 2024 |8. 4057 5 6 0 351 271 264 258 86 9. 7807 |7. ae 6. 4265 |7. 3097 |6. 4230 . 005 . 604 -000 | .002 | .000 265 186 359 353 1 @0s006 8. 6866 {8.0959 |8. 2350 |8. 3698 Soas 049 | .012 | .017 | .023 rates 280 273 268 95 7.7656 |7. 1547 -006 | .001 347 175 7.6613 |6. 4253 . 000 174 2 8. 2350 |7. 7656 |7. 6613 |_----- 7. 1202 -017 | .006 | .005 | ----- - 001 92 13 IRD |) ogee 8 8.3698 |7. 1547 |6. 4253 |7.1202 |____-- -023 | .00L | .000 | .001 |--_-_-- 265 185 358 352 |------ 252 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—-Continued SERIES 221 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) Disturbing constituents (B, C, etc.) Ji Ki Mi O1 0O Pi Qi 2Q S1 pl eg eo 7.4061 | 7.4052 | 7.8848 | 8. 2672 | 8.3590 | 7.7969 | 7.7322 | 8.5324 7. 6535 See . 003 . 003 . 008 . 019 - 023 . 006 . 005 . 034 005 Boa 356 353 324 28 318 321 317 247 334 Ta06 1) | ee ae 7.4061 | 8.0179 | 8.0179 | 9.2077 | 7.8848 | 7.7969 | 9.6969 | 7.7209 S003) || eee s—= . 008 . 010 . 010 . 161 - 008 . 006 . 498 . 005 4 aa 356 328 32 322 324 321 251 338 7.4052 | 7.4061 | __---- 8. 2672 | 7. 8848 | 8.4189 | 8.0179 | 7.8848 | 8.6172 7. 8313 . 003 OOS He | ene .019 . 008 ; 026 .010 -008 | .041 . 007 7 4 oS. 332 36 146 328 324 75 341 7. 8848 | 8.0179 | 8.2672 | _____- 7.9465 | 7.3352 | 7.4061 | 7.4052 | 8. 2991 7. 8800 . 008 . 010 SOLO), 1" s-=- 2 . 009 . 002 . 003 . 003 . 020 . 008 36 32 28 —™ 64 174 356 353 103 190 8. 2672 | 8.0179 | 7.8848 | 7.9465 | __-_.- 8. 2351 | 7.8628 | 7.7946 | 8.0778 7. 8188 . 019 . 010 . 008 0092 . 017 . 007 . 006 . 012 007 332 328 324 296 eas 290 292 289 219 306 8.3590 | 9.2077 | 8.4189 | 7.3352 | 8.2351 | _____- 6.7176 | 6.4350 | 9.6969 7. 5854 . 023 161 . 026 . 002 OLS |) . 001 . 000 - 498 . 004 42 38 214 186 70 ee 182 358 109 195 7.7969 | 7.8848 | 8.0179 | 7.4061 | 7.8628 | 6.7176 | _____- 7.4061 | 8.1112 8. 8310 . 006 . 008 . 010 . 003 . 007 SOLE pce ee= . 003 013 068 39 36 32 4 68 178 shea 356 107 13 7.7322 | 7.7969 | 7.8848 | 7.4052 | 7.7946 | 6.4350 | 7.4061 | __---. 7. 9748 8. 0073 . 005 . 006 008 . 003 . 006 . 000 S003 4|: Sees . 009 . 010 43 39 36 7 71 2 4 m8 110 17 8. 5324 | 9.6969 | 8.6172 | 8.2991 | 8.0778 |9.6969 {8.1112 |7.9748 |_____- 8. 1495 . 034 - 498 . 041 . 020 . 012 . 498 . 013 009% 3) “Sess . 014 113 109 285 257 141 251 253 250 maak 266 7.6535 | 7.7209 | 7.8313 | 7.8800 | 7.8188 | 7.5854 | 8.8310 | 8.0073 | 8.1495 | _...-. . 005 . 005 . 007 . 008 . 007 . 004 . 068 . 010 O14! ||) eee 22 19 170 54 165 347 343 94 cose 26 HARMONIC ANALYSIS AND PREDICTION OF TIDES 253 Table 29.— Ylimination factors—Continued SERIES 221 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, :C, etc.) pone soug Ke La M2 Na 2N Re Se Te re 2 v2 28M Kiqueewwen eo eee 8. 2872 |8.0179 |7. 8848 |7. 7969 |9.6969 |9. 2077 |8. 9831 |8. 3590 |7. 6535 |7.7209 |8. 2035 Bes . 019 . 010 . 008 . 006 - 498 . 161 . 096 . 023 . 005 . 005 . 016 peace 332 328 324 321 251 322 213 318 334 338 136 Tir ee oe 822672) |noce ee 7. 4061 |7. 4052 |7. 4037 |8.6189 |7. 8800 |8. 6448 |8. 8310 |6. 7176 17.3352 |7. 5854 O19? "ill as22 003 | .003 .003 | .042 | .008 | .044 | .068 . 001 .002 | .004 28 Spee 356 353 349 99 170 62 347 182 186 | 165 Vig eee ees 8.0179 |7. 4061 |_----- 7.4061 |7. 4052 |8. 2991 |7. 3352 |8. 3038 |7. 8800 |7. 3352 |7. 8800 17. 3333 . 010 003.) oS . 003 . 003 . 020 . 002 . 020 . 008 . 002 . 008 . 002 32 4 ee 356 353 103 174 65 170 186 190 168 IN ger eee 7. 8848 17.4052 |7.4061 |____-- 7. 4061 {8.1112 |6. 7176 |8. 1229 |7. 3352 |7. 8800 |8. 8310 |7. 0677 . 008 . 003 OOS sete . 003 . 013 . 001 . 013 . 002 . 008 . 068 . 001 36 7 4 tes 356 107 178 69 174 190 13 172 ON Sa? sl oes 7.7969 |7. 4037 |7.4052 |7.4061 |_____- 7.9748 |6. 4350 |7.9996 |6. 7176 18.8310 |8. 0073 16. 7183 . 006 . 003 - 003 a (X08} |) eee . 009 . 000 . 010 . 001 . 068 . 010 . 001 39 11 7 4 aes 110 2 73 178 13 17 176 1 ea at Se 9. 6969 |8. 6189 |8. 2991 |8.1112 |7.9748 |_____- 9.6970 |9. 2076 |8. 5324 18.0145 |8. 1495 |8. 3038 - 498 . 042 . 020 . 013 S009 3 22 . 498 . 161 . 034 . 010 . 014 . 020 109 261 257 253 250 sae 251 322 247 263 | 266 65 Seater Sgt Se 9. 2077 |7.8800 |7.3352 |6. 7176 |6. 4350 |9. 6970 |_____- 9.6970 |7. 4061 |7. 3333 |7. 5854 |7. 3352 . 161 . 008 . 002 . 001 . 000 -498 | __--- - 498 . 003 002 | .004 . 002 38 190 186 182 358 109 Bee 251 356 192 195 174 uf Dy i pia 8.9831 |8. 6448 |8. 3038 |8. 1229 |7.9996 |9. 2076 |9.6970 |_____- 8.6172 |7.9704 |8. 0914 |8. 2991 . 096 . 044 . 020 013 . 010 . 161 -498 | LL . 041 . 009 .012 . 020 147 298 295 291 287 38 109 aa 285 301 304 103 DAP) 35 eat 8. 3590 |8. 8310 |7. 8800 |7.3352 |6.7176 |8. 5324 |7. 4061 8.6172 |_-_.-- 7. 5854 |7.8738 |6. 7176 . 023 . 068 . 008 . 002 . 001 . 034 . 003 OAT 5|| sees . 004 . 007 . 001 42 13 190 186 182 113 4 75 en 195 199 178 [Diageo eelpare e 7.6535 |6. 7176 |7. 3352 |7. 8800 |8. 8310 |8. 0145 |7. 3333 |7.9704 |7. 5854 |_____- 7.4061 |7. 3294 . 005 . 001 . 002 . 008 . 068 . 010 . 002 . 009 004 | ____- - 003 . 002 26 178 174 170 347 97 168 59 165 Seen. 4 162 UT eee ce ene 7.7209 |7. 3352 |7. 8800 |8. 8310 |8. 0073 |8. 1495 |7. 5854 |8.0914 |7.8738 |7.4061 |______ 7. 4945 . 005 . 002 . 008 . 068 . 010 . 014 . 004 . 012 . 007 003% |) 22c25 . 003 22 174 170 347 343 94 165 56 161 356 eee 159 2S Mes 8. 2035 |7. 5854 |7. 3333 |7.0677 |6. 7183 |8. 3038 |7.3352 |8. 2991 |6.7176 |7. 3294 .016 . 004 . 002 . 001 . 001 . 020 . 002 . 020 - 001 . 002 224 195 192 188 184 295 186 257 182 198 254 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 250 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) PE SOUS [se a en a a gD i ok Be Jy Ki M, 0; oo Pi Qi 2Q $1 pi age ete ats aces cs 2 | Plier 7.9011 | 7.8898 | 8.0302 | 8.3544 | 8.4768 | 7.9350 | 7.8438 | 8.3527.) 7.7796 eae . 008 . 008 . 011 . 023 . 030 . 009 . 007 . 023 . 006 Bee, 347 324 293 41 280 280 267 224 318 1 ee a TQOL Us| Se 7.9011 | 8.1489 | 8.1489 | 9.3286 | 8.0302 | 7.9350 | 9. 5900 7. 7661 4008s ||) £ee— = . 008 . 014 014 . 213 . O11 . 009 . 389 . 006 13 eas 347 306 54 294 293 280 237 331 Mige eh stee. 2d cameo 7.8898 | 7.9011 | ----~- 8. 3544 | 8.0302 | 8.5201 | 8.1489 | 8.0302 | 8.5519 7. 6982 . 008 OOS || iepe = . 023 .O11 . 033 . 014 . O11 . 036 . 005 26 13 ee 319 67 127 306 293 70 344 Ope Aten a eee 8.0302 | 8.1489 | 8.3544 | ______ 7.9171 | 7.6029 | 7.9011 | 7.8898 | 8. 2274 8. 2405 . O11 .014 O23 eee = . 008 . 004 . 008 . 008 . 017 .017 67 54 41 op EE 108 168 347 384 111 205 OO ms Ses HE tse 8.3544 | 8.1489 | 8.0302 | 7.9171 | __---- 8. 1443 | 7.7748 | 7.6181 | 6. 8959 7. 8514 . 023 . 014 011 . 008 ee See . 014 . 006 . 004 . 001 . 007 319 306 293 252 ase 239 239 226 183 277 1 ee ee eee 8.4768 | 9.3286 | 8.5201 | 7.6029 | 8.1443 | ------ 6. 2382 | 7.3397 | 9. 5900 7. 8955 . 030 . 213 - 033 . 004 sO145 4), 2-22 . 000 . 002 . 389 . 008 80 66 233 192 121 L's 359 346 123 218 Qyee 3s Se ceed 7.9350 | 8.0302 | 8.1489 | 7.9011 | 7.7748 | 6.2382 | ----_- 7.9011 | 7.9950 | 9. 2133 . 009 . O11 . 014 . 008 . 006 O00!) see . 00S . 010 . 163 80 67 54 13 121 1 oe 347 124 39 2QOje se wake 28 eae 7. 8438 | 7.9350 | 8.0302 | 7.8898 | 7.6181 | 7.3397 | 7.9011 | ------ 7. 7821 8. 3852 . 007 . 009 . O11 . 008 . 004 . 002 = OO82s | 2s 5== | .006 . 024 93 80 67 26 134 14 13 iss 137 52 Spree cece: ei ween e 8.3527 | 9. 5900 | 8.5519 | 8.2274 | 6.8959 | 9.5900 | 7.9950 | 7. 7821 | __--_- 8. 0954 . 023 . 389 . 036 .017 . 001 . 389 . 010 006, |) peecee . 012 136 123 290 249 177 237 236 223 per 275 Dist 2osssoe Stee 7. 7796. | 7.7661 | 7.6982 | 8.2405 | 7.8514 . 006 . 006 . 005 .017 . 007 42 29 16 155 83 HARMONIC ANALYSIS AND PREDICTION OF TIDES. 255 Table 29.—Elimination factors—Continued SERIES 250 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constituent sought (A) 2 La Ma No 2N Re S2 Ts 2 ya vo | 25M 1 Me a a ala) I es 8.3544 |8.1489 |8.0302 |7.9350 |9. 5900 |9. 3286 |8. 4129 |8. 4768 |7. 7796 |7. 7661 |8. 3023 Mea .023 | .014 | .011 | .009 | .389 | .213 | .026 | .030 | .006 | .006 | .020 ae 319 306 293 280 237 294 350 280 318 331 101 Wee acereese 8. 3544 |_____- 7.9011 |7.8898 |7.8703 |8. 5672 |8. 2405 |8. 3634 |9. 2133 |6. 2382 |7. 6029 |7. 8955 BOIS er |e see .008 | .008 | .007 | .037 | .017 | .023 | .163 | .000 | .004 | .008 41 zoe 347 334 321 98 155 31 321 359 192 143 Vie ae Oe 8.1489 |7.9011 |___-__ 7.9011 |7. 8898 |8. 2274 |7. 6029 |8.1376 |8. 2405 |7. 6029 |8. 2405 |7. 5928 . 014 ROOST 22 -= . 008 . 008 017 . 004 .014 017 . 004 .017 . 004 54 13 ast 347 334 lll 168 44 155 192 205 155 Nig Sect re re Se 8.0302 {7.8898 |7.9011 |____-- 7.9011 |7.9950 |6. 2382 |8.0259 |7. 6029 |8. 2405 |9. 2133 |7. 1697 011 | .008 | .008 | ____- .008 | .010 | .000 | .O11 | .004 | .017 | .163 | .001 67 26 13 a 347 | 124 1 58 168 205 39 168 CAN | ak ep enn 7. 9350 |7.8703 |7.8898 |7.9011 |_.__-- 7. 7821 |7. 3397 |7. 9414 |6. 2382 |9. 2133 |8.3852 |6. 2381 -009 | .007 | .008 | .008 | ____. .006 | .002 | .009 | .000 | .163 | .024 | .000 80 39 26 13 aes. 137 14 71 1 39 52 2 1 8a ee setae 9.5900 |8. 5672 |8. 2274 |7.9950 |7. 7821 |______ 9. 5901 9. 3286 |8. 3528 17.9596 |8. 0954 |8. 1376 .389 | .087 | .017 | .010 | .006 | ____- .389 | .213 | .023 | .009 012 =| .014 123 262 249 236 223 es 237 294 224 261 275 44 Soemed. Lae 2 2 9.3286 |8. 2405 |7.6029 |6. 2382 |7. 3397 9.5901 |______ 9.5901 |7.9011 |7. 5928 |7. 8955 |7. 6029 213 | .017 | .004 | .000 | .002 | .3889 | ____- .389 | .008 | .004 | .008 | .004 66 205 192 359 346 123 eee 237 347 205 218 168 (d Dyce meee 8.4129 |8. 3634 |8.1376 |8.0259 |7.9414 |9.3286 |9. 5901 |_____-_ 8.5519 |7. 7084 |7. 6345 |8. 2274 . 026 . 023 . 014 O11 . 009 - 213 SOS Da eee - 036 . 005 . 004 . 017 10 329 316 302 289 66 123 ae 290 328 341 111 ON ahd eee ae ee 8.4768 |9. 2133 |8. 2405 |7.6029 |6. 2382 |8. 3528 |7.9011 |8. 5519 |______ 7. 8955 |8.1962 16. 2382 030 | .163 | .017 | .004 | .000 | .023 | .008 | .036 | _____ -008 | .016 | .000 80 39 205 192 359 136 13 70 ie 218 231 1 Nee a Se 7.7796 |6. 2382 |7.6029 |8. 2405 |9. 2133 |7. 9596 |7. 5928 |7. 7084 |7.8955 |______ 7.9011 |7. 5759 .006 | .000 | .004 | .017 | .163 | .009 | .004 | .005 | .008 | _____ .008 | .004 42 1 168 155 321 99 155 32 142 =o 13 143 pe eS 7. 7661 |7. 6029 |8. 2405 |9. 2133 |8.3852 |8.0954 |7.8955 |7.6345 |8.1962 |7.9011 |______ 7. 7671 006 | .004 | .017 | .163 | .024 | .012 | .008 | .004 | .016 | .008 | _____ . 006 29 168 155 321 308 85 142 19 129 347 ABs 130 OAS [ae ere 8.3023 |7.8955 |7.5928 |7.1697 |6. 2381 |8. 1376 |7. 6029 |8. 2274 |6. 2382 |7.5759 |7. 7671 |______ .020 | .008 | .004 001 000 | .014 | .004 | .017 000 | .004 (11323) Paes 259 | 218| 205| 192/ 358| 316] 192] 249| 359] 217|° 230] _.. 256 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 279 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) 8. 0469 . 011 45 8. 0127 . 010 99 7. 5672 . 004 144 7. 9987 . 010 160 7. 8339 . 007 57 Disturbing constituents (B, C, etc.) O; oO P; Qi 2Q Si pl 8.0127 | 8.3961 | 8.3841 | 7.8250 | 7.5672 | 7.9987 | 7.8339 . 010 . 025 . 024 . 007 . 004 . 010 . 007 261 54 242 239 216 200 303 8.1800 | 8.1800 | 9.3172 | 8.0127 | 7.8250 | 9.4495 | 7. 7947 . 015 .015 . 208 . 010 . 007 . 282 . 006 284 76 265 261 239 222 325 8.3961 | 8.0127 | 8.5477 | 8.1800 | 8.0127 | 8.4890 | 7.5510 . 025 . 010 . 035 .015 . 010 . 031 4 306 99 108 284 261 65 348 aaa 7. 5571 | 7.7343 | 8.0816 | 8.0469 | 8.1523 | 8.3798 ae . 004 . 005 . 012 .O11 . 014 . 024 a 152 161 337 315 119 221 7 aeAl |). == 7. 3456 | 6.7358 | 7.1964 | 7.9211 | 7.7771 0047) anes . 002 . 001 . 002 . 008 . 006 208 as 189 185 342 326 249 Ue (O43) | de O40) | == 6.8592 | 7.5574 | 9.4495 | 7.9990 . 005 O02 elie ee . 001 . 004 . 282 . 010 199 171 =e 356 334 138 240 8.0816 | 6.7358 | 6.8592 | ______ 8.0816 | 7.8251 | 9.3242 . 012 . 001 001) +2265 . 012 . 007 . 211 23 175 4 wee 337 141 64 8.0469 | 7.1964 | 7.5574 | 8.0816 | ___-._ 7.3460 | 8.4421 .O11 . 002 . 004 SOL2 Ral! pases . 002 . 028 45 18 26 23 ee 164 86 8.1523 | 7.9211 | 9.4495 | 7.8251 | 7.3460 | ---__. 8. 0384 .014 . 008 . 282 . 007 0024 Nh eee . O11 241 34 222 219 196 wee 283 8.2798 | 7.7771 | 7.9990 | 9.3242 | 8.4421 | 8.0384] ______ . 024 . 006 010 211 . 028 OLE pS sae 129 lll 120 296 274 77 ner HARMONIC ANALYSIS AND PREDICTION OF TIDES SERIES 279 DAYS. Constituent sought (A) Migbes. 2: 222 8. 1800 -015 76 INewreoue seat Se 8.0127 010 99 2N2s S25 2ce5e 7. 8250 007 121 Rote secede 9. 4494 . 281 138 DaeeeeeSe acs s 9.3172 - 208 95 Tg oss sosenes ss 9.0421 110 52 Agua ase lo oes. 8. 3841 . 024 La M2 8.3961 |8. 1800 £025 015 306 284 ses 8.0816 raed .012 es 337 8.0816 |____-_ OU ts 23 Se 8.0469 |8. 0816 O11 -012 45 23 7.9866 |8. 0469 .010 .O11 68 45 8.5211 |8. 1523 . 033 .014 264 241 8.3798 |7. 7343 . 024 -005 221 199 6.9057 |7. 8491 - 001 . 007 359 336 9.3242 |8.3798 .211 . 024 64 221 6. 8592 |7. 7343 - 001 . 005 4 161 7.7343 |8.3798 -005 . 024 161 139 7.9990 |7. 7105 -010 . 005 240 218 Table 29.—Elimination factors—Continued SEMIDIURNAL CONSTITUENTS Disturbing constituents (RB, C, etc.) No 2N Re 8.0127 |7.8250 |9. 4494 . 010 . 007 . 281 261 239 222 8.0469 |7.9866 |8. 5211 . O11 -010 . 033 315 292 96 8.0816 |8.0469 |8. 1523 .012 -O11 .014 337 315 119 ae 8.0816 |7. 8251 alice .012 . 007 BS 337 141 8.0816 |------ 7.3463 LOL2 i S-= . 002 23 bass 164 7.8251 |7.3463 |_----- . 007 S002 <2 as= 219 196 made 6.8592 |7.8721 19.4496 - 001 . 007 . 282 356 294 137 7.9108 |7.8885 |9. 3172 - 008 - 008 . 208 314 291 95 7. 7343 |6.8592 |7. 9987 - 005 -001 - 010 199 356 160 8.3798 |9.3242 |7.9102 - 024 5 PALI . 008 139 296 100 9.3242 18.4421 |8. 0384 . 211 . 028 O11 296 274 U0 7. 2354 |6.8588 |7.8491 . 002 - 001 . 007 195 352 336 6. 8592 -001 4 7. 8721 223 8. 0816 012 123 7.7105 . 005 142 7.9990 -010 120 7. 7343 - 005 199 9.0421 - 110 308 6.9057 - 001 1 7.8491 - 007 24 7.9108 - 008 9.3172 - 208 265 9. 4496 "993 7.9987 - 010 200 8. 0816 - 012 7.8339 |7. 7947 007 303 6. 8592 -001 356 7. 7343 -005 199 8.3798 - 024 221 006 325 7. 7343 - 005 199 8. 3798 024 221 9.3242 211 64 9.3242 |8.4421 - 028 - 211 64 7.9102 - 008 260 7.7105 -005 218 6. 8703 - 001 355 7.9990 - 010 240 86 8.0384 -011 283 7.9990 - 010 8. 2553 -018 263 8. 0816 -012 23 257 258 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 297 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constituent sought ) Ji Ki Mi O1 oo Py Q: 2Q 81 pi aD gest: Sekar cational, opps iN) geal sr 8. 2770 | 8.1622 | 7.9899 | 7.5338 | 8.3896 | 7.7726 | 7.1486 | 8. 4204 7. 5223 pote . 019 015 . 010 . 003 . 025 . 006 - 001 . 026 . 003 Z SEE 220 260 266 173 287 306 346 253 206 NT ee ae 8527703) sae 2 8.2770 | 8.0269 | 8.0269 | 9.2565 | 7.9899.) 7.7726 | 9.3360 7. 3907 SUE Sees .019 O11 . O11 . 181 .010 . 006 .217 . 002 140 ese- 220 227 133 247 266 306 214 346 VE et a th 8516223 |°8527;70)) |e eee 7. 53838 | 7.9899 | 8.2044 | 8.0269 | 7.9899 | 7.5392 8. 1019 -015 BOLO || epee . 003 . 010 . 016 O11 . 010 . 003 . 013 100 140 eget Ee 187 94 27 227 266 174 306 © jess eee et ee 7.9899 | 8.0269 | 7.5338 | _____- 7. 8638 | 7.7467 | 8.2770 | 8.1622 | 7.5336 8. 4724 .010 . O11 3003! ‘lees t= . 007 . 006 . 019 . 015 . 003 . 030 94 133 173 ae 87 21 220 260 167 300 OO) 2 Sanaa 7. 5338 | 8.0269 | 7.9899 | 7.8638 | ______ 8.0954 | 7.6320 | 6.7805 | 8.1433 7. 5129 . 003 O11 - 010 BOO Pi een . 012 . 004 . 001 . 014 . 003 187 227 266 273 avee 294 313 353 260 213 Bs aot oe Sethe a yl 8.3896 | 9.2565 | 8.2044 | 7. 7467 | 8.0954 | ______ 7. 5300 | 7.8163 | 9.3360 8. 0286 . 025 . 181 . 016 . 006 “OZR lps . 003 , 007 Bale . O11 73 113 333 339 66 one 199 239 146 279 (oe eee eee 7.7726 | 7.9899 | 8.0269 | 8.2770 | 7.6320 | 7.5300 | ______ 8.2770 | 7.9031 9. 3366 . 006 . 010 - O11 . 019 . 004 O0Sien |peree= .019 . 008 217 54 94 133 140 47 161 See 220 127 80 YA 0 ee See eee 7. 1486 | 7.7726 | 7.9899 | 8.1622 | 6.7805 | 7.8163 | 8.2770 | ______ 7. 8741 8. 2220 . 001 . 006 .010 . 015 . 001 . 007 JOI9R tees . 007 . 017 14 54 94 100 7 121 140 wee 87 40 [Sete ene et Sere 8. 4204 | 9.3360 | 7.5392 | 7.5336 | 8.1433 |9.3360 |7.9031 |7.8741 | _____- 7. 8897 . 026 avaly/ . 003 . 003 .014 A PAC) . 008 S000 or Sae= . 008 107 146 186 193 100 214 233 273 wane 312 (aoe aoe eee 7. 5223 | 7.3907 | 8.1019 | 8.4724 | 7.5129 | 8.0286 | 9.3366 | 8.2220 | 7.8897 |} —--___- . 003 . 002 . 013 . 030 . 003 . 011 .217 017 1008-4)" eaee 154 14 54 60 147 81 280 320 48 aoe HARMONIC ANIALYSIS AND PREDICTION OF TIDES 259 Table 29.—Elimination factors—Continued SERIES 297 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) @ovistiteien tau | meee eas en ee ee ee ee ts Bletie Racee bie sought (A) 2 La M2 Na 2N Re 82 Ta 2 p2 v2 25M 1D eee a ro eat 7. 5338 |8.0269 |7.9899 |7.7726 |9.3359 |9.2565 |9.1076 |8.3896 |7.5223 |7.3907 |8. 2357 aa . 003 O11 -010 . 006 .217 - 181 . 128 . 025 . 003 . 002 . 017 a 187 227 266 306 214 247 281 287 206 346 88 1) Bae eee Se Woussies eee a 8.2770 |8.1622 |7.9326 |8.1514 |8.4724 |8.5711 |9.3366 |7.5300 |7. 7467 |8.0286 -003, | ----. .019 015 - 009 - 014 - 030 . 037 57Alye - 003 - 006 - O11 We |} cons 220 260 300 27 60 94 280 199 339 81 Vio}: Seo: tee ay 8.0269 |8. 2770 |------ 8.2770 |8.1622 |7. 5339 |7. 7467. |8. 1268 |8. 4724 |7. 7467 |8.4724 |7. 7179 -O11 O19 We |= 22 -019 -015 - 003 - 006 - 013 - 030 - 006 - 030 . 005 133 140 | ---- 220 260 167 21 54 60 339 300 41 INGE se ee 7.9899 |8. 1622 |8.2770 |----_- 8.2770 |7.9031 |7.5300 |7.4214 |7. 7467 |8.4724 19.3366 |6. 2014 -010 .015 S019 Ma c= 2 = .019 - 008 - 003 - 003 - 006 - 030 ay} W - 000 94 100 140 eae 220 127 161 14 21 300 80 1 IN bak oh nae 7.7726 |7.9326 |8. 1622 |8.2770 |----__ 7.8741 |7.8163 |7.5240 {7.5300 |9.3366 |8. 2220 |7. 5050 - 006 - 009 -015 SOLO) 21) 2e2- . 007 . 007 - 003 - 003 .217 -017 . 003 54 60 100 140) 2252 87 121 155 161 80 40 142 18} ee a oles Bee 9.3359 |8.1514 |7.5339 |7.9031 |7.8741 |-.-_- 9.3362 |9. 2566 |8.4204 |7.0150 |7.8897 |8. 1268 . 217 -014 . 003 - 008 4007). ae 5 PANT . 181 . 026 - 001 - 008 - 013 146 333 193 233 273 ae 214 247 253 352 312 54 Se ee. er =e 9. 2565 |8.4724 |7. 7467 |7.5300 |7.8163 |9.3362 }_--___ 9.3362 |8.2770 |7.7179 |8.0286 |7. 7467 - 181 . 030 - 006 . 003 . 007 R21 74, = (eee = .217 - 019 - 005 - O11 - 006 113 300 339 199 239 146 sie 214 220 319 279 21 pig 9k bean. 8S 9.1076 |8. 5711 |8.1268 |7.4214 |7. 5240 |9. 2566 |9.3362 |_____- 7. 5885 17.8920 |8.0039 |7. 5339 . 128 . 037 - 013 . 003 . 003 .181 QUT. 1 | st 22 . 003 - 008 . 010 . 003 79 266 306 346 205 113 146 tseet 186 285 245 167 NOS ye af yk a 8.3896 |9.3366 |8.4724 |7. 7467 |7.53800 |8. 4204 |8.2770 |7.5385 |_____- 8.0286 |8. 1647 |7. 5300 - 025 . 217 - 030 . 006 - 003 . 026 - 019 BOOS ene -O11 .015 . 003 73 80 300 339 199 107 140 1A Vs gee 279 239 161 pee oe BEES Ne 7. 5223 |7. 5300 |7. 7467 |8.4724 |9.3366 |7.0150 |7. 7179 |7.8920 |8.0286 |_--__- 8.2770 |7. 6680 - 003 . 003 - 006 . 030 . 217 - 001 -005 . 008 SOM S| esses - 019 . 005 154 161 21 60 280 8 41 75 81 eer 140 62 We ees ase oS 7.3907 |7. 7467 |8. 4724 |9. 3366 |8.2220 |7. 8897 |8.0286 |8.0039 |8. 1647 |8.2770 |_-___- 7. 7984 . 002 . 006 . 030 217 .017 - 008 - 011 - 010 . 015 S019 tiie eens . 006 14 21 60 280 320 48 81 115 121 220 sues 102 OSM Eee ae, 8. 2357 18.0286 |7.7179 |6.2014 |7.5050 |8.1268 |7.7467 |7. 5339 |7. 5300 |7. 6680 |7. 7984 |__---. -O17 | .Ol1 | .005 | .000 | .003 | .013 | .006 | .003 | .003 | .005 | .006 | ----. 272 279 319 359 218 306 339 193 199 298 ORS) || snes 260 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued SERIES 326 DAYS. DIURNAL CONSTITUENTS Disturbing constituents (B, C. etc.) Ji Ki Mi O;1 oO P; Qi 2Q Si pl i RL OSS 8.1340 | 8.0698 | 7.8631 | 7.4352 | 8.3392 | 7.8244 | 7.6841 | 8.2809 | 7.0934 Siwes .014 .012 . 007 . 003 . 022 . 007 . 005 .019 - 001 ee 210 241 235 6 249 265 296 230 190 Kay hes 2s 23) Deas 8.1340 | __--_- 8.1340 | 7.7427 | 7.7427 | 9.0470 | 7.8631 | 7.8244 | 9.0723 | 7.5061 TOT4- lh Bese .014 . 006 . 006 -111 . 007 . 007 - 118 - 003 150 ets 210 204 156 219 235 265 199 340 5M FE Se Se ee 8.0698 | 8.1340 | _____- 7.4352 | 7.8631 | 7.6587 | 7.7427 | 7.8631 | 7.7472 | 8.0423 .012 ROLES ies = 2 . 003 . 007 -005 . 006 . 007 . 006 O11 119 150 nee S 354 125 8 204 235 169 310 Opeth SORe es eee 7.8631 | 7.7427 | 7.4852 | ______ 7.7018 | 7.5483 | 8.1340 | 8.0698 | 7.0956 | 8.3386 . 007 . 006 0030) |Ree= 2. - . 005 - 004 .014 . 012 -001 . 022 125 156 6 ee 131 14 210 241 175 315 (0) 0 ah re Dea 7. 4352 | 7.7427 | 7.8631 | 7.7018 | ------ 8.0443 | 7.7204 | 7.6227 | 7.9496 | 6.6245 . 003 . 006 . 007 A005 vale 5 . O11 . 005 . 004 . 009 . 000 354 204 235 229 Bae 243 259 290. 224 184 Pypsoee. Soe as ee 8.3392 | 9.0470 | 7.6587 | 7.5483 | 8.0443 | _____- 7.4186 | 7.7040 | 9.0723 | 7.9254 . 022 .111 . 005 . 004 ob |) Smee . 003 . 005 .118 . 008 lll 141 352 346 117 ae 196 227 161 301 Qie tsk Sexe Ae See L 7.8244 | 7.8631 | 7.7427 | 8.1340 | 7.7204 | 7.4186 | _____- 8.1340 | 7.7258 | 9.2882 . 007 . 007 . 006 . 014 . 005 SOO3Ke) |. Pe . 014 . 005 . 194 95 125 156 150 101 164 bee 210 144 105 2Qh 2 ees eae 7.6841 | 7.8244 | 7.8631 | 8.0698 | 7.6227 | 7.7040 | 8.1340 | _____- 7.7948 | 8.3594 . 005 . 007 . 007 .012 . 004 . 005 OLE a he ee . 006 - 023 64 95 125 119 70 133 150 a 114 75 Spe 2 ees eee 8.2809 | 9.0723 | 7.7472 | 7.0956 | 7.9496 | 9.0723 | 7.7258 | 7.7948 | _.__:_ 7. 7844 . 019 . 118 . 006 - 001 . 009 .118 . 005 sO0GKe | fae . 006 130 161 191 185 136 199 216 246 Be 321 eee 7.0934 | 7.5061 | 8.0423 | 8.3386 | 6.6245 | 7.9254 | 9.2882 | 8.3594 | 7.7844 | _____- . 001 - 003 O11 . 022 . 000 . 008 . 194 . 023 000s | emeeenes 170 20 50 45 176 59 255 285 | 39 =aee HARMONIC ANALYSIS AND PREDICTION OF TIDES 261 Table 29.—Elimination factors—Continued SERIES 326 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) Constituent sought (A) 2 Lo Moe N2 2N Ro Se Ts 2 pe vy. |2SM XO) aon Se ele | ok aan 7.4852 |7.7427 |7.8631 |7.8244 |9.0720 |9.0470 |9.0037 |8.3392 |7.0934 |7.5061 |8.0971 Sone . 003 . 006 . 007 . 007 . 118 lll . 101 . 022 . 001 . 003 . 013 ee 354 204 235 265 199 219 238 249 190 340 53 Ops ie eae 7.4352 |_.---. 8.1340 |8.0698 |7.9526 |8.0858 |8. 3386 8.4852 |9. 2882 |7.4186 |7. 5483 |7.9254 OOS ma eee - 014 . 012 . 009 . 012 . 022 . 031 . 194 . 003 . 004 . 008 6 Bie ps 210 241 271 25 45 64 255 196 346 59 Viger eee 7.7427 18.1840 |------ 8.1340 |8.0698 |7.0956 |7. 5483 |7.9190 |8.3386 |7. 5483 |8.3386 |7. 5347 . 006 .014 | ----- .014 . 012 -001 . 004 . 008 . 022 - 004 .022 - 003 156 150 Aske 210 241 175 14 34 45 346 315 28 IND ees cesenee 7.8631 |8.0698 |8.1340 |_-.--- 8.1340 |7. 7259 |7.4186 |6. 7213 |7. 5483 |8.3386 |9. 2882 16. 3062 . 007 - 012 SOLAS ees 014 - 005 . 003 . 001 . 004 . 022 . 194 - 000 125 119 150 ae 210 144 164 3 14 315 105 178 INES eee eS 7.8244 17.9526 |8.0698 |8.1340 |_____- 7.7948 |7. 7040 |7.5123 |7.4186 |9. 2882 |8.3594 |7. 4010 - 007 . 009 .012 nOU4 | Se. 3. . 006 . 005 - 003 - 003 . 194 . 023 . 003 95 89 119 150 ome 114 133 153 164 105 75 148 Rome a ee 9.0720 |8.0858 |7.0956 |7. 7259 |7.7948 |_--_-- 9.0725 |9.0472 |8. 2809 |7.0444 |7. 7843 |7.9190 -118 .012 - 001 - 005 006) 22-22 .118 .112 - 019 - 001 . 006 . 008 161 335 185 216 246 pre 199 219 230 351 321 34 gees mela 9.0470 |8.3386 |7.5483 |7.4186 |7. 7040 |9.0725 |-____- 9.0725 |8.1340 |7. 5347 |7.9254 |7. 5483 -1l1 . 022 . 004 - 003 . 005 WARS Agee . 118 .014 . 003 . 008 - 004 141 315 346 196 227 161 ace! 199 210 332 301 14 HN Seem eal ra 9.0037 |8.4852 |7.9190 |6. 7213 |7.5123 19.0472 19.0725 |______ 7. 7468 17. 7359 |7.9960 |7.0956 - 101 . 031 . 008 . 001 . 003 . 112 LS a ree - 006 - 005 .010 . 001 122 296 326 357 207 141 161 eee 191 312 282 175 Nowe eee cece 8.3392 |9. 2882 |8.3386 |7.5483 17.4186 |8. 2809 |8. 1340 |7. 7568 |__-__- 7.9254 |8.1912 |7.4186 . 022 . 194 . 022 . 004 . 003 .019 . 014 -006 | _---- . 008 .016 . 003 111 105 315 346 196 130 150 169 Seay 301 271 164 [Tp \erae ates ee ee 7.0934 17.4186 |7.5483 |8.3386 |9. 2882 |7.0444 |7.4347 |7.7359 |7.9254 |_____- 8.1340 |7. 5118 - 001 . 003 . 004 . 022 . 194 - 001 . 003 . 005 A008 | eases .014 . 003 170 164 14 45 255 9 28 48 59 ele 150 43 Yeo cscs oo sson eS 7.5061 17.5483 18.3386 |9. 2882 |8.3594 |7. 7843 |7.9254 |7.9960 |8.1912 |8. 1340 |------ 7. 7477 . 003 . 004 . 022 194 . 023 . 006 . 008 . 010 . 016 J014_-||| 222s - 006 20 14 45 255 285 39 59 78 89 210 Zeke 73 a 8.0971 17.9254 |7. 5347 |6.3062 |7.4010 |7.9190 |7.5483 |7.0956 |7.4186 |7.5118 |7. 7477 |------ .013 | .008 | .003 | .000 | .003 | .008 | .004 | .001 | .003 | .003 | .006 | ----- 307 301 332 182 212 326 346 185 196 317 22 || ecco 262 U. S. COAST AND GEODETIC SURVEY Table 29.—Elimination factors—Continued DIURNAL CONSTITUENTS SERIES 355 DAYS. Constituent sought (A) 7. 5794 . 004 26 Disturbing constituents (B, C, etc.) 7. 5111 - 003 7. 7393 - 005 191 6. 7064 - 001 178 7. 5111 7. 9839 - 010 47 O1 oO Pi 7. 8888 | 8.0444 - 008 011 19 211 6. 7064 | 8. 4581 - 001 . 029 178 190 7.5111 | 7.7393 - 003 - 005 157 169 6. 7060 | 7. 2500 - 001 - 002 175 8 aseses 7.3914 ap ee . 002 =2=4 192 (OLAS |e eae 002) Nie -2=== 168 Pa 7. 3284 | 7. 2957 - 002 - 002 154 167 7.4740 | 7.5554 . 003 . 004 133 146 7. 1838 | 8. 4598 . 002 029 173 185 7.3105 | 7.7296 - 002 - 005 24 36 Q: 2Q St 7. 6331 | 7.6506 | 8.0032 . 004 004 -010 224 245 206 7.5111 | 7.6331 | 8.4598 . 003 . 004 . 029 203 224 185 6. 7064 | 7.5111 | 7.8651 . 001 . 003 - 007 182 203 164 7.9464 | 7.9167 | 6.7729 . 009 . 008 - 001 201 222 3 7. 3284 | 7.4740 | 7.1838 . 002 . 003 . 002 206 227 187 7. 2957 | 7.5554 | 8. 4598 . 002 . 004 . 029 193 214 175 aoe aes 7. 9464 | 7.4212 pp . 009 . 003 ee 201 162 198 219 2525 9. 1482 | 8.3129 | 7.6607 141 021 - 005 230 251 31 HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 29.—Elimination factors—Continued SERIES 355 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) 263 Constituent sought (A) 3 Le Me Na 2N Ra S2 T. 2 1 yes a co 1 co i Qo — wo Ll cel ser el eth coe al cl ce cl ll oe el cl or > for) 3 I [Se] oOo rN co oO oo for) oOo ee ~ oO 7) Ree — oO w [ool ll ol w for) oo [etl a ell cool a cl coll Py 2.000 eee et bo i o ~ oo on bd a a i ee ey for) & a ~I & J _ 2.034 | 4.568 | 66 HARMONIC ANALYSIS AND PREDICTION OF TIDES 267 Table 30.—Products of amplitudes and angular functions for Form 245—Continued So CON OOF Whe 6 v) 8 9 esd met ° Sin Cos Sin Cos Sin Cos Sin Cos 0. 000 6. 000 0. 000 7. 000 0. 000 8. 000 0. 000 9. 000 90 105 5. 999 122 6. 999 . 140 7. 999 157 8.999 89 209 5. 996 244 6. 996 . 279 7.995 314 8. 995 88 314 5. 992 366 6. 990 . 419 7. 989 471 8. 988 if 419 5. 985 . 488 6. 983 . 558 7. 980 628 8. 978 86 523 5.977 - 610 6. 973 . 697 7. 970 784 8. 966 85 627 5. 967 » 732 6. 962 . 836 7. 956 941 8.951 84 731 5. 955 . 853 6. 948 -975 7. 940 1.097 8. 933 83 835 5. 942 . 974 6. 932 1.113 7. 922 1. 253 8.912 82 939 5. 926 1.095 6. 914 1, 251 7. 902 1. 408 8. 889 81 1.042 5. 909 1. 216 6. 894 1. 389 7. 878 1. 563 8. 863 80 1.145 5. 890 1. 336 6. 871 1. 526 7. 853 1.717 8. 835 79 1. 247 5. 869 1. 455 6. 847 1. 663 7. 825 1. 871 8. 803 78 1.350 846 1. 575 6. 821 1. 800 7. 795 2. 025 8. 769 77 1. 452 5. 822 1. 693 6. 792 1. 935 7. 762 2.177 8. 733 76 1. 553 5. 796 1.812 6. 762 2.071 Uo CPX! 2. 329 8. 693 75 1. 654 5. 768 1. 929 6. 729 2. 205 7. 690 2. 481 8. 651 74 1. 754 5. 738 2. 047 6. 694 2. 339 7. 650 2. 631 8. 607 73 1, 854 5. 706 2. 163 6. 657 2. 472 7. 608 2. 781 8. 560 72 1. 953 5. 673 2. 279 6. 619 2. 605 7. 564 2. 930 8. 510 71 2. 052 5. 6388 2. 394 6. 578 2. 736 7.518 3. 078 8. 457 70 2. 150 5. 601 2. 509 6. 535 2. 867 7. 469 3. 225 8. 402 69 2. 248 5. 563 2. 622 6. 490 2.997 7.417 3. 371 8. 345 68 2. 344 5. 523 2. 735 6. 444 3. 126 7. 364 3. 517 8. 284 67 2. 440 5. 481 2. 847 6. 395 3. 254 7. 808 3. 661 8. 222 66 2. 536 5. 438 2. 958 6. 344 3. 381 7. 250 3. 804 8. 157 65 2. 680 5. 393 3. 069 6. 292 3. 507 7. 190 3. 945 8.089 64 2. 724 5. 346 3.178 6. 237 3. 632 7. 128 4. 086 8.019 63 2. 817 5. 298 3. 286 6. 181 3. 756 7. 064 4. 225 7. 947 62 2. 909 5. 248 3. 394 6. 122 3. 878 6. 997 4. 363 7. 872 61 3. 000 5. 196 3. 500 6. 062 4. 000 6. 928 4. 500 7. 794 60 3. 090 5. 143 3. 605 6. 000 4.120 6. 857 4. 635 7. 715 59 3. 180 5. 088 3. 709 5. 936 4. 239 6. 784 4. 769 7. 632 58 3. 268 5. 032 3. 812 5. 871 4.357 6. 709 4. 902 7. 548 57 3. 355 4.974 3. 914 5. 803 4. 474 6. 632 5. 033 7.461 56 3. 441 4.915 4.015 5. 734 4. 589 6. 553 5. 162 7.372 55 3. 527 4. 854 4.115 5. 663 4. 702 6. 472 5. 290 7. 281 54 3.611 4.792 4. 213 5. 590 4.815 6. 389 5. 416 7. 188 53 3. 694 4. 728 4.310 5. 516 4. 925 6. 304 5. 541 7. 092 52 3. 776 4. 663 4, 405 5. 440 5. 035 6. 217 5. 664 6. 994 51 3. 857 4. 596 4. 500 5. 362 5. 142 6. 128 5. 785 6. 894 50 3. 936 4. 528 4. 592 5. 283 5. 248 6. 038 5. 905 6. 792 49 4.015 4. 459 4. 684 5. 202 5. 353 5. 945 6. 022 6. 688 48 4.092 4. 388 4.774 5.119 5. 456 5. 851 6. 138 6. 582 47 4. 168 4. 316 4. 863 5. 035 5. 557 5. 755 6. 252 6. 474 46 4, 243 4. 243 4. 9£0 4.950 5. 657 5. 657 6. 364 6. 364 45 Cos Sin Cos Sin Cos Sin Cos Sin 6 a 8 9 268 Table 31.—For construction of primary stencils ———— Difference Hour 0 +23 | —1 +22 | =2 +21 —3 +20 —4 +19! —5 +18 | —6 +17 —7 +16 —8 +15 | —9 +14 | —10 +13 | —11 +12 | —12 +11 | —13 +10 | —14 +9 |} —15 +8 | —16 +7 | —17 +6 | —18 +5} —19 +4 | —20 +3 | —21 +2 | —22 +1] —23 Difference Hour 0 +23 -1 +22 —2 +21 —3 +20 —4 +19 —5 +18 —6 +17 —7 +16 —8 +15 —9 +14 | —10 +13 | —11 SO} || i +11 | —13 +10 | —14 +9 | —15 +8 | —16 +7 | —-17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 | —22 +1 | —23 U. S. COAST AND GEODETIC SURVEY Constituent 2Q 71 72 73 74 75 76 77 d.h. d.h. d. h. 7 21 14 21 21 20* 8 4 15 4 22 3* 11 11 10* 18 18 17* il 16 1 23 0* 8 8 a 15 15 14* 22 21* 21* 10 5 17 4*| 24 4* 12 sD 11* 19 18* 18* 11 2 IS al Pas Il 9 8* 8* 16 15* 15* 23 22* 22* 12 6 IG) hl 2} 13 12* 12* 20 19* 19% 13 3 20; 2*| 27 2* 10 on os 17 16° 16" 14 0 23 23 7 21 6*| 28 6* 14 13* 13* d.h. d.h. d. h. 77 19 84 18*| 91 18* 78 2 85 1*| 92 1" 9 8* 8 16 15* 15* 23 22* 22% 79 5*| 86 5*) 93 5* 12* 12* 12* 19* 19* LO® 80) (2) 487" 25)" 949 12% Oe Oe 9* 16* 16* 16* 23* 23* 23 Si 6%) "88" 6%] 95" 6 13* 13* 13 20* 20* 20 82 3*| 89 3*| 96 3 10* 10* 10 17* Gf 17 83) (08) 90 0%) 5. 9780 7 lie 7 14* 14* 14 2i* 21* 21 84 4*| 91 4*) 98 4 11* 11* 11 d.h. d. h. 28 20* 35 20 29 3*| 386 3 10* 10 17* 17 30 0*| 37 0 if 7 14* 14 21* 21 31 4*| 38 4 11* 11 18* 18 32s ool 8 8 15 15 22 22 33 (5 40 5 12 12 19 19 34 2 41 2 9 9 16 16 23 23 i Bi 2h 0G 13 13 Constituent 2Q d.h. d. h. 98 18 105 18 99 1 106 1 8 8 15 15 22 22 LOON 5! 1070S 12 12 19 19 101 2 108 2 9 9 16 16 23 23 102 6 109 6 13 13 20 20 18} 83 |} Tbk). 383 10 10 17 16* 104 0 23* U 111 6* 14 ig} 21 20* 105 4 1} By» ll 10* 45 46 47 48 49 114 115 118 119 122 HARMONIC AN'ALYSIS AND PREDICTION OF TIDES 269 TaBLeE 31.—For construction of primary stencils—Continued Difference Constituent 2Q Hour d. h.| d. h. d. h. d. h. d. h. d. h. d. h. d. h. We Iie d. h, 0 140 17 | 147 16*| 154 16*| 161 16 168 16/} 175 15*) 182 15*| 189 15 196 15 | 203 15 +23 —1/| 141 0 23* 23* 23 23 22* 22* 22 22 22 +22 |) —2 7 | 148 6*| 155 6*| 162 6] 169 6 176 5*| 183 5*| 190 5] 197 5 | 204 4* +21 —3 14 13* 13* 13 13 12 12* 12 12 iit +20; —4 20* 20* 20* 20 20 19* 19 19 19 18% +19 —5 | 142 3*| 149 3*| 156 3*) 163 3] 170 3 177 2*| 184 2*| 191 2] 198 2) 205 1* +18 —6 10* 10* 10* 10 10 9* 9* 9 9 8* +17 —7 a Wy ilef 17 17 16* 16* 16 16 15* +16 —8 | 143 O* 150 0*} 157 O*; 164 0} 171 0 23* 23% 23 23 22* +15 —9 ee 7* of ra 7 178 6*| 185 6*| 192 6 199 6 | 206 5* +14 | —10 14* 14* 14 14 14 13* 13* 13 13 12* +13 | —1l 21* 21% 21 21 21 20* 20* 20 20 19* +12 | —12| 144 4*| 151 4*) 158 4 165 4 172 4 179 3*| 186 3*| 193 3 200 3 207 2* +11 | —13 11* 11* 11 li 11 10* 10* 10 10 9* +10 | —14 18* 18* 18 18 18 ipa ie 17 17 16* +9 | —15 | 145 1*/ 152 1*| 159 1 166 1 173 0*| 180 O*| 187 O*| 194 0] 201 0 23* +8 | —16 8* 8* 8 8 Ue ea Uf 7 7 | 208 6* +7) —17 15% 15* 15 15 14* 14* 14* 14 14 13* +6 | —18 22* 225 22 22 21> 21* 21* 21 21 20* +5} —19 | 146 5*| 153 5*| 160 5] 167 5] 174 4*| 181 4*| 188 4*| 195 4] 202 41] 209 3* +4 | —20 12* 12* 12 12 11* 11* 11* 11 11 10* +3 | —21 19* 19* 19 19 18* 18* 18 18 18 es +2 | —22 | 147 2*| 154 2*| 161 2 168° 2 175 1*| 182 1*| 189 1 196 1 203 1] 210 O* +1 | —23 9* 9* 9 9 8* 8* 8 8 8 ee Difference Constituent 2Q Hour d.h.| d.h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. Gales 0 210 14*| 217 14*| 224 14 231 14 238 13*| 245 13*| 252 13 259 13 266 13 | 273 12* +23 —1 21* aie 21 21 20* 20* 20 20 19* 19* +22} -—2 | 211 4*| 218 4*)| 225 4 |] 232 4) 239 3*| 246 3*| 253 3] 260 3) 267 2*| 274 2* +21 —3 11* 11* ll 11 10* 10* 10 10 9* 9* +20 —4 18* 18* 18 18 i7* ile 17 17 16* 16* +19 —5 | 212 1*| 219 1*| 226 1 233 «1 240 0*| 247 O*| 254 0 261 0 23* 23* +18 —6 8* 8* 8 8 7* Te 7 7 | 268 6*| 275 6* +17 —1/ 15* 15* 15 15 14* 14* 14 14 13* 13* +16 —8 22* 22 22 22 21* ies 21 21 20* 20* +15 —9 | 213 5*| 220 5) 227 5 | 284 5 | 241 4*| 248 4*| 255 4) 262 41] 269 3*| 276 3* +14 | —10 12% 12 12 12 11* 11* 11 11 10* 10* +13 | —11 19* 19 19 19 18* 18* 18 18 17* 17* +12 | —12 | 214 2*| 221 2 228 2 235 2 242 1*| 249 1*| 256 1 263 «1 270 O*| 277 0* +11 | —13 9* 9 9 9 8* 8* 8 8 ea lie +10 | —14 16* 16 16 15* 15* 15* 15 15 14* 14* +9 | —15 23* 23 23 22% 22* 22% 22 22 21% 21* +8 | —16 | 215 6*| 222 6 | 229 6| 286 5*| 243 5*| 250 5*| 257 5 | 264 5| 271 4*| 278 4* +7 | —17 13* 13 13 12* 12* 12* 12 12 11* 11* +6 | —18 20* 20 20 19* 19* 19* 19 19 18* 18* +5 | —19 | 216 3*| 223 3 230 3 237 2*| 244 2*| 251 2 258 2 265 2 22 lel 279) le +4 | —20 10* 10 10 g* 9* 9 9 9 8* 8* +3 | —21 17* 17 17 16* 16* 16 16 16 15* ilisy +2 | —22 | 217 0*| 224 0 231 0 23* 23* 23 23 23 22 22* +1 | —23 7* 7 7 | 238 6*)| 245 6*| 252 6] 259 6) 266 6} 273 5*| 280 5? 1 270 U. S. COAST AND GEODETIC SURVEY Table 31.—For construction of primary stencils—Continued d.h.| doh 280 12*) 287 12 19* 19 281 2*| 288 2 9* 9 16* 16 23* 23 282 6*| 289 6 13 13 20 20 283 3 | 290 3 10 10 17 17 284 0 | 291 0 7 7 14 14 21 21 285 4 | 292 4 11 11 18 18 286 1 | 293 1 8 8 15 15 22 22 287 5 | 294 5 300 0* Constituent 2Q Difference Hour 0 +23 —1 +22 —2 +21 —3 +20 —4 +19 —5 +18 | —6 +17 —7 +16 —8 +15 -9 +14 |} —10 +13 } —11 +12 | —12 +11 |] —13 +10 | —14 +9 | —15 +8 | —16 +7 | —17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 | —22 +1 | —23 Difference Hour 0 +23 —1 +22 —2 +21 —3 +20 —4 +19 | —5 +18 | —6 +17 | —7 +16} —8 +15 | —9 +14 | —10 +13 | —11 +12 | —12 +11 | —13 +10 | —14 +9 | —15 +8 | —16 +7 | -—17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 | —22 +1 | —23 dhs |asuh 350 10 | 357 10 17 17 351 0 | 358 0 7 7 14 14 21 21 352 4 | 359 4 11 11 18 18 353 1 | 360 1 8 8 15 14* 22 21* 354 5 | 361 4* 12 11* 19 18* 355 2 | 362 1* 9 8* 16 15* 23 22* 356 6 | 363 5* 13 12* 20 19* 357 3 | 364 2* 307 0* 308 4* Constituent 2Q 311 312 313 314 315 317 5 318 2 319 6 320 3 321 0 14 21 322 4 21 329 4 Constituent Q HARMONIC ANALYSIS AND PREDICTION OF TIDES De TaBLeE 31.—For construction of primary stencils—Continued Difference Constituent Q Hour d d d. h d. h. d. h d. h d. h ih, lie \) Gh Ip 0 66 9*| 75 18* 85 3% 94 12 103 21 113 6 122 14*| 1381 23*| 141 8 | 150 17 +23 = 19 76 4 12* 21*| 104 6* 15 123 0 132 9 17") Tot 2* +22 —2| 67 4* 13* 2 95 7 15*| 114 O* g* 18 142 3 12 +21 —3 14 22* 86 7* 16*} 105 1 10 18*| 133 3% 12* 21 +20 —4 23 77 8 7 96 1* 10* 19*| 124 4 13 PAGAL GP (the +19 —5 | 68 8* 17* 87 2 11 20 115 4* 13* 227) 143 907 16 +18 —6 18 | 78 3 11* 20*) 106 5 14 134 7* 167) 153) 1% +17 —7 69 3* 12 21 97 6 14* 23*| 125 8 17 144 2 10* +16 —8 12* 21 88 6* 15 107 0 116 8* VW") 135) 2* 11 20 +15 —9 22 79 7 15* 98 0* g* 18 126 3 11* 20*| 154 5* +14 | —10 70 7* 16 89 1 10 18*| 117 3* 12* 1 145 6 14* +13 | —11 17} 80 1* 10* 19 108 4 13 21*| 1386 6* 15*| 155 0 +12 | —12 71 2 ll 20 99 4* 13* 22 IP 2 16 146 0* 9* +11 | —13 ike 20* 90 5 14 23 is) ely 16*; 137 1 10 1 +10 | —14 21 81 5* 14* 23*| 109 8 17 128 2 10* 19*| 156 4 +9 | —15 72 6* 15 91 0 100 8* Weay wile) oh 11 20 147 5 13* +8 | —16 15*| 82 0* 9 18 110 3 11* 20*| 188 5* 14 23 +7 | —17 73°41 10 18*| 101 3* 12 21 129 6 14* 23*| 157 8* +6 | —18 10* 19 92 4 13 21*; 120 6* 15 139 0 148 9 17* +5 | —19 19*| 83 4* 13* 22 111 7 16 130 0* 9* 18 | 158 3 +4 | —20 74 5 14 22min OZ Mardi 16*} 121 1 10 19 149 3* 12* +3 | —21 14* 23* 93 8 17 112 1* 10* 19*; 140 4 13 22 +2) —22 |) 75 O 84 8* 17*| 103 2* 11 20 131 4*| 13* 22*| 159 7 +1 | —23 9 18 94 3 11* 20a 22) ton 14 23 150 7* 16* Difference Constituent Q Hour (hs Pe | GE lb d. h d. h d. h d. h d. h d. h d. h.| d. h 160 2 | 169 10*| 178 19*| 188 4*| 197 13 206 22 | 216 6*| 225 15*| 235 O*| 244 9 +23 —l 11 20 179 5 13* 225 200 lig 16 226 1 g* 18* +22 —2 20*; 170 5* 14 23 198 8 16*| 217 1* 10* 19 | 245 4 +21 —3 | 161 6 15 23*| 189 8* 1 208 2 11 19*| 286 4* 1B} +20 —4 15*;| 171 0 180 9 18 199 2* 11* 20 PPA,’ {5 14 oie +19 —5 | 162 0O* g* 18 190 3 1 21 218 5* 14* 23 | 246 8 +18 —6 10 19 181 3* 12* 21*; 209 6 15 228 0| 237 8* fg +17 —7 19*| 172 4 13 22 | 200 6* 15*} 219 0* 1 247 2* +16 —8 | 163 5 13* 225) LOI, 16 210 1 hy 18*| 2388 3* 12 +15 —9 14 23 182 8 16*| 201 1* 10 19 229 4 12% 21* +14 | —10 23*| 173 8* 17 192 2 11 19 220 4* 13 22 | 248 7 +13 | —11 | 164 9 17*| 183 2* 11* 2) | Bak ih 14 225\\ 239) wie 16 +12 | —12 18*| 174 3 12 205 |) 202) os 14* 23 230 8 17 | 249 1* +11 | —13 | 165 3* 12* 21*| 193 6 1 OBS PAL (Sh4 17*| 240 2 11 +10 | —14 13 22 184 6* 15*| 203 O*| 212 9 18 | 231 2* ilies 20* +9 | —15 22*| 175 7 16 194 1 g* 18*| 222 3 12 21250 noe +8 | —16 | 166 7* 16*;| 185 1* 10 19 213 4 12% 21*| 241 6 15 Sell |) ile 17 | 176 2 10* 19*| 204 4* 13 22 | 232) 7 15*| 251 0* +6 | —18 | 167 2* 11* 20 195 5 13* BP} GPRY oe 16 242 1 10 +5 | —19 12 20*| 186 5* 14* 23 214 8 16*| 233 1* 10* 19 +4 | —20 1 | 177 6 15 23*| 205 8* 17*| 224 2 11 19*| 252 4* +3 | —21 | 168 6* 15*| 187 0 196 9 18 | 215 2* 11* 20*| 243 5 14 -~ +2) —22 16 | 178 1 9* 18*| 206 3 12 21 234 5* 14* 23* +1 | —23 | 169 1* 10 19 197 3* he 21*) 225 6 15 | 244 0 | 253 8* 272 U. S. COAST AND GEODETIC SURVEY TaBLeE 31.—For construction of primary stencils—Continued Difference Constituent Q Hour d. h d. h. d. h. d. h. d. h d. h. d. h d. h. d. h. | d. h. 0 253 18 | 263 3 272 11*| 281 20*| 291 5 300 14 309 23 319 7*| 328 16*| 338 1* +23 —1 | 254 3* 12 21 282° 6 14* *| 310 8 17 329 2 10* +22 —2 944 21*| 273 6* 15 292 0 301 9 el 320) 2 11 20 +21} —3 22 | 264 7 15*} 283 0* g* 18} 311 3 12 20*| 339 5* +20 —4 | 255 7* 16*| 274 1 10 18*| 302 3* 12* 21 330 6 15 +19 —5 17 | 265 1* 10* 19*| 293 4 13 21*| 321 6% 15*| 340 0 +18 | —6 | 256 2 11 20 | 284 4* 13* 9B) || BHI, 7/ UGE rool Om 9* +17 —7 11* A 27(as 1G 14 23 303 4* 16*| 322 1 10 19 +16 —8 21 | 266 5* 14* 23*| 294 8 17 313 2 10* 19*| 341 4 +15 | —9 | 257 6* 15 76 0 | 285 8* 17*| 304 2* 11 20 | 332 5 13* +14 | —10 15*| 267 0* 9* 18 | 295 3 11* 207 |ozonnon 14 23 +13 | —11 | 258 1 10 18*| 286 3* 12% 21 314 6 14* 23*| 342 98% +12 | —12 10* 19 | 277 4 13 21*| 305 6* 15*| 324 0] 333 9 17* +11 | —13 20 | 268 4* 13* 22 296 7 16 ST5mOe 9* 18*| 343 3 +10 | —14 | 259 5 14 23 281) eal 16*| 306 1 10 19 334 3* 12% +9 | —15 14* 23*| 278 8 Tlf |) Pah alk 10* 19*} 325 4 13 22 +8 | —16 | 260 0 | 269 8* 17*| 288 2* 11 20 315 4* 13* 22*| 344 7 +7 | —17 9 18 | 279 3 i 20*| 307 5* 14 23 | 335 7* 16* +6 | —18 18*| 270 3* 12 21 298 6 14* 3*| 326 8* 17 | 345 2 +5 | —19 | 261 4 13 21*| 289 6* 15 | 308 0} 317 9 fl) BBW) 2 ifs +4 | —20 13* 22 280 7 16 299 0* 9 18 327 3 12 20* +3 |} —21 OPIN OY AN fe 16*| 290 1 10 19*| 318) 3% 12* 21 | 346 6 +2 | —22 | 262 8 17 281 le 10* 19*; 309 4 13 21*| 33 6% 15* +1 | —23 ef) 272 2 11 20 | 300 4* 1S 22*| 328 7 16 | 347 0* Difference Constituent Q Constituent Hour d. h d. h. dah: d. h. d. h. d. h d. h dah (hy Ide WG Tis 0 347 10 | 356 19 366 3* 10) 10 15* 20 11 30 6* 40 2 49 21*| 59 17 +23 -—l 19*| 357 4* 13 on Wk al 20* 16 12 a) ey 0) 8 +22 —2 | 348 5 13* 22* 15* 11 21 6* 31 92 21* 17 12* +21 —3 14 23 367 8 73 le 21 16* 12 41 7* 51 3 22* +20} —4 23*| 358 8* 17 11 2G | ee2inee Zi 17 12*| 61 8* +19 —5 | 349 9 18 368 2* 21 16* 12 ay Yh 42 3 22* 18 +18 —6 18*| 359 3 12 a} Gy By OF. 22 eh 13 52 8*| 62 4 +17 —7 | 350 3* 12% 21* 16* 12 PR. A 33 3 22% 18 13% +16 —8 13 22 369 6* 425 22 igs 13 43 8* 53 4 23* +15 —9 22*| 360 7 16 12 14 7* 24 3 225 18* 14 63 9* +14 | —10 | 351 8 IGF BY AN) ke 22 17* 13 34 8* 44 4 BY 19 +13 | —11 17M SOLS 52 tee as 5 8 15 3* 23 18* 14 54 9*| 64 5 +12 | —12 | 352 2* 51) (63 eee igh 13 258s 35 4 23% 19 15 +11 | -—13 12 208 ee eee 6 3* 23 18* 14 45 9* 55 5 65 0* +10 | —14 Pal || BUY) @: ecco necee 13 16 9 26 4* 36 0 19* 15 10* +9 | —15 | 353 6* V5R EER ae 23 18* 14 g* 46 5 56 0* 20 +8 | —16 TG || BBW) eens oe 7 9 7 4S 270) 19* 15 10*| 66 6 +7 | —17 | 354 1* LOM eS as ee 18* 14 9* By By él 20* 16 +6 | —18 10* LOS eee oer 8 4* 18 0 19* 15 10* 57 66 67 1* +5 | —19 20 3645 159) eae 14* 10 28508 38 1 20* 16 iii +4 | —20 | 355 5* AU i 9 0 19* 15 10* 48 6 58 2 Pale +3 | —21 15 Pye | eee OS 10 19 5* 29 #1 20* 16 1 6SiG +2 | —22 | 356 Shor OME eek Sues 19* 15* 11 39 6* 49 2 21% 17 +1 |) —23 g* iF ie beers TOMES 4 | 920 20* 16 Tb Gt) 7 |) Gt) Be HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 31.—For construction of primary stencils—Continued Difforence Hour 0 +23 —1 +22 —2 +21 —3 +20 —4 +19 | —5 +18 —6 +17 —7 +16 —8 +15 | —9 +14 | —10 +13 | —11 +12 | —12 +11 | —13 +10 | —14 +9 | —15 +8 | —16 +7 | -17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 | —22 +1 | —23 Difference Hour 0 +23 | —1 +22) —2 +21 —3 +20 —4 +19 —5 +18 | —6 +17 | -—7 +16 —8 +15 —9 +14 | —10 +13 | —11 +12 | —12 +11 | —13 +10 | —14 +9 | —15 +8 | —16 +7 | -17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 | —22 +1 |) —23 Constituent p 273 274 U. S. COAST AND GEODETIC SURVEY Table 31.—For construction of primary stencils—Continued Difference Constituent p Hour d. h d. h d. h d. h d. h d. h d. h d. h d.h 0 265 19 | 275 14*| 285 10 295 5*| 305 1 314 20*| 324 16 334 12 | 344 7* +23 —1 | 266 5 | 276 O* 20 oe 11 als GH) Shi 9 PA 17 +22 —2 14* 10 286 6 296 1* 21 16* 12 355 7*| 345 3 +21 —3 | 267 0* 2 oe 11 306 6*| 316 2 PAL 17 125 +20 —4 10*| 277 6 287 1* 21 ani 1D YA: deal 22* +19 —5 20 15% 11 297 6*) 307 2* 22 es 13 346 8* +18 —6 | 268 6 | 278 1* 21 16* 12 Sian oe Camo: op: 18 +17 —7 16 11*| 288 7 298 2* 22 igh 13 337 8*| 347 4 +16 —8 | 269 1* 21 6* 12 308 7*| 318 3 18* 14 +15 -—9 11*| 279 7 289 2* 22 Whe 13 328 8*| 338 4 IBY +14 | —10 21 16* 12*| 299 8 309 3* 23 18* 14} 348 9* +13 | —11 | 270 7 | 280 2* 22 17* 13 319 8*| 329 4 23% 1 +12 | —12 1 12*| 290 8 300 3* 23 18* 14 339 9*| 349 5 +11 | —13 | 271 2* 22 17* 13 310 9 320 4*| 330 0 19* 1 +10 | —14 12*| 281 8 291 3* 23 18* 14 *| 340) 5 350 0* +9 | —15 Pp 18 13*} 301 9 oulily Zh) Spal (1) 19* 15 10* +8 | —16 | 272 8 | 282 3* 2 18* 14 OF 330 (5%) 3405 ol 20* +7 | -17 18 13*| 292 9 302 4*| 312 0 19* 15 10*| 351 6 +6 | —18 | 273 3* 2am 19 14* 10 B22) oOsle ooceL 20* 16 +5 | —19 13*| 283 9 293 4*| 303 O 19* 15 10*| 342 6 352 2 +4 | —20 Zon 1 14* 10 S13 Mos Olona 20* 16 ily +3 | —21 | 274 9 | 284 4*| 294 0 19* 16F-! 11 333 6*] 343) 2 21* +2 | —22 19 14* 10 304 5*) 314 1 20* 16 iA BER 7 +1 } —23 | 275 5 | 285 O* 20 15 11 324 (6*) 334 2 21% 1 Difference p - Constituent O Hour d. h d. h d. h d. h. d. h. d. h d.h d. h d. h 0 363 22* iL 14 22* 29 3 43 7* 57 12 71 16* 85 21 100 2 +23 —1 | 364 8 8 15 12* 17 ORK! 58 2 123 tf 86 11* 16 +22 —2 18 22 16) .2* 30 7 44 12 16* 21 Sirs | 10teG +21 —3 | 365 4 2 12 17 21s 45 2 59 6* 73 11 16 20* +20 —4 13* By Ph. zl 7 By Bp Ui bee 16 21 ed ie 88 6 102 10* +19 —5 23* 16* 21 32 2 46 6* 60 11 15* 20 103 1 +18 —6 | 366 9 4 7 18 11* 16 20* 61a wom 89 10* 15 +17 —7 19 21 191s 33 «6 47 11 15* 20 90 O*} 104 5 +16 —8 | 367 5 5 11 16 20* 48 1 62 5* 76 10 15 19* +15 —9 14* Gis 20 6 34 10* 15 20 7 10% 91 5 105 9* +14 | —10 | 368 0* 15% 20 ni al 49 5* 63 10 14* 19 106 0 +13 | —11 10* v6 PA Kas 15 19* 64 0 (Seno) 92 9* 14 +12 | —12 20 2205 36 5 50 9* 14* 19 23*| 107 4 +11 | —13 | 369 6 8 10 14* 19* 51 0 65 4* 79 9 93 13* 18* +10 | —14 15* 9 0O* 23 5 3¢ 9% 14 18* 235 94 4 108 8* —9)|) —15 9370 14 14* 19 23* Fp} tli 66 9 80 13* 18 22* are. || SG esa se 10 4* 24 9* 38 14 18* 23 Sigs 95 8*| 109 13 art |) Sale ee ee 8 19 23* 39 4 53) 8* 67 13* 18 22*| 110 3 OMe — 18) | tee 11 9 OAS) IBY 18* 23 68 3* 82 8 96 12* iiss co tial cI Ke) ge eS on 26 4 40 8* 54 13 izes 22* 97 3 ie +4 | —20 |__-______ 10) Tek 1 22* it) BY 69 8 83 12* 1 2A 8. | = 21h ee 13) 3* at 8* 41 13 17* 22 84 2* 98° 37%) S212 =E2) | —22) ie 18 22* 42 3 56 7* 70 12* 17 215 eilomes, Gril. || OR} ee 14 8 28nl27 ihe 22 Til Be Shinai 99 11* 16 HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 31.—For construction of primary stencils—Continued Difference Hours 0 +23) —1 +22} —2 +21] —3 +20) —4 +19 | —5 +18 | —6 +17 | —-—7 +16 | —8 +15 | —9 +14 | —10 +13 | —11 +12 | —12 +11 | —13 +10 | —14 +9 | —15 +8 | —16 +7 | —17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 | —22 +1 | —23 Difference Hour 0 +23 | —1 +22/ —2 +21); —3 +20} —4 +19 | —5 +18 | —6 +17 | —7 +16 | —8 +15 | —9 +14 | —10 +13 | —11 +12 | —12 +11 | —13 +10 | —14 +9 | —15 +8 |} —16 +7 | —-17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 | —22 136 3* Constituent O 168 2 CTO Ge | RT rk Tp 199 10 | 213 14%) 22719 | 242 0 200 0| 214 5| 228 9* 14 14* 19 23*| 243 4 201 4*| 215 9 | 229 14 18* 19 23*| 230 4] 244 8* 202 9| 216 13* 18 23 217 4| 231 8%) 245 13 203 13* 18 22*| 246 3 204 3] 218 8) 23213) | ITs 18 22*| 233 205 8| 219 12* 17 2i* 22 | 220 2*| 234 7*| 248 12 206 12* 17 21*| 249 2 Pane Pie) OPT ee) als TEE 16* 16* 21*| 236 2| 250 6* 208 7 | 222 11* 16 20* ai || 293 1*| 2377 6%} 251 11 209 11* 16 2081 252 i 210 1s] 224 6 | 238 10"| 152 15 20*| 239 1| 253 5 211 6 | 225 10* 15 19 226 0% 240 5 | 254 10 212 10 15 19*| 255 0 213 0*| 227 5| 241 9* 14 otis | Gh Te Ge Be WB Uh. 341 8| 355 12*| 36917| 1 0 22h ase) 13 | a70 ae 8 SOTO TG Ihceaesie a 22 S43 026 |b doz, 7)|eemmeiees 2 12* 17 Pi aE eh Bee Bez) 359112 |bsael 17 OA goa) P79 Raeaeees: 47 345 11* 16g Heat ae 2i* 346) Fe fe 360) 6p eatunmel 5 1l* 15* 20s |b: ois 6 2 347 6*| 361 10%) - 16 20) | s62; 502 Een ee Taner 348 10* (ein Rees 20* BG) To) BY) 16) he 8 11 14* 199 /R eae oma S50 Mann NS6 (Ome ieee 15* 19 ses es 10 5* Beit) (OE |er3G5 14 | eomeneee 20 sl 366) 4a Eee eee 11 10 BOP) TEE ie Te | eae 12 _0* a53, A/F sey She lene ee 14* 18 77 eens 13 5 254) (Sill) 268 13) ewme cS 19 22%|) 1369) (ahi niael Sei 14 9* 275 276 U. S. COAST AND GEODETIC SURVEY Table 31.—For construction of primary stencils—Continued Difference Constituent 2N Hour Ch ide \) Ws to Ne Ob ide d. h. d. h. d. h. 0 29 6 43 12 57 18 72 0 86 6 100 12* +23 —1 20 44 2 58 8* 14* 20*| 101 2* +22 —2 30 10* 16* 22% or 4s 87 10* 17 +21 —3 31 0*| 45 6* 59 13 19 88 1 102 7 +20 —4 15 21 60 3 74 9 15 21* +19 —5 32) 5 46 11 17 23% 89 5*} 103 11* +18 —6 19*} 47 1* 617 (owes 20 104 2 +17 —7 33 9 15* 22 76 4 90 10 1 +16 —8 34 0 48 6 62 12 18 91 O*| 105 6* +15 -9 14 20 63a25 hin 8% 14* 20* +14 | —10 35 4*| 49 10* 16* 22* 92 5 106 11 +13 | —11 18*} 50 O* 64 7 78 13 19 107 1 +12 | —12 36 9 15 21 79 3 93 9* 15% +11 | —13 23 i 65 11* ee 23%) 08" 5% +10 | —14 37 13* 19* 66) 15 80 7* 94 14 20 +9 | —15 Bedi fay ak es 16 22 95 4 109 10 +8 | —16 18 53 0 67 6 81 12 18*} 110 O* +7 | —17 39 8 14 20* 82 2* 96 8* 14* +6 | —18 22554 As 68 10* 16* 23 TTT 5 +5 | —19 40 12* 19 69 1 Said 97 13 19 +4 | —20 41 3 55 9 15 21 98 3*| 112 9 +3 | —21 17 23* Bs 84 11* Ugh 23 +2 | —22 42.7") 56" 13* 19* Gk: 99 8 113 14 +1 | —23 Atal! Gy ~%! 71 10 16 22 114 4 Difference Constituent 2N Hour ad ha\|\aids he d. h. (ys lb d. h d. h. 0 171 19 | 186 1 200 7 214 13 228 19 243 1* +23 —1{|172 9 15 QGP 2150 rom) 2295 9% 15* +22 —2 23°) 187 54) 20K 11> 17* 23*| 244 6 +21 —3 | 173 13* 97 2026 ce 216 8 230 14 20 +20 —4| 174 4 | 188 10 16 22 231 4 245 10* +19 —5 18 | 189 0 2037672 ele* 18*| 246 0O* +18 —6 | 175 8* 14* 205) 218" 2%) 2325 9 15 +17 —7 22*| 190 4*| 204 11 17 23 247 5 +16 | -—8 | 176 13 TOW 205) | 219 ee 2338013" 19* +15 -—9]177 3); 191 9 15* 21*| 234 3*| 248 9* +14 | —10 ef 234) 206° 52) 220) 10% 18 249 0 +13 | —11 | 178 7*) 192 13* 20 221972 235 8 14 +12 | —12 22} 193 4 207 10 16 22*| 250 4* +11 | —13 | 179 12 18 208) OF) 222) 6510 236012* 18* +10 | —14] 180 2*| 194 8* 14* 205 238 13 251 9 +9 | —15 16* 22*| 209 5 223 11 17 23 +8 | —16 |] 181 7] 195 13 19 224 1 238) | weo2los +7 | —17 21 | 196 3 210 9* 15% PAE) OER Biv +6 | —18 | 182 11* 17% 23min 22Do| zoo 18 +5 | —19 | 183 1*| 197 8 211 14 20 240 2 254 8 +4 | —20 16 22 212 4 226 10 16* 22* +3 | —21 | 184 6] 198 12* 18*}| 227 O*| 247 6*)) 255 12* +2 | —22 20*| 199 2*) 213 8* 14* 21 256 3 +1 } —23 | 185 10* 17 23 228 5 242 11 17 HARMONIC ALY ANALYSIS AND PREDICTION OF 17 TIDES 2 Table 31.—F .—For co [ nstruction of primary stencil cils—Continu ed Difference Constituent 2N Hour eae Consti 0 ann LP d.h. d.h eer ituent L Lg} |] al 5 328 14 | 342 20 d.h. ‘ip +29 | —2 | 315 22) 329 4,| 343 10° 357 2 ha) deh. | dh +21| —3 | 316 at] 18*| 344 OF 16* cae ee CRUE | COTE el +20 | —4 2*| 330 8* 3 358 6* 33 16 2* Bt 45 0*| 59 19 d. h. Bg | es 17 3| 3 21 i 17*| 31 11* 15 60 9* 74 13 317 7 | 331 13 45 5] 359 11 2 13 17 8 ll 46 6 9*| 75 4 +13 | —6 | Sep el oe Sa eae so lieu ced ls mache BE eae’ Pagal ee| eae Gi seseck eh ale 15 | 76 9° pO arg 310 = 17*| 347 15*| 4 8) 48 2" ony 77 0* ay ma ee oF 333 8 0] 361 <6 : ae 19 4* es 0 15 tas 22 20 19 17 4113 unt 320 6*| 334 19* 348 4*) 362 10* 15*| 20 10 ae 13*| 49 8 oa 11*| 78 6 20*| 335 2*| 3 18*| 363 0* |) Bik Oe 4 22" 2 20* Beto} || a 49 9 15 21 15* 19 | 50 13+ 17| 79 11* +11 cae 321 11 17 712) 22 6 ae 10| 51 4 8 ies 80 2° es 21* 9* Fa) abl oe oath eS on Srpeelagestags| oe ot soca ee Oa OE ai |) Say 20 | 338 2| 3 18} 366 0 8| 24 2* 53 O* 22* 324 10 16* 52 8 14 23 17 21 15 19 | 82 13 45) —1 | 2s ee eee ea +5 | —19 0*| 339 6*| 353 12* 29* 1 20* 1 18* +4 | — 1s 2 12 18* 7| 55 11* 5| 84 9 ag on 326 5 | 340 11 354 aS 368 9/ 12 th 26 13*| 41 8 70 6) 8 0 +2 | =22 | 327 19|| 341 1s) 355 72) 369 FM ES a7 ee pre Pepste as 23* 21* 15* 42 13* 71 11*| 86 5* | 342 6 | 356 12 Bee 3") 14 6 29 i 43 4 Ze ope ee 2: 20* i 21 15* 19 58 13* IW eet ake Difference 44 g* 59 4 73 7*| 88 2 22+ 17 Hour aL Constituent u 0 ote d.h. d gee esa ate 7*| 104 2 hei) duh. 38 | Sh lh egee lina oe A ee ety | ty li | 23 (een soe llerite CoS aa ena dat aaa aS) MGT ea, ad ohe. | bie. Ne +20| —4 ieee ae feel tase oe roel Me cabal 18 Oz ours horl 298 +19} —5] 92 9* re oi 121 7* ae ie 150 5* be 179 3* al 208 1* 19* +18 ee 16* 20 | 165 14* Hall tod cael agus |e +16} —8| 94 15 | 108 9" ee By ey Z| Tip} ae os 18 | 210 io 2 16 +15 | —9 5*| 109 0 IR 22 16* 20 | 181 14* BBS +14 | —10 9 20* 15 124 8 138 13 153 7 167 11 182 5 196 8*| 211 3 Hig} =u] 96-2] 7° | Ba ase eal Be 168 i*| 20 | 197 14*| 212 8] 297 oi * +12 eh el vob) Ee ep 165 | 88 10, ee el 3 +i | 3 7/1111! 126 5* peeing Pe el PON re rullena 3 cant ees ae 6 5 141 0 , 16 | 199 10} ait 5 | oe roe as easel ae, Sees a iee| tee aba AeO Aaa else Ad ae eel Areutea @ fo on) Ee ep lets Oca 200 1*} | _ 19%| 229 14 aba 18*] 114 a2 16*| 143 a 14*| 172 aS 186 12*| 201 “ 215 10*| 230 5 46 | —18 3 129 7*| 144 1* 158 5* m 187 3 21* 216 1 19* sapere 18+] 130 13,) 145 16*| 159 11) 174 5 m8 ait) 36 _— * +2 | —22 102 5*| 117 0 131 i 22 aw ae 20 | 189 ae 18 | 218 12*| 233 6° 41 | —93 20 15 g*| 146 12*| 1 175 10*| 19 204 g*| 219 3 6 103 11 | 118 5* ee 9*| 147 3* 61 sf 176 1* She 23* a F 217 0 18*} 162 12* 16} 191 10* 205 14 | 220 8* ee 12 Wie 7 || 3 SC 206 5 *| 235 3 , 19] 201 14 Te 14 | 236 8* 278 U. 8S. COAST AND GEODETIC SURVEY Table 31.—For construction of primary stencils—Continued Difference Hour d. h.| d. h. d. h 0 236 23 | 251 17*| 266 12 +23 | —1 | 237 14 | 252 8*| 267 2* +22 | —2/| 238 5 23 17* +21 —3 19*| 253 14 | 268 8* +20 | —4 | 239 10*| 254 4* +19 | —5 |} 240 1 19 | 269 14 +18 —6 16 | 255 10 270 4* +17 | —7 | 241 6*| 256 1 19* +16 | -—8 21* 15*| 271 10 +15 —9 | 242 12 | 257 6*| 272 1 4-14 | —10 | 2438 3 21* 15* +13 | —11 17*| 258 12 PAB) (fe +12 | —12 | 244 8*) 259 3 21 +11 | —13 23 17*| 274 12 +10 | —14 | 245 14 | 260 8*) 275 2* +9 | —15 | 246 4* 23 ie fe +8 | —16 19*| 261 14 | 276 8 +7 | —17 | 247 10 | 262 4* 23 +6 | —18 | 248 1 LOSI Zia Wor +5 | —19 16 | 268 10 | 278 4* +4 | —20 | 249 6*| 264 1 19* +3 | —21 pit 15*} 279 10 +2 | —22 | 250 12 | 265 6*; 280 1 +1] —23 | 251 3 21 15% Difference Flour (iy tis) ia 12s dais 0 10) 19 20* 39 2 +23 —1 10*; 20 16 2 +22 —2 PA» fel Pa ral 40 16* +21 —3 3) ol 22) 6" 41 11* +20 —4 20 ORY ity 42 7 +19 —5 4 15* 20* 43 2 +18 —6 5 10*| 24 16 21% +17 —7 Gos) (25 44 16* +16 —8 Caan 26 6* 45 12 +15} —9 207 27 Gee, +14 | —10 8 15* 21 CY) De +13 | —11 9 10*; 28 16 21* +12 | —12 10 6 29k 48 17 +11 | —13 ugk al 30 6* 49 12 +10 | —14 Atal Bylo OOMed +9) —15 12 15* 21 ail Os +8} —16} 1311] 3216 oie +7 | —17 14 6 Bhs 8 led 52) 17 +6! —18|] 15 1] 34 6*| 5312 +5 | —19 20") 35) 2 SAN Te +4 | —20 16 15* 21 Ny p43 +3 | —21 17 11} 36 16* 22 +2} —22| 18 6| 3711*| 5617 +1] —23} 19 1* 38 7 yf 1044 Constituent u d. h d. h. d. h. d. h 310 19 | 325 13%) 340 8 | 355 2 311 10 326 4 22* 312 0 19 | 341 13*; 356 7* 15*| 327 9*| 342 4 22* 313 6 | 328 0* 19 | 357 13 21 15*| 343 9*) 358 4 314 11*} 329 6| 344 O* 19 315 2% 21 15 | 359 9* 17 | 330 11*| 345 6] 360 O* 316 8} 331 2* 20* 15 22% 17 | 346 11*) 361 6 317 13*; 332 8) 347 2 20* 318 4* 22 17 | 362 11* 19 | 333 13*| 348 8 | 363 2 319 10 | 334 4 22* 17 320 0* 19 | 349 13*) 364 7* 15*| 335 9*) 350 4 22* 321 6] 336 0* 19 | 365 13 21 15 | 351 9*) 366 4 322 11*| 337 oe 352 0* 18* 323 2* 0 15) SG7AN 9% 17 | 338 11*} 353 6] 368 0 324 8 | 339 2* 20* 15 22* 17 | 354 11*; 369 6 d. h. d. h. d. h d. h. 96 18*) 116 0} 1385 5*) 154 11 STASS 19 | 1386 O*| 155 6 98 9| 117 14* 20 | 156 1* 99 4} 118 9*| 187 15 20* 23*/ 119 5} 138 10*| 157 15* 100 18*; 120 0; 139 5*| 158 11 101 14 19*} 140 0*) 159 6 102 9} 121 14* 20} 160 1* 103 4) 122 9*| 141 15 20* 23*) 123 5] 142 10*| 161 16 104 18*} 124 0} 143 5*| 162 11 105 14 19*| 144 1] 163 6* 106 9} 125 14* 20] 164 1* 107 4*| 12610] 145 15* 21 23*| 127 5} 146 10*| 165 16 108 19 | 128 O*| 147 6] 166 11 109 14 19%) 1485S 5 167aon 110 9*) 129 15 20; 168 1* 111 4*| 13010)| 149 15* 21 23*| 131 5] 150 10*| 169 16 11219) | 2320") Fok 16%) SOs 113 14 19%) set2 aly |e gos 114 9*| 133 15 20*| 172 2 115 4*| 13410] 153 15* 21 d. h. 369 20* 370 11* HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 31.—For construction of primary stencils—Continued Difference Hour 0 +23 -1 +22 —2 +21 —3 +20 | —4 +19 —5 +18 | —6 +17 | —7 +16 —8 +15 | —9 +14 | —10 +13 | —11 +12 | —12 +11 |} —13 +10 | —14 +9 | —15 +8 | —16 +7 | —17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 | —22 +1 | —23 Difference Hour 0 +23 —l +22 —2 +21 -3 +20) —4 +19 —5 +18 | —6 +17| —7 +16 —8 +15 | —9 +14 } —10 +13 |} —11 +12 | —12 +11 | —13 +10 | —14 +9 | —15 +8 | —16 +7 | —17 +6 | —18 +5 | —19 +4 | —20 +3 | —21 +2 |) —22 a = COND PP wr He’ _ et bho Qmornen SOIC Tle * 227 8% Constituent N Gs Ie 269 19* 270 15 271 10 280 5* 306 16 120 19 322 16* 323 12 324 7 325 2* 21* 326 16* 361 3*! 22* 362 18 363 13 364 8*! 365 al d. h. 161 23 162 19* 163 15* 164 11* 165 8 166 4 167 0 20* 168 16* 169 12* 170 9 171 5 172 1 21” 173 17* 174 13* 175 10 176 6 177 2 22* 178 18* 179 14* 180 11 181 7 279 280 U. S. COAST AND GEODETIC SURVEY Table 31.—For construction of primary stencils—Continued Difference Hour 0 +23 —1 +22 —2 +21 —3 +20 —4 +19 —5 +18 —6 +17 —7 +16 —8 +15 -—9 +14 —10 +13 -l11 +12 —12 +11 —13 +10 —14 +9 —15 +8 —16 +7 —17 +6 —18 +5 —19 +4 —20 +3 —21 +2 —22 Constituent v d. h. 262 19* 263 15* 264 12 265 8 266 4 267 0* 20* 268 16* 269 13 270 9 271 5 272 «1* lig 273 17* 274 14 275 10 276 6 277° «(2 22* 278 18* 279 15 280 11 281 7 282 3 HARMONIC ANALYSIS AND PREDICTION OF TIDES 281 Table 31.—For construction of primary stencils—Continued Difference Constituent 2MK Constituent MN Hour d. hej d. h. d. h. d. h. d. h.| d. h. d. h ie te Is |) Ob Ips 0 217 16 | 239 9 261 2 282 19 304 12 326 5 347 22 369 15 il” (1) 23 20 +23 —1 | 218 14 | 240 7 262 0 283 17 305 10 327 3 348 20 370 13 12*| 24 19* +22 —2 | 219 11*| 241 4* 21*| 284 14*| 306 7*| 328 O*| 349 17*|________- 2 Il 25 18* +21 —3 | 220 9 | 242 2 263 19*| 285 12*| 307 5* IRIN Be) Wap 3 11 26 18 +20 —4 | 221 7] 248 0 264 17 286 10 308 3 329 20 Stil 18} |e nn 410*| 2717 +19 —5 | 222 4* 21*| 265 14*| 287 7*| 309 O*| 330 17*| 352 10*|_________ 5 9*| 28 16* +18 —6 | 223 2*| 244 19*| 266 12*| 288 5* 22% | ool Lbs |e oDannon| pee eee eee 6 9 29 16 +17 —7 | 224 0 | 245 17 267 10 289 3 310 20 332 13 354 6 |__---..-- 7 8 30 15 +16 —8 22 | 246 15 268 8 290 1 311 18 333 11 300) 42h” ae 8 7*| 31 14* +15 —9 | 225 19*| 247 12*| 269 5* PPA Bi) alas) BREE Ee) Sita) abe ees 9 6*| 32 13* +14 | —10 | 226 17 | 248 10 270 3 291 20 313 13*| 335 6* PNA Ae MS a eee 10 6 33 13 +13 | —1l | 227 15 | 249 8 271 1 292 18 314 11 336 4 ewe PAL | 11 5*) 34 12* +12 | —12 | 228 12*| 250 5* 22*1 293) 15%!) 35) (8%) 337 1%\ 1358 1842-2 12 4*| 35 11* +11 | —13 | 229 10*| 251 3*) 272 20%; 294 13*| 316 6* ae) Gel 13 4 36 11 +10 | —14 | 230 8 | 252 1 273 18 295 11 317 4 338 21 360 14 |__-_-____ 14 3 37 10 +9 | —15 | 231 5* 22*| 274 16 296 9 318 2 339 19 36le1 25 ee 15 2*| 38 9* +8 | —16 | 232 3*| 253 20*} 275 13*| 297 6* 23*| 340 16*} 362 9*/____._____ 16 2 39 8* -++-7 | —17 | 233 1 | 254 18 276 11 298 4 319 21 341 14 SOS Lee eee lyf al 40 8 +6} —18 23 | 255 16 277 9 299 2 320 19 342 12 364 56 |__-_____- 18 0*| 41 7* +5 | —19 | 234 20*) 256 13*} 278 6* 23%) (321516%| 343) 9%) 365) 2%|2 eee 23*| 42 6* +4 | —20 |} 235 18*| 257 11*} 279 4*) 300 21*) 322 14*) 344 7*| 366 O*|_________ 19 23 43 6 +3 | —21 | 236 16 | 258 9 280 2 301 19 323 12 345 5 221 Pee 20 22 44 § +2 | —22 | 237 13*| 259 6* 23*| 302 16*| 324 10 346 3 30/e20 5 een 21 21*| 45 4* +1 | —23 | 238 11*| 260 4*| 281 21*} 303 14*| 325 7*| 347 O*| 368 17*|_________ 22 21 46 4 Difference Constituent MN Heur d. h. d. h. d. h. d. h d. h. d. h. d. h. d. h d. h d.h 0 47 3 70 10 93 17 117 0 140 7 163 14 186 21 210 4 233 11 | 256 18 +23 —1 48 2*| 71 9* 94 16* 23 141 6 164 13 187 20 211 3 234 10 | 257 17 +22 —2 49 1*| 72 8* 95 15*} 118 22*| 142 5*| 165 12*) 188 19*| 212 2*)| 235 9*| 258 16* +21 -—3 50 1 73 8 96 15 119 22 143 5 166 12 189 18*| 213 1*| 236 8*| 259 15* +20 —4 51 0 74 7 97 14 120 21 144 4 167 11 190 18 214 1 237 8 | 260 15 +19 —5 23*| 75 6* 98 13*} 121 20*) 145 3*| 168 10*) 191 17*) 215 0*| 238 7*| 261 14 +18 —6 52 23 76 6 99 12*| 122 19*) 146 2*| 169 9*| 192 16* 23*| 239 6*) 262 13* +17 —7 53 22 77 5 100 12 123 19 147 2 170 9 193 16 216 23 240 6 | 263 13 +16 -—8 54 21*} 78 4*| 101 11*} 124 18%) 148 1 171 8 194 15 217 22 241 5 | 264 12 +15 —9 55 20*| 79 3*| 102 10*) 125 17*| 149 O*| 172 7*| 195 14*| 218 21*) 242 4*) 265 11* +14 | —10 56 20 80 3 103 10 126 17 150 0 173 7 196 14 219 20*| 243 3*| 266 10” +13 | —11 57 19 81 2]. 104 9 127 16 23 174 6 197 13 220 20 244 3 | 267 10 +12 | —12 58 18*| 82 1*| 105 8*! 128 15*| 151 22*| 175 5*| 198 12*| 221 19*| 245 2*) 268 9* +11 | —13 59 18 83 1 106 8 129 14*| 152 2i1*] 176 4*) 199 11*| 222 18*) 246 1*| 269 8? +10 | —14 60 17 84 0 107 7 130 14 153 21 177 4 200 11 223 18 247 1] 270 8 - +9} —15 61 16* 23*| 108 6*] 131 13*| 154 20*} 178 3 201 10 224 17 248 0 | 271 7 +8 | —16 62 15*] 85 22*| 109 5*| 132 12*| 155 19*| 179 2*/ 202 9*/ 225 16* PAA OPA (ihe +7 | —17 63 15 86 22 110 5 133 12 156 19 180 2 203 9 226 16 249 22*| 273 5* +6 | —18 64 14*| 87 21 lll 4 134 11 157 18 181 1 204 8 227 15 250 22 | 274 5 +5 | —19 65 13*} 88 20*} 112 3*| 135 10*} 158 17*| 182 O*| 205 7*| 228 14*| 251 21*| 275 4* +4 | —20 66 13 89 20 113 3 136 10 159 16* 23*| 206 6*| 229 13*| 252 20*| 276 3* +3 | —21 67 12 90 19 114 2 137 9 160 16 183 23 207 6 230 13 253 20 | 277 3 +2 | —22 68 11*} 91 18*/ 115 1*| 188 8*| 161 15*} 184 22*| 208 5*| 231 12 254 19 | 278 2 +1 | —23 69 10*} 9217 116 O*| 139 7*| 162 14%} 185 21*} 209 4*| 2382 11*| 255 18*) 279 1* 282 U. S. COAST AND GEODETIC SURVEY Table 3i.—For construction of primary stencils—Continued Constituent MN Difference Hour AL5B} || =i +22 —2 ALM || 8} +20 —4 +19) =5 +18 | —6 +17 —7 +16 —8 +15} —9 +14 | —10 +13 | —11 SEI) |} 115) ALi} | A8} +10 | —14 +9 | —15 +8 | —16 +7 | —17 +6} —18 +5 | —19 +4 | —20 418) fr HL5) |] — 99) +1 | —23 Difference Hour 0 +23 —1 +22 —2 +21 —3 +20 —4 et OMl m= +18 | —6 +17 —7 +16 —8 +15 -9 +14 | —10 aiiigy |] alll +12 | —12 Sei || 72 |/nameaenne 324) (0*|)) 35a 13% ee oe d. h. d. h. 89 0} 118 13 90 5*| 119 18* 91 11 |) 121 0 9217 | 122 5* 93 22*| 123 11 95 4] 124 16* 96 9*) 125 22 97 15 | 127 3* 98 20*) 128 9 100 2] 129 14* 101 7*| 1380 20 102 13 | 1382 2 103 18*| 133 7* 105 0] 134 13 106 5*| 135 18* 107 11 | 137 0 108 16*| 138 5* 109 22 | 139 11 111 3*| 140 16* 112 9*) 141 22 113 15 | 143 3* 114 20*| 144 9 116 2] 145 14* 117 7*| 146 20 d. h. 148 1* 149 7 150 12* 151 18 153 0 154 5* 155 11 156 16* 157 22 159 3* 160 9 161 14* 162 20 164 1* 165 7 166 12* ~167 18 168 23* 170 5 171 10* 172 16* 173 22 175 3* YAS Y Constituent MK d. h. d. h 1G 46 5* 2 0 48 3* 3 22 50 2 5 20 52 0 7 18*| 53 22 9 16*) 55 20* 13 13 59 16* 15 11 61 15 17 9 63 13 19 7*| 65 11 25 2 lee 27 (0 73° 4 28 22 75 2 30 20*} 77 0 38 13 84 17 40 11 86 15 42 9 88 13 HARMONIC ANALYSIS AND PREDICTION OF TIDES 283 Table 31.—For construction of primary stencils—Continued Difference Constituent MK Constituent > Hour d. h. | d. h. d. h. d. h. d. h. d. h. d. h. 5 We WCE. We Nab. Wie 0 138 13*] 184 17 | 230 21 | 277 1) 323 5] 369 9 10 55 0] 110 2*| 165 5 +23 | —1/ 140 11*) 186 15%} 232 19*) 278 23 | 325 3] 371 7 2 4*| 57 7} 112 9*| 167 12 +22 | —2| 142 9*| 188 13*) 234 17*| 280 21*) 327 1 |__-----_- 411*| 5914] 114 16*| 169 19* +21 | —3] 144 8] 190 11*} 236 15*) 282 19*} 328 23*|_-______- CIS si Gl 21) lls, 201) 1722% +20} —4/] 146 6] 192 10} 238 14 | 284 17*| 330 21*|__-_____- 9 1*| 64 4*| 119 7] 174 9* +19 | —5 | 148 4] 194 8 | 24012] 28616] 332 19*)_-_______ 11 8*| 6611*) 121 14 | 176 16* +18 | —6| 150 2*| 196 6| 24210) 288 14] 334 18 |_-.----_- 13 16 68 18*| 123 21 | 178 23* +17 | —7 | 152 0*) 198 4*| 244 8 | 29012) 336 16 |_--_-_--- 15 23 71 1*) 126 4] 181 7 +16 | —8 | 153 22*) 200 2*| 246 6*| 292 10*| 338 14 |____-_-_- 18 6 73 8*) 128 11*| 183 14 +15 | —9 | 155 21 | 202 0*} 248 4*) 294 8*| 340 12*|________- 20 13 75 16 | 130 18*) 185 21 +14 | —10 | 157 19 | 203 23 | 250 2*| 296 6*| 342 10*}________- 22 20*| 77 23) 133 1*| 188 4 +13 | —11 } 159 17 | 205 21} 252 1] 298 5 | 344 8*|_________ 25 3*| 80 6] 135 8*/ 190 11* +12 | —12 | 161 15*) 207 19 | 253 23 | 300 3) 346 7 |____----- 27 10*| 8213] 137 15*| 192 18* +11 | —13 | 163 13*) 209 17*) 255 21 | 302 1) 348 6 }____----- 28 17*| 8420) 139 23 | 195 1* +10 | —14 | 165 11*| 211 15*| 257 19*| 303 23*) 350 3 |_---_---- 32 O*} 87 3*| 142 6 | 197 8* +9 | —15 | 167 10 | 213 13*| 259 17*) 305 21*| 352 1*)________- 34 8 89 10*| 144 13 | 199 15* +8 | —16 | 169 8 | 21512 | 261 15*) 307 19*| 353 23*|________- 36 15 91 17*| 146 20 | 201 23 47 | —17| 171 6 217 10| 263 14} 30918] 355 21*|________- 38 22 94 O*| 149 3*| 204 6 +6 | —18 | 173 4*) 219 8| 26512] 311 16] 357 20 |__------- 41 5 96 8] 151 10*| 206 13 +5 | —19 |] 175 2*| 221 6*| 26710 | 313 14] 359 18 |___-__-_- 43 12*| 9815] 153 17*| 208 20 +4 | —20 | 177 0*| 223 4*| 269 8*) 31512] 361 16 |_____---- 45 19*| 100 22| 156 0*| 211 3 +3 | —21 | 178 22*| 225 2*| 271 6*| 317 10*| 363 14*|________- 48 2*| 103 5] 158 7*| 213 10* +2 | —22 | 180 21 | 227 1] 2738 4*| 319 8*} 365 12*|____.___- 50 9*| 10512) 160 15 | 215 17* +1 | —23 | 182 19 | 228 23 | 275 3) 321 6*| 367 10*|__-.-.--- 52 16*| 107 19*| 162 22 | 218 0* Difference Constituent » Constituent MS 250 19*) 309 21 | 368 22* 253 6*| 312 8 | 371 9* 255 etm 314) 19) Pees aes +15 | —9 | 240 23*) 296 2*| 351 5 21 23 81 0 258 4*| 317 6 |_.------ +14 | —10 | 243 7 | 298 9*| 353 12 24 10 83 11* 260 15*| 319 17 |-_---_-- +13 | —11 | 245 14 | 300 16*| 355 19 26 21 85 22* 263 3) 322 4 |__-.---- 265 14 | 324 15*|_______- 268 le ioe (aon | eee 270 12 | 329 13*|---____- +12 | —12 | 247 21 | 302 23*| 358 2 29 8 88 9* +11 | —13 | 250 4 | 305 6*| 360 9*| 3119 90 20* +10 | —14 | 252 11 | 307 14 362 16*} 34 6 93 7* +9 | —15 | 254 18*| 309 21 | 364 23*| 36 17 95 18* +8 | —16 | 257 1*| 312 4| 367 6*| 39 4 98 5* +7 | —17 | 259 8*| 31411] 369 14 41 15*) 100 16* +6 | —18 | 261 15*) 316 18*| 371 21 44 2*| 103 4 0 8 ON beseonss +5 | —19 | 263 22*} 319 1*)________- 46 13*| 105 15 282 19*| 341 20*/_-_.-._- +4 | —20 | 266 6 | 321 8*|________- 49 0*| 108 2 285 6* 8) 22-2 +3 | —21 | 268 13 | 323 15*)________- 51 11*| 110 13 287 17*| 346 19 |_------- +2 | —22 | 270 20 | 325 22*|_________ 53 22*| 113 0 200 4*) 349 6 |_------- +1 | —23 | 273 3 | 328 6 |_-------- 56 9*} 115 11 284 U. S$. COAST AND GEODETIC SURVEY Table 31.—For construction of primary stencils—Continued Difference Hour 0 +23 -1 +22 —2 +21 -3 +20 —4 +19 -5 +18 —6 +17 —7 +16 —8 +15 —9 +14 —10 +13 —11 +12 —12 +11 —13 +10 —14 +9 —15 +8 —16 +7 —17 +6 —18 +5 —19 +4 —20 +3 —21 +2 —22 +1 —23 Constituent L 158 18* 161 10 164 1* 166 17 169 9 172 0* 174 16 Me ths 179 23* 182 15 185 6* 187 22 d. h. 190 14 193 5* 248 21* 251 13 d. h. 254 5 256 20* 259 12 d. h. 317 20 320 11* 323 3 325 18* 328 10* 331 2 333 17* 336 9 339 1 341 16* 344 8 346 23* 349 15* 352 7 354 22* 357 14 Constituent P ComeHt d. h. d.h d. h. 1 0] 358 16 10 8 15*| 373 21 16 6 23207 | ESE ee 46 16* B92 |Seeee hee 77 3 4). 17 eels ee 107 13 124) ee 138 0 84 hisses See 168 10* 9923) | eae eae 198 21 S45 | See ae 130: (OF): 2225 Se 259 18 145014% | Soke 290 4* 160520) | 2a 320 15 176) 1) | Ses Ss shit te 191) 16"| Se.) eee 381 12 206 711% | 252) SA eee 221 17) || Sees es | eee Con- HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 31.—For construction of primary stencils—Continued OOS. “SS S—W—Wasuwuw 285 Difference Constituent 28M Constituent J Hour dake dak a hilds hall ed andl Ghali ahaa de hp d. 0 207 3 | 23616| 266 4*| 295 17*| 325 6| 354 19 10) 1g 26/83 Hee BL Le i 2 HH =23 208 2 237 21" 267 10, 296 23 326 11*| 356 O* 13*| 27 4*| 52 19*| 78 10* cy 15*] 298 4*| 32717] 357 6 215| 28 6 +3 | —21 | 210 19*/ 240 8*| 26921 | 29910] 328 22%] 35811*| 317/| 29 7* oe be igs +4 | —20 | 212 1} 241 14 | 271 2*| 300 15*! 330 4 | 359 17 4 18*] 30 9*| 56 0*| 81 15 +5 | —19 | 213 7 242 19*/ 272 8] 301 21| 331 9*/ 360225) 520] 3111] 57 2| 8217 +6 | —18 | 214 12%) 244 1] 27314] 303 2*/ 33215] 362 4 6 21*] 3212*| 58 3*| 83 18° +7 | —17 | 215 18 | 245 6*| 274 19*| 304 8| 333 21| 363 9*/ 723%] 3314] 59 5| 84 20 +8 | —16 | 216 23*| 24612] 276 1] 305 13*] 335 2*| 364 15 9 1} 3416] 60 6*| 85 21° +9 | —15 | 218. 5 | 247 17*| 277 6%) 306 19 | 336 8 | 365 20°| 10 2*| 35 17*| 61 8*| 86 23° +10 | —14 | 219 10*| 248 23 | 27812) 308 O*| 337 13*| 367 2/ 11 4/ 3619/ 6210] 88 1 +11 | —13 | 220 16 | 250 4*) 279 17*| 309 6| 33819] 368 7*| 12 6| 3720*| 63 11*/ 89 9* +12 | —12 | 221 21*/ 25110 | 280 23 310 11*| 340 0*| 36913| 13 7*| 3822*| 6413] 90 4 +13 | —11 | 223 3 | 25216 | 282 4*|/ 31117] 341 6| 37018" 14 9| 40 0| 6515/ 91 6 +14 | —10 | 224 8*] 253 21*| 283 10 | 312 23 | 342 11%) 15 10*| 41 i*| 66 16*| 92 7* +15 | —9 | 22514 | 255 3 | 28415*] 314 4*| 343.17 Joo 16 12*| 42 3| 6718]! 93 9 +16 | —8 | 226 19*| 256 8*/ 285 21 | 315 10) 344 22") 1714] 43 5| 68 19%] 94 10* +17 | —7| 228 1| 25714 | 287 2*| 316 15*| 346 4|._____.. 18 15*| 44 6*| 69 21*| 95 12* +18 | —6 | 229 6*| 258 19") 288 8| 31721 | 347 9*| 1917] 45 8] 7023 1 Bos 3 230 12, 260 1 280 13° 319 2*| 34815 |._...___ 20 18*| 46 9*| 72 0* e ts 42 | —4 1 19| 320 8| 349 20°)... 21 20*| 4711*| 73 2| 9817 421 |} —3 | 232 33*| 26212 | 292 0*| 321 13*| 351 2|__...__.. 2222] 4813| 74 4] 99 18* +22| —2]| 234 5 | 263 17%) 293 6*| 32219] 352 7*|__...___- 23 23*] 49 14*| 75 5*| 100 20* +23 | —1| 235 10*| 264 23 | 29412] 324 O*| 353 13*|....___. 2 1! 5016! 76 7/| 101 22 Difference Constituent J Hour d.h.| dh dh awh id vk Gehl hd: hdl Sap ds halla: ih: 0 102 23*| 128 14*| 154 5*| 179 20*| 205 11*/ 231 2| 25617] 282 8! 307 23 | 333 14 +1 | —23 | 104 1/ 12916] 155 7] 18022] 20613] 232 4/| 257 18*| 283 9*| 399 o*| 334 15* +2 | —22| 105 3] 13018] 156 8*| 181 23*| 207 14*| 233 5*| 258 20*| 284 11*| 310 2 335 17 +3 | —21 | 106 4*| 131 19*| 157 10*| 183 1| 20816] 234 7| 25922| 28513] 311 4 | 336 18* +4 | —20| 107 6 | 132 21] 15812] 184 3/| 20918| 235 8*| 260 23*| 286 14*| 312 5*| 337 20" +5 | —19 | 108 7*| 133 22*| 159 13*| 185 4*| 210 19*| 236 10*] 262 1| 28716| 313 7 | 338 22 +6 | —18 | 109 9*| 135 0*| 16015/ 186 6| 211 21 | 23712) 263 3] 28818] 314 8*| 339 23* +7 | —17] 110 11] 136 2] 16117] 187 7*| 212 22%] 238 13*| 264 4*/ 289 19*| 315 10*| 341 1 +8 | —16 | 111 12*| 137 3*/ 162 18*| 188 9*| 214 0*| 23915 | 205 6| 29021 | 316 12| 342 3 +9 | —15 | 112 14| 138 5/ 16320] 18911! 215 2| 24017! 266 7*| 201 29*| 317 13*| 343 4* +10 | —14 | 113 16 | 139 6*| 164 21*| 190 12*| 216 3*| 241 18*| 267 9*| 293 0*| 31815 | 344 6 +11 | —13 | 114 17*| 140 8*| 165 23") 19114] 217 5| 24220] 26811 | 294 2] 31917] 345 7° +12 | —12| 115 19 | 14110 | 167 1] 19216] 218 6*| 243 21*| 269 12*| 295 3*| 320 18*| 346 9* +13 | —11 | 116 20*| 142 11%] 168 2*| 193 17*| 219 98*| 244 23*| 27014] 296 5| 321 20 | 347 11 +14 | —10 | 117 22*| 14313 | 169 4] 19419] 22010] 246 1| 27116] 297 6*| 322 21*| 348 12* +15} —9] 119 0 | 14415 | 170 6] 195 20%] 221 11*| 247 2*| 27217*) 298 8*| 323 23*| 349 14 +16 | —8| 120 1*| 145 16%] 171 7*| 196 22*| 22213] 248 4| 27319| 29910] 325 1 | 350 16 +17] —7| 121 3] 14618] 172 9| 198 0| 22315] 249 6] 274 20%] 3°) 11*/ 323 2*| 351 17° +18 | —6| 122 5 | 147 19*/ 173 10*| 199 1*| 224 16*| 250 7*| 275 22*/ 301 13| 327 4 | 352 19 +19 | —5| 123 6*| 148 21*/ 174 12*| 200 3| 22518] 251 9| 277 0] 30215| 328 6 | 353 20° +20] —4| 124 8 | 14923] 17514] 201 5| 226 19*| 25210*| 278 1*| 303 16*| 329 7*| 354 29* +21 | —3 | 125 9*| 151 0*| 176 15*| 202 6*| 227 21*| 253 12*] 279 3| 30418| 330 9| 356 0 +22 | —2/ 126 11*| 152 2| 17717] 203 8| 228 23| 25414] 280 5| 305 19*) 331 10*| 357 1° +23 | —1]| 12713] 153 4| 178 18*| 204 9*| 230 n*| 255 15*| 281 6*| 306 21*| 332 12*| 358 3 286 U. S. COAST AND GEODETIC SURVEY Table 31.—For construction of primary stencils—Continued Difference | Con. J Constituent OO Hour d.h.| dh d. h d. h d. h d. h. d. h d. h d. h d. h 0 359 5 10 13 22 27 2 40 6* 53 10* 66 14* 79 18* 92 22*| 106 2* +1 | —23 | 360 6* 14 11* 15* 19* 23* 67 3* 80 7* 93 11* 15% +2 | —22 | 361 8 20* 15 0* 28 4* 41 8* 54 12* 16* 20* 94 1) 107 5 +3 | —21 | 362 9% 2 9* 13* im 22 55 2 68 6 81 10 14 18 +4 | —20 | 363 11* 23 16 3 29° 7 42 11 15 95 3] 108 7 +5 | —19 | 364 13 3.12 16 20 43 0 56 4 69 8 82 12 16* 20* +6 | —18 | 365 14* 4 1 17 5 30 9* 13* 17* 21* 83 1* 96 5*;| 109 9* +7 | —17 | 366 16 14* 18* 22* 44 2* 57 ~6* 70 10" 14* 18* 22* 20 +13 | —11 | 126 9* 13* 18 22 | 179 2] 192 6] 205 10 14 18 22 +14 | —10 23 | 140 3] 153 7] 166 11 15 19 23 | 219 3] 232 7 | 245 11 +15 | —9 | 127 12 20 | 167 0} 180 4] 193 8 | 206 12* 16* 20*| 246 O* +16 | —8| 128 1] 141 5} 154 9* 13* gs 21*| 207 1*| 220 5*| 233 9* 13* +17 | —7 14* 22*| 168 2*; 181 6*| 194 10* 14* 18* 22*| 247 2* +18 | —6| 129 3%) 142 7*| ‘155 11* 15* 19% 23*| 208 4] 221 8] 234 12 16 Saiki) tt) 16* 20*; 156 1] 169 5| 182 9| 195 13 17 1] 285 1] 248 5 +20} —4 | 130 6 | 143 10 14 18 22} 196 2] 209 6] 222 10 18 +21 | —3 19 23 | 157 3) 170 7) 183 11 19* 23*| 236 3%) 249 7* +22} —2/| 181 8 | 144 12* 16* 20*; 184 O*} 197 4*| 210 8*) 223 12* 16* hs +23) —-1 21*| 145 1*) 158 5*} 171 9* 13* 17* 21*| 224 1*| 237 5*| 250 9* HARMONIC ANALYSIS AND PREDICTION OF TIDES 287 Table 31.—For construction of primary stencils—Continued Difference Hour 0 +1 | —23 +2 | —22 +3 | —21 +4 | —20 +5 | —19 +6 | —18 7 | —17 +8 | —16 +9 | —15 +10) —14 +11 | —13 +12 | —12 +13 | —1l1 +14 | —10 +15 -—9 +16 —8 +17 —7 +18 —6 +19 —5 +20 —4 +21 —3 +22 —2 +23 —1 Constituent OO Hh We Gb tie 356 7 | 369 11 20*| 370 0* 288 U. S. COAST AND GEODETIC SURVEY Table 32.—Divisors for primary stencil sums CONSTITUENT J Series 29 58 87 105 134 163 192 221 250 279 297 326 Hour ON 30 59 87 106 134 164 192 221 250 279 298 326 1°: 3 ee 31 59 89 106 135 164 193 222 250 280 298 327 2:3 ee apd 28 58 86 104 134 162 192 220 250 278 296 326 one Ee ed 30 59 88 106 135 165 192 222 251 280 299 326 ea 29 59 88 104 135 163 193 222 250 280 297 327 bia oe Me 3 28 59 87 105 134 163 193 221 251 278 297 326 652 eee 30 57 88 106 134 165 192 222 250 280 298 326 SA ee ek 28 58 87 104 134 163 193 221 250 279 297 327 822. ee Bee 29 58 88 106 134 164 193 222 251 279 298 326 92. okt Mad 29 57 87 105 134 163 192 222 250 280 297 326 AN 5:5 eee 28 58 86 104 134 162 193 220 250 278 297 326 5 De ae oe 30 59 88 107 134 164 193 223 251 280 299 327 12) ees 29 57 87 104 134 162 191 221 250 279 296 326 13: a SESE 28 58 85 104 133 162 191 220 250 278 297 325 5 ee oe Sea 30 58 88 106 134 164 192 223 250 280 297 327 15s 2 See ee 29 58 87 105 135 162 192 220 251 279 296 327 A eee pe 28 58 86 105 133 163 191 220 250 279 297 325 i aes ae 30 57 87 105 134 163 192 221 250 280 296 326 185 Sea 28 58 86 104 134 162 192 220 250 278 296 325 1 esp 29 58 87 106 133 163 191 221 249 280 297 325 20 2 Beene a) 29 57 87 104 134 162 191 220 249 279 296 326 21S eee 28 58 85 104 133 162 191 219 249 277 296 325 Py james = Bee 30 58 88 106 134 164 192 222 249 279 298 326 23 ee 28 57 86 104 134 161 191 219 249 277 295 325 CONSTITUENT K Series 14 29 58 87 105 134 163 192 221 250 279 297 326 Hour (= 15 30 59 88 106 135 164 193 221 250 279 297 326 ieee 14 30 59 88 106 135 164 193 222 251 279 297 326 yee 14 29 59 88 106 135 164 193 222 251 280 298 327 See 14 29 59 88 106 135 164 193 222 251 280 298 327 p Vases 14 29 57 87 105 134 163 192 221 250 279 297 326 Hie s ne 14 29 58 88 105 134 163 192 221 250 279 297 326 ee 14 29 58 87 106 135 163 192 221 250 279 297 326 eee 14 29 58 87 105 135 164 193 221 250 279 296 325 tee = 14 29 58 87 105 135 164 193 222 251 280 298 327 Qpeasy 14 29 58 87 105 134 164 193 222 251 280 298 327 1Oys== 14 29 57 86 104 133 163 192 221 250 279 297 326 ie. 14 29 58 87 105 133 162 192 PPA 250 279 297 326 VASE 14 29 58 87 105 134 163 192 221 250 279 297 326 he}, = 14 29 58 87 105 134 163 192 222 250 279 297 326 i es 14 29 58 87 105 134 163 192 222 251 280 297 326 Nien 13 28 GY/ 86 104 133 162 191 220 250 279 297 326 1G. 14 29 58 86 104 133 162 191 220 249 279 297 326 La 14 29 58 87 105 133 162 191 220 249 279 297 326 18cs=3 14 29 58 87 105 134 163 191 220 249 278 297 326 LOLs 14 29 58 87 105 134 163 192 221 250 278 297 326 20 14 29 58 87 105 134 163 192 221 250 279 297 326 21s 14 28 57 86 104 133 162 191 220 249 278 296 325 DAMS eae: 14 29 58 86 104 133 162 191 220 249 278 296 325 Peli 14 29 58 7 105 134 162 191 220 249 278 296 325 Table 32.—Divisors for primary stencil sums—Continued HARMONIC ANALYSIS AND PREDICTION OF TIDES CONSTITUENT L Serics 29 58 87 105 | 1384 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour (QReerte eee 29 59 87 | 105 | 1383) 163 | 191) 221 | 250] 279) 297) 326) 355 369 eet AD we 29 59 87 106 134 164 192 222 251 279 297 326 355 369 Dee ee, 29 58 87 | 106 | 1384] 163) 192) 221 | 250] 280] 298) 326) 356 370 Sat Phe. TM 30 58 87 | 105 | 1384) 163) 192] 221 | 250] 279] 298) 326) 356 370 Ape A ON 30 58 88 106 135 164 192 222 250 279 297 326 355 370 Be es. wee 29 58 88} 106 | 1384] 164] 192) 222; 250) 280] 298 | 327] 356 369 GE eee 29 57 86 | 105} 133 | 163) 191 | 221 | 249 | 279} 297) 3825) 355 368 eR ROS aD 30 59 88 106 135 164 198 222 250 279 298 326 356 369 Rc sk OE es 30 58 88 | 105 | 185} 164] 193 | 222) 251 | 280) 298 | 327) 357 370 Oink ee eA 29 57 87 | 104} 133} 163 | 191 | 221 | 250) 279) 296 | 326] 355 369 Obs 2 see 30 58 87 105 134 164 192 221 249 279 296 326 354 368 Debs 29 58 87 105 134 162 192 222 250 280 297 326 355 369 eS oe eee ae 29 58 87 | 104] 134] 162] 192] 221 | 250) 279) 297 | 326) 355 369 Sak Sees Oe 29 58 88 | 105 | 135} 163) 192] 220] 250} 279) 296) 326] 354 368 1 Be Teas 29 58 88 105 134 163 193 221 250 280 297 327 355 370 15 Ese Sees aes 28 58 86 | 105 | 1384] 163; 192) 221 | 250) 279) 297 | 327) 355 369 IGEt Bie 28 58 86 104 134 162 191 220 249 278 296 325 353 367 ie Se ees 28 57 86 | 104} 184] 162} 192) 220 | 250) 278 | 297 |) 326) 355 369 neta, Ses eee 29 58 87 | 105 | 134) 162) 192) 220) 250] 278) 296 | 326) 355 369 Qt ae. Ste Loe 29 58 R7 105 135 163 192 221 250 279 297 326 354 369 OO aay: oe 28 58 86} 105 |) 134) 163 | 192) 221 | 250] 279) 297) 327) 355 369 Oo ee eee 28 58 86 104 132 162 191 219 249 277 296 324 354 368 DD saat ee 29 58 87 105 134 163 193 221 251 279 297 325 355 369 2S eee ee 29 58 87 | 105 | 1384] 163] 193 | 221 | 251 | 279) 298! 326) 355 370 CONSTITUENT M Series | 15 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour (yes 15 29 59 87 | 105} 185 | 164] 192} 222) 250) 279) 297) 325] 355 369 y (ee 15 29 57 87 105 134 163 192 221 250 279 296 326 354 369 Pe ee 15 28 58 86 | 105 | 1384] 162] 192} 221 | 250] 279| 296) 325) 354 369 Sree ee 16 29 59 88 | 107 | 185] 165 | 198 | 222) 251 | 281) 299) 328) 357 371 (, eres 16 30 58 87 106 135 164 193 222 251 280 297 326 355 370 iRaese ee 15 28 57 86 | 104] 184] 168) 192] 221] 250} 278) 296) 325) 354 368 6222-8 15 29 58 87 |} 106} 134] 163} 192] 222} 250) 280} 297 | 326} 355 369 (pee 16 29 58 87 105 134 163 192 221 250 279 296 325 354 369 82-28 16 29 59 87 106 135 164 193 221 251 280 298 326 355 370 (Oe 15 29 58 87 106 135 165 193 223 251 280 298 327 357 371 Me 15 29 57 87 105 134 163 192 221 250 279 296 326 354 368 ieseue 15 28 57 86 104 133 162 192 221 250 278 296 325 354 369 2s 15 29 58 87 105 133 162 191 220 250 280 297 326 355 368 Ug} 15 30 59 88 105 134 163 192 221 250 279 298 327 355 369 4s 15 29 58 87 | 105 | 134) 163 | 192] 220 | 250] 278) 297) 326| 356 369 15 Le se 14 29 58 87 | 104] 1384] 163 | 192} 222] 250] 279] 298] 326 | 356 369 16-225 15 29 57 87 | 104] 1383 | 162] 191 | 220} 249) 278} 296) 326) 354 368 ily fees 15 29 59 87 105 134 162 192 220 250 279 298 326 355 369 Ses Le 14 29 58 87 |} 105} 1383 | 163 | 191 | 220} 249 | 278} 297 | 326] 355 368 1 Ct al 15 30 58 88 105 135 163 192 221 250 279 297 326 356 369 20: 2-8 14 28 57 86 103 133 162 191 220 249 277 296 325 354 368 lee es 14 29 58 87 105 133 162 192 221 250 280 298 327 356 369 Py jee. 15 30 59 88 105 134 163 192 221 249 279 298 327 355 369 7a yee ie 15 29 58 87 105 134 163 192 22)) 250 278 296 325 355 369 290 U. S. COAST AND GEODETIC SURVEY Table 32.—Divisors for primary stencil sums—Continued CONSTITUENT N Series} 15 29 58 87 105 | 184 | 163 | 192 | 221 | 250 | 279 | 297 | 326 Hour (eae 16 29 58 87 | 105| 184] 163 | 191] 220] 250] 279} 297) 327 i eee 16 29 58 88 | 106] 1385; 165] 194) 223] 252] 281 | 299) 327 eer 15 29 57 87 | 105] 183] 162] 191 |) 220] 248} 278} 296] 324 Sisest 16 30 58 88 106 134 163 192 221 249 279 297 326 Ale 3s 16 30 58 87 | 105 | 1385] 164] 193] 223} 252) 282] 299| 328 Gases 15 30 59 88} 106) 134] 164] 192} 222) 250] 279] 297] 326 (a 15 29 58 87] 105}; 133] 163} 191 | 221 | 249] 278} 296] 324 Mie as 15 29 58 87 | 105} 133 163 | 191 | 220} 250) 279) 298] 326 See 14 29 58 88 | 107 | 185] 164] 194) 223] 251! 281} 299) 327 () Seer 15 30 58 88 105 134 163 192 221 249 279 297 325 Osean 15 30 58 88 105 134 163 191 221 249 279 297 326 Li Fc 15 30 58 86 106 | 185] 165] 193 | 224] 252} 281 | 299} 328 ipa 15 28 59 87 | 106; 1384] 164; 192]; 220} 250; 278) 297] 325 1k eee 15 28 58 86 | 104] 1383] 161 191 | 219] 249 | 277] 295] 324 14_____ 14 28 57 86 | 104] 133 161 | 191 | 220} 250) 279] 297} 326 sae 14 29 58 88 | 105] 1385] 164/ 194] 222} 251] 280] 298] 327 16.22 15 29 58 86 104 134 162 191 220 249 277 295 325 1 eee 15 28 58 86 104 134 162 191 220 249 278 296 327 1S: 15 28 58 87 | 105 | 1384] 164] 198 | 222) 252) 280) 298] 327 i} 15 29 59 87 105 134 163 192 220 250 278 296 325 7.) ete 14 28 57 86 104 133 161 191 219 249 277 295 325 OA aes 14 28 57 86 103 133 161 192 220 249 279 297 326 2p ae S 16 30 59 88 106 137 165 194 223 252 281 298 328 23._.-- 15 29 58 86 | 104; 183} 162] 191} 220 | 249] 277) 295} 325 CONSTITUENT 2N Series 29 58 87 105 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 Hour (ses Beate B 28 58 86/105)" 135 1 163 |" 193) 2225251 220n 299 327, 11 eae Sh 2 30 58 88 | 106] 185} 165] 194) 223] 252 | 281) 299] 329 Cpe eee ee 28 58 87 105 134 164 193 222 250 279 297 325 ee SE cle Me 30 59 88 | 106] 186} 164) 193] 221 251 | 280] 298 | 326 Ce? SRT let 29 57 86 | 104] 132] 161 190 | 220] 249} 278] 295} 325 aah ae are 28 58 86} 105) 134) 163) 192) 222) 251} 280 || 298) 326 Gt 2 ee 30 58 88} 106) 135] 164 194 | 222] 252] 281} 298] 328 Uae ae os 29 59 88 106 135 165 193 223 251 280 297 325 BE ah SE | 29 59 88 106 135 163 192 220 249 279 296 325 (aoe See ee 29 57 86 104 133 162 191 220 250 278 296 326 1 sees She ee 29 58 87 106 135 164 193 223 251 280 298 327 AT: = RR 29 58 87 106 135 164 194 222 251 280 298 326 29 58 88/5/1068 /R135 e165 el 93t |e 22 al 249 i277 aie 29 omleo2D) 29 59 87 105 134 162 190 219 248 278 296 325 29 57 86 104 133 161 191 220 250 278 297 326 29 58 87 105 | 133] 163] 192} 222] 250) 280; 298] 327 29 58 87 104 134 163 192 221 251 279 297 325 29 58 88 105 134 163 192 220 249 278 296 326 1 Bia: es 29 59 87 104 132 161 189 219 248 278 296 325 OE ai eae ake 29 57 86 103 133 161 191 220 249 278 297 326 20 Ee ee 30 59 88 106 134 164 193 222 251 280 299 328 Ze se SORE eS 28 58 87 104 134 163 192 221 250 279 297 325 PPG vam IE ee 30 58 87 106 135 163 192 220 249 278 296 326 pee ere 28 56 85 103 131 161 189 219 248 277 295 325 Table 32.—Divisors for primary stencil sums—Continued CONSTITUENT O HARMONIC ANALYSIS AND PREDICTION OF TIDES 291 Series] 14 29 58 87 105 | 184 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hours ean 13 29 58 87 | 106] 135] 164] 192} 222) 251 | 279} 298 | 327 | 355 369 j| Seas 14 29 59 88 | 105 | 184| 164] 192] 221] 251) 280 | 298) 327 | 355 369 saa 14 28 57 86 | 105 | 1383] 162] 192] 221 | 250 | 279) 296) 3825) 354 368 Ree 14 30 57 87 | 105 |) 134 | 164 | 193 |) 221 | 251 |) 280 | 297) | 326) 356 370 4k. 14 29 58 87 106 135 163 193 222 250 280 297 325 354 369 eee 14 29 59 87} 105 | 135] 163} 192) 222) 251] 280 | 297) 326] 355 369 Gr .28 14 29 58 87} 105 | 1384] 164] 193) 222] 251 | 279 | 297 | 326] 355 369 Wee 14 28 58 87 | 105 | 185] 164} 192) 222| 251 | 280) 297] 327) 355 369 Seat 14 29 58 88 | 106 | 134] 164] 193] 221 | 250) 278) 296) 325} 355 368 G) eee 15 30 58 87 | 106) 134] 163] 193 | 221] 250] 279) 297] 326) 355 369 LORS 14 29 58 87 | 105 | 135] 164] 193 | 221 | 249 |) 279) 297] 326) 356 370 iy ae 14 30 59 88 | 107 | 186] 164] 192 | 222) 251] 280 | 299) 327] 356 370 eee 14 29 59 87 | 105 | 135 | 163 | 192} 221 | 250 | 279} 297 | 3827) 355 369 gues 13 28 57 87 | 104|) 182) 161 | 189] 219 | 248 | 276) 295) 324] 353 367 1: eee 14 29 58 87 | 105 | 133] 163 | 192] 220) 250] 279 | 297) 327] 356 369 5s oe. 14 29 58 87 | 105 | 183 | 162) 192] 221 | 250) 280} 297 | 326| 355 370 TS oe 14 30 58 87 | 104) 134] 163} 192] 221} 250) 279 | 298 | 325) 355 369 ig eae 14 29 58 86 | 104] 133 | 161} 191 | 220| 248] 277) 296] 325] 354 369 JR. 13 28 58 87 | 104] 1384] 163 | 191 | 221 | 250] 279 |) 297) 327) 355 369 TGRes ae 14 29 58 88 | 104] 183 | 163] 192 | 221 | 250] 278 | 297) 326] 355 368 20h 8 15 29 58 87} 105 | 134] 163] 192] 220] 250} 279 | 296 | 326] 355 369 Oi ee 14 29 58 87 | 105 | 1383] 163 | 192] 220) 249 | 279 | 297 | 326] 356 370 Db 15 30 57 86 | 105 | 134] 162] 191 | 221 | 250] 279 | 298] 326] 355 369 Done 14 28 58 86 | 104} 134] 162] 192 | 221 | 249] 279 | 297) 326] 355 369 CONSTITUENT OO Series 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hours AUS = a 29 58 86 | 104] 134] 163] 192] 221 |} 250] 280) 298] 326 | 355 369 TES Ske” saree 30 60 88 | 107 | 136] 164} 193 | 221 | 250] 279) 297] 326] 355 369 hs aa) EN 29 57 86 | 103 | 133] 162] 192} 220) 250] 280} 297) 327] 355 369 3 ieee Stee, 31 60 89 | 107 | 137] 166] 194] 223) 251] 281 | 298 | 327] 355 370 Aree ein 30 58 87 | 104] 1382] 162] 191 | 220] 249 | 278 | 297) 326) 355 369 Gieeeee s e ae 29 59 88 | 106} 135] 166| 194} 223 | 251 | 280} 298 | 326] 355 369 GR See oe 28 58 86 | 104] 132] 161 | 190 | 219 | 249 | 277) 297] 325) 355 368 Ee ok I 8 29 58 88 | 105} 135] 165 | 194] 223} 251] 280) 298] 327) 355 369 Cees ree 29 59 88 | 105 | 134] 163] 191 | 220] 249] 278 | 297] 326] 355 369 ee ee ae 29 58 87] 105} 134] 163 | 193} 223 | 251 | 280} 298) 326) 355 369 11) ee ees ee 28 58 87 | 105} 133 | 162] 191} 219 | 248] 277] 296) 325) 355 368 11 eee Sea 28 57 87 | 104] 134] 162] 193} 222) 251 | 280] 299 | 327] 355 369 Dy ee ee 29 58 88 | 106} 135) 163 | 193 | 221 | 250) 278] 296} 326 | 355 369 1 (6 ee ont ae 29 57 87 | 104] 133] 162} 192| 221 | 250] 280 | 297] 328) 356 370 142) Oe 30 59 88 | 107 |} 135] 164] 192 | 222} 250| 278 | 296 | 325 | 354 369 1 eect ene 28 57 85 | 104] 132] 162] 191 | 220) 250] 279] 297 | 327 | 356 369 jee ase 29 58 87 | 106] 135] 164] 193] 222] 251 | 279 | 297] 326] 354 369 1 (Rg EI 29 57 86 | 104] 133 | 161] 190 | 220} 249] 278 | 296 | 326] 355 369 16 ieee 30 58 88 | 106] 135] 165] 193 | 224] 252] 281 | 298} 327] 356 370 19) ese ek. 28 57 85 | 104] 132] 161} 189 | 218 | 248 | 277 | 295] 323 | 354 368 Os Se See | 9 58 87 | 106} 135 | 164] 193] 222) 252) 280] 298} 326| 356 369 7 Cs ae ian 28 58 86 | 104] 133] 161] 190] 218] 248) 277] 295 | 324{ 354 368 22e te See 29 57 87 | 105] 135} 163] 193] 222} 251) 281] 298) 327 | 356 370 72 epee ee aes, 29 58 87 | 105 | 134] 163] 191 | 220] 249) 278} 295] 325] 354 369 292 U. S. COAST AND GEODETIC SURVEY Table 32.—Divisors for primary stencil sums—Continued CONSTITUENT P Series 29 58 87 | 105 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour (ee ee 29 58 87 | 105 | 1385] 164] 193 | 222} 251 | 280] 298 | 327] 356 369 Te 29 58 87 | 105 | 154] 163] 192] 221 | 250] 279 | 297 | 297 | 354 368 2 ae: 29 58 87 | 105| 1384] 163] 192] 222] 251] 280} 298 | 327 | 355 369 Gime Soa Y 29 59 88 | 106] 135] 164] 193 | 222) 251 | 280) 298) 327] 356 370 eeres Se Sea 1 29 58 87 | 105 | 1384] 164] 193 | 222] 251} 280) 297) 326] 355 369 Se Leas 29 58 87 | 105 | 134] 163 | 192 | 221 | 250] 279) 296) 325] 354 368 6ssen eS 29 58 87 | 105 | 134! 163] 192] 221 | 251 | 279 | 297] 326) 355 369 (sey AS 29 58 88 | 106] 135; 164] 193 | 222] 251 | 279) 297! 326] 356 370 Sic eR SL 29 58 87 | 105) 134] 163] 192] 221 | 249] 278] 296| 325] 354 368 O55 see S29 58 87 | 105 | 134; 164] 193] 221 | 250] 279 | 297 | 326) 355 369 10. ee 29 58 87 | 105 | 1384) 163] 192] 220] 249] 279 | 297 | 326) 355 369 1S 29 58 88 | 106) 1385) 164] 192) 221 | 250! 279] 297] 326| 356 370 ND 29 58 87 | 105] 1384] 163 | 191 | 220} 249] 278) 296} 325) 354 368 1G een ares ae Bh 29 58 87 | 105) 1384) 162] 192] 221 | 250] 279 | 297 | 326) 355 369 142 2 we 30 59 88 | 106 | 185] 163] 192] 221 | 250] 280] 298 | 327] 356 370 1552. 29 58 87 | 105 | 1383] 162] 191 | 220} 249] 27 296 | 325 | 354 368 GS 2k PARTS | 29 58 87 | 106) 1384) 1623] 192 | 221 | 250] 279] 297 | 326] 355 370 i oe Re 29 58 87 | 104) 183) 162) 192] 221 | 250] 279] 297} 326) 355 369 1 QoS. 30 59 87 | 105 | 134} 163 | 192] 221] 250] 279 | 298} 327) 356 370 Ose RL 29 58 86 | 104] 133] 162] 191 |} 220] 249} 278 | 296 | 325) 354 368 202 RSS 29 57 86 | 104] 134] 163} 192] 221} 250] 279 | 297) 326) 355 369 7A ies Bes 29 57 86 | 104] 133] 162] 191 221) |)250) |) 279) |) 29% |) 326) i855 369 DOES SM 28 57 86 | 104] 133] 162] 191 | 220} 249) 278) 296 | 325] 354 368 7 alee eee LOLS 28 58 87 | 105 | 434] 163 | 192) 221 | 250] 279] 298] 327 | 356 370 CONSTITUENT Q Series 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 1 369 Hour (eee et ee 29 59 88 | 106] 136] 164] 194] 222] 250] 280 | 297 | 326] 355 368 fe ee oe 8 29 58 86 | 104] 133] 162} 191] 221 | 250] 280 | 298 | 327 | 357 370 22k ee AT 29 59 88 | 106] 135] 165] 193] 222] 251} 280] 298 | 326] 354 368 Bi eee 28 56 86 | 103] 132] 161] 190; 220} 249 | 278 | 297 | 326| 354 369 CSN Se) eee 30 69 89 | 107] 136] 166] 195] 225] 253 | 282 | 299 | 328] 356 370 se See TS 29 58 87 | 105 | 1383] 162] 191 | 219] 249] 277] 296 | 325 | 354 369 (ee ee? eee 30 59 88 | 107] 136] 165} 195 | 224] 254] 281 | 300] 328] 356 371 pets eee at 30 58 8Z |) 104 |) 133!) 162 |) 190} 219) 248 1 277 || 295) 324) 354 369 Sia eee re 28 58 87 | 105] 135] 164] 194] 223] 251} 280] 298 | 326] 356 369 ee oe eee oe 29 59 88 | 106 | 185] 163 | 192] 221 | 248] 278] 296] 325] 355 369 10. Sah 28 58 86 | 104] 1384] 163] 192] 221} 250} 280} 298 | 327] 355 370 i Pe Be tae 29 58 88 | 106] 134] 164] 192] 220] 249 | 277 | 295) 324] 353 368 iD. 5a A. 3 2 29 57 86 | 104] 133] 163] 191 | 220] 250] 279] 297 | 327] 356 370 1G ae ies 30 59 89 | 107] 136] 165] 192] 221 | 250} 278 | 296 | 325] 354 369 14 ane 29 58 87 | 104] 1383] 161] 191 | 220} 250] 279 | 297 | 327] 356 371 523.885. 25- 30 59 88 | 107 | 186} 164] 194] 222] 251 | 280] 297 | 326] 355 368 162 3388-525 29 58 86} 104] 133] 161] 190} 219 | 248] 278 | 296 | 325] 355 368 U7-c5.24 ke 29 59 87 | 106 | 135] 164] 193 | 223] 251] 280] 298 | 326| 355 368 18: 5-988. <5 28 57 85 | 103) 1381] 160] 188] 218 | 247] 277] 295} 324] 354 367 19s 252 ae se 29 58 87 | 105 | 1384) 164] 193 | 223 | 252] 281 | 299} 328] 356 370 20522. Fee 29 57 86 | 104) 182] 161] 190] 218 |) 247) 276] 294) 324) 353 367 QoS Soeee see 30 57 87 | 105 | 184] 164] 193 | 222) 252) 281 | 299) 328) 356 370 22 BOR 29 58 87 | 105] 1384] 162] 191 | 220) 249) 27 295 | 325] 354 368 95 ee Se 27 56 85 | 103) 183) 162] 192] 221) 251) 280] 298) 327) 357 370 Table 32.—Divisors for primary stencil sums—Continued HARMONIC ANALYSIS AND PREDICTION OF TIDES Series 29 58 87 105 | 134 | 163 | 192 Hour Qige): eee3 25 50 83} 1138 | 142] 167] 192 ile BE ele ee 25 59} 101 | 116] 141 | 166] 191 7 iad Ree ae 36 CN eLOZ Na 42) 167, 1) 192 Osis wets egal 39 64 89 104 129 154 196 Cle Eee ear a 25 50 75 90 115 159 192 ie ae en se 25 50 75 90 136 167 192 Gigk ote 25 50 83 | 113 | 142] 167] 192 Thee 25 60 | 102) 117] 142) 167) 192 {Rae ae 36 76 | 101] 116] 141 166 |} 191 (2) eee te 39 64 89 104 129 154 197 LO Sis ere! 25 50 75 90} 115 | 159! 191 sli te ee ee 25 50 75 90 | 136 | 167) 192 gD ees eee 25 50 83 | 113} 142] 167] 192 Ee See ees 25 60 101 117 142 167 191 TAO cyte 3 37 77 101 117 142 167 191 15ee eee oo 38 63 87 | 103 | 128) 153 | 194 N62 ee 25 50 74 90} 115] 159} 191 Vga. 2 ee! 25 49 74 90 | 136} 165] 190 Sie pat ee 2 25 49 81 113 142 166 191 Qe cee at 25 58 | 101 | 117] 142] 166] 191 Qe Lie ae 36 76 | 101 117 141} 166} 191 PANS ae Cane 37 62 87 | 103] 127] 152] 194 PAE Cos eee Se 24 49 74 90 114 158 191 Doe at ce oi 24 49 74 90} 1385} 166} 191 CONSTITUENT R | Series 29 58 87 105 | 134 | 163 192 Hour OLS Aaa cee 30 59 88 106 135 164 193 HF js RE 29 59 88 | 106 | 185] 164] 193 PA ES ae 29 58 88 106 135 164 193 1s a Se ae 29 58 87 105 135 164 193 CGR alt os Ap 29 58 87 105 134 163 192 £5}. RRO on 8 a 29 58 86 | 104] 183 | 162] 192 28 57 86 | 104} 1383 |) 162] 191 29 58 87 105 134 163 192 29 58 87 105 134 163 192 29 58 87 | 105| 1384] 163} 192 29 58 87 105 134 163 192 29 58 87 105 134 163 192 120s mere ak 29 58 87 | 105] 134] 163 | 192 Ue See 2s cee 29 58 8&7 105 134 163 192 148. a 29 58 87 105 134 163 192 Lis ees ae 29 58 87 | 105 | 1384] 163 | 191 VG Het ee 22 29 58 87 105 133 162 191 ees ees 29 57 86 | 104] 1383 | 162] 191 nF ey eer ee 29 58 87 105 134 163 192 1! ee eee nee 29 58 87 105 134 163 192 20a eee 29 58 87 105 134 163 192 7S Se 29 58 87 | 105 | 1384] 163] 192 22 he 29 58 87 | 105} 134] 163 | 192 23 29 58 87 | 105} 134] 163] 192 CONSTITUENT 2Q 293 355 294 U. S. COAST AND GEODETIC SURVEY Table 32.—Divisors for primary stencil sums—Continued CONSTITUENT T Series 29 | 58 | 87 | 105 | 134 | 163 | 192 | 221 | 250 | 279 Hour (Dy eee bat 29| 58) 88] 106| 135) 164] 193] 222] 251] 280 Ore HE 29| 58] 87] 105) 134] 163] 192] 221] 250] 279 Oe Tea Sn es 29| 58| 87] 105| 134] 163] 192] 221] 250] 279 Si ee 29| 58| 87] 105) 134] 163] 192] 221 | 250] 279 ee ES 29| 58) 87] 105| 134] 163] 193] 222] 251] 280 iS Pyle 30} 59] 88] 106] 135] 164] 193] 222] 251 | 280 (ian rp rule 29| 58] 87| 105] 134] 163] 192] 221| 250] 279 ea Vi ine PP 29| 58| 87] 105| 134] 163] 192] 221] 250) 279 Sheil of dae 29| 58| 87] 105| 134] 163] 192] 221] 250] 279 (aie Ss Se 29! 58] 87] 105| 135] 164] 193] 222] 251 | 281 LO ek ees 29| 58] 87] 105| 134] 163] 192] 221] 250] 279 fie a a 2 29| 58] 87] 105| 134] 163] 192] 221] 250) 279 517) a ts 9p 29| 58| 87] 105| 134] 163] 192] 221] 250] 279 13s Oe 29/ 58] 87] 105| 134] 163] 192} 221] 250] 279 1 eer 29) 59| 88] 106| 135] 164] 193| 223] 252] 281 Thee eee: 29| 58| 87] 105| 134] 163] 192] 221] 250| 278 165. eee 29| 58| 987] 105| 134] 163] 192] 221] 249| 278 17 OO 29| 58| 87] 105| 134] 163] 192] 220] 249] 278 29| 58| 87] 105] 134] 163] 191 | 220] 249] 278 29| 58) 87| 105] 134] 163] 192] 221| 250] 279 29/ 58| 87] 105} 133] 162| 191] 220] 249] 278 29| 58] 86] 104] 133] 162] 191] 220| 249] 278 29| 57| 86] 104] 133] 162] 191 | 220| 249] 278 28| 57| 86] 104] 1383] 162] 191| 220] 250] 279 CONSTITUENT A Series 29 | 58 | 87 | 105 | 134 | 163 | 192 | 221 | 250 | 279 Hour (Veen ae 29| 58] 89] 107| 135 | 164) 194] 223] 252} 280 18. + Oe 29| 57| 87] 106| 134] 162] 191] 221 | 250] 278 One oS aa 29| 57| 86] 104] 134] 162] 191| 219] 250] 278 Shh eee 31| 59] 88] 105] 136) 165] 194] 222] 252] 282 AM 2k SR 31| 59| 88] 105| 134] 164] 193| 221 | 250] 279 fied eae 29| 59/ 88] 105| 134] 162] 193| 221] 250] 278 (eee ee 29| 58] 88] 105| 134] 162] 193| 221] 250) 278 29| 57] 88] 106| 135] 163] 192| 223] 252] 280 299| 57| 86] 104| 134] 162] 191] 219] 250] 278 30} 58] 87] 104] 135] 163] 192] 220] 250] 279 31| 60| 88] 106] 135] 166] 195] 223] 252) 282 28} 59| 87] 105| 134] 162] 193| 221 | 249] 278 HOW 2.1 SRR 28| 57] 987/| 105) 134] 162] 191] 221] 249] 278 138. Bae 28| 57] 87/| 105| 134] 162] 190] 221] 249] 278 1A US Pa 28| 57] 85| 105| 134] 163] 191] 220] 251] 280 Ti eae 29| 58| 86] 104| 134] 163] 191 | 220] 249] 279 16.22 f BEE 30| 59] 87] 105) 133] 164] 192] 221] 249] 280 17 ee 28| 60] 88]| 106| 134] 164] 194] 223] 251] 280 th eee 28 | 57| 87] 105| 133] 162] 191 | 221] 249] 278 TO. 200 Bae 28 | 57] 86] 105] 133] 162] 190] 221] 249] 278 11) ee Oe 28| 57| 85] 104] 133] 162] 190] 219 | 249] 278 21 3 Re 299/ 58| 86| 104] 134] 164] 192] 221] 250] 281 Fp) eae 2 TA 30| 59! 87] 105| 133] 164] 192] 221] 249] 279 23 ied See ee 28} 58] 87] 105| 133] 163} 192| 221] 249| 277 326 326 Table 32.—Divisors for primary stencil sums—Continued HARMONIC ANALYSIS AND. PREDICTION OF TIDES CONSTITUENT yu 295 Series 27 87 105 | 1384 | 163 | 192 | 221 89 105 135 163 192 223 88 107 135 164 194 223 88 105 134 164 193 221 87 104 134 163 190 219 87 106 135 162 192 222 86 106 133 163 193 220 86 103 133 163 190 220 88 106 136 163 193 223 87 107 134 164 194 222 87 104 134 163 191 219 88 105 134 163 191 221 86 | 105 | 134] 162} 192| 222 86 105 133 163 193 220 87 104 134 164 191 221 88 106 136 163 193 222 87 107 134 164 193 222 87 104 133 162 191 219 86 103 133 162 190 220 86 106 135 163 193 222 87 106 134 164 193 221 87 104 134 163 191 221 88 106 136 164 193 222 87 105 132 162 192 220 85 101 131 161 190 219 CONSTITUENT v 87 105 134 163 192 221 86 103 135 165 193 221 83 103 134 161 189 216 89 107 135 162 190 224 89 106 134 161 195 222 90 107 135 167 196 223 85 102 135 164 192 219 83 104 134 161 189 217 90 107 135 162 191 224 89 106 134 161 195 222 90 107 134 169 197 224 84 101 134 162 190 217 85 107 134 162 190 219 89 106 133 161 190 223 90 107 134 164 195 223 88 105 134 167 194 222 83 101 133 161 188 216 86 107 134 162 189 221 88 106 133 161 192 223 89 107 134 165 195 223 86 103 134 166 193 221 82 101 133 161 188 216 87 106 134 162 189 222 88 105 133 160 193 223 89 106 134 165 195 223 296 U. S. COAST AND GEODETIC SURVEY Table 32.—Divisors for primary stencil sums—Continued CONSTITUENT p Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour Qi: Ree. 30 59 89 107 135 164 193 222 251 279 297 326 355 369 AL a SR 29 59 88 105 134 164 193 222 251 280 299 328 356 372 7} pe ete 28 57 87 105 134 162 191 220 249 278 295 324 353 367 Oise aes 29 58 87 106 135 165 194 224 252 281 299 328 357 371 Ais = 5 Bee 1 30 58 87 106 135 163 192 221 250 279 297 326 355 370 Be pe AE 29 58 87 106 134 163 192 220 250 278 296 325 354 368 GSes So aee 28 58 87 106 136 165 193 223 252 282 299 328 357 371 Ciscoe 29 58 87 104 134 163 192 221 249 278 296 325 354 369 [ee ee 29 58 87 105 135 165 194 222 251 280 298 326 355 370 Que es Rae 29 58 87 105 133 162 192 222 251 280 298 | 327 357 371 QMS Sea 29 57 86 104 133 162 191 220 249 278 296 325 353 366 i a eo ae 28 58 87 105 134 163 193 223 252 280 298 327 356 369 2a lee Ue 29 58 86 104 133 162 190 219 249 279 297 326 355 369 Bis hee 30 59 88 106 135 163 192 221 250 280 297 326 355 368 Ay 2 te spears 29 58 87 104 134 163 193 221 250 280 299 328 357 370 1 ay Rees ge ce 28 Lyf 86 104 132 162 190 219 248 276 295 324 354 367 Gee 2 eee 8 30 59 88 106 135 164 193 222 250 279 298 327 355 369 17 Sear ee) es 28 57 86 104 133 161 191 220 250 279 298 327 356 370 18¥ a 2 eek 29 58 87 104 133 162 191 220 248 277 295 325 354 367 192. 2 eo 30 58 87 106 135 164 193 222 251 280 297 326 356 369 O21 ae 28 57 86 104 132 161 190 219 249 277 295 325 354 368 ys aid cigar 30 59 87 105 134 163 191 220 249 278 296 324 353 368 D7 ee ee ee * 29 58 88 105 135 164 193 222 251 281 298 328 357 371 OA eee ee 29 58 86 104 133 162 191 219 248 277 295 323 352 367 CONSTITUENT MK Series 29 58 | 87 | 105 134 163 192 221 250 279 297 326 355 369 Flour Qe) 2 eS 30 59 88 105 135 164 192 222 251 279 297 325 355 368 ES ee re ee 29 58 88 106 135 163 192 222 251 280 298 327 356 369 es a 5 Se 29 58 88 105 134 164 192 221 249 278 297 326 355 369 Se Ls aee o 30 58 86 104 133 163 191 221 250 278 296 325 355 368 Ae 2 mgt 29 58 87 106 134 163 192 222 251 280 298 326 356 369 Sees Sse oe 29 58 87 105 134 164 192 220 250 279 298 327 356 369 ‘Gis; 5. 5. Saeed 30 58 86 105 134 164 193 221 250 279 297 326 356 369 (ae es 30 58 87 105 133 163 192 221 251 279 297 326 355 369 {| eee a ee 30 59 88 106 135 164 193 221 251 280 299 327 355 369 Qi ae yea 8 28 57 86 105 134 164 192 220 249 278 296 325 354 369 DX) ee Sere eas 29 58 86 104 133 163 192 221 251 279 297 326 355 369 POSS Be Brean 29 59 88 106 134 162 192 221 250 279 298 326 354 369 De A. see 28 58 86 105 134 163 192 220 250 279 297 326 355 369 1 es ee 29 58 86 105 133 162 192 221 250 278 297 326 355 369 i ina ae ee eae 29 59 88 106 134 163 193 222 250 280 297 326 355 370 1 area tee ee 29 59 87 106 135 163 192 221 250 280 297 327 355 369 ee Bem eoaree 28 57 86 105 133 162 192 220 249 278 296 326 355 369 AN Fiero hs apse = 2 29 59 87 104 134 163 193 222 250 279 296 325 354 369 te een ee 29 58 88 105 134 162 191 220 249 279 296 326 354 368 Qe = fo eee 2 28 57 87 105 135 163 193 221 250 279 297 327 356 370 Q0REE ees 29 57 86 103 133 162 191 221 249 278 296 325 354 369 Bleese at ae Sa 29 58 88 105 134 162 191 221 250 280 297 326 355 369 Ae mane eee | 28 57 87 105 135 163 191 221 250 280 297 327 355 369 5 se hae 29 57 87 104 134 163 192 221 249 278 297 325 355 370 Table 32.—Divisors for primary stencil sums—Continued HARMONIC ANALYSIS AND PREDICTION OF TIDES CONSTITUENT 2MK 297 Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 30 59 87 106 134 164 193 221 251 280 298 326 355 369 29 59 87 106 134 164 193 221 250 280 298 327 356 370 29 59 87 106 134 163 192 221 250 279 297 325 355 368 29 58 86 105 133 163 193 222 251 280 298 326 355 369 30 59 87 105 134 164 192 221 250 279 297 326 355 369 29 59 88 106 135 164 192 222 251 279 298 327 357 370 (jee Tals sae 29 58 86 104 134 163 191 221 250 278 297 325 354 368 USO ine Same, 30 59 87 105 135 164 192 221 251 280 299 327 356 370 Cee ete 30 58 87 105 134 163 192 221 250 279 296 326 355 368 (0) A eae man a 29 57 87 | 104} 134] 164] 192] 222] 251 | 279] 297! 396] 356 369 QE eer 30 58 88 105 135 164 192 221 251 279 297 326 355 369 1 Lega eae oe 30 59 88 | 106] 135] 164} 193 | 222] 251 | 280] 297] 3271] 356 370 aye eee se 29 57 87 105 134 162 192 221 249 278 296 326 356 369 1G YR ape ae 29 57 87 | 104) 134] 162] 191 | 221 | 249] 278 | 296] 3251 354. 368 TYAS, ca 29 57 86 104 133 162 191 220 249 279 297 326 355 369 V5 cee ee 29 58 87 105 134 162 192 221 249 279 296 326 355 369 UG se tee oe 28 57 87 105 135 163 192 222 250 279 297 326 354 369 Ns ee ence aes 28 57 86 104 133 162 191 220 249 278 296 325 353 368 SW e te 29 59 88 106 135 163 193 222 250 280 298 328 356 371 IN) SRE Bs trash NS 28 58 87 105 134 162 192 221 249 278 296 325 354 368 20 erent 28 57 87 104 133 162 191 220 250 279 297 326 355 369 7A ee se pa 29 58 87 105 133 163 192 220 250 279 297 326 354 369 PP ACE oe ow 28 58 87 106 134 163 193 221 250 279 297 326 355 370 Pee Aiea 28 57 87 104 133 162 191 219 249 278 296 325 354 368 CONSTITUENT MN Series 29 58 87 105 134 163 192 221 250 279 297 326 355 269 Hour OPS eee ta 28 56 85 104 134 163 193 223 253 283 301 328 356 370 ape cote Se 30 60 90 109 139 166 194 222 250 277 296 325 355 370 Be Seedy 28 56 84 101 129 158 188 218 248 278 297 326 355 369 ee eee 30 60 91 109 139 168 198 226 254 281 299 326 355 370 Cheb Boo case 30 59 87 104 132 159 187 217 248 277 296 | 325 355 369 Lae Eee 28 57 88 106 136 165 195 225 256 283 301 328 356 369 (es aaa 30 61 90 109 136 164 192 220 247 277 295 325 355 369 le a2 aera 28 56 83 101 130 160 190 221 250 280 298 327 355 368 See eon coh 30 61 90 109 138 168 196 224 251 279 296 325 355 369 (igen Sens Ne 29 57 84 102 129 157 187 218 247 277 295 325 355 369 Oe 1a eee 30 60 89 108 | 137 167 198 228 255 283 300 328 356 369 ii ee Pee ee Le 31 61 89 107 134 162 190 21 247 277 295 325 356 370 28 55 84 102 132 162 193 222 252 282 299 327 355 368 30 59 89 107 137 165 193 220 248 276 294 324 355 369 28 55 83 100 128 159 189 218 248 278 296 327 355 368 30 59 89 107 137 168 198 225 253 281 298 326 356 370 30 58 86 103 131 159 187 216 246 276 294 325 355 369 if faeeee one ee 28 56 86 104 135 165 195 224 254 282 299 327 355 368 a ene ss 0 ae 29 59 89 107 135 163 190 218 246 276 295 325 354 369 LOE =e Seen 27 55 83 100 131 161 190 220 250 280 299 328 355 369 20 see 29 59 89 108 138 168 195 223 251 279 296 325 354 369 QE sae aed 28 56 84 101 129 157 186 216 246 276 295 325 354 369 222 ae ene 28 58 88 107 137 167 196 226 254 282 299 327 354 368 7 Bye eh 29 59 88 105 133 161 188 216 246 276 295 325 354 369 298 U. S. COAST AND GEODETIC SURVEY Table 32.—Divisors for primary stencil sums—Continued CONSTITUENT MS Series 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour Oo cE ase. 3s 30 59 88 | 106 | 1385 | 164 | 192] 222] 250] 279] 297] 326] 354 369 y ee tha. 30 58 88 | 106 | 1384] 164] 192] 221 | 250] 279 | 296) 326] 355 369 Deak Maa ae 29 58 87 | 105 | 1384] 163 | 192] 222) 250) 280] 298 | 327) 355 369 Beg, ey 30 58 88 | 107 |. 185] 165 | 194) 223] 252] 281 | 298] 328] 356 370 7, ape reo ees 8 29 58 87 105 134 163 191 221 249 279 296 325 354 368 Be cakes 2 oe 30 58 88 | 105 | 1384] 164] 192] 221 | 250; 280] 297] 327) 256 370 (iene Sn ae 29 58 88 105 135 164 194 223 252 281 299 328 356 371 (feet Seed 30 58 87 | 105} 184] 162] 192} 220 | 250} 278 | 296 | 326) 354 368 fant wie Cale 29 58 87 104 134 162 191 220 249 278 296 326 355 369 (epee ee eee 29 57 87 104 133 162 192 220 250 279 297 326 355 369 108 ee 30 59 88 | 106 | 136 | 164 |) 193 | 222); 251 | 279 | 298 | 327) 355 370 DGS A ara 38 30 58 88 105 134 163 192 220 250 278 296 326 354 368 iy een ie. eee 1 28 58 87 104 134 162 192 221 250 279 298 326 356 370 Wes eee 28 57 86} 105} 1384] 163} 193 | 221 | 251 | 279 | 297 | 325) 355 369 1 ie eee Se 29 59 87 105 135 163 192 221 250 278 297 325 354 369 he) oes ae 29 58 87 105 134 163 192 220 250 278 296 325 355 369 1622 ey sie oe 28 58 86 | 104] 134] 163 | 193 | 222 | 251 | 280 | 299 | 327) 356 371 if. eee 29 58 S7e pel OG) eld) Oo) iOS) lee Me e216 le 279) Ove oon |eooD 369 28 58 86 104 133 162 190 220 248 278 296 324 354 367 28 57 86 104 132 162 191 220 249 279 297 326 356 369 29 59 87 106 134 164 192 222 250 279 298 326 | 355 369 29 58 86 | 105 | 133 | 162] 191 | 220| 249] 278 | 296] 325); 354 367 28 58 86 104 133 162 190 220 248 278 296 325 355 368 28 57 86 | 105 | 133 | 163 | 192) 221 | 250] 280] 297] 326] 356 369 Series 29 58 87 105 | 1384 | 168 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour OS eae 28 57 87 | 106 | 186} 164] 192] 220] 250) 280 | 299} 329] 356 369 ) Nees ae ee Sees 30 60 88 106 133 163 193 223 252 280 297 325 355 370 Ojai AE EY See 28 56 86 105 135 165 193 220 249 279 297 327 356 370 Baa eS 30 60 89 107 135 163 193 223 253 281 298 326 355 370 Cee eo eee 28 55 84 103 133 163 191 219 247 276 294 324 353 367 (ee Se PS 30 60 90 107 135 163 193 223 253 282 299 326 355 370 (seek epee 28 56 84 | 103 | 133 | 163} 192] 220] 248) 277 | 295 | 325) 355 370 lis ae 30 60 90 109 137 164 193 223 253 283 300 328 356 370 {eae ee eS 29 57 85 103 133 163 193 221 249 277 295 325 355 370 O33 Soe 29 59 89 108 137 165 193 222 252 282 300 328 356 370 i ee ee 30 59 86 104 133 163 193 223 251 279 295 325 355 370 DT See SS 28 57 87 106 136 164 192 220 249 279 298 327 355 369 DRE CE aac Se 30 59 87 104 132 161 191 221 249 277 295 323 353 367 ee se eee 28 57 87 105 135 164 191 219 249 279 298 328 356 369 14s 5 See 30 60 88 105 133 162 192 222 251 279 | 297 325 355 369 5S Ce eee 27 55 85 103 133 163 191 219 247 277 296 326 355 368 NG oe ee 30, 60 89 106 134 162 192 222 252 281 298 326 355 369 ize: = 2: Sea 28 56 84 | 102] 132] 162] 191] 219] 247] 276] 295] 325] 355 368 ae See 30 60 90} 108] 135] 163 | 192] 222] 252] 282) 300] 328] 355 369 192 See ee 29 57 85 102 132 162 192 220 248 276 295 325 355 369 DQG eee Ee 29 59 89 107 135 163 190 220 250 280 298 326 354 367 Zi Se 29 57 85 102 132 162 192 221 249 276 295 325 355 369 7 ae Se eae A 28 58 88 106 135 163 191 220 250 280 299 327 355 368 D854 Ae oe 30 58 86 103 132 162 192 222 250 278 295 325 355 369 HARMONIC AN'ALYSIS AND PREDICTION OF TIDES 299 Table 33.—For construction of secondary stencils Con- stituent J s L Con- stituent oo 2SM. K and P R and T MS r Differ- Differ- Differ- g Differ- Differ- Differ- Page ence, ence, ence, ence, ence en 8 hours | pours | Hours | pours | hours | pours | hours hours | 2°Urs | pours | hours mae + — bk + - = 3 0-23 0 0-23 0 0-23 0 0-23 0 0-23 0 9 0-23 1 0-23 1 0-23 0 0-23 0 0-23 1 15 0-23 2; 0-23 1 0-23 1 17-21 0 0-23 1 21 0-23 3 0-23 2 0-23 1 0-23 1 0-23 1 3 0-23 4 0-23 2 0-23 1 0-23 il 0-23 2 10 0-23 § 0-1 2 0-23 1 0-23 iL 0-23 2 16 0-23 6 0-23 3 0-15 1 0-23 1 0-23 3 22 1-11 6 0-23 3 0-23 2 0-23 2 0-23 3 4 0-23 a 0-23 4 0-23 2 0-23 2 0-23 3 10 0-23 8 0-23 4 0-23 2 0-23 2 0-23 4 17 0-23 9 0-23 5 0-23 2 0-23 2 0-23 4 23 0-23 10 0-23 5 0-23 3 0-23 2 0-23 5 5 0-23 11 0-23 6 0-23 3 0-23 3 0-23 5 11 0-23 12 0-23 6 0-23 3 0-23 3 12-20 5 17 0-23 13 0-23 7 0-23 3 0-23 3 0-23 6 0 6- 3 13 0-23 7 0-23 4 0-23 3 0-23 6 6 0-23 14 0-23 8 0-23 4 0-23 3 0-23 7 12 0-23 15 0-23 8 0-23 4 0-23 4 0-23 7 18 0-23 16 0-8 8 0-23 4 0-23 4 0-23 8 0 0-23 17 0-23 9 0-23 4 0-23 4 0-23 8 7 0-23 18 0-23 9 0-23 5 0-23 4 0-23 8 13 0-23 19 0-23 10 0-23 5 0-23 4 0-23 9 19 4-9 19 0-23 10 0-23 5 0-23 5 0-23 9 1 0-23 20 0-23 11 0-23 5 0-23 5 0-23 10 8 0-23 21 0-23 11 0-23 6 0-23 5 0-23 10 14 0-23 22) 0-23 12 0-23 6 0-23 5 0-23 10 20 0-23 23 0-23 12 0-23 6 0-23 5 0-23 11 2 0-23 0 0-23 13 0-23 6 0-23 6 0-23 11 8 0-23 1 0-23 13 0-23 7 0-23 6 0-23 12 14 0-23 2 0-23 14 0-23 7 0-23 6 0-23 12 21 8-1 2 0-23 14 0-23 7 0-23 6 0-23 12 3 0-23 3 0-15 14 0-23 v 0-23 6 0-23 13 9] 0-23 4 0-23 15 0-23 7 0-23 a 0-23 13 15 0-23 5 0-23 15 0-23 8 0-23 7 0-23 14 22 0-23 6 0-23 16 0-23 8 0-23 7 0-23 14 4 0-23 7 0-23 16 0-23 8 0-23 U. 2-21 14 10 0-23 8 0-23 17 0-23 8 0-23 7 0-23 15 16 6- 8 8 0-23 17 0-23 9 0-23 8 0-23 15 22 0-23 9 0-23 18 0-23 9 0-23 8 0-23 16 5 0-23 10 0-23 18 0-23 9 0-23 8 0-23 16 11 0-23 11 0-23 19 0-23 9 0-23 8 13-15 16 17 0-23 12 0-23 19 0-23 10 0-23 8 0-23 17 23 0-23 13 0-23 20 0-23 10 0-23 9 0-23 17 5 0-23 14 0-23 20 0-23 10 | 0-23 9 0-23 18 12 0-23 15 0-23 20 0-23 10 0-23 9 0-23 18 18 10-23 15 0-23 21 Q-23 10 0-23 9 0-23 19 0-23 16 0-23 21 0-23 11 0-23 9 0-23 19 6 0-23 17 0-23 22 0-23 11 0-23 10 0-23 19 12 0-23 18 0-23 22 0-23 ial 0-23 10 0-23 20 19 0-23 19 0-23 23 0-23 ll 0-23 10 0-23 20 1 0-23 20 0-23 23 0-23 12 0-23 10 0-23 21 7 0-23 21 0-23 0 0-23 12 8-16 10 0-23 21 12 0-23 21 0-23 0 0-23 12 0-23 ll 0-23 21 300 U. 8. COAST AND GEODETIC SURVEY Table 33.—For construction of secondary stencils—Continued Constit- uent L A Constit- uent MK B L Differ: Page ene2, 8 hours | poure lee oat ae 23-10 0 eae ee 3 20- 8 1 Gite ee E 17- 5 2 Ars arere: 15- 3 3 5): 12-1 4 (ase: 2 9-22 5 (eee 7-20 6 Soe 4-17 7 ()Reees fi 2-15 8 NO ess 23-12 9 11) CRS 20-10 10 2 18- 7 11 IG essere 15- 5 12 Ne ea) 4 13 15S 10-70) 14 iow 7-21 15 5 eee 4-19 16 PS eeeae 2-17 17 19) eee 23-14 18 bee & 21-12 19 21 18- 9 20 Dy eons 5 15- 7 21 phe a 13- 5 22 Oy eS 10- 2 23 OM ieee 7-0 0 26a 5-21 1 Oy Seeley 2-19 2 Oe 23-16 3 2b aan 21-14 4 Beeson 18-11 5 oleae 15- 9 6 S2e seis 13- 6 7 8B segs 10- 4 8 St Sees 8-1 9 Omer eee 5-23 10 36t2 a= 2-21 11 BY (aera es 0-18 12 Oe saeee 21-16 13 6 18-13 14 40 kre 16-11 15 Ble er Sas 13- 8 16 Oe 3 10- 6 17 AQWe nt aes 8- 4 18 fe ap 3 5- 1 19 (ae 3-23 20 4Geee ee 0-20 21 CV (aaa! 21-18 22 48h sane 19-15 23 Q9w isa} 16-13 0 HORE Eee 13-10 1 Bes 1l- 8 2 iy ee 8- 5 3 (63) 0-23 4 M MN Differ- ae PUES) hours 0-23 1 0-23 2 0-23 4 0-23 5 0-23 7 0-23 8 0-23 10 5- 0 ll 0-23 13 18- 8 14 0-23 16 7-15 17 0-23 19 19-22 20 0-23 22 0-23 0 0-23 1 0-23 3 0-23 4 0-23 6 0-23 7 0-23 9 0-23 10 0-23 12 0-23 13 0-23 15 0-23 16 0-23 18 6- 0 19 0-23 21 19- 7 22 0-23 0 7-15 1 0-23 3 20-22 4 0-23 6 0-23 8 0-23 9 0-23 11 0-23 12 0-23 14 0-23 15 0-23 17 0-23 18 0-23 20 0-23 21 0-23 23 18-16 0 0-23 2 6-23 3 0-23 5 19- 7 6 Q-23 8 2MK Differ- Differ- N N ence, ence, hours | hours | 2OUFs | hours + + 20-7 0 0-23 0 11-23 1 0-23 1 2-14 2 0-23 1 17- 6 3 0-23 1 9-21 4 0-23 2 0-13 5 0-23 2 15- 4 6 0-23 3 6-19 7 0-23 3 22-11 8 0-23 3 13- 2 9 0-23 4 4-18 10 0-23 4 20- 9 11 0-23 5 1l- 1 12 0-23 5 2-16 13 2-10 5 17- 2 14 0-23 6 9-23 15 0-23 6 0-15 16 0-23 7 15- 6 17 0-23 7 6-21 18 0-23 8 22-13 19 0-23 8 13- 4 20 0-23 8 4-20 21 0-23 9 19-11 22 0-23 9 1l- 3 23 0-23 10 2-18 0 0-23 10 17-10 1 0-23 10 8-1 2 0-23 11 0-16 3 0-23 11 15- 8 4 0-23 12 6-23 5 0-23 12 22-15 6 0-23 12 13- 6 7 0-23 13 4-22 8 0-23 13 19-13 9 0-23 14 1l- 5 10 0-23 14 2-20 11 2-20 14 17-12 12 0-23 15 8-3 13 0-23 15 0-18 14 0-23 16 15-10 15 0-23 16 6- 1 16 6-9 16 21-17 17 0-23 rs 13- 8 18 0-23 17 4-0 19 0-23 18 19-15 20 0-23 18 10- 6 21 0-23 19 2-22 22 0-23 19 17-13 23 0-23 19 8- 5 0 0-23 20 0-20 1 0-23 20 15-12 2 0-23 21 6- 3 3 0-23 21 0-23 4 0-23 21 O 2N Differ- O Differ- ence, ence hours hours hours + + 0 0-23 0 1 0-23 0 1 0-23 0 2 0-23 0 2 0-23 0 2 0-23 0 3 0-23 0 3 0-23 0 4 0-23 0 4 0-23 0 5 0-23 1 5 0-23 1 6 0-23 1 6 0-23 1 7 0-23 1 7 0-23 1 8 0-23 1 8 0-23 1 8 0-23 1 9 0-23 1 9 0-23 1 10 0-23 1 10 0-23 1 11 0-25 1 11 0-23 1 12 0-23 1 12 0-23 1 13 0-23 1 13 0-23 1 14 0-23 2 14 0-23 2 14 0-23 2 15 0-23 2 15 0-23 2 16 0-23 2 16 0-23 2 17 0-23 2 17 0-23 2 18 0-23 2 18 0-23 2 19 0-23 2 19 0-23 2 20 0-23 2 20 0-23 2 20 0-23 2 21 0-23 2 21 0-23 2 22 0-23 2 22 0-23 3 23 0-23 3 23 0-23 3 0 0-23 3 0 0-23 3 HARMONIC ANALYSIS AND PREDICTION OF TIDES 301 Table 33.—For construction of secondary stencils—Continued (“a aww NNN TTF (oo Constituent A-..-.- Constituent B_---- oO Page hours 4 Le ee re Ue 18- 1 Jae Spa ee ers) 0-23 i ea oe ees 18-15 SD gre rc er 7-19 Bie eee eeed eee 19-22 Gpsteeast eect eer 0-23 (ee ee eee 19-12 [ee eee ae 7-16 Que asauseaneauee ee 19 VOSS ateice Ana She) 8- 5 1b sae er ae Sn 20- 9 LD ea Beast 8-13 U3 Ge cemeeys eee ames 0-23 11, Mae sh tt eae 8- 2 A 5 eae, Be sed: 20- 6 Gee eh eek 9 1 fea EEE eee 21-19 NS se Sse ee esses 9-23 VQ ett mee cette 21- 3 1) OL SOs See 0-23 a Ns ah 21-16 Op] aoe ae Oe ee ee apes 10-20 DR pas i Pote e 22- 0 PP ie ol a tec 0-23 Sy Ea a a rae 22-13 DG ert ride Spay ees 10-17 Die en reas aoe 22- 6 DS ieee aerogenes 10- 6 PAD eee et Meh Ee opera aaa 23-10 BO ee ste a yet steel 11-14 Sl axe sere ae 0-23 yee eee 11- 3 Boren saw Ses eters 23- 7 1a, apnea eee <7 aren ae 11 Sb. Shee ee ge 0-20 BO meee eerste oes 12- 0 Shan eee So eoee lo see Q- 4 Gls ee eee ae 0-23 OOM soe oe 0-17 40) Soc ieee ee eee 12-21 Cy Legian a RED oela 1 CD aaa eee es 13-10 LS ee Or Oe io 1-14 CV Lei eects aes eee 13-16 45h = te Si Se 0-23 4G) = Meee aes 13- 7 LY San eee oer Sees 2-11 CE ae ps ES ret lS 14-15 AQ rece soe eset 2-0 HOR: steels er 14-4 fii beh igh ee = re 2-8 Gp eee eae en ee 0-23 | (Gi) Seats ee REE 4-16 and _ NNwOowW NITRO WwW 2Q O Differ- O Differ- ence, ence, hours hours hours hours 18-22 5 23-11 6 6- 8 17 9-20 18 18- 6 6 7-17 7 7-16 18 17- 5 19 19- 1 6 2-14 7 7-11 18 12- 0 19 19-21 6 22-10 7 7-19 19 20- 6 20 19- 5 rd 6-18 8 8-15 19 16- 3 20 20- 0 i 1-13 8 8-10 19 11-23 20 20 7 21- 8 8 8-18 20 19- 7 21 20- 4 8 5-17 9 9-14 20 15- 2 21 21-23 8 0-12 9 9 20 10-22 21 21-7 9 8-20 10 9-17 21 18- 6 22 21- 3 9 4-16 10 10-12 21 13-1 22 22 9 23-11 10 10-21 22 22- 9 23 22- 6 10 7-19 11 10-16 22 17- 5 23 22- 2 10 3-14 11 10-11 22 12-0 23 23-10 11 11-22 12 11-19 23 20- 8 0 23- 5 11 6-18 12 11-15 23 16- 4 0 23- 0 11 1-13 12 11-23 0 0-10 1 0- 9 12 10-21 13 12-18 0 19- 7 1 0- 4 12 5-17 13 12-14 0 15- 2 1 0-12 13 13-23 14 12-22 1 23-11 2 1-7 13 8-20 14 13-17 ] 18- 6 2 1- 3 13 4-16 14 13 i 14-1 2 1-11 14 12- 0 15 13-21 2 22- 9 3 2- 6 14 7-19 15 14-16 2 17- 5 3 2 14 3-15 15 14- 0 3 1-13 4 2-10 15 11-23 16 14-20 3 21-8 4 4 13 5-16 1t O Differ- ence, hours | pours sta nn 21- 5 19 pages Cola hrs Spt 55 15-18 8 1- 6 20 11-18 8 rs 21 14-19 9 0- 7 21 9-19 9 rs 10)) |inanne 10 3- 8 22 13-20 10 23- 8 22 itouch eae 23 17-20 11 2-9 23 12-21 11 202210 |e 12 6- 9 0 15-21 12 1-9 0 CSO 1 19-22 13 5-10 10 14-22 13 00"55 1 eae 14 8-11 » 18-23 14 3-11 2 21- 0 15 7-12 33 17- 0 15 2-12 3 10-12 4 20-1 16 6-13 4 16-1 16 ees Rew ca 17 9-13 5 17- 3 i 302 U. S. COAST AND GEODETIC SURVEY Table 34.—For summation of long-period constituents ASSIGNMENT OF DAILY SUMS FOR CONSTITUENT Mf Constituent division Days of series eres ea eer Beare talents uy Sie desires eet 1 28 55 82* 110 137 164* 192 219 246 274 301 328 356 1 Oe eae Smee pia ee se 2 29 56 84 111 138 166 193 220 248* 275 302 330* 357 Qua e kts. BEM Neh Ae ee 3 30 57* 85 112 189 167 194 221 249 276 303 331 358 Be eee SS os I A oe AE eT Se 4 31 59 86 113 141* 168 195 223* 250 277 304* 332 359 Greets es ee ee ee Sea ee eS 5 32 60 87 114 142 169 196 224 251 278 306 333 360 Sie eee oy el et RP Re oe 6 34* 61 88 115* 143 170 197* 225 252 279 307 334 361 (i) oe ee et Sea Ae ee eee ee Ud Be OY 89 117 144 171 199 226 253 281* 308 335 363* Te te ee Mathie 2 ae LM ee 8* 36 63 90* 118 145 172 200 227 254 282 309 336 364 CSS Se Ree sedan ea ail 10 37 64 92 119 146 174* 201 228 256* 283 310 337* 365 (3 aes ace Sk ch ooh SB RES en ee Dm ll 38 65 93 120 147 175 202 229 257 284 311 339 366 10: 3 be ne OR ee 12 39 67* 94 121 149* 176 203 230* 258 285 312* 340 367 i ace eee Pa eee eee ae | hee eee See 13 40 68 95 122 150 177 204 232 259 286 314 341 368 a Rl I A Pe, er ia) Seng 14 42* 69 96 123* 151 178 205* 233 260 287 315 342 369 0S ees aD Se Be ee ee eee, ees 15 43 70 97 125 152 179 207 234 261 289* 316 343 ___- 1 Tabara, Palen Nt we pete ay Oe OS A Ee 16* 44 71 98 126 153 180 208 235 262 290 317 344 ___. DS 2 osm eee Be Ne Ee he 18 45 72 100* 127 154 182* 209 236 263* 291 318 345* ____ TG iss 2 eR eS as ae ee 19 46 73 101 128 155 183 210 237 265° 292 319) 347 2222 1 RR Ss ONE WR SARC IE Dp he SR 20 47 75* 102 129 156* 184 211 238* 266 293 320 348 —-__- 5 aan SS a Se OE Be ee el 21 48 7 103 130 158 185 212 240 267 294 322* 349 ____ 1G as ee 1 Me or See Se ee Be 22 49* 77 104 1381* 159 186 213 241 268 295 323 350 —-___ PI pet es J eee, SMe ger ee Spon gee 23) 51 78 105 133 160° 187 215* 242 §269) \297* 324-351 ase 7) SRE es ae es eS Eee ee 24 52 79 106 134 161 188 216 243 270 298 325 352 -__- DIANE DOE Wi PaaS SE tae SS. oe SD 26* 53 80 108* 185 162 189* 217 244 271* 299 326 353 --_-- OB Sees: She AES Be eee aa 27, U54a 81) 9109136) 9163" 191 218) 245) 273 00m o2acoo =e ASSIGNMENT OF DAILY SUMS FOR CONSTITUENT MSf Constituent division Days of series 148 178 207 237 266 296 325 355 149* 179 208* 238 268* 297 327* 356 151 180 210 239 269 298 328 357 152 181* 211 240* 270 300* 329 359* 153 183 212 242 271 301 330 360 154 184 213* 243 272* 302 332* 361 156* 185 215 244 274 303 333 362 157 186 216 245 275 304* 334 364* 158 188* 217 247* 276 306 335 365 159 189 218 248 277 307 336* 366 160* 190 220* 249 279* 308 338 367 162 191 221 250 280 309 339 368* 163319255222 9252528 oes 40 164 194 223 253 282 312 341 ____ 165* 195 224* 254 284* 313 343* ____ 167 196 226 255 285 314 344 ____ 16819779227 25602860 Ono+ oe 169 199 228 258 287 317 346 ____ 170 200 229 259 288* 318 348* ____ 172* 201 231* 260 290 319 349 ____ 2() ee ERS ce REY oN BE Dre Wee 25) 55) 84) 4S) 1439739202) 2325) 26 29132020350 eee Dee OER ee Bae a Ee Re 26 56 85* 115 144* 174 204* 233 263* 292 322 351 —--_- DN pte ere eee Pe et a eek Q8*°57 87) WIG) 146) 175) 205) 234) 254 9293) 3230 sole Qo tp enn hae Meme ce wen Meo Ser ae Sa ees od 29 58 88 117* 147 176* 206 236* 265 295% 324 354 ____ HARMONIC ANALYSIS AND PREDICTION OF TIDES 303 Table 34.—For summation of long-period constit uents—Continued ASSIGNMENT OF DAILY SUMS FOR CONSTITUENT Mm Constituent division Days of series (ee ee fee ee oes 1 28 56 83 111 188 166 193 221 249* 276 304 331 359 oe a ee aoe 2 29 57 84 112 139* 167 195* 222 250 277 305 332 360 Oh oi SA OI Seales Sea eee eye 3 30 58 85* 113 141 168 196 223 251 278 306 333* 361 apeetioeets Lane Ab toto Sere 4 32* 59 87 114 142 169 197 224 252 280* 307 335 362 Mapai is nti ap ig be cnn ns 5 33 60 88 115 143 170* 198 226* 253 281 308 336 363 116* 144 172 199 227 254 282 309 337 364* 118 145 173 200 228 255 283 311* 338 366 119 146 174 201* 229 257* 284 312 339 367 120 147% 175 203 230 258 285 313 340 368 121 149 176 204 231 259 286 314 342* 369 122 150 177 205 232* 260 288* 315 343 __- 123 151 178* 206 234 261 289 316 344 ___ 125* 152 180 207 235 262 290 317 345 __. 126 153 181 208 236 263* 291 319* 346 127 154 182 209% 237 265 292 320 347 ___ 128 156* 183 211 238 266 293 321 348 _ 129 157 184 212 239 267 294* 322 350* _. 130 158 185 213 240* 268 296 323 351 ___ 131 159 187* 214 242 269 297 324 352 133* 160 188 215 243 270 298 325* 353 ___ 134 161 189 216 244 271* 299 327 354 ___ 135 162 190 218* 245 273 300 328 355 __ 136 164* 191 219 246 274 301 329 356* ___ 137 165 192 220 247 275 302* 330 358 —__- ASSIGNMENT OF DAILY SUMS FOR CONSTITUENT Sa 2 eee Days of . Pare Days of Constituent division aaa Constituent division aa taS 0 SES) ol eitade ee es eae ee se RE 3 ee 176-190 Esty SR TS ane ee aT eS SOG 09 | Eels Semees rs eee Lok ee eek 9 es me eS 191-205 Tl saat peg eS ee op a Ge PAS e TN id We bach SO i Safes Bl cee a en nee eng YE 206-221 (2 RE SER Baer oo eee a ht eee DARE SS All Wepete coe yee ian suse UNE eee ns SE 222-236 Sean ek AE en ee Wek oe ee 39- 53 SUG Sc rig Soe eet aap Vel aoe 237-251 Cha Se RO Da 2 a oe a ah A SG bal lly Sontime OE LS RRS A 252-266 Tene a Bee Titre = Hi Beat eee RS nd We Be 1 Qe S 4a |B tes Soe ee ee ik ON eae as ea aS 267-282 (a a i a i en a Se 85-99)" tel Osertee nel ete. yilares Wee Eee ihe a 283-297 eee Se PEE NEES ek Se wt Se Se 100-114 2) RO IS ONS Sa LA aN hee 298-312 EO ae SSS eR AS ea Le i ee PV 5 = 129" || 82 eee heen en pn ea yk 313-327 eRe d I ea os ERLE. SE SE UES T3054 bill C22 eee SE Ae a eae Se ee ae eae 328-342 1 eae) ee OO ER a BIR ee ea P4AG=1 6014 ||| 523 tate ese 2 SEE Aa Re eg ives ae 343-358 1s phe gc ee a RR et em 161-175 304 U. S. COAST AND GEODETIC SURVEY Table 35.—Products (a =) for'rorn 4aL Time meridian in hours=S+15 Constituent 1.000 | 2.000 | 3.000 | 4.000 5.000 | 5.500 6.000 6.500 Products, in degrees A ae ee eee a eee 28. 98 57.97 86.95 115. 94 144.92 159. 41 173.90 188. 40 Sone So epee ee 30. 00 60. 00 90. 00 120. 00 150. 00 165. 00 180. 00 195. 00 Nome epee ae ae 28. 44 56. 88 85. 32 113. 76 142. 20 156. 42 170. 64 184. 86 Koes osy Saget eee 2 15. 04 30. 08 45.12 60. 16 75. 21 82.73 90. 25 97.77 Mian sn eee emer 57.97 115. 94 173. 90 231. 87 289. 84 318. 83 347.81 16. 79 Oise ee ee eee ss 13. 94 27.89 41.83 55.77 69. 72 76. 69 83. 66 90. 63 Mig. oa: see eee 86.95 173.90 260. 86 347.81 74.76 118. 24 161.71 205. 19 (MEK) oe fae e tee ae 44.03 88. 05 132. 08 176. 10 220.13 242.14 264. 15 286. 16 Sie see ee Saue Ze 60. 00 120. 00 180. 00 240. 00 300. 00 330. 00 0. 00 30. 00 CNEIN) ae 22 sae = 57. 42 114. 85 172. 27 229. 70 287.12 315.83 344. 54 13. 25 eens eee es ae se ee 28. 51 57. 03 85. 54 114. 05 142. 56 156. 82 171.08 185. 33 Stic sso esep oe 90. 00 180. 00 270. 00 0.00 90. 00 135. 00 180. 00 225. 00 27.97 55. 94 83. 90 111. 87 139. 84 153. 83 167. 81 181.79 27.90 55. 79 83. 69 111. 58 139. 48 153. 42 167. 37 181. 32 16. 14 32. 28 48. 42 64. 56 80. 70 88.77 96. 83 104. 90 29. 46 58. 91 88. 37 117. 82 147, 28 162. 01 176. 73 191. 46 15. 00 30. 00 45.00 60. 00 75. 00 82. 50 90. 00 97. 50 14. 50 28.99 43.49 57.99 72. 48 79.73 86. 98 94. 23 Jysssevewossstssecese 15. 59 31.17 46. 76 62. 34 77.93 85. 72 93. 51 101. 31 IMG See eo eee eee 0. 54 1.09 1.63 2.18 2.72 2.99 3. 27 3. 54 OSaeso eee ce 0. 08 0. 16 0. 25 0. 33 0. 41 0.45 0.49 0. 53 Bass ee 0. 04 0. 08 0.12 0.16 0. 21 0. 23 0. 25 0. 27 AVIS Fore ee eect a 1.02 2.03 3. 05 4.06 5. 08 5. 59 6.10 6. 60 Mie ecw os eeeceeseee 1.10 2. 20 3. 29 4.39 5.49 6. 04 6. 59 7.14 Dies ee os San sees 13. 47 26. 94 40.41 53. 89 67.36 74. 09 80. 83 87. 56 Qpceere eee coo ee 13. 40 26. 80 40. 20 53. 59 66. 99 73. 69 80. 39 87.09 Metcelne: one mace 29.96 59. 92 89. 88 119. 84 149. 79 164. 77 179. 75 194. 73 Toe Aue eA oe eens 30. 04 60. 08 90. 12 120. 16 150. 21 165. 23 180. 25 195. 27 (2Q) est asec aeoe es 12.85 25. 71 38. 56 51.42 64. 27 70. 70 77.13 83. 55 14. 96 29.92 44. 88 59. 84 74. 80 82. 27 89.75 97. 23 31.02 62. 03 93. 05 124. 06 155. 08 170. 59 186. 10 201. 60 43. 48 86.95 130. 43 173.90 217. 38 239. 12 260. 86 282. 60 29. 53 59. 06 88. 59 118.11 147. 64 162. 41 177.17 191. 94 42.93 85. 85 128. 78 171.71 214. 64 236. 10 257. 56 279. 03 30. 08 60. 16 90. 25 120. 33 150. 41 165. 45 180. 49 195. 53 115. 94 231. 87 347. 81 103. 75 219. 68 277. 65 335. 62 33. 59 58. 98 117.97 176. 95 235. 94 294. 92 324. 41 353. 90 23. 40 HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 35.—Products (a 5) for Form 444—Continued Constituent (2MUKG) 35. eee J RG e Eh = Esa ele 8.000 | 231. 87 240.00 227. 52 120. 33 103.75 111. 54 335. 62 352. 20 120.00 99. 39 228. 10 0.00 223. 75 223. 16 129. 11 235. 65 120.00 115. 97 124. 68 4,35 0. 66 0. 33 8.13 8. 78 107.77 107. 19 239. 67 240. 33 102. 83 119. 67 248. 13 347. 81 236. 23 343. 42 240. 66 207. 49 111. 87 305 Time meridian in hours=S~+15 9.000 260. 86 270.00 255. 96 135. 37 161.71 125. 49 62. 57 36. 23 180.00 156. 81 256. 61 90. 00 251.71 251. 06 145. 25 265. 10 135. 00 130. 47 140. 27 4.90 0. 74 0. 37 9.14 9.88 121. 24 120. 59 269. 63 270. 37 115. 69 134. 63 279. 14 31. 29 265. 76 26. 34 270. 74 323. 43 170. 86 10.000 Products 289. 84 300. 00 284. 40 150. 41 219. 68 139. 43 149. 52 80. 25 240. 00 214. 24 285. 13 180. 00 279. 68 278.95 161.39 294. 56 150. 00 144.97 155. 85 5. 44 0. 82 0. 41 10. 16 10. 98 134.72 133.99 299. 59 300. 41 128. 54 149. 59 310. 16 74. 76 295. 28 69. 27 300. 82 79. 36 229. 84 10.500 | 11.000 | 11.500 | 12.000 in degrees 304. 33 318. 83 333. 32 347.81 315. 00 330. 00 345.00 0.00 298. 62 312. 84 327.06 341. 28 157. 93 165 45 172.97 180. 49 248. 67 277.65 306. 63 335. 62 146. 40 153. 37 160. 34 167. 32 193.00 236. 48 279.95 323. 43 102 26 124. 28 146. 29 168. 30 270. 00 300. 00 330. 00 0. 00 242.95 271. 66 300. 37 329. 09 299. 38 313. 64 327.89 342. 15 225. 00 270. 00 315.00 0.00 293. 67 307. 65 321. 63 335. 62 292. 90 306. 85 320. 80 334. 74 169. 46 177. 53 185. 60 193. 67 309. 28 324. 01 338. 74 353. 47 157. 50 165. 00 172. 50 180. 00 152. 22 159. 46 166. 71 173.96 163. 65 171. 44 179. 23 187. 03 5. 72 5. 99 6. 26 6. 53 0. 86 0.90 0.94 0.99 0.43 0. 45 0.47 0. 49 10. 67 11.17 11.68 12.19 11. 53 12.08 12. 63 13.18 141.45 148.19 154.92 161. 66 140. 69 147.39 154. 08 160. 78 314. 57 329. 55 344, 53 359. 51 315. 43 330. 45 345. 47 0. 49 134. 97 141. 40 147.82 154. 25 157.07 164. 55 172. 03 179. 51 325. 67 341.17 356. 68 12.19 96. 50 118. 24 139. 98 161. 71 310. 05 324. 81 339. 58 354. 34 90. 73 112. 20 133. 66 155. 13 315. 86 330. 90 345. 94 0.99 137. 33 195. 30 253. 27 311. 24 259. 33 288. 83 318. 32 347. 81 306 Constituent (2IMiaKS) 53222 eee Kip aS 2 Se U. S. COAST AND GEODETIC SURVEY Table 36.—Angle differences for Form 445 fees eA a Feb. 1, 04, to Dee: 31, 24) Jan. 1, 04, to Feb. 1, 04 ° ° +324.2 | —35.8 0 0 +279 —81 +31 |—329 +288 —72 +294 —66 +253 |—107 +355 —5 0 0 +243 |—117 +333 —=27 0 +288 —72 +234 |—126 +127 |—233 +315 —45 0 +342 —18 +76 |—284 +45 |—315 +61 |—299 +31 |—329 +36 |—324 +97 |—263 +303 —57 +249 |—111 +329 |— 31 +31 |—329 +204 |—156 +329 —31 +36 |—324 +306 —54 +9 |—351 +258 |—102 +61 |—299 +217 |—143 +324 —36 Common year ° +136. 6 0 +93 +329 +274 +167 +49 +106 0 +230 +317 0 +274 +49 +131 +316 0 +248 +12 +44 +299 ° —223.4 |+112.2 0 0 —267 —31 —86 —193 Leap year ° +56 +330 +225 +142 +336 +82 0 +168 +281 0 +225 +359 Jan. 1, 04, to Dee. 31, 24h Common year ° +100. 8 0 +12 0 +202 +101 +302 +101 0 +113 +290 0 Leap year ° ° ° —259.2 | +76.4 | —283.6 0 0 0 —348 +335 —25 0 atl —359 —158 +153 —207 —259 +76 —284 —58 +229 —131 —259 +77 —283 0 0 0 —247 +51 —309 —70 = |-+254 —106 0 0 0 —158 |+153 —207 —77 +233 —127 —102 |+286 —74 —89 |+258 —102 0 0 0 —130 +218 —142 —272, |--103 —257 —271 +102 —258 0 1 —359 0 Spt —359 —101 |+284 —76 —101 |+285 —75 —69 |+254 —106 —348 |+334 —26 0 +259 —1 0 Stal —359 —76 +232 —128 0 +359 —1 —101 +284 —76 —29 |+294 —66 —171 /|+178 —182 —158 +152 —208 0 +1 —359 —317 |+306 —54 —259 +76 — 284 4 HARMONIC ANALYSIS AND PREDICTION OF TIDES 307 Table 37.—U. S. Coast and Geodetic Survey tide-predicting machine No. 2 GENERAL GEARS AND CONNECTING SHAFTS Face Number Period Gears and Shafts or of Pitch of Remarks diameter| teeth rotation Dial Inches hours Sa eee OMS Greta see Ee 4 | Hand crank shaft for operating machine. GN a 0. 56 40 24 4 | Spur gear on shaft 1. GaDHneemetes int 0. 50 120 24 12 | Spur-stud gear. (Ga ee Sos 0. 50 120 24 12 | Spur gear on shaft 2. Saver ses aya OO ise Ae eee 12 | Short horizontal shaft. Gut Saas 0. 41 72 24 12 | Bevel gear on shaft 2. (Cb eee 0. 41 72 24 12 | Bevel gear on shaft 3. Si8). 3 eee See OV50R Renken ere ts 12 | Diagonal shaft connecting with middle section. Gao ees so ee Lou 0. 38 75 30 12 | Bevel gear on shaft 3. (Gi uanses eae 0. 38 75 30 12 | Bevel gear on shaft 4. Se Aired neo ONSSaleeeetoncea sonra os 12 | Short vertical shaft through desk top. (Gh See S Ree eee 0. 38 75 30 12 | Bevel gear on shaft 4. Gao 0. 38 75 30 12 | Bevel gear on shaft 5. Sie are aaa OF38n [Sao 2 ees ee 12 | Short horizontal shaft. GAO ee es 0. 27 75 30 12 | Bevel gear on shaft 5. Gale 0. 27 75 30 12 | Bevel gear on shaft 6. Se Obeee eee OSSS |Z ateeee a era os 12 | Main vertical shaft of dial case. GAO Se: 0.17 60 48 12 | Releasable bevel gear on shaft 6. Ge13 8 eee 0.17 120 48 24 | Bevel gear on shaft 7. S27 eae ee (Albi Bee ee eee ee 24 | Intermediate shaft to hour hand. 0.17 84 48 24 | Bevel gear on shaft 7. 0.17 84 48 24 | Bevel gear on shaft 8. OGU SE East ees ea ere a 24 | Hour-hand shaft. 0.17 180 48 12 | Releasable bevel gear on shaft 6. 0.17 60 48 4 | Bevel gear on shaft 9. CO) Hay) fete eras |e 4 | Intermediate shaft to minute hand. 0.17 240 48 4 | Bevel gear on shaft 9. 0.17 60 48 1 | Bevel gear on shaft 10. O15 Boke Weis 3) eee 1 | Minute-hand shaft. 0.17 60 48 12 | Releasable bevel gear on shaft 6. 0.17 120 48 24 | Bevel gear on shaft 11. CO) TNS [pees sete Ns ee 24 | Intermediate shaft to day dial. 2 se yp Fe es es 24 | Worm screw, 0.56 inch diameter, 18 threads to inch on shaft 11. (CRORE PEs See) eens Soaps B66RlEse 3-28 24366 | Worm wheel, 6.47 inch diameter, on shaft 12. Saloes=s5 22-2 See COS | i ete 2 Ss 24366 | Day dial shaft. 0. 25 46 40 12 | Spur gear at top of shaft 6. 0. 25 60 AQ) || Maoaes ease Spur-stud gear. 0. 25 60 AQ is ee ce Spur-stud gear connected with gear 25 by ratchet wheel and pawl. 0. 25 46 40 12 | Spur gear at lower end of feeding roller. 0. 41 72 24 12 ,; Bevel gear on shaft 3. 0. 41 72 24 12 | Bevel gear on shaft 13. Ase ee ee 12 | Main vertical shaft of middle section. 0. 38 110 30 12 | Spur gear on shaft 13. 0. 38 110 30 12 | Spur stud gear. 0. 38 110 30 12 | Spur stud gear on shaft 14. 2383] see eeee eel hase eer 12 | Front vertical shaft of rear section. 0. 28 75 30 12 | Bevel gear on shaft 14. 0. 28 75 30 12 | Bevel gear on shaft 15. O50 R Sac Sew ees 12 Mein connecting horizontal shaft of rear sec- ion. 0. 28 75 30 12 | Bevel gear on shaft 15. 0. 28 75 30 12 | Bevel gear on shaft 16. ONS SH ees Ses 12 | Rear vertical shaft of rear section. 308 USES: COAST AND GEODETIC SURVEY Table 38.—U. S. Coast and Geodetic Survey tide-predicting machine No. 2 CONSTITUENT GEARS AND MAXIMUM AMPLITUDE SETTINGS Theoretical Constit- Vertical speed uents per hour shafts I ° eee aie 15. 5854433 107 1 Ge ee 15. 0410686 61 Kip ees 30. 0821372 122 | pane te eee 29. 5284788 104 SMireeseeee 14, 4920521 103 Mo. eee 28. 9841042 103 IMIj- eos) ZBL CONGR 86 Mie asses 57. 9682084 118 Migecyean = 86. 9523126 140 IVIg: Seem 115. 9364168 118 Nga 28. 4397296 65 2N 3 ae 27. 8953548 68 Oras 13. 9430356 92 OOS 16. 1391016 134 Pie eae 14. 9589314 91 (Oyiae aes 13. 3986609 84 20) see 12. 8542862 127 3p as 30. 0410667 85 Sieeeene eee 15. 0000000 63 S2--=---_--| 3030000000 70 Sass nee 60. 0000000 75 See aeeewes 90. 0000000 90 sf Ris Fae 29. 9589333 81 Noe Seen eae 29. 4556254 131 Ti Serene een 27. 9682084 125 Vgee— ees 2 28. 5125830 89 Divas ta 13. 4715144 69 MK angen 44. 0251728 120 2a 42. 9271398 81 NN 57. 4238338 135 IMIS aeetene 58. 9841042 118 25 Mena 31. 0158958 69 IM ie Cee 1. 0980330 84 IMSS fo ee 1. 0158958 149 Vin eee 0. 5443747 93 Sa FF ees 0. 0410686 51 Ssazcee 2 ae 0. 0821372 51 Teeth in gear wheels : Crank Intermediate shafts shafts II III IV 90 52 119 73 51 85 80 96 146 61 56 97 85 59 148 74 59 85 62 70 67 74 103 85 62 86 67 37 103 85 46 53 79 58 46 58 89 58 129 131 71 135 73 50 125 88 51 109 114 50 130 50 43 73 75 50 84 70 70 70 45 60 50 48 80 50 50 45 73 65 57 117 82 74 121 69 70 95 70 41 90 81 105 106 52 79 86 42 53 89 61 62 61 47 50 71 45 1 51 80 1 55 41 1 125 { 149 ha ee 125 60 120 149 1 125 *Designed for one-half of speed of Mo. Gear speed per dial hour ° 15. 5854342 15. 0410959 30. 0821918 29. 5284773 14. 4920509 28. 9841017 43. 4761675 57. 9682035 86. 9523351 115. 9364070 28. 4397358 27. 8953627 13. 9430363 16. 1391009 14. 9589041 13. 3986656 12. 8542510 30. 0410959 15. 0000000 30. 0000000 60. 0000000 90. 0000000 29. 9589041 29. 4556213 27. 9681516 28. 5125858 13. 4714286 44. 0251572 42. 9271020 57. 4237560 58. 9841440 31. 0158825 1. 0980392 1. 0159091 0. 5443902 0. 0410738 0. 0821477 Maximum Error per] amplitude year i s settings of cranks Units SNOW FOOD opeeS prwty ORO S DORDO WHDOSCD ROOCORO SCHOO beh G2 ESS OO. eri oo fo CORD NROWSO HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 39.—Synodic periods of constituents DIURNAT. CONSTITUENTS 309 Ji Ki Days. Days days ec 21 ODD | Pane eee Mie ta ce 13. 777 27. 555 Ore 5 = 9. 133 13. 661 OOn ees 27. 093 13. 661 Ring 2 23.942 182. 621 (Oye ae See 6. 859 9. 133 2Qhn =F Soe, 5. 492 6. 859 Sis 5 a2 25. 622 365. 243 eee See 7.096 9. 557 Ke Lz M2 Days. Days Days Tee Coes aes se leee eee M2____| )13. 661 PY. de) oso SSS No 9.133 13.777 | 27.555 2N_.--| 6.859 9.185 | 13.777 Ro_____] 365. 225 29. 263 | 14.192 Soe 182. 621 31.812 | 14.765 Monee ez 121.748 | 34.847 | 15.387 Notte 23.942 | 205.892 | 31.812 pea bese 7. 096 9.614 | 14.765 pe ae 9.557 | 14.765 | 31.812 2SM___| 16.064 10. 085 7. 383 M; O; O0O Pi Qi 2Q Si Days Days Days Days Days Days. Days poe 5 acs 2 aI eee ate (GR Eee Pine cin a Relig ae 9.133 62830 Fl so eae |e ee ee ell ae ee alee ee ee 32. 451 14. 765 EIA) IR SP A oe ee ee ee | | eee, ae 13. 661 27. 555 5. 474 EGU il |.on cerned Al iN ke all tare cle ae aE 9. 133 13.777 4. 566 7.127 ZOD: || Saee a2 ee oe ee 29. 803 14. 192 13. 168 365. 243 9. 367 62990 see 2 14. 632 31. 812 5. 623 10.085 | 205.892 24. 302 9.814 SEMIDIURNAL CONSTITUENTS Ne 2N Re S2 Ts 2 p2 v2 Days Days Days Days Days. Days Days. | Days MOOT RES e (at me 1 REG. i Ge Cm Ll ON 9. 367 CST ee fs aN a I SN ee BE Si ad 9. 614 LUPE SO LE | sess te ie Aer ce ll aes ey Ls 9.874 9G) || WE GAD |) AR CGO) Joo see a || os ase essesene|[eo=355 14. 765 9. 614 25. 622 Destine) SRY es ee ee el 31.812 | 205.892 7. 236 7. 383 Oso wl OX O85 ae ea 205. 892 24. 302 9.814 10.085 | 10.371 | 15.906 | 27.555 | .__--- 5. 823 4.807 | 15.387 | 14.765 | 14,192 | 9.614 | 4.922 | 5.992 i) 1 Table 40.—Day of the common year corresponding to day of month [For leap year increase all numbers after February 29 by 1 day] Day of month. Jan. Feb. Mar. | Apr. May 1 32 60 Oil. AL Zoo Ol 92 22, 3 34 62 93 128 4 35 63 94 124 5 36) 664 95 125 6 37 65 96 126 (et .) ee 27, 8 39 67 98 128 9 40 68 99° 129 10 41 69 100 ~=180 Wt di} 7) 101-181 12 43 “71 102 1382 13 44 72 103 =: 138 14 45 73 104 «134 15, 46 74 105-135 IG e/a 106 «136 17 48 76 107.137 18 49 77 108 138 19 50 78 109 139 20o len 110. =©140 21 52) 80 111141 22 OMSL 112 «©1142 23 54 82 113 (143 24 55 83 114 «(144 25 56 84 115) 145 261) 57 85 116 =146 2 258) SO 117.— «(147 28) 59) 87 118 «148 29 88 119 «149 30 89 120 = 150 31 90 151 June 152 153 154 155 156 157 158 159 160, 161 162 163 July 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 Aug. 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 Sept. 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 Oct. Nov. Dee. 274 «9305 = 335 275 «6306 «= 3336 276 «6307 )«=—:3337 277 «3308 =S 338 278 «©9309 = 3389 279 «4310 340 280 «311 «341 281 312 342 282 «313 «343 283 «314 «(344 284 315 = 345 285 316 346 286 317 347 287 318 348 288 319 349 289 320 350 290 321 361 291 322 352 292 323 353 293 324 354 294 325 48355 295 326 356 296 327 357 297 4328 358 298 329 359 299 330 360 300 «6331 == 3361 301 332 362 202 333363 303 334 = 3364 304 365 310 USS: CC’ST AND GEODETIC SURVEY Table 41.— Values of h in formula h=(1+r?+ 2r cos x) S88 850° 0.0 0.00 SOS) SS OSS Sls S oo ooo oo oo VS SS = = rol o Ee OD 00 noo WRC CNN oR tS qin wom momo oon wanint Com oanNr 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ° 1. 300 1. 400 1. 500 1. 600 1. 700 1. 800 1. 900 2.000 } 360 1. 296 1. 396 1. 495 1. 594 1. 694 1. 793 1. 893 1.992 350 1. 286 1. 383 1. 480 1.577 1. 675 es C0 1. 871 1.970 340 1, 269 iL BYail 1. 455 1. 549 1. 644 1.739 1. 835 1, 932 330 1, 245 1. 331 1. 420 1.510 1. 601 1. 693 1. 786 1. 879 320 1. 215 1. 294 1. 376 1. 460 1. 546 1. 634 1. 723 1. 813 310 1.179 . 249 a23 1. 400 1. 480 1. 5A2 1. 646 1. 732 300 1. 138 1. 197 1. 262 1.331 1. 403 1. 479 1. 567 1. 638 290 1.093 1. 140 1. 193 1, 252 1.316 1.385 1. 457 1. 532 280 1. 044 1.077 TLS 1. 166 1. 221 1. 281 1. 345 1. 414 270 0. 993 1.010 1. 037 1. 072 Te JUNE 1. 167 1, 224 1. 286 260 0. 941 0. 941 0. 953 0. 974 1. 006 1.045 1. 093 1. 147 250 0. 889 0. 872 0. 866 0. 872 0. 889 0.917 0. 954 1.000 240 0. 839 0. 804 0.779 0. 767 0. 768 0. 782 0. 808 9. 845 230 0. 794 0.740 | 0.696 0. 664 0. 646 0. 644 0. 657 0. 684 220 0. 755 0. 684 0. 620 0. 566 0. 527 0. 504 0. 501 0. 518 210 0. 725 0. 639 0. 557 0. 482 0. 418 0. 369 0, 344 0. 347 200 0. 706 0. 610 0.515 0.422 | 0.334 0.254 | 0.193 0. 174 190 0. 700 0. 600 0.500 | 0.400 0. 300 0. 200 0. 100 0. 0CO 180 ; r sin x Table 42.— Values of k in formula k=tan-’——_— 1+rcosx [When z is between 180° and 360°, tabular values are negative] Tt; =. a5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ° ° ° ° ° ° ° ° ° 0. 00 0. 00 0.00 0. 00 0. 00 0. 00 0. 00 0. 00 360 2. 30 2. 85 3.33 3.75 4.12 4.45 4.73 5. 00 350 4. 58 5. 68 6. 63 7.47 8. 22 8.88 9. 47 10. 00 340 6. 78 8.45 9.90 11.17 12. 30 13. 30 14. 20 15. 00 330 8.92 Wie 1 13. 08 14. 80 16. 32 17. 68 18. 90 20. 00 320 10. 99 13. 70 16.17 18. 35 20. 30 22.03 23. 60 25. 00 310 11725783 16. 10 19. 10 21. 78 24, 18 26. 33 28. 27 30. 00 300 14. 33 18. 30 21. 87 25. 07 27.95 30. 55 32. 88 35. 00 290 15. 68 20. 22 24. 37 28. 15 31. 58 34. 67 37.47 40. 00 280 16. 70 21.80 26. 57 30. 96 34.99 38. 66 41.99 45. 00 270 WAR 22.95 28. 33 33. 42 Beale 42.45 46. 42 50. 00 260 17. 43 23. 53 29. 55 35. 35 40. 85 45. 98 50. 70 55. 00 250 17.00 23. 42 30. 00 36. 58 43.00 49.10 54. 78 60. 00 240 15. 90 22. 42 29. 45 36. 80 44. 27 51. 60 58. 57 65. 00 230 14.05 20. 33 27. 52 35. 52 44.13 53. 02 61.77 70. 00 220 11. 45 17. 02 23. 80 31.98 41. 63 52. 48 63. 88 75. 00 210 8.13 12. 37 17. 88 25. 20 34. 98 47. 82 63. 38 80. 00 200 4. 23 6. 53 9.70 14, 28 21.37 33. 22 53. 98 85. 00 190 0. 00 0. 00 0. 00 0. 00 0. 00 0. 00 0500) | == 180 EXPLANATION OF SYMBOLS USED IN THIS BOOK Although the following list is fairly comprehensive, some of the symbols given may at times be used in the text to represent other quantities not listed below, but such application will be made clear by the context. A o Ci ve) ae} (1) General symbol for a tidal constituent or its amplitude. It is some- times written with a subscript to indicate the species of the constituent (par. 52). (2) General symbol with an identifying subscript for a constituent term in the development of the lunar tide-producing force (par. 66). (3) The particular tidal constituent being cleared by the elimination process (par. 245). (4) Azimuth of tide-producing body reckoned from the south through the west (par. 80). (5) Azimuth of horizontal component of force in any given direction (par. 85). (1) Speed or rate of change in argument of constituent A. (2) Mean radius of earth. (1) Tidal constituent following constituent A in a series. (2) General symbol with an identifying subscript for a constituent term in the development of the solar tidé-producing force (par. 117). (3) General symbol for disturbing constituents in elimination process (par. 245). Speed or rate of change in argument of constituent B. (1) Mean constituent coefficient (par. 74). (2) Comer symbol for coefficients of cosine terms in Fourier series (par. 187). Reciprocal of mean value of 1/d. Reciprocal of mean value of 1/d). Declination of moon or sun. Distance from center of earth to center of moon. Distance from center of earth to center of sun. (1) Mass of earth. (2) Argument of tidal constituent (same as V+). Eccentricity of moon’s orbit. Eccentricity of earth’s orbit. Reduction factor, reciprocal of node factor f (par. 78). Horizontal component of tide-producing force in azimuth A. When numerals are annexed, the first digit (3 or 4) signifies the power of the parallax of the moon or sun involved in the development and the second digit (0, 1, 2, or 3) indicates the species of the terms included in the group. ‘Thus F439 represents that part of the horizontal component in azimuth A that comprises the long-period terms depending upon the cube of the parallax. South horizontal component of tide-producing force. (See F, for explana- tion of annexed numerals.) Vertical component of tide-producing force. (See F, for explanation of annexed numerals.) West horizontal component of tide-producing force. (See F, for explana- tion of annexed numerals.) Node factor (par. 77). (1) Greenwich epoch or phase lag of a tidal constituent (par. 226). (2) Gear ratio of predicting machine (par. 396). (1) Mean acceleration of gravity on earth’s surface. (2) Modified epoch of tidal constituent, same as x’ (par. 225). Mean amplitude of a tidal constituent (par. 1438). 311 312 U. S. COAST AND GEODETIC SURVEY Hy Mean water level above datum used for tabulation. h (1) Mean longitude of sun; also rate of change in same. (2) Height of tide at any time. hg Height of equilibrium tide involving cube of moon’s parallax. A second digit in the subscript limits the height to that due to terms of the single species indicated by this digit (pars. 97 and 101). hg Height of equilibrium tide involving 4th power of moon’s parallax. A sec- ond digit in the subscript has the same significance as in the case of hs. I Obliquity of lunar orbit with respect to earth’s equator. 1 Inclination of lunar orbit to the ecliptic. 1 Tidal constituent. ] Longitude of moon in its orbit reckoned from lunar intersection. i, Kk, Tidal constituents. KJ, KP;, KQ; Tidal constituents. k Difference between mean and true longitude of moon (par. 59). L Longitude of place; positive for west longitude, negative for east longitude. L,, LP, Tidal constituents. M Mass of moon. M;, Ms, Ma, My, Me, Mg Tidal constituents. Mf Tidal constituent. MK;, 2MK:, MK, Tidal constituents. Mm _ Tidal constituent. MN, 2MN,, MNS, Tidal constituents. MP, Tidal constituent. 2MS,, MS,, 2MS., 3MS8s, 2(MS)s Tidal constituents. MSf Tidal constituent. MSN¢, 2MSN;_ Tidal constituents. m Ratio of mean motion of sun to that of moon (par. 62). N Longitude of moon’s node; also rate of change in same. No, 2Ns, NJ; Tidal constituents. O,, OO, Tidal constituents. 2, Mean longitude of lunar perigee reckoned from lunar intersection (par. 122). P, Tidal constituent. p (1) Mean longitude of lunar perigee; also rate of change in same. (2) Numeral indicating species of constituent, frequently written as the subscript of the constituent symbol. In special case used with long- period constituents to show number of periods in month or year. Dp Mean longitude of solar perigee; also rate of change in same. Q Term in argument of constituent M, (par. 123). Qa Factor in amplitude of constituent M, (par. 122). Qu Term in argument of constituent M; (par. 122). Q:, 2Q; Tidal constituents. R (1) Amphtude of constituent pertaining to a particular time (par. 148). (2) Term in argument of constituent L, (par. 129). itv Factor in amplitude of constituent L2 (par. 129). Ry, RP; Tidal constituents. r Distance of any point from center of earth. S (1) Mass of sun. (2) Longitude of time meridian; positive for west longitude, negative for east longitude. (3) General symbol for coefficients of sine terms in Fourier series (par. 187). (4) Working seale factor of predicting machine. S’ Solar factor U,/U (par. 118). S1, S:, Ss, Si, Ss, Ss Tidal constituents. Sa Tidal constituent. Sk 3 Tidal constituent. 2SM, Tidal constituent. HARMONIC ANALYSIS AND PREDICTION OF TIDES 313 SO,, SO; Tidal constituents. Ssa ‘Tidal constituent. s Mean longitude of moon: also rate of change in same. Sy True longitude of moon in orbit referred to equinox (par. 59). a (1) Number of Julian centuries reckoned from Greenwich mean noon, December 31, 1899. (2) Hour angle of mean sun. (3) Time expressed in degrees of constituent reckoned from phase zero of Greenwich argument (par. 439). T> Tidal constituent. t (1) Hour angle of tide-producing body. (2) Time reckoned from beginning of tidal series. U Basie factor (M/E) (a/c)8. U; ‘Factor (S/E) (a/e)3. u Part of constituent argument depending upon variations in obliquity of lunar orbit (par. 71). V (1) Principal portion of constituent argument (par. 71). (2) Velocity of current (par. 330). (V+u) Constituent argument at any time. (V.+u) Constituent argument at beginning of a tidal series. Ve Potential due to gravity at earth’s surface (par. 96). V3 Tide-producing potential involving cube of moon’s parallax (par. 94). Vi Tide-producing potential involving 4th power of moon’s parallax (par. 94.) Longitude of observer reckoned in celestial equator from lunar inter- section. Ww Latitude of observer. When combined with a subscript consisting of a letter and numerals, it represents the latitude factor to be used with the tidal force component similarly marked (par. 79). zZ Geocentric zenith distance of tide-producing body. a (Alpha) General symbol for the initial phase of tidal constituent A. B (Beta) Initial phase of constituent B. y (Gamma) Initial phase of constituent C. 5 (Delta) Initial phase of constituent D. e (Epsilon) Initial phase of constituent E. ¢ (Zeta) The explement of the initial phase of a constituent (par. 221). 6, (Theta) ‘Tidal constituent, same as \Q). « (Kappa) Local phase lag or epoch of tidal constituent (par. 144). x! Modified epoch of tidal constituent (par. 225). d» (Lambda) Tidal constituent. uw (Mu) Attraction of gravitation between unit masses at unit distance. Be Tidal constituent, same as 2MS8p. v (Nu) Right ascension of lunar intersection (par. 24). py! Term in argument of lunisolar constituent K; (par. 133). Qe Term in argument of lunisolar constituent Ke (par. 135). V2 Tidal constituent. — (X72) Longitude in moon’s orbit of lunar intersection (par. 24). m (P2) An angle of 3.14159 radians or 180°. Ty Tidal constituent, same as TK. pi (Rho) Tidal constituent, same as vK,. o, (Sigma) ‘Tidal constituent, same as vJ}. rt (Tau) Length of series in mean solar hours (par. 248). ¢ (Phi) Tidal constituent, same as KP). x1 (Ch?) Tidal constituent, same as LP;. vy, (Ps?) Tidal constituent, same as RP). w (Omega) Obliquity of ecliptic. aq Vernal equinox. 8% Moon’s ascending node. BAR. Wk. aon Seer at ie Ne e. INDEX A Page ANOIEEG OSs i Coes aa es a 1 Avi) GEOTRe Bytes ee Pe lk 1 Amplitude of constituent-----_--- 2, 49 Analysis of high and low waters___ 100 Analysis of monthly sea level__ 98, 114 Analysis of observations__________ 49 Analysis of tidal currents___----__- 118 Anomalistic month, year__-__-____ 4 Approximation, degree of_________ 8 Argument. (See Equilibrium argu- ment.) Astressiietiige 2252 22 88 ae Page Equilibrium argument___-_______ 22 50, 75, 108, 124, 157, 204 Equilibrium theory 3 28 Eig uilaorinnas tic ema 2 ae meee 28, 38 EA CUT OX Ob es 2B sions ns RE SNE ae UG OXaS i ts Nk AS eins ORR E IN pa 1 Hvection= += 7 ose meee Et SPE hee 4 Explanation of tables] 222-222" 72 153 Explanation of tidal movement___ 2 Extreme equilibrium tide_________ 33 Extreme tide-producing force____- 13 F Astronomical data_______-_- 3, 153, 162 Astronomical day - ~~ ------_-_--- 3) Factor F. (See Reduction factor.) Astronomical periods _____---__~- 163 | Factor f. (See Node factor.) Astronomical tide. _ -- --.--_.---- el) | eral, Wallen, 8 2 1b 1127) Augmenting factors___- 71, 91, 157, 228| forms for analysis of tides________ 104 Forms for predicting machine_____ 143 ‘ B Mourleniseniesmaee css. eae 62 SSS ECAC COT eee eee ea 24) Fourth power of moon’s parallax__ 34 c Fundamental astronomical data__ 153, Wallon elise ae eee 4) Fundamental formulas___-_______ a0 Cinliday 2 aa ee pay Ba 3 Woehicienmtset = ooo. by hela 24 G Ce (See Constituent tides. : Component of force, horizontal___ 26 cone cena af sieeve: 24 Component of force, vertical______ 15 pee IRL IRS abt AONE 2 Wompound! tidesx225 25254. 47, 167 splaptiae ca Ms ag ee eae ew a Wonstiuent dayeie as. Se 3 caaieuensl tidesvs2 56. eee 30 F reatest equilibrium tide_______- 33 Constituentmour. (2) eee eee, 4 (Gheavinds (El al f 13 Constituent tides____________ ALG, Sir ae eee ucing force-___- reenwich argument_____________ 76 Formulas Rapie y eran ee eee eer Greenwich epoch 77 DDLCS ears eee ah 6 RN res RRR aN PST, eC Sy Oa Currents vanalysisee 6) — 558 esos 118 Gregorian calendar __---________- 4 Currents; sprediction_2-— 2225 =e 147 HW D Dewan Geb cee nao gel 1 | Harmonic analysis____--____ 3, 49, 112 Datum for prediction_______ _ 124, 144] Harmonic constants__-_____- 3, 49, 143 Day, several kinds_______________ 3 | Harmonic prediction____-______ 3, 123 Waysot years tables ss. 0 slo) 309 | Harris, Rollin A___----_-- ———— 1 Declinational factor_____________ 17| High and low water analysis______ 100 Degree of approximation_________ g | Historical statement -_----__.___-- 1 Development of tide-producing Horizontal component, tide-pro- HORCEUO EE ee lem eo eee 10) ducing forces == 22 Seas eae 6, 37 Diurnal constituents_____________ 16 Hour pouera eins -------------- ad ourlyshelghtsess = Ge0ee 7o aie E iydrawhercurrent =a) ee ee 148 ene eee maty amines 66 a Hecentricity, otonbibes= = sesame Fi zy deceraphicsdatun : Eclipse; year= 200i La Nes ed I Blimination22 4) 255 = 84, 116, 158, 236 Hip tic factor === so eee Inclination of moon’s orbit_______ 6 Epoch of constituent_________-_- 49, 75 155, 173 Equations of moon’s motion -_-_-_-_- 19! Inference of constants____-_-__-_ 78, 114 316 INDEX J Page Page| Predicting machine. (See Tide- Juliancalenda see 4 predicting machine.) Prediction of tidal currents_______ 147 K iPredictionyol tides=as. == aaa 123 iGvanduko tides: =. ssss2ee> see ne 44 | Principal lunar constituents_______ 21 Kelvin, Word. 22-235. 02 5 1, 126| Principal solar constituents_______ 39 L R 1 Gre mei an mC oO ete Clan Ty 2 1 HEE REET WOE he tase” ¢ | Record of observations___________ Latitude factors______- 17, 24, 154, 168] Reduction factor _____ 25, 111, 156, 186 bength of series. = 22 2_ 2-2) 2a 51 Lesser lunar constituents_________ 35 S Lesser solar constituents_________ 40 Lesser tide-producing force____- Wuonentudes = esse te et ns Longitude, lunar and solar ele- MmentS- = a. sae ee se ee 162, 170 Long-period constituents___ 16, 87, 302 ] FES 10 (aaa eg es oO 43, 156, 177, 192 Lunar constituents____________ Zila MGuNarXC a = yer ts oe aoe 3 ena ears hy @ nes eee ee ee 4 unar intersections =a = ses 6 MIR TOO Cee se ee ee eee 6, 8 bhunisolar tidesss-=—- — 4-56 s sno 44 M Mean constituent coefficient ______ 24 Mean Jongitude®= 22210! Lo vena 7 Meteorological tides_____________ 46 Month, several kinds___________- 4 Monthly sea-level analysis_____ 98, 114 Moon’s motion, equations________ 19 Moon’sinodest! _ 1eltaiahess Len 6, 8 Moon’s parallax, 4th power_______ 34 IM, =tide.3 22-2288 41, 156, 179, 192 N Node; Juana. 5-0 ess gn ois jan pees 6,8 INodeltactor 22. 5 ay eee pene 25 Compound tideste es 252 ae 47 Constituent Kaas 2222 --22--- 45 Constituent Kes 222 aeee 46 Constituent Epes. oe ee 44 Constituent Migjse5.—2 3252 43 Lesser tide-producing force___ 36 Predictions a2 == aes 124 BSN 0) VEE) is a a SS S| PN = ea 199 INodical montha== 35542. =24 a 4 O Obliquity factor. =.= 5 _- 254e2 4=22 24 Obliquity of eclip ic___-_._______ 6 Obliquity of moon’s orbit_________ 6 Observational data______________ 50 Overtides cn 25 oe cp Ai by heh singles ted 47 P Period of cons/ituent____________ 3 Periods, astronomical____________ 163 Phase lap. <2. 99) tonne te once 49, 75 Phasejoficonstituent_ 20 5-2-2) __ Poor, Charles Lane_____________- 1 Potential. foe Pee ee ae 30 Secondary stencils________ Semidiurnal constituents_________ Settings for tide-predicting ma- Chines SUI Cl Bee ae 145, 306 Shallow-water constituents____ 46, 167 Shidy, -L..P..--.-.--..--- oe 53 Siderealday .--=--. =) 23eniiae 3 Sidereal hour, month, year_______ 4 Solar day__..=- . 299 3. Le 3 Solar factor... 95) sao ae 40 Solar hour)! _.-./20.8. Sees 4 Solar tides... ==... _\ 39 force. -+=-.se<+2+=2-- eee 26, 37 Species of constituent____________ 16 Speed of constituent_2 25 saa ——— 3, 23 Stationary wave_.-------=-as0 2 Stencil sums-=-==-----= === eee 107 Stencis24 «222-2208 53, 106, 158, 268 Summarized formulas: Equilibrium tide__-__-___-2__ 33 Lesser tide-producing force___ 36 Principal tide-producing force. 26 Summation for analysis__________ 52 Surface of equilibrium_________ 30, 32 Symbols used in book____________ 311 Synodical month: ===") = eae 4 Synodic periods of constituents___ 161, 309 T Talbles= = === - == 250i 0s Mahe See 162 ixplanation= == -- =. 25= === 153 Terdiurnal constituents__________ 34 Thomson; ‘Sir Williams a 1, 126 dhidalvcurrentse a2 == 118, 147 Tidalsmovement===2 "==. ===as 2 Tide-predicting machine__________ 126 Adyustmentsaa=. ==) =a 139 Automatie stopping device___ 135 Base eet. ee 127 Constituent cranks__-______- 130 Constituent dialss=) =] === 131 Constituent pulleys__________ 132 Constituent sliding frames____ 131 Datum of heights___________ 141 Day dial_2..---._-. sae 128 Dial hour. ==. .2.. 22 ee esaee 128 IDimensions==5 2 === ae 127 Doubling gears= 23] ee 132 Formsiuseds 2422 ==") eee 4= 143 Gear speeds... 2.2224 5a ee 129 Gearing see 128, 160, 307 Graphiscales 2222229 =e 137 INDEX 317 Tide-predicting machine—Con. __ Page | Tide-predicting machine—Con. Page Heizbtitormulas—===—- re - 126 Summation wheels__-_------- 133 Height predictions_-_--------- 134 hide cunye = ee ea eae 136 Height seales! 2. === 2-22 134, 141 dlimetdials === eee 128 Meishipsidens. 22 ob eee 128 Mime formulae sa> sae = 126, 132 High and low water marking ‘hime: predictiona=2-=2 25-2 5=— 135 GeviCen severe hoe ee he 139 Aime SiGe as aes ees oe 128 Hour marking device- - - ----- 139 Verification of settings___--__ 142 Marigram gears--------- 137, V4i\|\ide-producing force=== 525-22 —— == 10 Manieram scales? 92-5200 22 137 | Tide-producing potential____--_-- 31 Nonreversing ratchet __---_-_- 136) Bropical month years. 222-2 52=— 4 Operation of machine- ------- 142 Raperee ress ewan) Oe ee 136, 142 Vv SEG Ta See one Se eae Tah coe 138, 142 Ret Dect A Plane of reference__---_____- 141 | Variation inequality ------------- 4 Positive and negative direc- Vernal equinox___--_- air eee 6 Te as MOM ate 131.| Vertical component, tide-produc- predic tines 242 C U2 = ane 142 WS NONCOs saeseseasseoass= -- 15, 34 Releasable gears____-_-_--__- 130 w ae Dapbude seunes- 132, 140 cale, height dial________ 134, 141 STEN : Scale, marigram Wie ars oe See tes 137 Ace’. component tide producing Scale Stable. Gee) oe ALPS ee ee crsar rate peregrine y Neales working. = 20) 2) aes 135 Y Settimesmachines ===.) 5222 _ 140 Stopping devices ==. 25227 - = 135| Year, several kinds__-_....-_---- 4 Summation chains____-_____- 3a Young, “bhomass asa e= eee 1 © *U, S. 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