w 4 od on lat Ma ot bt Sintaytnaen: bese Peclintae yn vg pteteten Yan rier ror cna rea SI rine cery oe tote! Sorcrer tore moerie rn Gants % re reser error Noe berber bar yre errr ere ? oo tetra maton rh ; WW sesh rancitoah Dp aeanial vi Eogee u st etic or pyeectare tbe Sey ‘ c ord a per bo ery HOE : ss 4 Bar trtiboun Srveircierewr gurl a} wry ecimente oerfere atta sry zie ert etieyverset 3 eat Races eimai See ee age St Pirie tery, Kea < S bhEELOO TOEO O | MUTATION lOHM/181N " A 7 HENRY STOMME! Serial No. 244 DEPARTMENT OF COMMERCE . U. S. COAST AND GEODETIC SURVEY E. LESTER JONES, DIRECTOR / g TIDES A MANUAL OF THE HARMONIC ANALYSIS AND PREDICTION OF TIDES By PAUL SCHUREMAN Mathematician U.S. Coast and Geodetic Survey Ss 2 aan PN OS intl SI t ian Hoi A Nan annrvanhi wilmeen j L: | "00GS Hoe Oceanographic Institution | 2 COLLECTION i v4 Woods Hole Oceanographic Institution Purchase Order No. att PRICE, $1.00 / au 1973 Sold only by the Superintendent of Documents, Government Printing Office Washington, D. C. WASHINGTON GOVERNMENT PRINTING OFFICE 1924 i PREFACE. This volume was designed, primarily, as a working manual for the tidal computers of the U. S. Coast and Geodetic Survey, and the aim has been to produce a convenient work of reference on the subject of the harmonic analysis and prediction of the tides. It is based largely upon the works of Sir William Thomson, Prof. George H. Darwin, and Dr. Rollin A. Harris. The tidal components. scheduled in this volume are those which have been in general use in the harmonic prediction of the tides for many years, and they are sometimes called the Darwinian components. It may be added that Dr. A. T. Doodson, of the Tidal Institute, University of Liverpool, in a recent development of the tide-producing force by a method differ- ing somewhat from that used by Professor Darwin, has obtained a new schedule of components which include some elements which were: not contained in the Darwinian schedule. Although the additional components are generally of small theoretical magnitude, it is possible that some may be found by further investigation to be of sufficient importance to be taken into account in the analysis and prediction. of the tides. The volume includes a collection of tables used in the analysis and prediction of the tides, these tables being prepared with the aim of facilitating the work of the computer. At the end of the volume there is given a table of the principal tidal harmonic constants for many stations throughout the world, thus providing the data required for the harmonic prediction of the tides at these stations. A table of this kind 1s, of course, subject to growth and revision, and it is hoped that the present compilation may serve as a basis for future publications in which the entire maritime world is more comprehen- sively represented. II CONTENTS. PART I.—DESCRIPTION. Paeecneralexplanation. 20 oe NE eth hee tneey cpt A SPECULUM: GE OT yee ue EA aloe a (NN ge Aya el ase mips pay A AMeAStROnomiIcal datat we be Bes bleh eer ual 1 ila dap ad he A eT De Deoree Of approxima tom. Sho.) o5 iy oe el) papas eo op yeaa ng yD Miettinen MONeM tS! kb Lo peer ee he i ep tbat ie teh oy orig ae Oe: | Gmmhide=producing forces. 2.4 Be wee ND he pas ae ues Caley RL onl de=pLOGUGING MOCEMCIAl s/s 27EN eye aye phe AN da ies a Seaoumace Of EquillloriGm 200 Pa ke yeu RA oobi et Dae 92 Development.of equilibrium tides: 2). 440228 8 kee ee HOSE Gullit nium ATO UIM emt 2 So ee ee he aa ae ee Sh ee HSER Getincrenitisi ako 210 POs ee oS oo age eee eR ahah Sep ILSS\H PT aVe Big kG Kevaieoe i ates Pitney Ngee WON eM ya O}eNm [Oy Pace Yemen anaes PnP DLy Were me EY a fey BAe ANE IN iste AAT CL ee ee cg es AW py td a ee a rare yn alga Sie ig pda eo ea a VS 15. Tides depending upon the fourth power of moon’s parallax___:__ Mame entices. 228 uta) oh pale fF yd i vedas iT Gah heh na OE eae uimisolan: Kay ande WK tides So) cele pepe eee ayaa geese The op) yh SMO eT GG CG, wpupeciuiieg thle cop eks oe. a nave el Ry Bl) Boy ee’ DD IN). Ckoiaray cxonou ave lah aie re si pee Oe a gS I eee Als Ao) eae Uso ey teead Seer er Ns ZOseWVLeheorolo gical tid ess iirsen 2.tys< ay iye ep Mp bg ee DAD PeeviySispan Cen CIC LI OMS a0 ae Ree eh i a il Wadena year pies 2 2 wlanmonic constants: 2720). — asueweregse by esac er 29: ae SHH a6) USS SS SN ig a eyo OL a A DM OUIELeT SETI CSE eset e werden Bie oe nue tlre AD ERA AAUEME ORO NREL STEN PiemeAUl OTM eT GIN Eka CTOTS His te ee Sela eee eee Se oo 28. Reduction of epochs and amplitudes to mean values__________-_ Pome linkerenGerOk constaimtsin = 2a cael a i AD MeO. aCe Demme TT) AKG UC Tne cpr eae Srp pa py LR i ere ie tee BE MN Sleetone- periods componentgu tases eke due toile). De ae eee ie eas > Critical logarithms for Horm: 2452-2 ee eee . Component speed differences (b—a)___.______~_. LL. Ll 222 Lee ~ Euhimination: faetoOnse rss ck ke LU Sa eke AS ary a aa . Hor construction of primary stencils) See ee eee Py DivASOrs fOr priMeatays SbCl SUMING ees epee ge ee For, construction ofssecondary stencils... 2-2 1s 2 2 eee Assignment of daily page sums for long-period components___-__-_ Products: for Form: 444.) oe ae ni , Angle .differences:for Form 445.000. 22202 Ue ae ee . U.S. Coast and Geodetic Survey tide-predicting machine No. 2— general: sears... Ss he . U.S. Coast and Geodetic Survey tide-predicting machine No. 2— Component wears. ee ‘Synodie periods of components. 2.2 2 ee 2 ee . Day of common year corresponding to day of month__________-_ PART III._TIDAL HARMONIC CONSTANTS. Introductory, note and references. 442232 22 Soe Ae ee ee Ametic Te gT@ns <2 ks ee aN aL eee a a British United America: east /GOaSt bi 5 0 ee la Ie aes oe el OR States east COs ee ole aN SIU aN seca Ue a ae Mexico, east coast iets 28 0's ee oo RR A i sae ee a Banama Canal Zone; east coast.) 2204. = ee eee ee ae ees Wiestr Tm Gles fj. c5 cece SN uh he aS yea sh ETE ire Ieee Sounblny Aimer cy 087m COE i oo Mg Ua i TOL yr Ee) Mn 0 9 LGR oy Panamai@anal’ Zone, west, coasts ek! 1 eee ae es eee Mexico; west:coashes! 0 0. eee en ee ee ee Oe United British Alaska States, Wese cOast.i 0 ke wick TR eae es ae America west Coastuse 22 2b oe Nees Reon Sek (exclusive! of Arctic Zone)2+ a2 28) ea Re ee oe Siberiat Casts coastos 2 Me iio iels BAU RNAP IAN Ms i DOME eI Se eta alee Japan Tslamdee si ke ee. daw ee ae a ee ee ee ee Chosen (GTA F2 9 YStan ear a IDR a PU ATEN OP SA Cl JUNE) ARN eS (Gd aisha: ene sa ds AEM ca AN INR Poe SE NON AR Uae he LS PARAM MRE GTS) Ph French Siam __ Malay Indo’) China’ (Amami) 000s © aT a ew OTT SLES AS eee Pemims lay ei he a Oe a RS CSE SR DTT Ren Philippine’ Esler si 2 2 a's ee Ee DE eee PTET ee Borneo @elebesrandovilermitry fe Cee ERM SNA Sea eeabie WY CURL TG BAUS Stima tra ‘arid svi hm ysl oe TN NR RG TANS Jena. amet viet tty So hf AES NT ERD TE REISE TELS, i pe en New 'Gulimedie 2 Who on a SER ESOS RS RTE eS nea ye pea Hawaiian Islands and miscellaneous islands in Pacific Ocean___________- AUIS Grate Wie eae eR RLU RD NUM SRR ROA 0 2c oA Nie we Liesaen rn chip ea hy ENR SACI ge ea et CONTENTS. PSST a a US Rg RS yr ee Maran icdterraneams Sem! Ga SY aR La AS EL Sha Ps Ba ee eh Riropemwesu coasu (exclusiveior Arctic Zone) 2! oss sf 2a eee CESS Ee SIGS UR AMT AS Ea RT ly 8s pa ee eR ip sR PMG GNC VC OL OMS uate mec remy. NSPE eet: Sse ks ee ina TI Re etl at Site ara lae rc cee ones eye ee tyerge MPS ele LSA) | e e ae D khe el Ya SCI Ae TM eSE eo te ae ee gE Nenad Mee ORDA Ba ee hs Se Compound and long-period! components.22 Vee oe ee TDGUSSS 2b et Ra a a RE 2 dea Ok Bk MS a YW eels ciae Dg Re PART ET oe ta oe, ILLLUSTRATIONS. MIO NED UNI Ce CURVES eas ene) tee yl IN Sey Eu he . Equilibrium tide with moon on the equator, section through poles_ __ . Equilibrium tide with moon on the equator, sections perpendicular 1G GHC Eee a lee SE ee eRe Selly teat eaten | amie Wk es bell PUR Ee poles_- ene pore ye een os ORM em Sea RET SB hh VAMOS, ERLE LAC SNe We)! Mi RP ND. he Ny SOM OP cM alg eat HW APR pega sire fee ON lies Fee OL aye Ri EL epee Da ale Bras ie lg . Intersections of ecliptic, celestial equator, and moon’s onpit Soe fed ay el OD Pena TS OUICI Te OECE Gk = alu sgh aN Dag Oe nol ta Se ep 2 SPL ESn ES, TUK RG IOI TH 09 I Be ca ap A SRI We ps LIN ance sen Ne aoe OE ae mveveclopmentéot equilibrium (tide so We ye TE a ea Oe . Relation of elements in equilibrium tide__________________________ Be OMS Ol ase si, ANG) Ke ees Boy ee ne ee ee 1 . Tide-predicting machine, view of dial case and height side__________ Ealide-predicting machine. view of timesidets 1 Wl a bn ee ee ee Tide-predicting machine, view of rear end and height side___________ Tide-predicting machine, main driving gears_____________________-_ . Pide-predicting machine, dial case, side view___._____-___-.._-__.- 4 Tide-predicting machine, dial case and front component section, time side_ Pp IRN) een MRL ANON Te MO fc PN ENR AN VAL IT TUN 2 Sh Hn CGE ceeds Ree eee Le SETI Ls EAA ae A aes SR PR Oe en Ure aT Id a SI RIES . Tide-predicting machine, details of releasable gears________________ . Tide-predicting machine, details of component cranks______________ mics oe hourly Welehts: fio eur ige! Takka e yim Te Vi eS Re SHemen enor Coma pomemb NVI ele Cle pie ec A Be A) Sah ts Meo a PP MPR TCHONLOL SECMGN ss Bia ue eed tN A ea ee ee eg Peon OPUS LEM CUS IINIS tr wm ere eS NR Sen EES ie ee a monn 142) computation of hourly means.i< Ye eet) see) te Se mboera 2445 ‘conputation of Viesputs wel’ fk Ee ey Pe ee ey . Form 244a, Log F and arguments for elimination__________________ RORMELO4 Hharmonievanaliysisios,5 see ut = oN os Cee eS ee . Form 452, R, x, and ¢ from analysis and inference, diurnal components_ . Form 452, R, «x, and ¢ from analysis and inference, semidiurnal com- PAG TC TGS PR eiene Le Pa EA: 2) ERIE Le eek De ear eye re SIN Pg ry aye I -. Horm 245, elimination of component effects_____________-2.22222_+ . Form 444, standard harmonic constants for predictions_____________ . Form 445, settings for tide-predicting machine____.______________- Rib aheer HARMONIC ANALYSIS AND PREDICTION OF TIDES. Part I.—DESCRIPTION. INTRODUCTION. 1. HISTORICAL STATEMENT. 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.t- 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. 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 Teport 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 committee, 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. 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. 2. GENERAL EXPLANATION. A simple harmonic function is a quantity that varies as the cosine of an angle that increases uniformily 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 any initial epoch. A Hiamienie analysis as applied to the tides is a process by which the actual observed tide at any place is separated into a number of partial or constituent tides of which it is composed, the rise and fall. of each partial tide being a simple harmonic function of time. Harmonic prediction of the tides consists in reuniting the partial tides in accordance with the relations which will prevail at the time for which the predictions are to be made. The partial tides are called components and are usually repre- sented by letters either with or without subscripts, as M,, K,, Mm, and Sa. Theoretically, the tides consist of innumerable components of various magnitudes, but only a comparatively few are of sufficient size to be of practical importance in the prediction of the tides. The predicting machine used by the Coast and Geodetic Survey is de- signed to take account of a maximum of 37 components. Each component represents an elementary periodic cause produc- ing or affecting the tide. The principal component, designated as M,, represents the mean effect of the moon. Another component. S,, represents the mean effect of the sun. Other components take account of the various inequalities in the motions of the moon and. the sun, such as changes in parallax and declination, and also inequali- ties resulting from shallow water and seasonal meteorological changes. The amplitude of a component, commonly designated by the letter H, is the semirange between the maximum and minimum heights of the tide due to that component. The amplitude of any component varies with the locality, but for any particular place it is practically constant for all time. The epoch of a component, commonly designated by the Greek letter kappa (x), is an angle whose value depends upon the interval between the time of the maximum of the component as determined. theoretically from the equilibrium theory and the actual time as determined from the analysis of the observations. The epoch of a component varies with the locality but, like the amplitude, is con- stant for any particular place. The harmonic constants are the numerical values of the amplitudes. and epochs of the components for any place. The determination of these constants from the records of tidal observations is the purpose of the harmonic analysis. The rise and fall of the tide may be graphically represented by a curve, with the ordinates representing the height of the tide and the abscissas the time. The tidal record as traced by an automatic tide gauge is such a curve. The general equation of this curve, giving ie height of the tide as a function of time, is usually written in the orm y=H,+A cos (at+a)+B cos (bt+8) + C cos (ct+y)+ete. (1) in which y is the height of the tide at any time t, H, is a constant depending upon the datum from which the heights are reckoned, and. each cosine term represents the height of a component tide. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 3 A single component tide referred to its mean level as the datum is expressed by the equation y,=A cos (at+ a) (2) in which y,, the height of the component tide, is a function of t, the time reckoned from some initial epoch. The coefficient A is the amplitude or semirange of the component. The angle at--a changes at the rate of a@ units of angle per unit of time, and this rate of change is called the speed of the component. The period of the component is the time required for the angle 360 a when a is expressed in degrees. The phase of the component at any time t is the value of the angle at-+a, with multiples of 360° rejected, or it may be defined as the angular change in the component since the time of the preceding maximum or high water of the com- ponent. The initial phase is the phase at the instant from which the time is reckoned; that is, when t=0, and 1s equal to a in the above angle. A component tide is also expressed in the following forms: at+e to go through a cycle of 360° and is therefore equal to y,=R cos (at—¢) (3) y,= fH eos {(V+u) —«] (4) y,— fH cos lat+ (V.+u) —«] (5) In an analysis theoretically perfect the coefficient A of formula (2) must be an absolute constant; but in practice it has been found convenient, in order to take account of the effects due to the changes in the longitude of the moon’s node, to consider this coefficient as subject to certain variations. These variations are, however, so slow that for a series of observations not exceeding a year in length the coefficient may be treated as a constant, but factors are applied for reducing the results from different years to a mean value. The coefficient R of formula (3) represents the unmodified ampli- tude applying to a particular series. The mean value of the ampli- tude for all years is represented by the H of formulas (4) and (5). The f is a factor, usually near unity, which gives the theoretical rela- tion between the observed amplitude from any series of observations and the mean amplitude, this relation depending upon the longitude of the moon’s node. ; The angle ¢ of formula (3) is the equivalent of —a of formula (2). The angle (V+) of formula (4) is the theoretical phase of the com- ponent for any time ¢ as derived from the equilibrium theory, and the epoch « 1s the difference between the theoretical and actual phase as determined from the tidal observations. The complete angle (V+) —x« is the equivalent of the angle at+a in formula (2). The angle (V,+4) of formula (5) is the value of the-angle (V+) when ¢ equals zero. The change in (V+) being the speed of the component, its value at any time ¢ is equal to at+(V,+u), and formula (5) is therefore equivalent to formula (4). The values of the mean amplitude H and the epoch « of the above formulas are the harmonic constants which are to be determined by the analysis. 4 U. S. COAST AND GEODETIC SURVEY. Figure 1 is a graphic representation of a component tide and illus- trates certain relations of quantities given in the above formulas. In this figure the full horizontal line represents the mean level of the component, and distances along this line correspond to time as meas- ured by an angle which increases uniformly with time. The height of the tide at any time is represented by an ordinate to the curve perpendicular to the mean level line. The height of the maximum or high water of the component is the coefficient fH of the formulas. The point J in the figure indicates the instant of time at which high water would occur in accordance with the uncorrected equilib- rium theory. The epoch « is the angular expression for the interval between the time of the theoretical high water and the actual high water of the component as determined from observations. The in- terval between two consecutive high waters is the period of the com- ponent and is represented by the angular cycle of 360°. The interval measured backward from MM to the preceding high water may there- fore be expressed by 360°—« or —xk. Fig. 1. Let the vertical line through the point T indicate any instant of time ¢ under consideration. Then the interval between JM and this instant will correspond to the angle (V+) of formula (4). If this instant represents the initial epoch from which the time is to be reckoned, the (V+wu) becomes (V,+u) of formula (5), and the in- terval from 7' to the time of the following high water becomes the ¢ of formula (3). The interval measured backward from T to the preceding high water is the @ of formula (2). The epoch « equals the sum of the (V,+w) and ¢. 3. EQUILIBRIUM THEORY. 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. A surface of equilibrium, also known as an equipotential or level sur- face, is a surface at every point of which the potential has the same value; that is to say, the potential energy of a particle in such a sur- HARMONIC ANALYSIS AND PREDICTION OF TIDES. 5 face would be neither increased nor diminished by changing the posi- tion of that particle to any other poimt in the same surface. On a surface of equilibrium the resultant of all the forces at each point must be in the direction of the normal to the surface at that point. The equilibrium theory assumes that the solid part of the earth is covered with water of considerable depth; that the water has neither inertia nor viscosity and may move without friction. These ideal conditions, differing so greatly from the actual conditions, it is not to be expected that the liquid surface of the earth will attain the state.of equilibrium assumed under this theory. The presence of the great continental barriers, together with the inertia of the water, would make such a state of equilibrium impossible on the rotating earth. Nevertheless, the theory is of much service in the discussion of the harmonic analysis, because it affords a convenient and com- plete way of specifying the forces which act upon the ocean at each instant. The attraction of either the moon or the sun will tend to draw the earth out in the shape of a prolate ellipsoid of revolution with the longest axis in the direction of the attracting body. Figures 2, 3, 4, and 5 illustrate the forms of tides which may be expected under _ the equilibrium theory when either the moon or the sun is acting alone. In these figures the oblateness of the earth due to its centrif- ugal force is ignored. Figures 2 and 4 may represent any section | made by a plane passing through the center of the earth and the | center of the moon, but here they are supposed to represent especially © the section contaiming the earth’s axis. In Figure 2 the moon is | assumed to be in the plane of the earth’s Equator, and in Figure 4 | the declination is taken at about 28.5° N., which is approximately | yp the maximum declination reat show the mean undisturbed surface of the earth and the ellipses the surfaces as modified by the attraction of the moon. Im these figures | éd by the’moon. The great circles © eee i aia: the ellipticity of the modified surface has been made about a million ~ _ times greater than the theoretical ellipticity due to the attraction of the moon. If drawn to true scale, the disturbance due to the moon ~ could not have been detected with the eye. This magnification of the ellipticity will introduce some discrepancies in the figures when compared with the true theoretical form but which are unimportant at this time. In Figures 3 and 5 are shown sections of the undis- turbed and the modified surfaces made by planes perpendicular to the earth’s axis in latitudes 0°, 30° N., and 60° N. Let us now consider what tides may be expected under the equi- librium theory. We will suppose that the liquid surface of the earth retains its ellipsoidal form with the major axis always toward the center of the moon, and that the solid portion of the earth rotates on its axis. It is evident that every point of the solid part of the earth will describe a circle parallel to the Equator and with its center in the axis of the earth, and that in passing around this circle the water surface at the point will fluctuate in height. Referring to Figure 3, let us take a point P, of the undisturbed surface on the Equator and directly under the moon, the latter being in the plane of the Equator. At P, it will be high water. As the earth rotates the point P, will move to P,, the height of the water gradually diminishing until at P, it is a minimum or low water. In passing on to P, the water will rise to a maximum and then fall to another minimum at P, and then .ND GHODETIC SURVEY. fi COAST. A Ss. U. PoLe 5 < =} G my) Fig. 2. FiG. 3. OF TIDES. HARMONIC ANALYSIS AND PREDICTION © < > 3 Pa Fia. 4. Fig. 5. ted U. S. COAST AND GEODETIC SURVEY. rise until the point is again directly under the moon. The time required for the pomt P, to pass completely around the circle and back to its original position directly under the moon is approximately one day. During this period there have been two high waters of equal height and two low waters of equal depression. The height of the high waters above the mean level, however, is greater than the depression of the low waters, indicating that the average elevation of the water at the Equator has been raised above the normal by the presence of the moon. If we take a point P,’ in latitude 30° N., we again find two high waters of equal height and two low waters of equal depression occurring during the day, but the height of the high waters is more nearly equal to the depression of the low waters. In latitude 60° we again find two equal maximum and two equal mini- mum heights, but in this case the equilibrium surface is below the mean undisturbed surface durmg the entire rotation of the earth. From the foregoing we may conclude that under the equilibrium theory, when the moon is on the Equator, there will usually be in all latitudes two equal high waters and two equa! low waters during each day, and that the presence of the moon will tend to raise the average level near the Equator and to lower it near the poles. It will also appear evident that the amount of these variations will depend upon the distance of the moon from the earth; the nearer the moon is to the earth the greater will be its effects. Let us now examine Figure 5, which illustrates the tidal condition when the moon is near its greatest north declination. If we take a point P, in the Equator, as before, we find that in this case also we have two equal high waters and two equal low waters during the day, but the increase in the average height of water on the Equator is not as great as that shown in Figure 3. If we take the point P,’ and follow it around the small circle in latitude 30° N., we pass through. a low water at P,’, a high water at P,’, a low water at P,’, and return to the high water at P,’.. We still have two high waters and two low waters during the day, but it will be noted that one high water is somewhat higher than the other, while the two low waters are of equal height. In latitude 60° N. we have during a single day only one high water, which is at P,’’, ‘and a low water at P,’’. In this case the tide is said to be diurnal, while in the usual case of two high waters and two low waters each day the tide is called semidiurnal. If we were to take sections in the Southern Hemisphere corresponding to those for the Northern Hemisphere, with the moon still m its north declination, we would obtain ellipses similar to those in Figure 5, except that the centers of the ellipses stead of being on the side of the earth’s axis nearest the moon would be displaced by an equal amount on the opposite side. From these figures we may conclude that, according to the equilibrium theory, there will be a semidiurnal tide with equal high and equal low water heights at all places on the Equator for any declination of the moon. If the moon is in north declination, and we travel from the Equator northward, we should expect to find the semidiurnal tides continuing with equal low waters, but with an increasing difference between the heights of the two high waters, the higher high being on the side of the earth toward the moon and the lower high on the opposite side. After reaching a certain latitude the lower high and the two low waters should, according to this theory, blend into a single low water on the side of HARMONIC ANALYSIS AND PREDICTION OF TIDES. 9 the earth opposite the moon, the tide then becoming diurnal and remaining such for all latitudes north of this. If we were to proceed southward from the Equator while the moon is still in north declina- tion, we should find similar conditions prevailing, except that the unequal high waters of the semidiurnal tide and the single high and single low waters of the diurnal tide would have their positions reversed; that is, the higher high water of the semidiurnal and the single high water of the diurnal tide would occur on the side of the earth farthest from the moon instead of on the side nearest to that body. If the-moon is,in south declination, the conditions in respect to the Northern and Southern Hemispheres will, of course, be exactly re- versed. In this discussion only the moon has been considered. The sun alone should have an exactly similar effect, except that, on account of the greater distance, the magnitude of the tide would be only about one-half as great as that due to the moon. Theoretically, the height of the tide at any place due to each body can be computed separately and the sum taken to represent the height due to the com- bined effect. _- As already stated, the actual conditions that exist on the earth differ so greatly from the ideal conditions assumed for the equilibrium theory that the tides as derived from that theory are expected to differ greatly from tides as actually observed. It will be interesting to note here some of the agreements and differences. 1. Generally two high waters and two low waters occur during each day, but the high waters do not necessarily occur when the moon and sun are on the meridian. The interval between a transit of the moon and the eccurrence of a high water varies in different parts of the earth without any apparent regard for the equilibrium theory, and high water may occur at any hour between successive transits of the moon; but for any particular place the interval between the time of transit and the time of high water remains approximately constant. : 2. Usually the alternate high waters or the alternate low waters are nearly equal in height when the moon is near the Equator and have an increasing diurnal inequality as the moon’s declination increases north or south of the Equator. According to the equilibrium theory there should be a diurnal inequality in the high waters only, and with any given declination this inequality should depend upon the latitude. As an actual fact we find that at many places there is a much larger inequality in the low water heights cine in the high water heights, and that the magnitude of the equality apparently has no direct relation to the latitude of the place. 3. By the equilibrium theory the diurnal tides would be expected only in latitudes near the poles, but observations show that stations near the Equator as well as those near the poles have diurnal tides. In the following chapter a formula will be obtained which will represent the approximate height of the tide at any time and place, based upon the equilibrium theory. Although it is recognized that any calculations of the tide based solely upon this theory may give results entirely at variance with the real tide, because the actual conditions on the earth differ so much from the assumed ideal con- ditions, yet such a formula is very useful, inasmuch as we may intro- duce into it certain factors and differences determined from actual 10 U. S. COAST AND GEODETIC SURVEY. observations of the tide at any place and obtain a corrected formula which will generally represent very satisfactorily the true height of the tide at that place for any desired time. 4. ASTRONOMICAL DATA. The reader of this volume is presumed to have a knowledge of elementary astronomy, but it may be well to emphasize here some of the important details which pertain especially to the tides. Be- sides the earth the only celestial bodies with which we are directly concerned in this discussion are the sun and the moon. Because of the greater distance or smaller size of all the other heavenly bodies their direct effect upon the tides of the earth is negligible. The principal motions to be considered are the rotation of the earth on its axis, the revolution of the moon around the earth, and the revolution of the earth around the sun (or the apparent revolution of the sun around the earth). The earth rotates on its axis once each day. There are however several kinds of days—the sidereal day, the tropical day, the solar day, the lunar day, and the component day.—depending upon the object used as a reference for the rotation. Since the stars are the most nearly fixed objects we have for comparison, the sidereal day, which is the time between two successive passages of the same star across any given meridian of the earth, is usually considered as the true period of the earth’s rotation. The tropical? day is the time between two successive passages of the vernal equinox over a given meridian, and the solar and lunar days are the time between two successive transits of the sun and moon, respectively, over a given meridian. A component day is the time between two successive transits over a given meridian of a fictitious satellite which is assumed to represent the cause of a component tide. Each diurnal component will have its own component day. The solar and lunar days vary a little in length because of the lack of uniform motion of the earth and moon in their orbits, and for this reason the average or mean values of each is taken as a standard unit of measure. The mean solar day corresponds, of course, to the ordinary calendar day. Hach day of whatever kind may be divided into 24 equal parts, called hours, which are qualified by the name of the day which was subdivided, as sidereal hour, solar hour, lunar hour, or component hour. 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 ~ 2 The tropical day is also generally called a sidereal day, since its length differs from the true sidereal day within a hundredth part of a second. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 11 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. 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 center of gravity; but if we imagine the earth as fixed, the sun will describe an apparent path around the earth which is exactly the same in size and form as the orbit of the earth around the sun, and the effect upon the tides would be just the same. This orbit is an ellipse with an eccentricity that changes so slowly that it may be regarded as practically constant. The period of the revolution of the earth around the sun is one year, and, as with the day, we have several kinds of years—the sidereal year, the tropical year, and the anomalistic year, and also the calendar and the Julian years. The sidereal year is the time required for the earth to complete one reyo- lution, so that the sun will have returned to its same position among the stars. The tropical year is the time included between two successive passages of the vernal equinox by the sun. As the declina- tion of the sun, and consequently the changes in seasons, depend upon its relation to the equinox, this is the year with which we try to make our calendar approximately agree. The anomalistic year is the time between two successive passages of the perihelion by the sun. The calendar year is one consisting of an integral number of mean solar days, either 365 or 366 days, the average length of which is made to agree as nearly as practicable with the length of the tropical year. 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 distinguish them from the dates of the older calendars.’ Prior to the year 45 B.C. there was more or less confusion in the calendars, intercalations of months and days being arbitrarily made by the priesthood and mag- istrates to bring the calendar into accord with the seasons and for other purposes. The Julian calendar received its name from Julius Cesar, who in- troduced it in the year 45 B. C. By this calendar the true year is assumed to be exactly 365.25 days, and it was provided that the com- mon 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. 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. Its use is becoming more and more general, but it is not as yet univer- 72934—24}——2 t2 U. S. COAST AND GEODETIC SURVEY. sally accepted. By this calendar the true year is assumed to be 365.2425 days in length. It differs from the Julian calendar in haying the century years, which are 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 contimued. until 1700, which was a leap year, according to the Julian calendar, and a common year by the Gregorian calendar. The difference be- tween the two then became 11 days and in 1800 was increased to 12: days. Since 1900 the difference has been 13 days, which will remain. the same until the year 2100. Dates of the Christian era prior to October 4, 1582, will, in general, conform to the Julian calendar. Since that time both calendars have _been used. The Gregorian calendar was adopted in England by an. act of Parliament passed in 1751, which provided that the day fol- lowing 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 com- menced on March 25. Except for this arbitrary beginning of the year, the old English calendar was the same as the Julian calendar. In Russia the Julian calendar has contimued in general use up to the Present time, but for scientific and commercial purposes the dates. rom both calendars are frequently written together. 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 com- pared across the one hundred and eightieth meridian. Dates in the tables at the back of this volume refer to the Gregorian calendar. There are three celestial planes to be considered—one contaiming the earth’s equator, another the earth’s orbit, and a third the moon’s. orbit. The intersection of these planes with the celestial sphere gives. three great circles—the celestial equator, the ecliptic, and the inter- section of the plane of the moon’s orbit (see fig. 6). These three cir- cles intersect in six points—the celestial equator and the ecliptic at the: equinoxes, Y and ¥,; the ecliptic and the plane of the moon’s orbit at. the moon’s nodes §3 and Q, and the celestial equator and the plane of the moon’s orbit at the intersections A and A,. In the following dis— cussions references will usually be made to only three of these inter- sections, namely, the vernal equinox ¥, the moon’s ascending node Q,. and the ascending intersection of the plane of the moon’s orbit with the celestial equator. For brevity these intersections may be re- spectively referred to as “the equinox,” “the node,” and “the m- tersection.”’ The three angles made by the intersections of these great circles,. representing the angles between the corresponding planes, should HARMONIC ANALYSIS AND PREDICTION OF TIDES. 13 be noted. The angle between the ecliptic and the celestial equator # is known as the obliquity of the ecliptic. Its value is about 234° at the present time, but it is subject to a very slow secular change (see Tables 1 and 2). The angle 7, measuring the inclination of the moon’s orbit to the ecliptic, has a constant value of a little more than 5°. The angle J, measuring the melination of the moon’s orbit to the plane of the earth’s equator, varies in value from w—7% to w+7; that is, from about 184 to 284°. The complete cycle of this variation is approximately 19 years, so that if the angle is 183° in any year it will gradually increase for about 94 years until it reaches its maximum value and then diminish for about 94 years until it returns to its minimum value. The vernal equinox ¥ although subject to a very slow westward motion, known as the procession of the equinoxes, which amounts to only about 50 inches per year, is frequently taken as a fixed point Fig. 6. of reference for the motion of the ether parts of the solar system. The moon’s node 2 has a westward motion of about 19° a year, which is sufficient to carry it entirely around a great circle in approxi- mately 19 years. It is upon this motion that the variations in the value of the angle J depend, and it is of considerable importance in its effects upon the tides. 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 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 mane ascension, and the angular distance north or south of the celestial equator is called declination. The true longitude of any point referred to any great circle in the celestial sphere may be / | / [ 14 U. S. COAST AND GEODETIC SURVEY. defined as the are 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. The mean longitude of abody moving in an inclosed 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 pomt 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 ¥, which is commen to both circles. In order to have an equivalent origin in the moon’s orbit, we may lay off an arc 8, ¥’ (see fig. 6) in the moon’s orbit equal to 8 ¥ in the ecliptic and for convenience call the point ’ 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 ee Oe This will, of course, not be the case for the true longi- tude. Let us now examine more closely the spherical triangle Q ” A in Figure 6. The angles w and 7 are very nearly constant for long periods of time and have already been explained. The side 2 9, usually designated by N, 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 19 years, causes a constant change in the form of the triangle, the elements of which are cf considerable im- pertance in the present discussion. The value of the angle J, the supplement of the angle 2 A ¥, has an important effect upcn both the range and time of the tide, which will be noted later. The side A ¥v, designated by », is the right ascensicn cr longitude in the celestial equator of the intersection A. The are designated by ~ is equal to the side Q v-—side Q A and is the longitude in the moon’s orbit of the intersection A. Since the angles 7 and w are assumed to be ccnstant, the values of J, v, and é will depend directly ee N, the longitude: of the moon’s ncde, and may be readily obtained by the ordinary soluticn of the spherical triangle Q ¥ A. Table 6 gives the values of J, v, and & 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 cf a century in 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 medern time. Looking again at Figure 6, it will be noted that when the longitude of the moon’s node is zero the value of the inclination J will equal the HARMONIC ANALYSIS AND PREDICTION OF TIDES. 15 sum of w and 7 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 / will be always positive and will vary from w—7tow+%. 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 é will have positive values, as indi- cated in the figure, these arcs being considered as positive when reckoned eastward from ¥ and ¥’, respectively. For all positions of the node south of the Equator, as the longitude changes from 360 to 180°, » and é will each be negative, since the intersection A will then lay to the westward of Y and ’. Tables 1 and 2 contain a collection of astronomical constants and formulas to which reference will frequently be made in this work. 5. DEGREE OF APPROXIMATION. The problem of finding an expression for the equilibrium height of the tide in terms of time and place does not admit of a strict so- lution, but an approximate expression may be obtained which may be carried to as high an order of precision as may be desired. In or- dinary numerical computations exact results are seldom obtained, 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 numericall 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 334 per cent of the quantity itself, while the use of two significant figures will reduce the maximum error to less than 5 per cent 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 earth is popularly expressed by two significant figures as 93,000,000 miles. 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 xf the increase in the labor without a corresponding increase in util- ity, it will be usually found advantageous to limit the degree of pre- Gision in accordance with the prevailing conditions. 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- tamed. The retention of the smaller terms will depend to some ex- tent upon the labor involved, since their rejection would not serie ously affect the final results. The formulas for the moon’s true longitude and distance in Table 1 are said to be given to the second order of approximation, a frac- tion of the first order being considered as one having an approximate value of 1/20 or 0.05, a fraction of the second order having an ap- 16 U. S. COAST AND GEODETIC SURVEY. proximate value of (0.05)? or 0.0025, and a fraction of the third order having an approximate value of (0.05)? or 0.000125, etc. These formulas of the second order should, therefore, give the results cor- rect to the third decimal place. In Table 2 are given the numerical values of a number of astro- nomical relations which will appear in the following development of. the subject. A knowledge of these values will enable us to deter- mine what terms may be safely neglected in the development. TIDAL COMPONENTS. 6. TIDE-PRODUCING FORCE. As the sun and moon are similar in their action in the production of the tide, the force of either may be considered by itself, and the resulting forms of expression may then be readily adapted to the other. The tide-producing force of the moon is that portion of its gravita- tional force 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. The tide-producing force, being the difference between the attrac- tion for particles situated relatively near together, is small compared with the attraction itself. It may be interesting to note that, al- though 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. The tide-producing force may be graphically represented as in Figure 7. 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 will represent the direction of the attractive force of the moon upon a particle at P. Let the magnitude of the moon’s attraction at P be represented by the line PC. Now, since the attraction of gravitation varies im- HARMONIC ANALYSIS AND PREDICTION OF TIDES. 17 versely as the square of the distance, it is necessary, in order to rep- resent the attraction at O on the same scale, to take a line CQ of ‘such a length that CQ:CP =CP?:Co’. 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 moonat 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. FiG. 7. 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. We will now seek analytical expressions to represent the tide-pro- ducing force of the moon at any point P within or on the surface of ‘tthe earth. Referring to Figure 7, let r=OP =distance of P from center of earth, b=PC =distance of P from center of moon, d=OC =distance from center of earth to center of moon, 6= COP =angle at center of earth between OP and OC, M=mass of moon, w=attraction of gravitation between unit masses at unit distance apart. Since the force of gravitation varies directly as the mass and in- versely as the square of the distance, uM _ Attraction of moon for unit mass at point O in direction OC= P (6) Attraction of moon for unit mass at point P in direction PC a (7) 18 U. S. COAST AND GEODETIC SURVEY. Let each of these forces be resolved in directions parallel and per- pendicular to the radius through P, and let the direction from O toward P be taken as positive and the reverse as negative, and also the direction of the perpendicular to OP that most nearly conforms with the direction from O toward C as positive, and the reverse direction as negative. We then have from (6) and (7). Attraction at O in direction O to P= 4 cos 6 (8) Attraction at O perpendicular to opt sin 6 (9) Attraction at P in direction O to punts CPR (10) Attraction at P perpendicular to OP = a sin CPR (11) As the tide-producing force of the moon at the point P is measured by the difference between the attraction of the moon at P and at O, the following may be obtained from (8), (9), (10), and (11). Tide-producing force at P in direction O to P =iM | cos OPR—= cos | (12) Tide-producing force at P perpendicular to OP uM le sin OPR—5 sin | (13) By a solution of the plane triangle COP the following relations are obtained: 2 P=r’+d0— 2rd cos 0=@ | 1-2 5 cos 047] (14) ‘ : Giaaes sin 6 sin CPR =sin CPO=- sin 0= b ; Pak (1) 1-24 cos 0+ cos es cos CPR=./1—sin? CPR= - Ss = iL sin? CPR [1-25 cu ore p (16) In Figure 7 it will be noted that the value of 6, being reckoned from the line OC in any plane may vary from zero to 180°, and also that the angle CPR increases as @ increases within the same limits. Sin @ and sin CPR will therefore always be positive. As the angle OCP is always very small, the angle CPR will differ by only a very small amount from the angle @ and will usually be in the same quad- rant. In obtaining the square root for the numerator of (16) 1t was therefore necessary to use only that sign which would preserve this HARMONIC ANALYSIS AND PREDICTION OF TIDES. 19 relationship. The denominators of (15) and (16) are to be con- sidered as positive. Substituting (14), (15), and (16) in (12) and (13), and designating the forces in the direction OP and perpendicular to the same as the vertical and horizontal components, respectively, we have Vertical component of tide-producing force at P u cos 6 — - . = ie = 72\sR ~ COS 0 (17) (1-2 d COs 0475) Horizontal component of tide-producing force at P p p s in 0 ' _uM “= 72\s2 ~ SD 6 (18) 2 fice pe: iui d (1 2 7 cos 6+ >) By Maclaurin’s theorem the fraction rd r r2\3/2 oe ek (1 27 098 0455) may be developed into a series arranged according to the ascending powers of Ai which has a value of approximately 0.017 when r is taken as the mean radius of the earth and d as the mean distance of the moon from the earth. When d is taken as the mean distance of the sun from the earth, the value of "is considerably smaller. In d equation (19), which follows, the terms involving the higher powers of - are relatively unimportant and may be neglected. 1 r (2-2 je 6+ 2 72 s2—1+3 cos 0 7 +3/2 (5 cos? @—1) = ") 5) ip +5 (7 cos? @—3 cos 4) qa t ete. (19) _ Substituting (19) im (17) and (18) and neglecting all terms contain- ing powers of a above the fourth we obtain Vertical component 2 ae (3 cos? @—1) +3/2 oe (5 cos* 8—3 cos 6) (20) Horizontal component =3/2 "A sin 2 0+3/2 "42" (6 cos? @—1) sin 0 (21) 20 U. S. COAST AND GEODETIC SURVEY. As the moon’s parallax varies inversely as its distance d, the terms containing the reciprocal of d’ are said to depend upon the cube of the moon’s parallax and those containing the reciprocal of d‘ upon the fourth power of the moon’s parallax. Assuming the approximate numerical value of - as before, it is evident that the above terms in- volving the fourth power of the parallax will generally be only about 2 per cent of the entire tide-producing force and are therefore of little relative importance. . They will, however, be given further attention. For convenience, the force depending upon the fourth power of the moon’s parallax will be treated separately from the principal tide- producing force, which depends upon the cube of the parallax. They may be expressed separately, as follows: Tide-producing force depending upon the cube of moon’s parallax Vertical component =~ ve (3 cos? 6—1) (22) i ul . Horizontal component = 3/2 qs sin 28 (23) Tide-producing force depending upon the fourth power of moon’s parallax Ss ip? Vertical component = 3/2: re (5 cos® 6—3 cos 4) (24) ; u Mr? : Horizontal component = 3/2 EE (5 cos?@— 1) sin@ (25) Similar expressions for the tide-producing force of the sun may be obtained by substituting the mass of the sun for M and the distance of the sun ford. Because of the greater distance of the sun the terms depending upon the fourth power of its parallax will be negligible. The relation of the tide-producing force of the sun to that of the moon will be approximately Mass of sun (mean distance of moon) 3 Mass of moon’ (mean distance of sun) 3 = 0.46 (26) Examining formulas (22) and (23) for the principal tide-producing force it will be noted that the vertical component becomes zero when » cos @= + 74, and the horizontal component becomes zero when 6=0, 90, or 180°. The vertical component has a maximum positive value when 9=0 or 180°. and a maximum negative value when §=90°, the latter force being only one-half as great as the maximum positive value. The horizontal component is at a maximum in the positive direction when @=45° and a maximum in the negative direction when 6=135°. These forces all become zero at the center of the earth where 7 is zero. To express these forces in terms of gravity, let g=mean force of gravity on earth’s surface a@=mean radius of earth H=mass of earth 2 then ga, and an (27) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 21 The substitution of this value of » in equations (22) to (25) will give the forces in terms of gravity. If we assume 7 to be equal to the mean radius of the earth and d to be the mean distance of the moon, we may obtain numerical values from Table 2, which, when substituted in (22) and (23), will give the following expressions for the approximate tide- producing force of the moon: Vertical component = 0.000,000,056 (3 cos? @—1) g (28) Horizontal component = 0.000,000,084 sin 2 6g (29) The tide-producing force of the sun will be 0.46 times as large. 7. TIDE-PRODUCING POTENTIAL. 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 of work required to move a unit mass, against the force of gravity, from that pot to an infinite distance from the earth’s center where the force of gravity becomes zero. With the symbols as in the preceding section, we have according to the law of attraction Force of gravity on or above earth’s aparece (30) D Mi r The amount of work required to move a unit mass against this force through an infinitesimal distance dr =a dr (31) The total amount of work necessary to move this particle from a point 7 distance from the earth’s center to infinity is the gravitational potential at that point, and will be here designated by V,. Then, spe oll the IP Tr Ve= (32) The tide-producing potential at any point in the earth is the amount of work required to move a unit mass, against the tide-producing force, from that point to the center of the earth where the tide-produc- ing force becomes zero. If we assume the particle to be moved along the radius of the earth directly to the center, we will be concerned with only the vertical component of the tide-producing force, since the horizontal component would not affect the amount of work required along this path. Considering, first, the force depending upon the cube of the moon’s parallax, the amount of work necessary to move a unit mass against the vertical component, formula (22), through an infinitesimal dis- tance —dr toward the center of the earth equals =-4% (3 cos? 6—1) rdr (33) 22 U. S. COAST AND GEODETIC SURVEY. Then, designating the tide-producing potential due to this force by Vi, we have as the total amount of work necessary to move the particle to the center of the earth— ae (3 cos? @—1) (34) Vo - { (3 cos? 6—1) rdr=4 bs The same result will be obtained by assuming the particle to be moved again against the horizontal component of the tide-producing Pee ie 1 ‘ force until it reaches.a position where cos aS and the vertical com- ponent becomes zero. From this point it can be moved directly to the center of the earth without additional work. For that part of the tide-producing force depending upon the fourth power of the moon’s parallax let the potential be designated by V;. Then, assuming a unit mass to be moved against the vertical component, formula (24), directly to the center of the earth, we have hie di dt This potential may also be obtained by assuming that the particle is first moved against the horizontal component, formula (25), to a position where @=z/2 and the vertical component becomes zero. Similar expressions for the tide-producing potential of the sun may be obtained by substituting in the above formule the mass and distance of the sun for M and d, respectively. The tide-producing potential of the sun which involves the fourth power of its parallax is negligible. fo) 2 3 t= - [328 5c (5. cos? @— 3 cos 0) dr= ede (5cos?@—3cos@) (35) i 8. SURFACE OF EQUILIBRIUM. 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 spheroidal surface due to the tidal forces will not differ materially from the disturbances in a true spherical surface due to the same cause. Let us first consider the surface of equilibrium due to gravity and the principal tide-producing force of the moon. Designating the potential due to these two forces by V, we have as the condition of a surface of equilibrium, V=V,+ Vi=a constant (36) Substituting the values of V, and V; from (32) and (84), pt a, i V = +4 e = (3 cos? @—1) =a constant (37) HARMONIC ANALYSIS AND PREDICTION OF TIDES. M6 Equation (37) must be true for all points in the surface of equi- librium, so if a point be taken in the surface where the tide-producing potential is zero—that is, where cos 6= -/1/3—and let a represent the value of r at this point we have yak =the constant (38) Substituting this in (37) Mr? E Le +36 = (3 cos? 6-1) = (39) from which, by transposing and dividing, 3 2 4 Hes cos 0-1) -Sr—a) (40) Let r=a+u - (41) so that u will represent the equilibrium height of the tide due to the principal lunar force, referred to an undisturbed spherical surface of radius a. Substituting (41) in (40), we obtain M 3 2 2 3 s ca cos? @—1)= aa -__- 3( 2) a (7) — ete. (42) The fraction - is approximately the ratio of the semirange of tide to the mean radius of the earth, and if we assume a range of 40 feet, the numerical value of this fraction would be about 0.000001. It is evident, therefore, that we may neglect the powers above the first, and write 3 rabeee cos? @—1) (43) or : : 3 i=in cos? 6—l)a (44) as the equilibrium height of the tide due to the principal lunar force. In the preceding formulas a was taken as the radius of the earth along which the tide-producing potential of the force under con- sideration was zero. Let us now see whether this is the mean radius of the earth; that is to say, the radius of a perfect sphere having the same volume as the earth. It is evident that this volume must remain constant without regard to any deformation to which the surface may be subjected. Referring to Figure 7, consider the volume of the earth to be divided into infinitesimal solids by a series of planes with their common intersection in the line OC, the angle between two consecutive planes being designated by d @; a series of right conical surfaces with their common apex at O and common axis in line OC, the angle between the generating line and the axis being designated by 6; and a series of spherical surfaces with their 24 U. S. COAST AND GEODETIC SURVEY. common center at O and the radius designated by 7; then the volume of one of these infinitesimal! solids will be dr.rdé@.rsn6do=7" sin 6 dd dé dr (45) and the entire volume of the earth as included in the surface repre- sented by equation (44) wiil be "2a (7 (atu) Volume - J le sin 6 dd dé dr (46) 20 7 ; ~1/3 |, i) (Gg tay eine deco (47) From (44) we may obtain 3 3 (a+u)?=a3 144 7 ae cos? 6-1) 3 =«q3 E + 3/2 2 ~ (3 cos? @—1) + ete. | (48) the terms containing the powers of : above the third being neglected. Substituting (48) in (47), 2m (Pr : M a . ; Volume =1/3 a? 1+3/2 E pe cos’ 6—1) | sin @ d@¢ dé Qa — 2/3 a i scape (49) As equation (49) represents the volume of a sphere with radius a, it is evident that a is the mean radius of the surface represented by (44), and that u is the amount of the disturbance in the mean surface due to the force under consideration. In other words, wu is the equilibrium height of the tide as referred to mean sea level. One of the conditions of an equilibrium surface is that the resultant of all the forces at each point must be in the direction of the normal to the surface at that point. Let us see if this condition is fulfilled as to equation (44). In Figure 8 let P represent any point on the surface defined by equation (44), and let yw be the angle between the radius vector and the normal at this point. If we imagine the surface to be cut by a plane passing through the point P and the centers of the earth and moon, it is evident that the trace of the sur- face will intersect the are of a concentric circle drawn through the point P with radius 7 at an angle equal to the angle y, and that dr tan aaah (50) From (41) and (44), raa+h par (3 cos? 6—l1)a (51) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 25 Then i or Say (52) Substituting (51) and (52) in (50), Saag) 3 Ma’ sin 26 a : . | 1+4 ie cos? 6— 1) z f 3 ae sin 26 E =) aS (3 cos? @—1) + ete. | (53) Be Since = is very small compared with unity, we may neglect the higher powers in (53) and write 3 tan ye sin 26 (54) as the tangent of the angle between the radius vector and the normal to the surface at the point P. TOowAROS MOoN== Fig. 8. If we let y, represent the angle between the radius vector and the resultant of the forces under consideration at the same point P, we have from (22), (23), and (30), ae sin 26 tan y,= pone cos? @—1) em 3 3 = 3/29 sin 20 & (55) eee (3 cos? 8 — 1) 26 U. S. COAST AND GEODETIC SURVEY. Substituting the value of r from (41), U 3 7 1+%) . ( tan y,=3/2 _ g x Sin 20 = He 1— eee By (3 cos? 6 —1) (56) 3 The values of - and ie a are each very small compared with unity, and the value of the bracketed portion of (56) is therefore a very close approximation to unity. We may therefore write A tan i= ee sin 26 (57) as the tangent of the angle between the radius vector and the result- ant of the forces at the point P. Comparing this with (54), we find it to be the same as the angle made by the normal with the radius vector, indicating that the resultant force is normal to the surface at any point P. If we let d represent the mean distance of the moon and substitute the numerical values from Table 2 for the coefficient in (54) we obtain tan y =0.000,000,084 sin 2 6 (58) in which y as a maximum value of about 0.017’’ when 6=45°. The maximum deflection of the normal due to the tide-producing force of the sun is about 0.46 times as great, or 0.008’’, approximately. Let us now consider the disturbance in a spherical surface due to — the potential depending upon the fourth power of the moon’s parallax (35). This potential will become zero when @=90°, and also when cos 6=-/3/5. Letting a represent the radius vector at either of these points, we have as the equation for the equilibrium surface due to the potentials (32) and (35) we i 3 fe (5 cos? @—3 cos a = (59) from which may be obtained 4 3 ee (5 cos? @—3 cos 6) =< (r—a) (60) Letting : r=at+u’, (61) we have Ma a rh Pron au 2 fu’ 3 . 4 Ei pe cos? 6—3 cos 6) Gg ae -4[" +10{2 — ete. (62) Neglecting powers of . above the first, u! Ma‘ ‘ ant E gi © cos* 9-8 cos 6) (63) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 27 or u’=4 a (5 ae 6—3 cos @)a . (64) as the equilibrium height of the tide due to that part of the lunar force depending upon the fourth power of the moon’s parallax. In forming equation (59), a was taken at the radius vector along . which the potential due to the fourth power of the moon’s parallax was zero. To determine whether this is the mean radius, we find the volume included in the surface represented by (64). Qr 7 atu’ oe i if 73 citi 0 dik dade 2r (r =1/3 [ [(a+w’)*sinadg ds (65) From (64) M at (atu rma 1+3/2 E g@® cos? @— 3 cos 6) + ete. | (66) the terms containin: the powers of - above the fourth being neglected. Substituting (66) in (65) 2r (or Volume=1/3 «| | E + 3/2 Ee Edi (5 cos? 6—3 cos 6) |sinededs ~ 2a = 2/3a° { do =4/3n0° (67) As (67) is the volume of a sphere with radius a, it is evident that this is the mean radius of the volume included in the surface repre- sented by (64), and that w’ is the amount of the disturbance in the mean surface due to the force depending upon the fourth power of the moon’s parallax. Letting y’ equal the angle between the radius vector and normal ‘to any point P'in this surface, LOOT tan y aaa rah (68) and from (61) and (63)... _ as, T=a+9 a (5 cos? 6—3 cos 6) a (69) LPs: yuibcocgoh oid) | 8 Ye : cy ony ae (15 cos? 6—3) sin 6 (70) Substituting (69) and (70) in (68) ‘and neglecting’ powers’ of 7 above the fourth thod ssorlt 6 Lola YoorPhe “72934 pay ig Mat Eg (5 cos? 6—1) sind seit i) 28 U. S. COAST AND GEODETIC SURVEY... For the tangent of the angle between the radius vector and the resultant of the forces under consideration we have from (24), (25), (30), and (61) 2 3/2 ua (5 cos? 6—1) sin 0 Sr deck: peel as We = 3/2 eat (5 eos? @—1) sin 6 (72) ae 3/2 == (5 cos? @—3 cos 6) SiN (a+u’)* 4 se since | and i are each very small compared with unity. Comparing (72) with (71), it is found that the angle made by the resultant force with the radius vector is the same as the angle made by the normal at the same point. Taking d as the mean distance of the moon and substituting nu- merical values from Table 2 for the coefficient in (71) we obtain tan y’ =0.000,000,001,4 (5 cos? @—1) sin 0 (73) in which ¥’ has a maximum value of about 0.0004’’ when 6=31.1°. The maximum deflection of the normal due to the fourth power of the sun’s parallax is only about 0.000,000,5’’. Expressions similar to (44) and (64) may be formed for the equilib- rium height of the tide due to the sun, letting S=mass of sun, d,=distance from center of earth to center of sun, — §,=angle at center of earth between line to sun and to point of observation on earth. i Then for the height (y) of equilibrium tide due to combined action of moon and sun we have 3 ; y=4 8 - (3 c0s?)0 1) Que ol). 44% (approximate lunar tide). D ) +4 a 7 (5 cos? 6—3 cos @) a. . (tide depending upon 4th power of moon’s paral- lax). S a E : +4 Ed (3 cos? 6,-1)a.... (approximate solar tide). ; S a : : +4 Fd (5 cos? 6,—3 cos @,) a . (tide depending upon 4th 1 power of sun’s parallax). (74} In (74) it will be noted that M, FE, S, anda are constants. The dis- tance d and d, vary within certain limits because of the eccentricity of the orbits of the moon and earth. The angles @ and @,, which are practically the same as the zenith distances of the moon and sun, re- spectively, vary with the declinations and hour angles of these bodies and also depend upon the latitude of the place of observations. — HARMONIC ANALYSIS AND PREDICTION OF TIDES. 29 If we ascribe the mean values to the distance d and d, and substi- tute the numerical values: of the constants from Table 2, we obtain from (74) the following, the unit of height being the foot: 0-584 (a cass Oe Ve ce ie be (approximate lunar tide). +0.010 (5 cos? 6—3 cos #) .. . (depending upon 4th power of moon’s parallax). SUEZ LU Ke COS Gg Ly are a eee (approximate solar tide). +0.000,01 (5 cos* 6,—3 cos 6,). (depending upon 4th power of sun’s parallax). (75) From (75) it appears that the extreme ranges of the equilibrium lunar and solar tides are approximately 1.75 and 0.81 feet, respec- tively, and that the tides depending upon the fourth power of the par- allax of the moon and of the sun are of little or no importance. 9. DEVELOPMENT OF EQUILIBRIUM TIDE. In equation (74) an expression: was obtained for the equilibrium height of the tide in terms of the linear distances and the zenith dis- tances of the tide-producing bodies. It is now proposed to develo this equation in terms of variables which change uniformly il time. Let us consider, first, the term representing the principal lunar tide, which may be written At L y= 3/2 FE ~ (cos? 6— 1/3) (76) In Figure 9 let O represent the center of the earth and let projec- tions on the celestial sphere be as follows: Cc IAB, — the earth’s equator; IM, the moon’s orbit; CPA, _ the meridan of place of observation; CMM’, the hour circle of the moon; ja the intersection of moon’s orbit with the Equator; Be: the place of observation; M, the position of the moon. 30 U. S. COAST AND GEODETIC SURVEY. Then the Z MOP will equal the Z¢@ of equation (76). Now, let \=AP=latitude of place of observation, §= M’ M= declination of moon, C= ZAOM’ =arc AM’ =hour angle of moon, . I= Z MJA=inclination of moon’s orbit to the Equator, 1=IM=longitude of moon in its orbit reckoned from the intersection J, x= IA =right ascension of meridian of place of observations reckoned from the intersection J. In this discussion the radian is considered as the angular unit. In the spherical triangle CMP Cos P M=cos 6=cos CP cos CM+sin CP sin C&M cos 0 —sin \ sin 6+cos \ cos 6 cos A MW’ (77) In the right spherical triangle MM’T, sin 6=sin J sin l (78) and in the right spherical triangle M’A, cos 8 cos AM’=cos AM. (79) In the spherical triangle MAI ~ cos A M=cos 1 cos x+sin / sin x cos I (80) Substituting (78), (79), and (80) im (77) cos 6=sin \ sin J sin 1+ cos \ [cos J cos x + cos J sin / sin x] =sin \ sin J sin 1+4 cos X [cos (—x) + cos (1+ x) +cos I {cos d— x) —cos (1+ x) }] =sin \ sin J sin 1+ cos A [cos? + I cos (J— x) + sin? 4 ede (+x)] (81) Then cos? §=sin? \ sin? J sin? 1 +2 sin \ cos \ sin {snl [cos? 3 J cos (l—x) _+sin? $2 cos (1+ x)] + cos? d [cost 4 I cos? (l=x)-- +2 sin? 4 T cost 4 I cos (/—x) cos “UPx- +sin* $ 7 cos? T+x)] =4 oe d cost $ J cos (21— 2y) ~. +4 cos? A sin4 re ‘T cos (21+ 2x) | “Kg cos? A sin? [ cos 2x_ +4# sin 2\ sin J cos? $ J sin. (1 — x) +4 sin 2h-sin J sin? $ J sin (21x) +4 sin 2\ sin [cos 7 sin x + (4 cos? \ sin? J—4 sin? d sin? J) cos 21 +4 (cos? \ cost 4 [+ cos? d sin‘ 4: J+:sin? \ sin? J) (82) The last two lines of (82) may be written (4—3/2 sin? \) (4.sin? J cos 21) + (4—3/2 sin? \) (1/3 —4 sin? J) 41/3 (83) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 31 Substituting (82) and (83) in (76), we have y =3/2 se ZL cos? \ cost $ I cos (21—2x) +4 cos? \ sin‘ 4 J cos (21+ 2x) +4 cos? d sin? I cos 2x +4 sin 2d sin J cos? 4 I cos (2l—x—7/2) +4 sin 2\ sin J sin? $ I cos (21+x—7/2) +4 sim 2d sin 2 I cos (x—7/2) + (4—3/2 sin? \) (4 sin? J cos 21) + (4—3/2 sin? \) (1/3—4 sin? D | 7 (84) In the above formula d, the moon’s actual distance from the earth and 1, the moon’s true longitude in its orbit measured from the inter- section, although functions of time, do not vary uniformly because of certain inequalities in the motion of the moon. It is desired, therefore, to find expressions for these quantities in terms of elements that wil vary uniformly with time. Letting c=mean longitude of moon in radians measured from the intersection, then | being the true longitude from the same origin, we may obtain from Table 1 l=o+2e sin (s—p) + 5/4e? sin 2(s—p) +15/4 me sin (s—2h+ p) +11/8 m?-sin 2(s—h) (85) Letting c=mean distance of moon from earth we may also obtain from Table 1 — = a +a! e cos (s—p)+a’ é cos 2 (s— p) +15/8 a’ me cos (s—2h+p) +a’ m* cos 2 (s—h) (86) ; ; 1 in which a ae oul). A reference to Table 2 will show that the quantities e and m may each be considered as fractions of the first order and the product of the two or the square of either as fractions of the second order. In the following development terms smaller than those of the second order will be neglected. Substituting the value of a’ in (86) and multiplying by c, we obtain after neglecting terms containing powers of e above the second quite cos (s— p) +e? cos 2 (s— p) +15/8 me cos (s—2h+>p) . + m?.cos.2 (s—h) (87) Cubing (87), 3 . aait3 e cos (s—p) +3 é cos? (s—p) +3 € cos 2 (s—p) +45/8 me cos (s—2h+ p) +3 m? cos 2 (s—h) =1+4+3/2 e&+8e cos (s—p) +9/2 é cos 2 ea + 45/8 me cos (s—2h+p)+3 m? cos 2 (s—h) (88) ou U. S. COAST AND GEODETIC SURVEY. In equation (84) the functions involving / and x may be expressed in the general form cos (2/+ a) or cos a, in which a= + 2x, (+x—7/2), or zero. In equation (85) let k =2e sin (s—p) + 5/4 e? sin 2 (s—>p) +15/4 me sin (s—2h+p)+11/8 Pe sin 2 (s—h) (89) Then l=o+k (90) The maximum value of kf is small. If numerical values for e and m be substituted in (89), it is found that the maximum value of 2k is 0.273 of a radian, the sine of which is 0.270. It may therefore be assumed without material error that the sine of any angle not greater than 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) (91) and cos 2k=1—2 sin? k=1— 2k? =1-—8 é sin’ (s— p) =1—4e?+ 4e? cos 2 (s—p) (92) if terms smaller than those of the second order are neglected. From (90), (91), and (92) we may now obtain. cos (21+ a) =cos (20+2k+ a) =cos 2k cos (20+ a) —sin 2k sin (26+ a) =[1 — 4e’ + 4e? cos 2 (s—p)] cos (20+ a) —[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)] sin (20+ a) = (1 —4e?) cos (20 +a) ; +2e cos (26+a+s—p) —2e cos (20+a—s+p) +13/4 e cos (20+a+2s—2p) +3/4 é cos ecwal 2s+2p) +15/4 me cos (26+a+s—2h+ p) ~ 15/4 me Cos (260+a—s+2h—>p) +11/8 m?.cos (26+a+2s—2h) —11/8 m? cos (2a+a—2s+2h) (93) The general coefficient of (84) may be written ee a (4) | (94) and the variable ey, of each of the terms in (84) may then be ex- pressed by one of the following general forms: (3) + (G) 08 a or (G) cos (2a) The value of the first is given in equation (88). For the second we have from (88) (5) cos a=(1+3/2 e?) cosa +3/2 e cos (a+s—p)+3/2 e cos (a—s+>p) +9/4 e cos (a+ 2s—2p)+9/4 e? cos (a— BOND Dp) +45/16 me cos (a+ s—2h+p) + 45/16 me cos (a Y + 3/2 m? cos (a+ 2s—2h) + 3/2 m? cos (a—2s+ 2h) (95) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 33 For the third form we may obtain from (88) and (93) to the second order of approximation the following: (3) Gos Ql+ a) = (1—5/2 €”) cos (2o+a) +7/2 e cos (26+a+s—p)—1/2 e cos (Qo+a—s+>p) + 17/2 @ cos (20+a+2s— 2p) + 105/16 me cos (26+a+s—2h+p) — 15/16 me cos (26 +a—s+2h—p) + 23/8 m? cos (2o0+a+2s—2h)+1/8 m? cos (26 +a—2s+ 2h) (96) Substituting (88), (95), and (96) in (84), and letting a equal +2 (+x—7/2), or 0, as the case may be, we may obtain the fallowine equation for the equilibrium height of the lunar tide. For conven- ience in reference each term is designated by the letter A with a subscript. Following each term is a numeral giving its maximum value in feet when / has its mean value. These numerical values include the general coefficient and are calculated from the constants in Table 2. They serve to indicate the relative importance of each term. Ul =3/2 * hi a (A)y [1/2 cos? X cost 4 I{(1 —5/2 e?). cos(2o —2x)_ 24 ee (0.7869) Alas 11/2 en cos. (26-2. py eh (0.1524) (A)3 .~ —1/2 e cos (2a—2x—s potato lett oae ha ol andr dee rea yeaa) (0.0218) (A)4' +17/2 e& cos (20—2x+2s—2p)_--- ee (0.0203) (A) > +105/16 me cos (eSB Hews ohesp) BHUTTO mAr any eh (0:0214) (A)e —15/16 me cos (26 —2x —s+2h—p)_______---------- (0.0031) (A)7 23/81 m2.cos (24 —2%4+2s— 2h). 2 sey v8 ee (0.0128) (A)s +1/8 m? cos (26 —2x% —2s+2h)} = gt (0.0006) (A)o +1/2 cos? \ sint 4 I {(1— 5/2 e) cos (20 +2x)_______- (0.0015) (A) 10 +7/2 e cos Oe ox ae sg OMe pe NL a Ss SORE Mad AW (0.0003) We 12 ¢ tos Cotas py tN (0.00004) (A) 12 +17/2 & cos (20 -+2x%+-2s —2p)__ bt --- (0.00004) Woe FE LOS/1G ge cos .(2o-Hey-+s2-2h tp) (0.00004) (A) 14 —15/16 me cos (26+2x —s+2h—p)_________------ (0.00001) (A)is +23/8m? cos (20+2x+2s—2h)____1_____--------- (0.00002) (A) 16 +1/8° m? cos (26+2x% —2s+2h)}_________-______-_- (0.000001) (A) 17 +1/4 cos? d sin? I_{(4+3/2 @) PEC RI al, MO ee Ae (0.0686) A)18 Oe NCOs Wats pcs nse eee a LA Ce (0.0056) (A) is aE O/ AERCOSI(2y 3815p) 6 Seal sh ap ei ah he SP (0.0056) (A) 20 a-O/4we. cos! (2y 1 2s— 2p) Oe (0.0005) (A) a1 Et Of4._.62-eO8-(2x 28-2 Perec - ---£-------- 7%. (0005) (A) 22 +45/16 me cos Ox ts 2happ)- 228 PEPE. aye a (0.0008) (A) o3 -+45/16 me cos (2x —s+2h—p)__-_---- | ____u J ST>>=-400908) (A) 24 + 3/2 m? cos V5 diner eae a AS, Di Raa eat ad (0.0006) (A)os +3/2 m? cos (2x —2s+2h)} 2-2-2 Ad Sab Te (0.0006) (A) 26 +1/2 sin 2d sin J cos? § I{ (1 —5/2°e?) cos (Up as —'x/2)~ (0.3266) (A) 27 +7/2-e€ cos (ee aig a7 SE I eae, HRI uate ge (0.0632) (A) 2s —1/2 e cos (Qa— sR ea il NV EATER) Mig SUIT ar ae AT a (0.0090) (A) 29 +17/2 e? cos (26 —x+2s —2p—7/2)_-- 2 ee ee (0.0084) (A) 30 +105/16 me tos (26—x +s —2h-p —/2)__*% ---_--- (0.0089) (A) 31 —15/16 me cos (260 —x—s+2h—p—7m/2)_-_-_________- (0.0013) (A) 32 +23/8 m? cos (26 —x+2s —2h —7/2)_____-_____-_--_- (0.0053) (A) 33 els m? cos (26—x —2s+2h —7/2)}_______-____-_-_- (0.0002) (A) 34 +1/2 sin 2 sin J sin? 3 J {(1- nue e?) cos (20+ x—2/2)_ (0. oan (A)3s. +7/2e cos (otto. doy 4 Ff Ea: RAK OER ESE ESI A (0.0027) - (A) 6 —1/2 e cos (Q2o+x—s+p—7/2)__ 0 ee io ones} (A) 37 +17/2 e cos (26+ x%+2s —2p —7/2)__ 2 ee (0.0004) (A) 38 +105/16 me cos (26+x+s—2h+p—7/2)_______-___- (0.0004) (A) 39 — 15/16 me Cos (2o-+x —s+2h —p—7/2)__-__-_-__e- (0.0001) A) 40 +23/8 m? cos (2o+x+2s —2h —7/2)___--_- (0.0002) Aba ai lifsu mn COS: (Zot 2S ell — mp e)ip oe ba (0.00001) 34 U. S. COAST AND GEODETIC SURVEY. (A) 42 +1/4 sin 2 sin 2 J {(1+8/2 e?) cos (y—7/2)________ (0.3164) (A) 43 23/2 1é- COS) (x FESi—— P= TH] 2 ee ee A ae (0.0259) (A) 44 3/2 6. COSN(X Sp ST] 2 ee ee ee eee Ae ae (0.0259) (A) 45 +9/4 e? cos (x-+-2s —2p —a/2)_____________________- (0.0021) (A) 46 +9/4 e? cos (x -—2s +2p—7/2).__ bee bel elle (0.0021) (A) 47 +45/16 me cos (x+s —2h+p —7/2)_________________ (0.0036) (A) 43 +45/16 me cos (x —s+2h—p—7/2)_-____) 2 (0.0036) (A) 49 +3/2 m? cos (x +2s —2h —w/2)_.- 2 ek eth eee ek (0.0026) (A) 50 +3/2 m? cos (x.—28 12h =q/2)}. .. g40 a ee (0.0026) — (A)s: +1/2 (1/2 —3/2 sin? d) sin? I {(1—5/2 €2) cos 2-2-2 2. (0.1356) Y~ (A) 52 if CCOBS (Lae ct Ss a s)) ee oe liga alee SR ra (0.0263) (A) 53 = 1/2:e*ces) (Qo Sse p) swe TOR) ea Eee SE OLOO STA (A) 54 17/2: 62} cosi(2o4528 2p) sf wee Lue a ee & (0.00385) (A) 55 +105/16 me cos (26+s —2h+>p)____________________ (0.0037) (A) 56 — 15/16 me cos Qc —s 4-2h —p) e282. a ee (0.0005) (A) 57 “2/8 Mm COS (2a 25" 2h) eee. ot RAS ENO Cones (0.0022) (A)s58 4-1/8! m? cos (2c 2seE QA) Al SU Pe eel SO (0.0001) (A) 59 + (1/2 —3/2 sin? ») (1/3—1/2 sin? ZT) {(1+8/2 e?)______ (0.4404) (A) 60 +3) BeG8i(S py EO, OF ON TO Lee RRR (0/0722) "Mm (A) e1 +-9/2 7, cos (25, 2p)int 24 bY gen iuar dp eit ele (0.0059) (A) es -++-45/8: me"cos.\(s' = 2h ap) See oe eee (0.0101) (A) 63 Poi) COS M25 2h) il sae ooo AW le a Ce Sere ee (0.0074) (97) MCUs Referring to section 4 and Figure 6, it will be noted that the lon- gitude measured in the moon’s orbit from the intersection A equals the longitude measured from the referred equinox ’ less the dis- tance £, and that longitude measured in the Equator from the inter- section A equals the longitude measured from the equinox ¥ less the distance ». In Figure 10, let 8’ and P’ be the points where the hour circles of the mean sun and of the place of observations intersect the celestial Fia. 10. equator, and let TJ=the hour angle of the mean sun measured west- ward from the meridian of the place of observations. It is evident. that the difference between the mean longitude of the sun and the right ascension of the meridian of the:place of observation when reckoned from the same origin will be equal to the hour angle of the mean sun. ‘ HARMONIC ANALYSIS AND PREDICTION OF TIDES. 35 With other notations as before, we then have eee eee (98) Substituting (98) and (99) in (97), making all coefficients positive by adding or substracting z from the angle, since —cos x= +cos (x+7), and neglecting all the smaller terms whose numerical maximum val- ues as indicated are less than 0.003, we obtain the following: y=3/2 = (2) a@ cos? A X (A); [(1/2 —5/4 e?) cost $ J cos (27 +2h —2s+2¢ —2y)________-__-- M, (A)o +7/4e cos! 3 I cos (27+2h—3s+p+2é —2y)__________----- Nz (A); +1/4 e cos! 4 I cos (27+2h—s—p+2§—2v4+7)_____-_----- [Le] (A), +17/4 e cos! § I cos (27 +2h —4s+2p+2é —2y)_______-.-.-2N (A); +105/32 me cos! 4 I cos (27 +4h—3s —p+2é —2y)_________- V2 (A)e +15/32 me cos! 3 I cos (2T —s+p+2¢—2v+7)__---_------- Ae (A), +23/16 m? cos! 4 I cos (2T+4h—4s+2é —2v)_____--------- Be (A) 17 +(1/4+3/8 e) sin? I cos (27 +2h —2v)________-. Sayper OO SRET [Ka] (A) 18 +3/8 e sin? I cos (27+2h+s —p—2yp) (A) 19 +3/8 e sin? I cos (27+2h—s+p—2v)]____-_-___---------- [L2]? +3/2 = (2) asin 2 X (A) og [(1/2 —5/4 e) sin I cos? $ I cos (T+h—2s+2¢ —v +7/2)____- O; (A) 27 +7/4 e sin I cos? 4 J cos (T+h—3s+p+2§—v+7/2)__------ Q: (A) os +1/4e sin I cos? 4 J cos (T+h—s—p+2§ —v —#/2)___-_-- [M,]* (A) a9 +17/4 e sin I cos? 4 J cos (T+h—48s+2p+2§ —v+7/2)___-- 2Q (A) 30 +105/32 me sin I cos? 4 I cos (T+3h—388s —p+2§—v+7/2)__-ps (A) 32 +23/16 m? sin I cos? 4 J cos (T+38h—48 +2 —»y+7/2) (A) 34 +(1/2—5/4 e) sin J sin? } J cos (T+h+2s —2£—v—7z/2)__-OO (A) 42 +(1/4+3/8 e) sin 2 I cos (T +h—v—7/2)___---_--------- [Ky] (A) as +3/8 e sin 2 J cos (T+h+s—p—v—7/2)__.-_--------------- Ji (A) 44 +3/8 e sin 2 I cos (T+h—s+p—v—7/2)____------------ [Mil (A) a7 +45/64 me sin 2 I cos (T —h+s+p—v—7/2) (A) as +45/64 me sin 2 I cos (1 +3h—s —p—v—7/2)] M (/a™3 : +3/2 (2)° a (1/2—3/2 sin? ») x (A) 51 V2 ol 4.7) sine COS 2s eee nee ee een Mf (A) 52 +7/4 e sin? I cos (3s —p —2é) (A)s3 +1/4 e sin? I cos (s-+p —2é+7) (A) 54 +17/4 e sin? I cos (4s —2p —2€) (A)s5 +105/32 me sin? I cos (8s —2h+p-—2é) (A) 59 +(1/3+1/2 e) (1—3/2 sin? 1) (A) 60 Te bi ehi ee ;SMNGH I COB MSC he Lah ee el Mm (A) 61 +3/2 e? (1—3/2 sin? J) cos (2s —2p) (A) ez +15/8 me (1—3/2 sin? I) cos (s—2h+7p) (A) 63 +m (1 —3/2 sin? I) cos (2s —2h)___ 2. tL -- [MSf£] (100) 36 U. S. COAST AND GEODETIC SURVEY. If we disregard for the present the slow variations in the value of I, £, and v, which for a series of observations of a year or less may be considered as practically constant, each term in the above formula, excepting (A);9, is an harmonic function of an angle which changes uniformly with time. Hach term represents an harmonic component of the lunar tide, and, if the deal conditions assumed under the equilibrium theory (section 3) actually existed, each term including the general coefficient would represent the approximate true height of that component referred to mean sea level and the sum of all the terms the approximate height of the entire lunar tide. , The notation following each term in the formula is the generally recognized symbol for the component represented. The bracketed symbols indicate that the terms only partially represent the compo- nents designated. These will later be given special consideration. ‘The terms without symbols are of little practical importance and are generally neglected. Terms with coefficients e and ¢? represent the elliptic components, since they depend directly upon the eccentricity of the moon’s orbit. Terms with coefficients me, and m? represent the evectional and variational components, respectively, since they are derived from the corresponding inequalities in the motion of the moon. (See formulas for the true longitude and distance of the moon in Table 1.) 10. EQUILIBRIUM ARGUMENT. Although the actual height of the tide and the time of occurrence of the maxima and minima are greatly modified by conditions upon the ear'th’s surface, equation (100) furnishes us with important rep- resentations of the elementary periodic forces which tend to produce the lunar tide. _These forces may be defined by their periods which depend upon the varying angles of the several terms of the equation. Disregarding for the time being the slow changes in the functions of the angle J, the value of each term in general is the product of a con- stant coefficient and the cosine of a varying angle. This angle is called the equilibrium argument of the component. represented. The numerical value of the argument is constantly changing. The mean rate of change is called the speed of the component, and the time required for the argument to complete one cycle of 360° is the period of the component. Examining equation (100) it will be noted that each argument is composed of a combination of some of the following elements: T, hour angle of the mean sun at the place of observation; h, longitude of the mean sun; s, longitude of the mean moon; p, longitude of the moon’s perigee; p,, longitude of sun’s perigee (for solar tides only) ; £, longitude in moon’s orbit of intersection (fig. 6) ; v, right ascension of intersection (fig. 6). The hour angle J is zero at mean local noon at the place of obser- vation and increases uniformly at the rate of 15° per solar hour. At any given instant of time the value of T will be different for each meridian of the earth, but will be identical for all places on the same HARMONIC ANALYSIS AND PREDICTION OF TIDES. 37 meridian. Formulas forh,s,p, and p, are givenin Table 1. Although the rates of change in these elements are not absolutely uniform, the variations from uniformity are negligible in this work. The values for the beginning of the present century and the hourly rates of change will be found in Table 2. The values of — and v for each degree of N are given in Table 6. They vary slowly between small positive and negative limits, but do not affect the mean speed of an argu- ment, since an acceleration in the speed at one time due to the posi- tive values of these elements will be compensated for at another time by the retardation due to corresponding negative values. The argument of a component is divided into two parts designated _by V and wu, respectively, so that the entire argument may is eX- pressed by (V+w). The principal part V is composed of a combina- tion of the elements T, A, s, p, and p,, together with any constant, such as a multiple of 90°. The V changes uniformly throughout the entire cycle of 360° and determines the speed and period of the com- ponent. The w includes the elements é and », and alternately in- creases and decreases through small limits with a mean value of zero. The change in the wis very slow, and for the reduction of any series of observations not exceeding 369 days in length it is assumed to be constant with its value as of the middle of the series, but for the comparison of results from different years of observations the change in this quantity must be taken into account. The u, being a function of \, has a period of approximately 19 years. Of the elements that may enter into the V it will be noted that. T has a speed of 15° per hour, giving a period of one solar day for this element, while the speeds of the other elements (Table 2) are each less than 1° per hour. The approximate period of the elements s, h, p, and p, are t month, 1 year, 9 years, and 20,000 years, respectively: Ina combination of elements of which the speeds differ so greatly it is apparent that the approximate period of. the component will be deter- mined by the element of greatest speed and shortest period. Thus all the components which contain the element 7 in their arguments must have periods that will not greatly exceed the length of a solar day, but if the element of greatest speed in the argument is s the period will be approximately one month. | Tidal components are considered under two classes, short-period components with periods of approximately one day or less and long- period components with periods extending over a longer time. The former contain the element JT in their arguments, while the latter are independent of this element. The short-period components may be subdivided and_classed as diurnal, semidiurnal, terdiurnal, quarter-diurnal, etc. The diurnal components have periods approximately equal to a solar day, and they are distinguished by the presence of a single 7 in the argument. The actual period of such a component is called a component day. The semidiurnal components have periods approximately equal to one-half of a solar day and are distinguished by the presence of 27 in the argument. For these components the component day will be exactly twice the length of the period of the component. Terdiurnal and quarter-diurnal components will have three and four periods each component day and will be distinguished by the multiples 37 and 47 in their arguments. In formula (100) the only short-period components represented are the diurnal and semidiurnal components. 38 U. S. COAST AND GEODETIC SURVEY. The long-period components are of much less practical importance and only five are usually considered in the analysis. The lunar monthly Mm, with a period of approximately one month indicated by the single s in its argument, ate the lunar fortnightly Mf, and the lunisolar-syncdic fortnightly MSf, with periods of approximately one-half of a month, as indicated by the 2s in their arguments, may be found represented in formula (100). The annual and semiannual components will be referred to later. In order to visualize the equilibrium arguments of the short-period components, the periods of Sich depend primarily upon the rotation of the earth, it may be found convenient to conceive of a system of fictitious stars, or ‘‘astres fictifs,’’ as they are frequently called, which move in the celestial equator. Each diurnal component may be represented by such a star moving at a rate which will cause it to transit the meridian of the place of observation at the instant the argument of the component is zero, the interval between successive transits corresponding to the period of the component. We might conceive the motion of the star relative to the earth’s meridian to be strictly uniform corresponding to the rate of change in the V of the argument. In this case the intervals between successive transits will be equal and will determine the length of the mean component day, just as successive transits of the mean sun determined the length of the mean solar day. It may be more convenient, however, to assume that the motion is subject to the inequalities of the u of the argument. In this case the hour angle of the fictitious star will at each instant of time correspond exactly with the argument of the diurnal component that is represented. For the semidiurnal components the conception is a little less simple. Perhaps the best assumption is a system of two fictitious stars at 180° apart for each component, moving so that the argument of the component will always be equal to twice the hour angle of either star. Similarly, for the terdiurnal and quarter-diurnal com- ponents, systems of 3 and 4 fictitious stars moving so that the argu- ment of the component is always three or four times, as the case may be, the hour angle of the component star. The conception of the astres fictifs is not adapted to the long-period tides, as these do not depend upon the rotation of the earth for their periods. - Under the equilibrium theory the time of a component high water will correspond to the zero value of its argument, but under actual conditions the occurrence of a component high water will, in general, be delayed by an amount which is constant for a given place. The lag, expressed in angular measure, is called the epoch of the com- onent and is usually designated by the Greek letter x. The epochs or any place are determined from actual observations of the tide, and if applied to the equilibrium arguments will give corrected argu- ments which will correspond to the true phases of the component tides at that place. The general expression for the corrected arguments is V+u—k, which will equal zero at the time of the high water of the corresponding component. If we adopt some initial instant from which to reckon time, such as the beginning of any series of observations or predictions, and let ¢t =number of time units from the initial instant, V,=value of V when t=o, a =rate of change in V per unit of time, HARMONIC ANALYSIS AND PREDICTION OF TIDES. 39 then we may write (V+tu—x) =(at+ Votu-—k) (101) In (101) wu will be assigned a value corresponding to the middle of the series under consideration and will be assumed to hold that value as a constant for the entire series. Table 3 contains the formulas for the arguments of the principal components and also the hourly rates of change in the argument, and Table 15 gives the values of V,+ for the meridian of Greenwich for the beginning of each year from 1850 to 2000 as computed from the formulas. A graphical representation of the relations between V,+4, ¢, and k is aomn in Figure 11. The heavy horizontal line represents the time argument advancing to the right, the distance being expressed in angular measurement that increases uniformly with the advance of time. The figure takes account of a single typical short-period component with an hourly speed of a, the ratio of this speed to that ao s gi? ef ee Be ge bso RE Fe ep dae ab 5 adie ole at = Ge pee = $3 ce Fs 68 ge ese : 4 3 Be) eS rte % g ne ep 663 ! Gis divi View = of the mean sun being represented by c. In the figure the horizontal distance that corresponds to one hour in time is equivalent to a units of the angular measurement. The point indicated as ‘‘05 of time used for observations”’ is assumed to represent the exact beginning of the series of observations analyzed; that is, the time of the first hourly height of the tabulations. The interval between the beginning of the observations and’ the time of the first following component high water is indicated by ¢. The interval between the next preceding transit of the astre fictitf over the local meridian and the beginning of the series is designated as the local V,+u. The true epoch or x is the interval between the transit of the astre fictif over the local meridian and the time of the followmg component high water, and therefore equals the sum of the local V.+u and the ¢. | i 2 D The Greenwich V,+ wu as given in Table 15 is the interval between the transit of the astre fictif over the meridian of Greenwich and ‘the 0 hour of the following Greenwich day: The interval between the transit over the meridian of Greenwich and the transit over any other meridian is equal to the product of the subscript of the component and the difference in longitude, the subscript indicating the number of component periods in a component day. For east ‘longitude the transit would occur earlier and for west longitude later than the 40 U. S. COAST AND GEODETIC SURVEY. transit over the meridian of Greenwich. For the long-period com- ponents an initial epoch of the V,+v 1s reckoned from certain astro- nomical relations independent of the rotation of the earth and is consequently independent of longitude on the earth. Therefore, to adapt Figure 11 to the long-period component, the subscript p must be assumed to be zero. The change in the V,+¥ from the 0 hour of a Greenwich day to the 0 hour of the same calendar day defined by another time meridian pauals the product of the speed ratio ¢ and the difference in longitude of the time meridians. From the figure it is evident that a correction of (—pLZ+cS) must be applied to the Greenwich V,+u to obtain the local V,+u. If the epochs or «’s are referred to some standard time meridian instead of the local meridian at the place of observations, a correction equal to the product of the component subscript and the difference between the longitude of the local and standard meridians must be applied to reduce such ppoees to the local meridian, the subscript being taken as zero for all the long-period components. 11. COEFFICIENTS. Referring to formula (100), on page 35, it will be noted that the coefficients of the terms are made up of two parts—a general coefficient applying to all the terms within a group and an individual coefficient applying to a single term. In this formula three groups of components are represented—the semidiurnal, the diurnal, and ihe long-period tides. The general coefficient of each group includes the 3 common facter 3/2 (a) a, but differs from the othersin the factor involving the latitude (A) of the place of observation. The individual coefficients of the components of a single group are therefore directly comparable with each other and will give the relative theoretical importance of the components of that group. The relative import- ance of the components in different groups will depend also upon the latitude of the place of observation. For the semidiurnal compo- nents the general coefficient will have a maximum value for places on the Equator; for the diurnal tides in latitude 45° north or south; and for the long-period tides at the north and south poles. For convenience the term coefficient is frequently applied to the imdi- vidual coefficient, exclusive of the general factor, and may be so used in the following discussion of mean values. A general expression for each term in equation (100) is J cos (V+4) (102) in which the coefficient J is a function of J, and wu is a function of v and €. Since /, v, and é are all functions of the longitude of the moon’s node, which is usually represented by N, the values of J and u will also be functions of N. If we assume a succession of a great many short series of tidal observations to be analyzed, the mean of the resulting amplitudes for any component might be represented by the mean value of J in (102); but if a single very long continuous series is to be analyzed the resulting amplitude will be more accurately represented by the mean value of the product J cosu. The difference may be explained as follows: The inequalities due to the u in the HARMONIC ANALYSIS AND PREDICTION OF TIDES. 41 argument cause the intervals between successive maxima and minima to vary slightly in length. In the analysis of a succession of short series the amplitude obtained from each has the maximum value of the function which occurs when (V+) is 0 or. a multiple of 27; but in the analysis of a single long series covering a great many years the resulting amplitude represents the average of the values of the function when V equals 0 or a multiple of 2z. It will be readily seen that the value of (102) is J when (V+u)=0 or a multiple of 2z, and J cos vw when V=O or a multiple of 2z. The expression ‘‘mean value of coefficient,’’ as applied to the terms. in the formula for the equilibrium height of the lunar tide, is usually taken to represent the mean value of the product. J cos u. With the value of u small, cos u has a value near unity and the mean value of J cos u differs but little from the mean value of J. In the practical application of the equilibrium theory to the harmonic analysis’ and prediction of the tides it is of no consequence whether the mean value J alone or of the product J cos u be taken as the mean coefficient, but for the sake of uniformity in representing the results the practice heretofore adopted will be followed. With the factor cos w always near unity, the mean value of the product J cos wu can be shown to be approximately equivalent to the product of the mean value of each, and is so taken in the computa- tions that follow. | Referring to (100), the mean value of the: following variables will be required: cos‘ 4 J cos (26 — 2») for terms (A), to (A), sin? J cos 2v for terms (A),, to (A),, sin I cos? 4 I cos (2&—v) for terms (A),, to (A),, sin J sin? 4 J cos (26+ 7) for term (A) 5, sin 2 J cos v for terms (A),, to (A) 4, sin’ [ cos 2€ for terms (A),, to (A),, (1—3/2 sin? J) for terms (A),, to (A), The first step will be to express the functions of J, v, and € in terms. of N, the longitude of the moon’s node. The latter changes uni- formly with time, and it is in reference to time that the mean values are desired. Referring to Figure 6, the following formulas may be readily derived from the spherical triangle QYA. Noting that the side & ¥ =N, the longitude of the moon's node; side RA=QVvP’—&=QA¥V —€=N—€; and side Y A=»; and that the opposite angles are (z— J), w, and 2, respectively; we have cos J=cos 2 cos w—sin? sin w cos N (103) tan 4 [(N—€) += eh =} fein (104) tan 4 [(N—€) —= Seb eo tan 4 N ! (105) sini ¢ sin, N | san *~ GOS 7 SIN w+siN 2 Cos w cos N 106) tan (V—£) = sey (107) cot w sin 2+ cos 2 cos N 49 U. S. COAST AND GEODETIC SURVEY. Since tan (W—6) = Ae (108) we have from (107) and (108) tanta ‘sin? cot w tan ING (cos ial) sin NV sin 2 cot w+cos7 cos N+sin N tan N sin 4 cot w sin N—sin? #1 sin 2 N (109) ~ sin 2 cot w cos N—2 sin? 42 cos? N+1 For the computations of tables of the values of J, v, and €, with NV as the argument, formulas (103), (104), and (105) will be found especially convenient. Formulas (106) and (109) provide for the direct computations of v and € independent of each other. For the computations of the mean values now sought it will be found desirable to modify formulas (103), (106), and (109) and represent the values of I, v, and € in approximate forms that are more easily developed. By Table 2 it will be noted that 7 is small, being equal to 5.145°, or 0.090 of a radian; therefore, using the radian as the unit angle, the sine of 7, or of any fraction thereof, may be taken as approximately equal to the angle itself. Then sin 4=1 (110) cos 1=1—2 sin? 40=1-47 (111) Substituting (110) and (111) in (103), (106), and (109), and develop- ing to the second power of 2, we may obtain the following: Cos J=cos w—isin w cos N— }2? cos w (112) aa 7 sin N aoe Ga sin w +7.cos w cos N - =7 cosecw sin N — 42? cos w cosec? w sin 2 N (113) Gini ow i.cot w sin N—42? sn. 2 N ae 7 cot w cos N— 42? cos? N+1 : =i cot w sin N—}? [cot? o +4] sin 2N (114) From (112) ‘ cos? J=cos? w—72 sin 2w cos N+? (sin? w cos? N—cos? w) (115) cos? w— cos? NV Re Ae, mF a ‘ ry sin J=(1—cos? J)? =sin w+2 cos w cos N+ 44 ie (116) sin I cos J =4 sin 2w+7 cos 2w cos N+47? cot w [cos 2w— cos? N (1+ 2 sin’w)] (117) From (113) ' We NEGRO ULE Go 2 Ha 9) cos v 4 tan)? 1— 42? cosec?w sin? NV (118) sin v=tanycosy=icosecwsin N—47 coswcosec?wsin2N (119) cos 2v =2 cos? y—1=1—2/? cosec? w sin? NV (120) sin 2v =2 sin vy cos y= 22 cosec w sin N—7? cos w cosec?w sin2N (121) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 43 From (114) cos fF i rtar 1—47 cot? w sin? N (122) sin £=tan — cos £=7 cot w sin N—47 [cot? w+4] sin2N (128) cos 2 =2 cos? E—-1=1—2? cot? w sin? N (124) sin 2£=2 sin é cos £=21 cot w sin N—? [cot?w+4]sin2N (125) From (118) to (125) cos 2 cos 2v = 1 — 27? leosec? w+ cot? w] sin? N (126) sin 2~ sin 2v= 47? cosec w cot w sin? NV (127) cos2~ cos v=1—7? [4 cosec? w+2 cot? w] sin? N | (128) sin 2 sin v=27? cosec w cot w sin? NV (129) Since JN is an angle that changes uniformly throughout the entire circumference, it may readily be shown that the mean value of sin J, cos N,sin 2 N, and cos 2 NV is zero for each, since for each positive value there will be a corresponding negative value in the same period. Indicating the mean value of a variable by the subscript 0, we may now write [sin N],=[cos N],=[sin 2.N];=[cos 2N],=0 (130) [sin? N],=[$—4 cos 2N],=4 (131) since the mean value of the sum of several terms equals the sum of the mean value of each term. Substituting (130) and (131) in formulas (112), (115) to (118), (120), (124), and (126) to (129), and indicating the resulting mean values by subscript 0, the following may be obtained: and ~ [eos I], = (1 — 422) cos w : (132) [cos? J], = cos? w + 7? (4 sin? w — cos? w) py, =cos* w +7? (4 —3 cos? w) me ME ~ (133) [sin A,=sin w+ 42 29 (134) [sin J cos J], =sin w cos w + 47? cot w [cos? w — sin? w — 4— sin? a] =sin w cos w+42? cot w [$—3 sin? w] = ,'(135) [cos vy], =1—4 2? cosec? w (136) [cos 2v],=1—7? cosec? w (137) [cos 2£],=1—2 cot? w - (138) [cos 2 cos 2v], = 1 — 2? [cosec? w + cot? w] (139) [sin 2£ sin 2v], = 22? cosec w cot w (140) [cos 2€ cos vy], =1—47? [cosec? w +4 cot? w] (141) [sin 2£ sin v],=7? cosec w cot w (142) 72934— 2474 . 44 U. S. COAST AND GEODETIC SURVEY. Then, for the functions of J in the coefficients of (100), we may obtain from (132) to (135) the following: [cost 4 J], =4 [142 cos J+ cos? I], —1[14+2 cos w+cos? o+47 (1—2 cos w—3 cos? w)] =1[(1+cos w)?+4# (1+ cos w) (1—3 cos w)] 1—3 cosa =cost4w+47 cos* 4a © 2 Zs ES cosia .. 1—3 COs w (143) as 4 of le ge ee =cos bo| 1444 eee] | [sin? J], =[1 —cos? f], =sin? w»—4$7? (1—3 cos? w) = sin? w E +47 cS Bees “| (144) [sin J cos? 4 7], =4[sin 7+sin I cos J], : : ., cos? w—4+4 cosw—3 cos w sin? w =3| sino tsino cosa +47 a+ sin w | ne 4 : aphasia: rb 2 cos? w — 6 cos =] sin Ww =3] sin w+siN w COS w— Pa +cosw) (1 BOP eu 6 cos? w sin w £3, 2 =sin o cos $w[ 1-3 2 1+4 cos w—6 cos =| (145) sin? w [sin J sin? 4 7], =} [sin 7—sin J cos J], ‘ : .. cos?w—4—4 cosw+3 coswsin?w =1| sinw—sinweosw+4# SS =3| sin w—sin w cos w— 4} 7? 1—5cosw— 2 cos?w+6cos?w SIN w FT, —_— = 2 sIn w : é .. L—4 cos w—6 cos? w (146) oh 2 Be ag poe es es Weta ee =sIN w sin bolt + 2 Snes | [sin 2 I],=2 [sin J cos [],= =2 sin w cos w+? cot w[4—3 sin? w] 2 =sin 2 E +i ? oon] (147) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 45 oe 2 ann, 1 — 3/2 (144) —1—3/2 sin? »—3/4 7? (2—8 sin? w) = (1—3/2 sin? w) (1—3/2 77) : (148) From (136) to (142) we may obtain the following: [cos (2 — 2v)], =[cos 2 cos 2v+sin 2é sin 27], =1—7? [cosec? w+ cot? w—2 cosec w cot a] = 1—+7? [cosec w—cot w/}? .. | l—ecos w F .. | l=—cos'@ Le Fie be aU EAN OND FE aC Wa nie GO il, ates tenes? | sin w | rt E ++ cos 2] ee) to. 34 (150) Ferg. 3 2 RE ie SU iam [cos 2v],=1—7 cosec? w=1—7% ae, [cos (2&€—v)],=[cos 2 cos y+sin 2é sin 7], . =) ae ? [cosec? w+4 cot? w—4 cosec w cot x] =1-4¥7 pes, w—2 cot w]? (151) a po 1—2 cos =f sin w [cos (2+ 7)],=[cos 2€ cos y—sin 2é sin 7], =1—4 2? [cosec? w+4 cot? w+4 cosec w cot] . hl +2 cos w P (152) =1—}3 7 f———_ | sin @ eee [cos vy], =1—} #? cosec w=1— Bisa (153) [cos 2 |, =1—? co? w=1-?? or = (154) By taking the products indicated on page 41 the mean values for the variable factors of the coefficients of (100) may be obtained as indicated below. By} substituting cost } 7 as the equivalent of 1—4 2, and 1—3/2 sin? 7 as the equivalent of 1—3/2 2”, the results are obtained in the forms adopted by Professor Darwin. The numerical equivalents of the formulas are obtained by substituting the values of w and 7 from Table 2. For terms (A), to (A), [cos* 4 I], [cos (2€— 9 = (143) x (149) = cos* 4 w [1 — $77] = cos‘ $ w cost $1=0.9154 (155) For terms (A),, to (A),, [sin? 1], [cos 2 v],= (144) X (150) =sin? w [1 —3/2 27] (156) k =sin? w [1 —3/2 sin? 7] =0.1565 For terms (A),, to (A)., [sin J cos? } /], [cos (2 Ex No= (145) x (151) (157) =sin w cos? } w [1 —§ 7?]=sin w cos? $ w cost $1 =0.3800 For terms (A),, [sin J sin? 4 Hh, [cos (2 &+»)],= (146) x (152) (158) =si w sin’ 3 w [1—477]=sin w sin’ $ w cost $ 1=0.0164 46 U. S. COAST AND GEODETIC SURVEY. For terms (A),, to (A) 4, [sin 2 I], [cos v]o= (147) X (153) =sin 2 w [1—3/2 77] (159) =sin 2 w [1—3/2 sin? 7] =0.7214 For terms (A),, to (A),;, [sin? 7], [cos 2 £], = (144) x (154) =sin? w [1 —4 27] (160) = sin? w cost 4 7=0.1578 For terms (A), to (A) ¢5 [1-3/2 sin? I] — (148) = (1-3/2 sin? w) (1—3/2 #4) gy) = (1—3/2 sin? w) (1—3/2 sin? 2) =0.7532 The mean value of the coefficient of each term of (100) may now be readily obtained by substituting for the variable factor the corre- sponding mean value from formulas (155) to (161). Table 3 contains a compilation of such mean values. 12. FACTORS OF REDUCTION. For the analysis of a series of observations not exceeding 369 days, the coefficient may be considered as practically constant, with I having a value corresponding to the middle of the series, but in order that the results from several years of observations may be comparable it is necessary to take account of the changes in J and apply a factor of reduction to the amplitude’as obtained from any particular series of observations. For these factors it is assumed that the variations in the actual tidal components due to the changes in the longitude of the moon’s node will be proportional to the corre- sponding variations in the coefficients of the terms of (100). Representing the mean value of any component amplitude by and the amplitude for any particular time by # let F=*, or H= FR (1C2) and f—3i, or R-fH (163) Then F' will be the factor for reducing an amplitude for a particular series to its mean value and f the reciprocal of F, the factor for adapting a mean amplitude to a particular time. Using the notation of the preceding section, this factor may be expressed pote (164) The constant factors of the coefficients of (100) being common to both the numerator and denominator of (164) need not here *be considered. Making the substitutions of the mean values represented by J, [cos u], from formulas (155) to (161) the following factors of reduction for the components represented by the terms of (100) may be readily obtained. F'for terms (A), to (A),, including components M,, N,, 2N, v2, 2,, and by, __cos* 4m cos* $4 0.9154 RY GOS! ar. COST (165) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 47 F for terms (A),, to (A),, including component lunar (K,), _ sin? w [1-3/2 sin? 4] _ 0.1565 it sin? [ ~ gin? J (166) F for terms (A),, to (A),., including components O,, Q,, 2Q, and Bi _sin w cos? $a cos*$2_~—0..8800 (167) a sin J cos? 4 J “sin J cos? $ I F for term (A),, for component OO _sinw sin’ $wcost47_ _—*0.0164 (168) ee sin J sin? 4 J Jsin Pf sin?e 7 F for terms (A),, to (A) 4 including components lunar (K,) and J,, _ sin 2w[1—3/2 sin? 7] _0.7214 sin 2 [ ~ sin 2 I (169) F for terms (A),, to (A),,;, including component Mf, _ sin? w cost $2_ 0.1578 a su f)"5 — { sin? 'T Sch) F for terms (A),, to (A),,, including component Mm, a an Ss BAG a Clea) 2, sine a) (3/2 sin?w)i Ono 32 (171) 1—3/2 sin? J 1 —3/2-sin? I The last is also the factor of reduction for the equilibrium com- ponent MSf; but as there is also a compound component having the same argument and generally a greater amplitude, which unites with the equilibrium component, the factor of reduction is usually deter- mined from the compound part, which will be discussed in a later section. The factor of reduction for a number of other special cases will be treated separately in the text. The factor f may, of course, be readily obtained by taking the reciprocals of the above expressions for the factor F. Table 12 gives the logarithm of the factor F for the principal components corresponding to every tenth of a degree of J, and Table 14 gives the natural factor f for the principal components for the middle of each year from 1850 to 1999. 13. THE L, TIDE. The separation of the components from each other by the processes of the analysis depends upon the differences in the speeds of the components. If two components have speeds that are very nearly equal, the analysis of a series of observations, unless of a very lon period, will not separate such components from each other but wil give a single component that is a resultant of the two. Referring to equation (100), we note that the speeds of the terms (A), and (A),, are very nearly equal, the difference being twice the rate of change in p, the longitude of the moon’s perigee, and this changes oH about 41° in an entire year. 48 U. S. COAST AND GEODETIC SURVEY. Because of this fact it is customary to treat these two terms as a single component known as the L, tide and having the speed of (A),, since this has the larger theoretical amplitude. Omitting for the present the general coefficient applying alike to both terms, we have term (A),=1/4 e cost 4 J cos (27+ 2h—s—p+2€—2u+z) (172) and term (A),,=3/8 e sin? J cos (2T+2h—s+p—2v) (173) Let 6=2T+2h—s—p+2&€—2v+zn (174) and P=p—€ (175) Substituting (174) and (175) in (172) and (173) and combining the latter we have 1/4 e cost 4 J cos 6+38/8 e€ sin? I cos (0+2p—2€ —z) = 1/4 e cost 4 I [cos 0—6 tan? 4 I cos (04+ 2P)] =1/4 e cost 4 J[cos 6—6 tan? 4 J cos 6 cos 2P +6 tan? 4 J sin 6 sin 2P] = 1/4 e cost 4 I [(1—6 tan? 4 J cos 2P) cos 0+6 tan? 4 J sin 2P sin 6] =1/4 e cost $ [[1—12 tan? 4 [cos 2P+ 36 tan‘ 4 I]? y cos( @— tan 6 tan? 4 J sin 2P 1—6 tan? 4 I cos 2P cos? 4 I n which = =[1—12 tan? 4 I cos 2P +36 tan‘ 4 J]? (177) and yh : pre 6 tan’?3/ sin 2P Ase sin 2P (178) 1—6 tan’? 4/ cos 2P 1/6 cot? 4l—cos 2P The values of log R, and F corresponding to different values of J and P will be found in Tables 7 and 8, respectively. Formula (176) represents the composite L, tide. The V of the argument is 27'+2h—s—p+z7, with a speed identical with that of (172). The inequality wu of the argument is 2§—2v—R. For the mean value of the variable factors of the coefficient we have 41 Boos cos 2-2) | (179) =| cos‘ 4/ i = cos (2& — 27) sae = sin (26 — 2») I] R, Ra o From (177) and (178) > iE Spar rctee trae (180) sin R : R =6 tan? 4/ sin 2P (181) a substituting (180) and (181) to (179). HARMONIC ANALYSIS AND PREDICTION OF TIDES. 49 Mean value of variable factors of coefficient =[cost 4J {(1—6 tan? 4/ cos 2P) cos (2§ — 2») +6 tan? 4/ sin 2P sin (2€—27)], (182) =--|cost 4J cos (2€—2v) —6 sin? 4/ cos? 4J cos (2P +26 —27)], Substituting the equivalent of P from (175), the last term of (182) becomes 3/2 sin? I cos (2p — 27) (183) Now p increases uniformly throughout the entire circumference, while J and » are functions of N, the period of which is incommen- surate with that of p. It is evident, therefore, that in a series of infinite length, the sum of the positive values of (183) will equal the sum of the negative values, and the mean value of the term becomes zero. The mean value of the first term is given by formula (155). For the mean value of the variable coefficient of the composite L, tide, we may now write heats a cos (2 —2y— ) | =[cos* $1 cos (26 —2r)]. = cos‘ 4w cos* 44 =0.9154 (184) For the factor of reduction, cost 4w cost $1 0.9154 cos! 4 ~ cost $1 Ra F of L,= xR, (185) A comparison of (185) with (165) shows that F of L, = (fF of M,) x Ba - (186) 14. THE M, TIDE. In equation (100) we also have the terms (A),, and (A),, with a difference in speed equal to twice the rate of change in p. Neglecting for the present the general coefficient applying to both terms, we have term (A),,=1/4 e sin J cos? 4/ cos (T+h—s—p+2&—v—7/2) (187) term (A),,=3/8 e sin 2J cos (T+h—s+p—v—7/2) (188) A reference to (99) indicates that the coefficient of the term (A),, is only about one-third that of term (A),,. The latter will therefore redominate and determine the mean period of the composite tide ormed by the combination of the two, while the former will intro- duce certain inequalities in the resultant amplitudes and epochs. Let @= T+h—s+p—7/2—p (189; and P=p—€ as in (175) 50 U. S. COAST AND GEODETIC SURVEY. The sum of the terms (187) and (188) may then be written cos I 1/4 e sin I cos? 41 | cos (@—2P)+3 eos alge: | =1/4 e sin I cos? 11 cos 6 cos 2P+sin 6 sin 2P+3 <2 " cos | cos? 4] a ‘ Sais cos? I cos I Casto Pao 1/4 e sin I cos s1[ 9 cos iT 6 es iI te (190) : Rs akg sin 2P X cos (’ on 9 £08 V8 ii 9 =| ! CISA Selgin 1 : j 21 2e mes Bes (Pitas 4p af maly) where 1 cos? I cos I 4 eS is 0. | 9 cre te eres or 1| (191) 9 9 4 = Ea tan? 4] ina tant $/+3/2 (1—tan? 4J/) cos aP | and deh ae sin 2P toa sin 2P . Qu = tan cos I mae 300) tan? a) cose (192) 3 ec ee 2P Formula (190) represents the composite M, tide, the mean speed and period of which are determined by the V of the argument, which is T+h—s+p—*/,. The u, which equals —y—Q,, may be shown to vary between the limits of approximately +70°, and will therefore not affect the mean period as determined by the V of the argument. The period of this component is very nearly an exact multiple of the period of the principal lunar component M,, and for this reason the summations which are necessary in the analysis for the latter may be conveniently adapted to the analysis for component M,. Let 6= T+h—s—*/,+&-v (193) and P as before. The sum of terms (187) and (188) may then be written 1/4 e sin I cos? $1 E (@—P)+3 sles cos @+P) | cos? 4/ cos I cos? 41 ee COStL Nee : +(4 =a ANE 7) sin @ sin P| = 1/4 e sin F.cos? u| (143 ) e085 @ cos P HARMONIC ANALYSIS AND PREDICTION OF TIDES. 51 cos I cos? I 74 cos 2P +9 cos? 41 +9 Cos! 4] =1/4e sin I cos? ui +6 cos I cos? 41 cos I cos? 4] —3 x cos{ @—tan- tan P 1+3 ' gin J cos? 4 a a me (GEA SM cotta Sn ae (194) in which Q,=same as in (191) and cos I 24 af et cos? $f wl i. 2—3 tan?’ 31 Q = tan 5 £08 + Peni JEU irae gras men ale (195) cos? 41 The values of log @, and Q corresponding to different values of P and the mean value of J will be found in Tables 9 and 10, respectively. In formula (194) the V of the argument is taken as 7+h—s—7/, and the uas&—v+Q. Formulas (190) and (194) both represent the com- posite M, tide and are equal to each other, since each is the sum of the terms (187) and (188). It may also be shown from (192) and (195) that QutQ=P=p-é (196) p—Qu.=E+Q (197) The complete argument of (190) is therefore equivalent to the argument of (194), the distinction being that the uniformly varying element p in the V of the first argument has been transferred to the u in the latter, where it is assumed to be constant in the analysis of any given series of observations. The speed of the component as determined by the remaining part of the Vis then exactly one-half the speed of the component M, [term (A), of (100)]; and with this assumption the summations for component M, will be adapted to the analysis for the component M,. It will be noted, however, that the u in this case, unlike the w’s of any of the other components discussed, has a pro- gressive forward change that takes it entirely around the circumfer- ence (see Table 10 for values of Q). The true average speed of this component is therefore determined by the V of the argument of (190), the approximate average speed determined by the V of formula (194) bens assumed when the summations for component M, are to be used or M,. In obtaining an expression for the mean value of the variable factors of the coefficient of this component the u from formula (190) must be used. For this coefficient we have 3 wa [= oe i] ae +Q.) | and therefore . =| sin I cos? 41 cores cos yes sin r| | (198) 52 U. S. COAST AND GEODETIC SURVEY. From (191) and (192) a -4[2 eee cos 2P | (199) tad sin 2P (200) Substituting (199) and (200) in (198), the mean value of variable _ factors of the coefficient is as follows: E sin [ cos? 4J 3 ee vy+cos 2P cos y—sin 2P sin | z (201) = [3/4 sin.2/ cos v+4 sin I cos’ $1 cos (2P + )],_ For reasons similar to those given on page 49 the mean value of the last term in the above is zero for an infinite series. The mean value of sin 2/ cos v is given by formula (159), which when substituted in the above gives the following: [ge cos 0+Q,) | (202) = 3/4 sin 2w [1 —3/2 sin? 7] =0.5410 For the factor of reduction we have _ 3/4 sin 2w[1—3/2 sin? 4] (0.5410 F of M,= sin [ cos? 4 ~ sin I cos? 4/ xQa (203) Qa. The factor / for reduction of M,, originally adopted and now in eneral use for analysis made in accordance with the system of Sir eorge H. Darwin, is as follows: sin w cos? $ w cost} 4 Aa 0.38005 sin I cos’ 4 1 [5/2+3/2cos2P]* sin J cos?4 1[5/2+3/2 cos 2P}4 In the above the factor aot oe is the approximate (204) equivalent of the factor Q, in (203). The ratio of (203) to (204) is therefore approximately ee = 1.42 (205) This discrepancy appears to be due principally to the accidental omission of the factor 2.5, or 1.58, from the original formula. (See Scientific Papers by Sir George H. Darwin, vol. 1, p. 39.) The effect of this error has been that all the mean amplitudes for component M, obtained by the formula of Darwin are too small and should be increased by nearly 50 per cent in order to be theoretically correct. Since the primary purpose of reducing the amplitudes to their mean values is to render the results from different series comparable with each other, this purpose has not been frustrated by the introduction HARMONIC ANALYSIS AND PREDICTION OF TIDES. 53 of a factor error which has been applied alike to all amplitudes for this component. Neither has the use of these components in the prediction of the tides led to any error in that work, since the error in the factor for reduction to the mean value has been exactly compen- sated by a corresponding error in the reciprocal factor used for reduc- ing the mean value to the amplitude for the year of prediction. Therefore no practical difficulties have resulted from this error. To change now to the corrected formula, unless the change were uni- versally adopted, would lead to considerable confusion in the com- parison of the amplitudes as determined and published by different authorities. It seems wisest, therefore, for the time being to adhere to the present practice of using formula (204) or its approximate equivalent sin w cos? $w cos* $2 ee xQa= (F of O,) XQ, (206) for the reduction of the component M,. The resulting amplitudes may at any time be readily converted into the corrected means by the application of the factor 1.42 from (205). 15. TIDES DEPENDING UPON THE FOURTH POWER OF MOON’S PARALLAX. A reference to equations (74) and (75), pages 28-29, shows that the tide depending upon the fourth power of the moon’s parallax is very small, the maximum value being only about 2 per cent of the total lunar tide. In developing the term representing this tide we need, therefore, seek only a rough approximation to its true value, neg!ect- ing the elements which are relatively small compared with the entire term. As the angle J is never very large, the sine will always be smaller than the cosine, and for our approximation the powers of sin TI and sin 4 J above the first may be neglected in this development. Substituting the value of cos @ from (81) in the second term of (74) and neglecting powers of sin J and sin 3 J above the first, we obtain the following: 4 3/2 aS [5/3 cos? @—cos 6] 5 235 Pi ae [5/3{3 sin \ cos? \ sin J cos* 4 J sin J cos? (J—x) +cos® \ cos®4 J cos? (J—x)}—sin d sin J sin l —cos \ cos? 4 I cos (l—x)] 5 Se 2 ae [5 sin \ cos? \ sin J cos* 4 /{4 sin /—1/4 sin (J— 2x) +1/4sin (31 — 2x) } + 5/3 cos? \ cos® 4 1{3/4 cos (J —x) +1/4 cos 3 @—x)}—sin d sin J sin 1 —cos d cos? 4 J cos (l—x)] 5 = 3/2 ue [5/12 cos? \ cos® $ I cos 3 (I—x) + 5/4 sin \ cos? \ sin I cos* 4 I cos (81—2x—7z/,) + 5/4 sin d cos? \ sin J cos* $ J cos (L—2x+7/,) + {5/4 cos® \ cos® + J—cos Xd cos? 4 J} cos (— x) + {5/2 sin X cos? d sin J cost $ I —sin d sin J} cos @—7z/,)] (207) 54 U. S. COAST AND GEODETIC SURVEY. Neglecting the eccentricity of the moon’s orbit as being unim- portant for this tide, we may take the mean distance c of the moon as the equivalent of the actual distance d, and the mean longitude o measured from the intersection as the equivalent of the actual longi- tude / from the same origin. Substituting these equivalents and (98) and (99) in (207) we obtain: 5 3/2 Fa l5/l2 cos? cos® 4 J cos (87+3h—3s + 3€— 37) M, (0.0107) +5/4 sin \ cos? d sin J cos‘ $ J cos (2 T+ 2h —3s+3£— 2v+ w/,) | (0.0051) +5/4 sin \ cos? d sin J cost $ J cos (27+2h—s+é—2y—z/,) (0.0051) + {5/4 cos® \ cos® 4 [—cos \ cos? 4 I} cos (T+h-s+é—») [M,] (0.0100) + {5/2 sin d cos? d sin I cost $ —sin \ sin J} cos (s—§—z/5)] (0.0116) (208) The maximum theoretical value in feet of the amplitude of each term, when J has its mean value, is given after the term. For the first term the maximum amplitude will apply to the Equator, where cos X=1; for the second and third terms to latitude \=cos-' + 2/3, where sin \ cos? \ will have the maximum value of 2/3+/1/3; for the fourth Teaco eT a i 7 and for the last term to lat- itude 90°, where sin \=1. It will be noted that the first term containing 3 7'in its argument is a terdiurnal component with a speed exactly three halves that of the semidiurnal M, term (A), of (100). This component is usually designated as M,. The fourth term is a diurnal component with a speed exactly one half that of M,. This component might be appro- priately designated as M, but a distinction should be made between this and the composite M, described in the preceding section. The M, depending upon the fourth power of the moon’s parallax is usually ignored in the analysis, as its effects are negligible. All of the terms of (208) are so small that they are of no practical importance in the analysis and predictions of the tide. The component M,, however, being obtained with very little additional labor when analyzing for M,, is usually evaluated. The mean value of the variable coefficient of M, is [cos® 4 J], [cos (3& — 3) ], (209) Developing in a manner similar to that described in section 11, we find term to latitude \=cos7! .. L—2 cos w 6 yet 6 O} ie ENS [cos® 4 I], =cos® 4 w [1+3/2 Fa eae | (210) .. L—COS w es meee 2 [cos (8 -3v)],=1—9/4 2 Tren (211) [cos* 47], [cos (8€—3v)],=cos* 4w [1—3/4 i?]=cos® 4 w cos® 4 = 0.8758 (212) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 55 The factor F of M, is therefore cos’ 4 w cos'47_ 0.8758 cos' 4 { ~ cost £ J (218) Comparing (213) with (165) we find F of M,=[F of M,}? (214) 16. SOLAR TIDES. The development of the term in (74) that represents the approxi- mate solar tide will be similar to that for the lunar tide. By making the proper substitutions of the solar elements for the lunar elements in (100) a corresponding expression for the solar tide may be obtained. For mass of moon ( M) substitute mass of sun (S). For mean distance of moon (c) substitute mean distance of sun (¢,). For eccentricity of moon’s orbit (e) substitute eccentricity of earth’s orbit (e,). For inclination of moon’s orbit to Equator (J) substitute obliquity of ecliptic (w). For mean longitude of moon (s) substitute mean longitude of sun (/). For mean longitude of moon’s perigee (p) substitute longitude of sun’s perigee (p,). For longitude of intersection of moon’s orbit with Equator, in the Equator and in the moon’s orbit, vy and €, respectively, substitute zero, the longitude of vernal equinox. All terms in (100) representing the evection and variational ine- qualities in the moon’s motion may, of course, be omitted, and also because of the small eccentricity of the solar orbit all elliptical terms of the second power of e, and elliptical terms of the first power of e, when combined as a factor with a sine function of the angle w are negligible. In terms such as (A), and (A),,, where the second power of é, is a part of a larger coefficient, the entire coefficient is retained. With the above-named substitutions and omissions, the following formula is obtained for the equilibrium height of the solar tide: Say? =3/ = 2 (B), [@j2—5/4 72) cost 1/2 cos (27) ae 2 22 ee Se (0. 3716) (B)> +7/4 e, cost 1/2 w cos (2T —h+p;) _-________- T. (0. 0218) (B) 3 +1/4 e; cost 1/2 » cos (27 +h—pi+m) _______- Re (0. 0031) (B) 17 +(1/4+3/8 e.’) SimevonCOSn (2-22) yee [K.]}! (0. 0321) +3/2 = (+) a sin 2\X (B) 26 [(1/2 —5/4e:2) sin w cos? 1/2 w cos (T —h+7/2) __ P, (0. 1542) (215 (B)sg + (1/2 —5/4 e,?) sin sin? 1/2 woos (T +3h —7/2) - (0. 0066) ) (B) 4s +(1/4+3/8 ex”) sin 2 » cos Lah 2) on a [KK,]! (0. 1478) ip @) a) (i/2 3/2 ain? AaB) 1k SS (B) 51 2 5/4. 652) sim2ray COS) (2M) as Ge a ae Ssa (0. 0640) (B) 59 +-(1/3--1/2 ¢37)) (1 —3/2 sin? w)e ree eae eS (0. 2057) (B) 6 sip ea Clichy) SUA tea) COS (lyn) py gt ee es al (0. 0103) The subscript of the notation at the left of each term refers to the corresponding term in the development of the lunar tide. The nota- tion at the right gives the usual designation of the component repre- sented, the brackets indicating that the term only partially represents 56 U. S. COAST AND GEODETIC SURVEY. that component. The numerical value at the right of each term ives its maximum value in the foot unit. Term (B8),,, although aving a larger theoretical value than some of the lunar terms which were retained in (100), is usually neglected in the solar tide. Term (B) 9 is a constant and therefore does not affect the rising and falling of the tide but does cause a permanent deformation of the earth’s surface. Term (5),, has a period of an anomalistic year which differs very little from a tropical year, the latter being the period of the meteorological component Sa, to which later reference will be made. The coefficients and arguments of the terms of (215) are free from the quantities depending upon the longitude of the moon’s node. The u’s of the arguments of the solar tides may therefore be consid- ered as zero, and as all the coefficients are constant the factor F of reduction will be unity for each. It will be noted that the general coefficient of each group of terms of (215) differs from the corresponding general coefficients of (100) by 3 the factor (Z) - Inorder that the coefficients of the individual terms of the solar and lunar tides may be more conveniently compared 3 with each other, this factor i (2) is usually transferred from the general coefficient of the solar tide to each of the individual terms, thus leaving the general coefficients the same for both formulas. For brevity this factor is represented by G in Table 3. 17. LUNISOLAR K, AND K, TIDES. Comparing (100) and (215), we find that the terms (A),, and (A),, have the same speeds as (B),, and (B),., respectively, the small ine- qualities represented by 2v and v not affecting the mean speeds of the terms in which they occur. In the analysis and predictions of the tides the components of equal speeds are combined into single components, known as the lunisolar tides, and designated as K, for the diurnal component and K, for the semidiurnal component. For brevity in the following discussion let 62 3/2 a(*) a (216) C, = C sin 20 (217) C,= C cos? d (218) e-7,(5) (219) A=(1/4+3/8 e) (220) A,=A sin 21 (221) A,=A sin? I (222) B=(1/443/8 e2) G (223) B,=B sin 2 (224) B,=B sin? w (225) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 5T The lunar K, term (A),, of (100) may then be written CA, cos (T+h—v—7/2) (226) and the solar K, term (B),, of (215) C.B, cos (T+h—7/2) (227) Taking the sum of (226) and (227) we have Lunisolar K,= C,[A, cos (T+h—v—7/2) +B, cos (T+h—7/2)] = (, [A, cos » cos CIEE E12) +A, sinv sin (T7+h—7/2) +B, cos (T+h—7/2)] = C, [(A, cos v+B,) cos (T+Ah—7/2) +A, sin vy sin (T+hA—7/2)] = (C, (A’Z+B/7+2A,B, cos v)? cos (T+h—71/2—7’) =C( sin 2d [A? sin? 27+ B? sin? 2w +2AB sin 2/ sin 2u cos vt cos (T+h—z/2—7') (728) in which ; vy’ =tan ages, eed sin v 2+3e7 Gsin 2w Coser 2+ 3¢é? sin 2] sin v sin 27 cos v sin 2/+.0.3357 G2) =ftan— the values for the constants in (229) ine obtained from Table 2. Similarly, for the semidurnal component from A,, of (100) and B,, of (215), using the abbreviations of (216) to (225), we have Lunisolar K, = C, (A+ B,’+2A,B, cos 2v)* cos (27+ 2h — 2v'’) = C cos? \ [A? sint J+ B? sin* w (230) +2AB sin? I sin? w cos 2v]? cos (27+ 2h — 2y’’) in which A, sin 2p Lites =f ees A, cos 2v+B, sin 2y 24+3¢e/ G sin? w 2+8e¢, sin? I — bag cos 2v+ { “aie iS sin Qpisin? 7 ee cose = ules (231) Values of »’ and 2»’’ for each degree of N are given in Table 6. 58 U. S. COAST AND GEODETIC SURVEY. ‘ For the mean value of the variable part the coefficient of (228) we have [((A/?+B,?+2A,B, cos v)? cos v’], (232) From (229) He A, cosv+B, cos »' = (A 74+ Bi +2A,B, cosy)t a Therefore [((A’2+B,+2A,B, cos v)? cos v’], =[A, cos 7+ By], =[A sin 27 cos ¥+ B sin 2], (234) Substituting (159) in (234), mean value of variable part of coefficient of K, =A sin 2w [1—3/2 sin? 4] + B sin 20 =0.2655 (235) Referring to (228) and (235), . a ° 2 . . i FotK, A sin 2w [1—3/2 sin? 7]4+-B sin 2 (236) ~ [A? sin? 27+ B? sin? 20+2AB sin 27 sin 2w cos vp aie the spherical triangle QvA in Figure 6 it may be shown that cos 1—cos w cos I Coenen sin w sin [ (237) From which it follows that alt De eben ane Ais Oy 2 (cos 2—cos w cos J)?—sin? w sin? J (238) sin? w sin? J Substituting (237) and the numerical values of the constants from Table 2 in (236) we obtain ) Fof K,=[0.1009 +. 3.0073 cos J+ 0.8093 cos? I—3.5793 cos* I} (239) For the mean value of the variable part of the coefficient of (230), referring to (231) and (156), we have [((A?+B,?+2A, B, cos 2v)* cos 2u’’], =[A, cos? v+ By]. =[A sin? J cos 2u+B sin? w], = A sin? w (1—3/2 sin? 2) +B sin? w=0.0576 (240) Referring to (230), (239), and (240), and to Table 2, Fof K.= A sin? w (1—3/2 sin? 2) +B sin? w or 2 TA? sint 7+ B? sint a +2 AB sin? I sin? w cos? uf = [51.0453 — 63.9167 cos J— 5.8300 cos? 1+ 19.0186 cost J]-* (241) ———— HARMONIC ANALYSIS AND PREDICTION OF TIDES. a9 18.. OVERTIDES. In the development of the equilibrium theory the absence of friction and sufficiency of depth were assumed. Under these condi- tions each term of the result represented a simple harmonic wave. When a wave runs into shallow water, the trough is retarded more than its crest, so that the duration of rise of the tide becomes some- what less than the duration of fall and the wave loses its simple harmonic form. We may, however, represent this modified form of the wave by the introduction of a series of components whose speeds are simple multiples of the speed of the fundamental astronomical tides. ie are called overtides because of their analogy to over- tones in musical sounds. The only overtides usually considered in the analysis are those for the principal lunar and solar components M, and 8, [Terms (A), and (5), of formulas (100) and (215), and are designated by M,, M,, M,, and S, and S,, the subscript indicating the number of periods in a component day. The arguments of the overtides are taken as exact multiples of the argument of the fundamental tide. There is no theoretical expres- sion for the coefficients of these tides, but it is probable that the amplitudes as determined from observations will be subject to variations due to changes in the longitude of the moon’s node analogous to the variations in the fundamental tide. It is assumed that the variability of the overtides may be represented by the square, cube, fourth power, etc., of the fundamental tides, and the factors of reduction are taken accordingly. Thus, F of M,=(F of M,)? (242) F ot M,=(F of M,)8 (243) F of M,=(F of M,)4 (244) The F of S, and F’ of S, are taken as unity. 19. COMPOUND TIDES. Compound tides are components whose speeds are the sums or differences of the speeds of the elementary components. They were suggested by Helmholtz’s theory of sound waves, and, like the over- tides, are due to shallow water. The arguments of the compound tides are taken as the sums and differences of the elementary tides. Thus, Arg: (MS), "= Arg. M,+ Are. 8, (245) Arg. (MN), —Arg. M,+Arg. N, (246) Arg. (MK), = Ate. My Are. K, (247) Arg. (2MK),=Arg. M,--Are. K, (248) Arg. (25M), =Arg. S,— Arg. M, (249) Also, Arg: (2MS), = Are. M, —Arg.S,=Equilibrium Arg. 7, + (2&—2v) (250) Arg. MSf=Arg.S,— Arg. M,= Equilibrium Arg. MSf—(2—2v) (251) 72934—244—_5 60 U. S. COAST AND GEODETIC SURVEY. The mean period of the compound tide 2MS is identical with that of the equilibrium tide », of (100) and the mean period of the com- pound tide MSf is the same as that of the equilibrium tide MSf of (100). The arguments, however, differ by small quantities which are functions of the longitude of the moon’s node and do not affect the mean periods. In the analysis the compound and equilibrium tides of equal period can not be separated from each other, and there is no known theoretical relation in their magnitudes. For convenience in reduction such tides are considered as arising from a single source. Following the past practice of the Coast and Geodetic Survey, the component #, or 2MS will be treated as though it were entirely the variational equilibrium tide represented by the term (A), of (100) and the component MSf will be considered only as a compound tide. According to the system of Sir George H. Darwin, both of these com- ponents appear to be considered as compound tides. The differences resulting from the two methods of treatment are negligible compared with the probable errors in the final results. For the factor # for the reduction of the compound tides the ee of the corresponding factors for the elementary tides are taken. Thus, F of (MS), =Fof M, (252) F of (MN), =(fF of M,) x (Ff of N,) =(F of M,)? (253) F of (MK), =(F of M,) x (F of K,) 3 (254) F of (2MK),=(F of M,) x (F of K,) (255) F of (28M), =F of M, (256) FofMSf =FofM, (257) The component p, or 2MS being treated as an equilibrium tide, the factor F' of this component is given by formula (165). 20. METEOROLOGICAL TIDES. Meteorological conditions have a considerable influence upon the tides, but, in general, the effects are very irregular and do not admit of being represented by harmonic terms. There are, however, some conditions that occur with a rough periodicity which may be so represented. The land and sea breezes and the daily variation in the atmospheric pressure may give rise to a tide whose period is a solar day and the changes in the seasons to a tide with a period of a tropical year. The former is designated as the S, component and has a speed just one-half that of the principal solar components S, [((B), of (215)]. The latter is the Sa component, and, although it may be accompanied by a number of overtides, the only one generally sought in the analysis is the semiannual component Ssa, which is also a component of the equilibrium tide [(B),, of (215)]. Although the determination of meteorological tides from long series of observa- tions is valuable for some purposes, their recurrence is not generally certain enough to make them of much value in the tidal predictions. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 61 The argument of component S, is taken as one-half that of the component S,, and the argument of component Sa is one-half that of the component Ssa. The factor #' for the reduction of each of the meteorological components is taken as unity, since the magni- tudes of these components appear to be unaffected by changes in the longitude of the moon’s node. ANALYSIS AND PREDICTION. 21. HARMONIC CONSTANTS. In the preceding chapter there were found from a consideration of the tidal forces the principal components that may be expected to exist in the tide. Each component is represented by the product of a coefficient, which may include a variable function of J, depending upon the longitude of the moon’s nede and the cosine of an argument which defines the component and determines its period. The period, being independent of conditions upon the earth’s surface, is the same for every locality and gives to each component its identity. The coefficients and phases of the components of the true tide do not, however, agree with the corresponding coefficients and phases of the equilibrium tide, but vary in different localities. It is therefore necessary, In order to represent the true tide at any place, to sub- stitute for the coefficients of the equilibrium tide the amplitudes of the true components determined from actual observations, and also to find corrections for the phases of the equilibrium components which will make them conform to the true phases. These phase cor- rections are called the epochs of the components and are constants for any locality. The amplitudes (H) and the epochs (x) comprise the harmonic constants which are to be determined by the harmonic analysis of the tidal observations taken at the place for which the results are sought. The principal use of these constants is in the prediction of the tides. 22. OBSERVATIONS. The most satisfactory observational data for the harmonic analy- sis are from the record of an automatic tide gauge, which traces a continuous curve from which the height of the tide above any adopted datum plane may be readily obtained for any hour. This record is usually tabulated to give the height of the tide at each solar hour of the series, the kind of time used being that which is customarily used at the place. Where an automatic tide gauge is not available, hourly heights as observed directly upon a plain tide staff may be used for the analysis. 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. If the hourly heights have not been observed, but a record of the high and low waters is available, an approximate determination of the larger components may be obtained by a special treatment. The results, however, are not nearly as satisfactory as those obtained from the hourly heights. 62 U. S. COAST AND GEODETIC SURVEY. An interval greater or less than the solar hour might be used for the record of tabulated heights. A shorter interval would cause a considerable increase in, the work of the reduction without materially increasing the accuracy of the results for the components usually sought. However, if an eu were made to analyze 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 over- tides. The hour interval appears to be the most convenient and practicable for the usual analysis. The summations are most effective in separating a disturbing component from the component sought when the length of the series of aheae ane is an exact multiple of the synodic period of those components. The synodic period of two components is the time required for the difference in their phases to complete a cycle of 360°. If the speeds of two components in degrees per solar hour be repre- sented by a and 6 the synodic period will be360°/(a ~ 6) hours. If there were only two components in the tide, the best length of series would be easily fixed; but in the actual tide there are many components and a length of series most effective in the elimination of one disturbing component may not be best adapted to the elimination of another. It is therefore necessary to adopt a length that is a compromise of the synodic periods involved, weight being given according to the theoretical relative magnitudes of the components. Fortunately, the exact length of series to be used is not of essential importance, and for convenience all series may be taken to include an integral number of solar days. Theoretically, different lengths should be used for the different components sought, but practically it is more convenient to use the same length for all of the components. An exception to this is found desirable for the very short series which is taken as 14 days for components chiefly diurnal and 15 days for components chiefly semidiurnal. The longer the series the less important is the exact length, and the greater the number of synodic periods of two components included the more nearly complete will be the separation of those two components from each other. Two components like S, and K,, which have a very small difference in speed and a synodic period of about six months, can not be satis- factorily separated by the summation of a series of less than six months. On the other hand, two components with a large difference in speeds like a diurnal and a semidiurnal component 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, in selecting the length of series, no special consideration need be given to the effect of a diurnal and ‘a semidiurnal. component upon each. other. The length of series adopted for the harmonic analysis of the tides in the office of S. Coast and Geodetic Survey are as follows: 14-15, 29, 58, 03" 134, 163, 192, 221, 250, 279, 297, 326, 355, and 369 days. or the 14—15-day series—14 for diurnal and 15 for semidiurnal ponents—the length conforms to the synodic periods of the rincipal diurnal components K, and QO,, and the principal semidiurnal components M, and S,, and also to the synodic periods involving a few of the less important components. This is the shortest series for which the harmonic analysis is made. It can not be considered as a HARMONIC ANALYSIS AND PREDICTION OF TIDES. 63 very satisfactory length but will serve for obtaining the approximate constants for the principal components at a place where a longer series of observations is not available. The 29-day series conforms approximately in length to the multi- les of the synodic periods of nearly all the principal components. his may be considered as a standard length for short series, and whenever it is necessary to limit the observations to a short period of time, the minimum requirement should be 29 days. The 369-day series may be considered as the standard long series, and whenever there are sufficient observations available it should be used for the analysis. This length conforms very closely with the multiples of the synodic periods of practically all of the short-period components. The length is also well adapted for the elimination of the irregular meteorological effects. If tidal observations have been continued at a place for a number of years, it is desirable to have an independent analysis for each year in order that the results from the different years may be compared, and thus serve as a check on the work. Although not essential, there are certain conveniences in having such series commence on January 1 of each year. If observations for several successive years are analyzed, each may be made to begin on the first day of thé calendar year without regard to the fact that the last few days of a 369-day series will thus extend into the following year and become the first days of the next series. If the observations are for a period of less than 369 days, the standard length selected will usually be the greatest one that is entirely covered by the observations, extra days of the observations being rejected; but if the period of observations lacks only a few hours of being equal to the next larger standard length, it may be advan- tageous to extrapolate additional hourly heights in order to complete the larger series. 23. SUMMATIONS. The first approximate separation of the components of the ob- served tide is accomplished by a system of summations. For each component of independent speed that is sought a separate summation is required, but the overtides will be combined with their fundamental components, and these will not require a separate summation. A single summation will serve for any group of components with commensurate periods. | Let us assume that the entire series of observations is divided into periods, each equal to the mean period of the component sought, which, for convenience, may be designated as component A. Hach such division will include exactly one complete period of this com- ee but all the other components with incommensurate periods will e represented in each division by more or less than a whole period. Each division will include also certain irregularities due to meteoro- logical conditions. Starting with the same phase at the beginning of each division, component A will be exactly reproduced in each successive division throughout the entire series, but the other com- ponents and the meteorological irregularities will occur differently im each division. Now, suppose that each of these divisions which corresponds to the period of component A be subdivided into any number of convenient parts, and that the initial instants of the subdivisions of each original 64 U: S. COAST AND: GEODETIC SURVEY. division be numbered consecutively, beginning with zero at the initial instant of each original division. Because of the exact reproduction of the period of component A in each division, it is evident that the instants with like numbers in the different divisions will correspond to the same phase of component A but to unlike phases of each of the other components. At each such instant the height of the observed tide will equal the height of component A for the phase corresponding to that instant, plus the heights of all of the other components, together with any variations due to meteorological causes. Assuming, for convenience, mean sea level as a datum and adding the heights of the tide corresponding to the instants of like numbers in each division it is evident that there will be included in the sum a cer- tain number of equal heights corresponding to a particular phase of component A, together with the unequal heights corresponding to various phases of the other components and also the meteorological fluctuations. In a series of sufficient length the sum of the unequal heights of each of the disturbing components and of the meteorological fluctuations will become zero, since the positive heights will be offset by the negative heights, while the equal heights corresponding to the particular phase of component A will accumulate as the summing proceeds. The average height of the tide obtained from such a sum when a limited series of observations has been used will be equal to the height of component A corresponding to a particular phase, plus a small residual due to the imperfect elimination of the disturbing com- ponents and meteorological effects. If the average height correspond- ing to each subdividing instant of the division is obtained and plotted as an ordinate with the time subdivisions as abscisse, a curve may be drawn which will approximately represent the component A through- out its entire period. If the heights of the original tabulation are referred to any arbitrary datum having a definite relation to mean sea level, the summation will be equally effective in eliminating the dis- turbing elements, and the resulting heights for component A will be referred to that datum. If, instead of making each division of the series equal to a single period of the component sought, we let it include any exact integral number of periods, the same principles will apply for the elimination of the other components. In this case the resulting average heights when plotted will represent the corresponding number of periods of the component sought. As a matter of convenience in the harmonic analysis of the short period components it is customary to include in each division such multiples of the component period as will most nearly conform to the solar day. Such a division is a component day. If the component is diurnal, its component day will include exactly one period; if semidiurnal, exactly two periods; if terdiurnal, exactly three periods, etc. Each component day is subdivided into 24 equal parts, each part being designated as a component hour. The initial instants of the component hours of each component day are num- bered consecutively from 0 to 23. For the long-period components the original divisions are taken as the component month or the com- ponent year, and these are subdivided into 24 equal parts. As the long-period components require a different treatment than the short- period components, they will be given special consideration in a later section. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 65 If the summations for the short-period components were made strictly in accord with the principles outlined above, it would be nec- essary to have a separate tabulation for each component sought, in which the heights would strictly apply to the beginning of each com- onent hour. The labor involved m such separate tabulations would e too great to be considered. Instead of making independent tabu- lations for each component a single tabulation with the heights of the tide taken to correspond to each solar hour of the series is used. These solar hourly heights are then assigned to the component hours with which they most nearly coincide and the summations made as though the heights applied exactly to those component hours. Cor- rections are later applied to take account of any systematic error in this approximation. Two systems of distribution of the solar hourly heights differing only slightly in detail will be considered: In the system ordinarily adopted each solar hourly height is assigned to the nearest component hour. In the second system there is selected for each component hour the nearest solar hourly height. By the first system each solar hourly height is used once, and once only, in the summation for each component, but each component hour may not be equally represented by the solar hourly heights. If the component day is shorter than the solar day, some of the individual component hours will be unrepre- sented; but if the component day is the longer, some of the individual component hours may have two solar hourly heights assigned to them. By the second system the component hours are equally represented, but it will be necessary to reject some solar hourly heights or to use some of them twice in order to accomplish this purpose. The differ- ence in the final results obtained by using the two systems will be practically negligible, and as the first system affords a quicker method of checking the summations it is the one generally adopted. The distribution of the solar hourly heights was at first accom- plished by making separate copies of the heights for each component, using tables that had been prepared to show the assignment of each hour. The making of these separate copies involves much labor which has since been eliminated by various devices. The U.S. Coast and Geodetic Survey has adopted a system of stencils devised by L. P. Shidy, which will be described in the next section. The British have been using a set of movable strips devised by Sir George Darwin. These are made of xylonite, an artificial ivory, and each strip is 9 inches long and one-fifth of an inch wide and is divided by black lines into 24 equal spaces to provide for the 24 hourly heights of the day. Theset used by Darwin consists of 74 strips on which are to be written the hourly heights for 74 consecutive days. By the aid of printed forms these ::trips are arranged so that the heights correspond- ing to any particul:r component hour which are to be summed to- gether will be found in a vertical column. By rearranging the strips’ summations can be made successively for the different components. After the summation of the first 74 days of the series for all of the com- ponents has been completed the strips are cleaned off with a damp cloth and entries made for the next 74 days of the series, and these are then summed for all the components. These operations are repeated until the entire series has been summed. A more detailed descrip- tion of this apparatus will be found in Scientific Papers by Sir George Darwin, volume 1, pages 216 to 220. 66 g@ U. S. COAST AND GEODETIC SURVEY. Another device to accomplish the distribution of the hourly heights is a set of tracing-paper sheets designed by Doctor Borgen, of Ger- many. These sheets are prepared with lines so arranged that when the sheets are laid on the hourly heights that have been copied in a standard form the heights which are to be grouped under any particu- lar component hour will appear between a pair of les. A separate set of sheets 1s necessary for each component. In principle and in use these sheets are essentially the same as the stencils of the Coast and Geodetic Survey. 24, STENCILS. A system of stencils was devised and prepared by L. P. Shidy, of the U. S. Coast and Geodetic Survey, early in the year 1885, for the purpose of effecting the distribution of tabulated solar hourly heights according to the component hours to be represented by the sums.* Since that time these stencils have resulted in a very great saving of labor. For the use of the stencils it is necessary that a standard form be used for the original tabulations of the observed hourly heights (fig. 22). The standard form adopted by the Coast and Geodetic Survey is a sheet 8 by 104 inches, with spaces arranged for the tabulation of the 24 hourly heights of each day in a vertical column, with 7 days of record on each page. The hours of the day are numbered con- secutively from 0° at midnight to 23" at 11 p.m. Each day is indi- cated by its calendar date and also by a serial number commencing with 1 as the first day of series. The stencils (fig. 23) are prepared from the same standard forms, with days numbered serially to cor- respond to the serial numbers of the tabulations. They are thus applicable to any series of observations without regard to the calendar dates. A separate set of stencils is required for each component for which sums are to be obtained. For convenience in construction each set of stencils is prepared with two stencil sheets for each page of tabulated heights, one sheet taking account of the odd component hours and the other sheet of the even component hours. To provide for the summation of series up to 369 days in length, each set con- sists of 106 stencils for use on 53 pages of tabulations. When a shorter series is summed, only a portion of the stencils need be used. The openings in the stencils are numbered according to the com- ponent hours that correspond most closely with the times of the height values that show through the openings when the stencil is applied to the sheet of tabulations. Openings applying to the same component hour are connected by ruled lines which clearly indicate to the eye the heights which are to be summed together. These stencils are adapted to tabulations made in any kind of time, either local or standard, civil or astronomical, provided the time is uniform throughout the series of observations. In the tidal analysis made by the British authorities the records have generally been referred to astronomical time with the day beginning at noon; but for convenience the tabulations made by the Coast and Geodetic Survey generally conform to the mean civil time ordinarily used at the place of observations. The series to be reduced must, however, commence with the zero hour of the day. If the actual series of observations commences at any other time of day, the heights for * Report of U. S. Coast and Geodetic Survey, 1893, Vol. I, p. 108. HARMONIC ANALYSIS AND PREDICTION OF TYDES. 67 that day may be rejected and the following day adopted as the first: day of series, or the heights for the earlier hours on the original first day may be estimated. The zero solar hour of the first day of the series is also adopted as the zero component hour of the first com- ponent day for each component. Successive solar hours will fall either earlier or later than the corresponding component hours ac- cording to whether the component day is longer or shorter than the solar day. For the construction of the stencils it is necessary to calculate the component hour that most nearly coincides with each solar hour of the series. Let a=speed or rate of change in argument of component sought in degrees per solar hour. »=number of component periods in component day; 1 for diurnal tides, 2 for semidiurnal tides, etc. 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 component hour reckoned from 0‘at beginning of each component day. chs =number of component hour reckoned from 0 at beginning of series. Then 1 component period = = solar hours. (258) 1 component day = solar hours. (259) t 15p component hour = sa solar hours. (260): 1 solar hour mies component hours. (261) Therefore, a a (chs) sina = Tape (dos) —1} + (sh)] (262) The above formula gives the component 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 it is desired to obtain the integral component 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 component 68 U. S. COAST AND GEODETIC SURVEY. hour of the component day (ch) required for the construction of the stencils may be obtained by rejecting multiples of 24 from the (chs). In the application of the above formula it will be found that the integral component 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 another group of solar hours, etc. This fact is an aid in the prepara- tion of a table of component hours corresponding to the solar hours of the series, as it renders it unnecessary to make an independent calculation for each hour. Instead of using the above form ila for each value the times when the difference between the solar and com- ponent hours changes may be determined. The application of the CUferences to the solar hours will then give the desired componeat hours. Formula (262) is true for any value of (shs), whether integral or fractional. It represents the component 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 ths beginning of the series as the zero hour. The difference between the component and the sole pe of any instant may therefore be expressed by the following formula: one Ha@ ot 4 _a~ lop Difference =; ap (shs) = [5p (shs) (263) If the component day is shorter than the solar day, the speed a will be greater than 15p, and the component hour as reckoned from the beginning of the series will be greater than the solar hour of the same instant. If the component day is longer than the solar day the component 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 component 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 component 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 (263) equals (d—0.5). Substituting this in the formula, we may obtain Lop 2 ap (shs) = Tee (264) a ~ in which (shs) represents the solar time when the integral difference between the component and solar time will change by one hour from (d—1) to d. By substituting successively the integers 1, 2, 3, etc., for d in the formula (264) the time of each change throughout the series may be obtained. The value of (shs) thus obtained will 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 HARMONIC ANALYS{S AND PREDICTION OF TIDES. 69 part of (shs) but need note only the integral hours between which this value falls. If, however, the second system of distribution should be desired, . 1t should be noted whether the fractional part of (sks) is greater or less than 0.5 hour. With a component day shorter than the solar day and the differences of formula (263) increasing positively, the application of the differences to the consecutive solar hours will result in the jumping or omission of a component hour at each change of difference. Under the second system of distribution each ’ component 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 component 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 (264) 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 difference will be applied to the integral solar hour following the change. With a component day longer than the solar day and the differ- ences of formula (263) increasing negatively, the application of the differences to the consecutive solar hours will result in two solar hours being assigned to the same component 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. Table 31, computed from formula (264), 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 component hour of the component day rather than with the component hour of the series, and these differences may be applied directly to the solar hours of the day. For convenience eihintlent 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. The tabulated solar hour is the integer hour that immediately follows the value for the (shs) in formula (264). 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 component hours, or will be rejected altogether according to whether the component day is shorter or longer than the solar day. If the tabular hou 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 components with commensurable periods 70 U. S&S. COAST AND GEODETIC SURVEY. will have the same tabular values, and no distinction is made in the construction of the stencils. Thus, stencils for component M serve not only for component M, but also for M,, M,, M,, ete. For the construction of a set of stencils the standard forms designed for the tabulation of the hourly heights may be used. A preliminary set of such forms is prepared with the days of series entered consecutively, beginning with 1, and with each hourly height space designated by the number of the component hour to which the height is to be assigned. The component hours are readily derived by the application of the differences given in Table 31. Each difference applies to a group of solar hours, the first hour of each group being indicated by the table. Under the usual system of distribution each hourly space will be represented by a single component hour number. ter the preliminary set of forms has been filled out as indicated the odd and even competent hours on each page will be transferred to separate sheets of the form and the spaces marked cut out. In the Coast and Geodetic Survey this cutting is done by a machine with a. punch operated by a small hand lever. The openings corresponding to the same component hour are as far as practicable connected by ruled lines, which are numbered to accord with the component hours represented. Black ruling with red numbering is usually adopted. The use of the red numbers to indicate the component hours has the advantage that it emphasizes the distinction between these num- bers and the figures representing the hourly heights which are to be summed. Figure 23 illustrates one of the stencils used for the sum- mations for component M. In using the stencils they are placed one at a time on the forms containing the tabulated heights of the observed tides, and all the heights on a page corresponding to each component hour are summed separately, the grouping of the heights being indicated by the ruling on the stencil. For component S no stencils are required, since the component hours are identical with the solar hours in accordance with which the observed hourly heights have been tabulated. For components like K, P, R, and T, whose speeds differ little from the speed of component S, the lines joining the openings in the stencils will frequently become horizontal. Since the sum of the values in such a horizontal line will have previously been obtained and entered in the margin of the form, the resuming will be saved by having a corresponding opening in the margin of the stencil which will expose this sum. 25. SECONDARY STENCILS. After the sums for certain principal components have been obtained by the stencils described in the preceding section, which for con- venience will be called the primary stencils, the summations for other components may be abbreviated by the use of secondary sten- cils which are designed to regroup the hourly page sums already ob- tained for one component into new combinations conforming to the periods of other components. 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 HARMONIC ANALYSIS AND PREDICTION OF TIDES. ww 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 six months in length. In the primary summations there are obtained 24 sums for each page of tabulations, representing the 24 component hours of a com- ponent day. In general each sum will include 7 hourly heights, and the average interval between the first and last heights will be 6 component 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 component days. Let the component for which summations have been made by use of the primary stencils be designated as component A and the com- ponent which is to be obtained by use of the secondary stencils as component B. For convenience let it be first assumed that the heights included in the sums for component A refer to the exact component A hours. This assumption is true for component S but only approximately true for the other components. It is now pro- posed to assign each hourly page sum obtained for component A to the integral component B hour with which it most nearly coincides. Component A and component 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 hour on the fourth day, or 83.5 solar hours from the beginning of the page of record. | Let a and 6 represent the hourly speeds of the components A and B, respectively, and p and p, their respective subscripts, and let n equal the number of the page of tabulation under consideration, beginning with number one as the first page. The middle of page n will then be [168 (n—1) +83.5] or (168.n— 84.5) solar hours (265) from the beginning of the series. Since one solar hour equals a/15p component A hours (formula 261), the middle of page n will also correspond to (168 n—84.5) Be component A hours (266) from the beginning of the series. j As there are 24 component hours in each component day, the middle component A day of each page will commence 12 component A hours earlier than the time represented by the middle of the page, or at [(168 n— 84.5) a 12] component A hours (267) from the beginning of the series. be) U. S. COAST AND GEODETIC SURVEY. The 24 integral component A hours of the middle component day of the page will therefore be the integral component 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 component hour according to its order in the middle component day of each page. For each page m will haye successive values from 1 to 24. The integral component A hours falling within the middle component day of each page of tabulations will then be represented by the general formula Gy ae from the beginning of the series. The relation of the lengths of the component A and component B hours is given by the formula [1168 n— 84.5) —12—f+m] component A hours (268) ue component B hours. (269) P1@ 1 component A hour= The component B& hour corresponding to the integral component A hour of formula (268) is therefore [(168 n— 84.5) fae 12—f+m] = component Bhours (270) from the beginning of the series. The last formula will, in general, represent a mixed number. The integral component B hour to which the sum for the component 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. The assignment of the hourly page sums for component A hours to. the component B hours may now be represented as follows, multiples of 24 hours being rejected: [(168 n—84.5) isp -12-F+ m—m‘multiple of 24] component A aoy | i sum to be assigned to a [{(168 »—84.5) 7° —12—f+ my Pr +g—multiple of 24] compo- nent B hour. (272) The difference between the component A hour and the component B hour to which the A hour sum 1s to be assigned is b x [{ (168 n — 84.5) ree 12—f+ ma 1}+g—multiple of 24] (273) By means of the above formula Table 33 has been prepared, giving the differences to be applied to the component A hours of each page to obtain the component B hours with which they most nearly coincide. For the construction of secondary stencils the forms designated for the compilation of the stencil sums from the primary summations HARMONIC ANALYSIS AND PREDICTION OF TIDES. Le 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 each alternate line in the form for stencil sums is left vacant. As with the 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 component B hours as derived by differences from Table 33 applied to the compo- nent A hours. The odd and even component B hours are then trans- ferred to separate forms and the spaces indicated cut out. The open- ings corresponding to the same component & hour are connected with ruled lines and numbered to accord with the component hour repre- sented. The page numbering corresponding to the page numbering on the compiled primary sums and referring to the pages of the original tabulated hourly heights is to be entered in the column pro- vided near the left margin of the stencil. In using the stencils each sheet is to be applied to the page of com- piled 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 component B hour are added and the results brought together in a stencil sum form, where the totals and means are obtained. 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 component A primary sums the number of hourly heights included in each primary sum is entered in the space corres— poe to that used for such primary sum. ‘The secondary stencils. or component B are then applied and the sums of the numbers obtained and compiled in the same manner as that in which the com- ponent B height sums are obtained. The divisors having been once. obtained are applicable for all series of the same length. - In the analysis the means obtained by use of the secondary stencils. may be treated as though obtained directly by the primary summa- tions 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 components A and B. The smaller the difference in. the speeds the closer will be the agreement. To determine the extreme difference in the time of an individual. hourly height and of the component 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 compo-. nent A hours, and that on the middle day of the page of tabulated. hourly. heights one of the integral component B hours coincides. exactly with a component A hour. At the corresponding component A hour, one component A day later, the component B hour will have. wncreased by 24 a component B hours. Rejecting a multiple of 1 24 hours, this becomes 24 (27- 1), so that at the end of one compo-- 1 74 U. S. COAST AND GEODETIC SURVEY. nent A day after the coincidence of integral hours of components A and B the component, A hourly height will differ in time from the integral component B hour to which it is to be assigned by 24 (2. ) component B hours. At the end of the third component 1 A day this difference becomes 72 ( vi 1) component B hours. The 1 same difference with opposite sign will apply to the third component day before the middle day of the page. Now, taking account of the fact that the component 6 hour on the middle day of the page may differ by an amount as great 0.5 of a component B hour from the integral component A hour, and that the integral component A hour may differ as much as 0.5 of a component A, or 0.5 pb/p,a of a compo- nent 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 component 6 hour with which this height is grouped by the secondary stencils may be represented by the formula. ake [7 = 1)+0.5 (Pi+ 1) component 6 hours. (274 1 1 / The differences may be either positive or negative, and in a lon series it may reasonably be expected that the number of positive cad negative values will be approximately equal. 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 components for which it is proposed to use the secondary stencils, the extreme differences as indicated are obtained. The differences are expressed in component 8 hours and also in com- ponent B degrees. It will be noted that one component hour is equivalent to a change of 15° in the phase of a diurnal component, 30° in the phase of a semidiurnal component, etc. ontpPOnent A yee ee ee nea a= J s i WOmMpPonent Bryan sees eee (ox@) 2SM Ky Ke Re Ts Py Difference in hours..........-.- 3. 58 1.36 1.20 1.20 1.10 | 1.10 1. 20 Difference in degrees.-..-....-.- 54 41 18 33 33 1 Component Ag. jhe esos eee cee L 2MK Componente8 tas. 15. 34-26 Ss) 229. 45g U2 MS Aa MEK MN 9 Na Difference,in hours: 7i2-cceet Phe. 1.09 1.18 1. 43 1.24 1. 26 1.45 Ditference:in Gesrees:. on. on an ote Peele nes 65 35 64 74 38 44 a | CoraepopentAen ect, ate eke Ep Aor Re eg ea O CoMponentuBi xe ae ae eo eese renames eae cele cemioe 1) 2N re Q 2Q Difference imshours =e seb eee sees Soe eneisaeeretic 1.21 1. 02 3. 42 3.79 6. 58 IDifierence mmiGepreess seem aroseeses cece cscehie cece 36 31 | 51 57 99 HARMONIC ANALYSIS AND PREDICTION OF TIDES. 75 In the ordinary primary summation the extreme difference between the time of the observation of a solar hourly height and the intregal component hour to wnich it is assigned is one-half of a component hour and, represented by component degrees, it is 7.5° for diurnal, 15° for semidiurnal, 22.5° for terdiurnal, 30° for quarter diurnal, 45° for sixth-diurnal, and 60° for eighth-diurnal components. 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 component 2Q ‘when based upon the primary summations for O. Thisis a small and unimportant component, and heretofore no analysis has been made for it, the value of its harmonic constants being inferred from those of component 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 component which will be more satisfactory than the inferred constants. Although the general use of secondary stencils for series of obser- vations less than six months in length is not at present recommended, it is possible that future tests may indicate that these stencils may be used to advantage with shorter series. 26. THE FOURIER SERIES. 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 Batre, cos 0+ CO, cos 20+ CO, cos 30 + See te (275) +8, sin +8, sin 20+S8, sin 30+ -- : __ It can be shown that by taking a sufficient number of terms the Fourier series may be made to represent any periodic function of 8. This series may be written also in the following form: h=H,+A, cos (6+a,) +A, cos (26+ a,) +A, cos (30+a,) +--+» (276) an which Sm Orn m being the subscript of any term. Ap —[Cy?+S,2]) and om= —tan— From the summations for any component 24 component hourly Means are obtained, these means being the approximate heights of the component tide at given intervals of time. These mean com- ak hourly heights, together with the intermediate heights, may e represented by the Fourier series, in which H,=mean value of the function corresponding to the height of mean sea level above the adopted datum. @=an angle that changes uniformly with time and completes a cycle of 360° in one component day. The values of 6 corresponding to the 24 hourly means will be 0°, 15°, 30°, - - - - 330°, and 345°. Formula (275), or its equivalent (276), is the equation of a curve with the values of 6 as the abscisse and the corresponding values of has the ordinates. If the 24 component hourly means are plotted as ordinates corresponding to the values of 0°, 15°, 30°, - - - - for @, it is 72934—24}—_6 76 U. S. COAST AND GEODETIC SURVEY. possible to find values for H,, Cn, and Sn, which when substituted in (276) will give the equation of a curve that will pass exactly through each of the 24 points representing the component means. 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 / pertaining to the beginning of each of those parts is known. Let wu equal the interval between these ordi- nates, then N W=27, or 360° (277) Let the given ordinates be hy, h,, h,-- - - hm corresponding to the abscissae 0, u, 2u--- + (n—1) u, respectively. 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+ C, cos 20+ .... CG, coské@ eas 6+ 8, sin te Le) Sy sintblg (278) =H,+> Cn cos m 0+ 35) Sp sin m0 m=1 m=1 : : eis nN. : m=-1, : in which the limit k=5 if n is an even number, or k=—>— if n is an she n ee tee n— Lae, odd number; and the limit /=5—1 if n is even, or —5 if n is odd. By substituting successively the coordinates of the n given points in (278) we may obtain n equations of the form m=k m=1 ha=H,+ SS Cn cos maut+ 3S} Sy sin mau (279) 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 uantities may be obtained, including H, and the (n—1) values for ‘in and Sp. It will be noted that the sum of the limits & and J of (278) or (279) equals (n— 1) for both even and odd values of n The reason for these limits is as follows: A continued series 2 Cy cos m a u may be written C, cosau+C,cos2au+--+-+G,cosnau + Cay CoS (n+ 1) aU+ Cine.) Cos (N+2)auU+-++++C,cos2na t + C (n+) cos (2n +1) @u+ Coons) cos (2N4+2)a4U+-+-- +Cjy, cos3nau St ks Varian yal Lal as eed Bang) asa od 2 lol ae rrr Since n u =2z and a is an integer, the above may be written [C,+ Cary + Cont +++ +] cosau aCe Gane) Ctane2) + -+++]cos2au tf Or AUOE Ss Gish eae. BA eos Nn =A) + cos 2a u SLC a: Ce ohne ae NAD ; + Om) + Gon-1) + Coney t + + + > Jcoskau (282) The first term of the above is a constant which will be included with the H, in the solution of (279). From an examination of (282) it is evident that the cosine terms will be completely represented when =i : : k=5, or a according to whether n is even or odd. 2 Similarly, the continued series S$} S,, sin m au may be written US = Sone Sin de ua) dep], Sin/O +18, +841 + S(onty + a , =p O@a1) On) 2 OGr aay a JIS: +[8, + S(rs2) + S(ont2) + ote eye f Sa) Sn) Signy - > | SM 2a u 5P Si =pisGean an (Gnas ele Sc — S@—1) — Sogn) — S(gn—1) — Bak peEeS ] sin l au (283) The first term in the above equals zero. The remaining terms will take complete account of the series S38, sin m a u, if 1=5- 1 when nm— | n is even, or ao when n is odd. From the foregoing it is evident that the limit of m will not exceed a 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 Paeltaal) ts singénmu =. oe se del eli —1 284 = sin (a mu+a) saa ae (n—1) mu+a] (284) ing sin¢nmu Pete Sela eae — 285 >> cos (a m u+a) Si tnd cos [4 (n—1) m uta] (285) Bt) ; _, sin $n (p—m) u cos 3 (n=1) (Pm) u = snapu snamu=3% sin } (p—m) u sin $n (p+m) wu cos 4 (n—1) (ptm) u (286) ae sin 4 (p+m) u p= (lee) sin $n (p—m) u cos 4(n —1) (p—m) u = cosa pucosamu=y sin 4 (pm) sin $n (p+m) ucos 4 (n—1)(p+m) u oer +3 sin 4 (p+m) wu Cee) pao) , sin $n (p—m) wu sin 4 (n—1) (p—m) & = ees Te ee enamine y a 6 10 Og sin $ (p—m) wu fa SE oil pa Ae SI (p+m) u (288) sin 4 (p+m) wu 78 U. S. COAST AND GEODETIC SURVEY. If we let a=0 and v=, or n u=2z, then formulas (284) to (288) may be written as follows: : : m sin m z sin (m nar esp zx) a=(n—1) : SS sm amd — a (289) rae si 7 n : m > cosam sacha Rae ae a (290) ie sin — x é n pod ali sin (p—m) =z cos | (pm) zh” “| >> sm ap wsin a mu=s => = aoe sin (pom) m) i n sin (p—m) z cos | +m) z—bam r| oe _ ptm sin 7 n - payee sin (p—m) z cos | (om) z—P—™ r| > cosa pucosamu=s my ON sin ie ra nN sin (p+™m) z cos | +m) z—E=m “| +4 ay (292) sin ? a & athe) sin (p—m) z sin | em) —— x | SS) sma pu cos am u—s a=0 ramen Use £0) sin 1 n sin (p+m) =z sin | +m) 7 btm =| +4 (293) : +m sin 2™ "a m If p and m are unequal integers and neither exceeds the above (289) to (293) become equal to zero. Thus, a=0 a=(n—1) > cosamu=0 | a=0 | a=(n—1) 3 ; > smapusnamu=0 (294) a=0 a=(n—1) > cosapucosamu=0 | a=0 | a=(n—1) _ 2D) sina pucosamu=0 | a=o0 a=(n—1) Ff > smnamu=0 | ! HARMONIC ANALYSIS AND PREDICTION OF TIDES. 79. If p and m are equal integers and do not exceed m formulas (291), sin (p—m)z (292), and (293) will contain the indeterminate quantity i n => and also when p and m each equal ah the indeterminate quantity sin (p+m)z_0. sin (or my) 1 t nN Evaluating these quantities we have sin (p—m)zx zCos (p—m)z ; — 0 =m =n ‘ (295) sin 2 fea 7, 098 = and sin (p-+m)z _m cos (pt+m)z alge | ae Gog Pet 2 | aa ee”) n (ptm=n n. n (p+m) =n In (296) it will be noted that when the integers p and m each equal > nm must be an even number, and therefore cos nz is positive, while Cos z is negative. Assuming the condition that p and m are equal integers, each less than 5, we have by substituting (295) in (291), (292), and (293), ! Sees I 24> (n—1) 5 ~ S smapusnamu= 3S simamu=in (297) a=0 a=0 a=(n—1) a=(n—1) > cosapucossamu= 3S) covamu=jn (298) a=0O a=o a=(n—1) a=(n—1) | > smapucosamu= >) snamucosamu=0 (299) a=0 ‘ a=0 Assuming the condition that p and m are each equal to > we have by substituting (295) and (296) in (291), (292), and (293), | a=(n—1) : > sin? am u=4n+4 n cos r=0 (300) a=0 a=(n—1) > cos’ amu=sn—$n cosz=n (301) a=0 a=(n—1) > smamucosamu =0 (302) a=o 80 U. S. COAST AND GEODETIC SURVEY. Returning now to the solution of (279), by substituting the suc- cessive values of a from 0 to (n—1), we have h,=H,+ C, cos0+C,cosO+ .... +, cos 0 +S,sin 0+S, sin 0+ .... +8) sin 0 h,=H,+ C, cosu+C, cos 2u+ .... +0, cos ku +8, sin w+8, sin 2u+ .... +8, sin lu | h,=H,+C, cos 2u+ C, cos 4u+ .... +, cos 2ku +S, sin 2u+S, sin 4u+ .... +8) sin 2lu (303) ha) = Hot ©, cos (n—1)u+ 0, cos 2(n—1)u+ .... | + OQ, cos (n—1)ku | +S, sin (n—1)u+S8S, sin 2(n—1)u+ .... +8, sin (n—1)lu To obtain value of H,, add above equations a=(n—1) > ha=n Ieh, a=o0 a=(n—1) a=(n—1) a=(n—1) + C, >> cosaut+C, = COS 2caiae ee ee eae = cosaku a= ae a=(n—1) ao . . +8, = snau+S, >) sm2aut+.... +8; SS snalu a=0 a=o a=(n—1) m=1 a=(n—1) =n H, + 3 On = cosamut 2 Sm = sInamu (304) ene 1) a=(n—1) From (294), >) cosa muand 3) sin a mu each equals zero, a=o a= io since neither & nor J, the maximum values of m exceeds = 2 Therefore a=(n—1) i Es (305) a=0 and i a=(n—1) He = Se hs (306) NM a=o To obtain the value of any coefficient C, such as C,, multiply each equation of (303) by cosa pu. Then he cos 0= H, cos 0 +0, cos 0+ C, cas OFF) ..¢) 0+ Gy (eos: 0 +S, sin 0+S, sin 0+ .... +8); sin 0 ” cos p u= H, cos p u +¢, cos u cos pu+ C; cos 2ucosput...+, coskucos pu +S,sin u cos pu+S8, sin 2ucosput+...+8, sinlu cospu h, cos 2p w= H, cos 2p u + C, cos 2u cos 2p u+ C, cos 4u cos 2p ut... +06, cos 2k wu cos 2p wu +8, sin 2u cos 2p u+S, sin 4ucos2put+.... +S; sin 21 wu cos 2p wu —_e eee ew oe ewe ee ew ee ee eK ee we we wm we A = ew we fe mw we Be new we Bw ew ww ee ew ee ewe ee HARMONIC ANALYSIS AND PREDICTION OF TIDES. 81 hay cos (n— 1) p w=, cos (n—1) pu + C, cos (n— 1) wcos (n—1) put C, cos 2 (n—1)ucos(n—1) put.... +(C, cos (n—1) k u cos (n—1) pu +8,sin (n— 1) ucos (n—1) pu+S,sin2(n—1)ucos(n—1) put.... +8, sin (n—1) lu cos (n—1) pu (307) Summing the above equations a=(n—1) a=(n—1) Ph cos p u—H,) Ss (cosa p u a=0 a=0 a=(n—1) a=(n—1) | +0, S cosaucosaput+sS, D> sinaucosapu a=0 a=o0 ks a=(n—1) ~ a=(n—1) _ +C, > cos2aucosapu+S, 3) sin 2aucosapu a=o a=oO I a=(n—1) BS) +, SS cosakucosaput+sS; D> snalucosap a=0 a=0 a=(n—1) m=k a=(n—1) =Tf1p5 > cosa put boy Cn ee coSamUCOSa PU 2=0 m=1 a=0 m=1 a=(n—1) | + >S,°S> snamu cosa pt (308) = 2=0 Examining the limits of (808), it will be noted by a Telerones to _ page 77 that k, the maximum value of m for the C terms is 5 when n wat when n is odd; also, that J has a value of 57 1 when 2 n is even and a when nis odd. The limits of p, which isa partic- is even and ular value of m, will, of course, be the same as those of m. a=(n—1) By (294) the quantity >) cos a p u becomes zero for all the values of p, and the quantity ape cOSs @ MU COS a p u becomes zero for all values of m and p ifibne ston p equals m. By (294), (299), and (302) the quantity vaste sin am u cos a p u becomes zero for all values of m and p. tea Formula (308) may therefore be reduced to the form a=(n—1) a=(n—1) SS (ci Cos opal — Cons \ cas? apts (309) a=0 a=0o ; For any value of p less than a a=(n—1) > cos’ apu=sn (298) a=o 82 U. S. COAST AND GEODETIC SURVEY. but when D=5) this quantity becomes equal to n (301). Therefore for all values of p less than > 9 a=(n—1) Cp=— h, cosa pu (3 10) hh a=o : n but when p is exactly 3 | 1 a=(n—1) Co=- > hacosapu (311) TM a=o f Since in tidal work p is always taken less than = we are not espe- cially concerned with the latter formula. : To obtain, the value of any coefficient S, such as S,, multiply each. equation of (303) by sin a pw. Sum the resulting equations and. obtain a=(n—1) a=(n—1) (ceo. ha sinapy,.—Ho “> sin a pu =0 a= ‘ m=k Pees ‘ +>) Ga: 3. cos a mu sin & pu m=1 a=o0 ; m=1 a—(o= 1) : : au. mn Ss) SMamusnap we (312) m=! a=o ce Oa) Wa By (294), (299), and (302) the quantities >> sm @ p wu and a=o a=(n—1) > cosamu sina p u are zero for all the values of m and p; a=0 . a=(n—1) and > sinamu sin a p wu becomes zero for all the values of m and p except when m and p are equal. In this case the limit of J for a=(n—1) m and p is less than 5 and by (297), the quantity > sin? a pw a=) ai: Therefore, formula (312) reduces to the form a=(n—1) ; 4 S hzasnapu=$nS8y (313) a=o and Qa=(1) Sp=7 > Asinapu (314) HARMONIC ANALYSIS AND PREDICTION. OF TIDES. 83 By substituting (306), (310), (311), and (314) in (278), the follow- ing equation of a curve, which will pass through the n given points, will be obtained 1 a=(n—1) 2 a=(n—1) h== DS Agere ss hy cos aw |cos % a=o NM aso z 2)A-@-) j é +1/— D>) A,sin au| sin 6 N a=o 9 a=(n—1) sei —!<0BSD hy cos 2.0 | cos 2 8 1 a=o - DP Bre Be Ue tiel: +/—-, ha sin 2a | sin 2 0 tN a=o 9* a=(n—1) +15 > hs cos kau | cos k 6 a=0 y) a=(n—1) 4 a +{/— 3 hasinla u| sin 1 6 (315) 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 7 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 components having periods incommensurable with that of the component sought. It is desirable to include only the terms of the series which represent the true periodic elements of the component. With series of observations of sufficient length, the coefficient of the other terms, if sought, will be found to approximate to Zero. By a reference to formula (100), page 35, it will be noted that the short-period components as derived from the equilibrium theory are, in general, either diurnal or semidiurnal. If the period of 6 in formula (278), page 76, is taken to correspond to the component day, the diurnal components will be represented by the terms with coefficient C, and S,, and the semidiurnal components by the terms with coefficients C, and S,. For the long-period components, the period of 6 may be taken to correspond to the component month or to the component year, in which case the coefficients (, and S, will refer to the monthly or annual components and the coefficients C, and S, to the semimonthly or semiannual components. For most of the components the coefficients C,, S,, C,, and S, will be the only ones required, but for the tides depending upon the fourth power of the moon’s parallax (sec. 15) for the overtides (sec. 18), and the compound tides (sec. 19) , othen coefficients will be required. Terms. beyond those with coefficients C, and S,, for the overtides of the prin- cipal lunar component, are not generally used in tidal work. en it is known that certain periodic elements exist in a component, tide and that the mean ordinates obtained from observations include . F tad a 2 * Tf nis even and oe this fraction is = instead of ae 84 U. S. COAST AND GEODETIC SURVEY. 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 (279), that the most probable values of the constant H, and the coefficient C, and S, are the same as those given by formulas (306), (810), and (314), respectively. 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 (306) is unnecessary except for checking purposes. Formulas (310) and (314) are used for obtaining the most probable values of the coefficients C, and S, from the component Heunie means obtained from the summations. When 24 hourly means are used n = 24, and u = 15°, and the formulas may be written 1 a=23 Co== DD ha cos 15a p (316) 12 3=0 J 2=23 i Sp=75 h,sin 15 ap (317) a=0 in which the angles are expressed in degrees. If only 12 means are used, the formulas become 1 a=11 P=6 > hz cos 30 a p (318) a=o 1 a=11 : Sp== DD Aa sin 30a p (319) a=0 The upper part of Form 194 (fig. 29) is designed for the compu- tation of the coefficients C, and S, in accordance with formulas (316) and (317) to take account of the 24 component hourly means. - It is now desired to express each component in the form y= A cos (p 6+a) (320) or using a more specialized notation by y=A cos (p 6—5) (321) By trigonometry A cos (p 6—§) =A cos ¢ cos p6+A sin ¢ sin p 6 (322) = (, cos p6+S, sin p 0 in which -G=Acosg and S,=Asin¢ (323) Therefore, S t phar) an ¢ C. (324) and | Cy _ Sp ee 2 2 mie Rene V0? +S8p (325) By substituting the values obtained for O, and S, by formulas (316) and (317) in formulas (324) and (325), the corresponding values of A and ¢ for formula (321) may be obtained. In (321) we now have an harmonic expression, which, with its constants A and ¢ determined by the methods already described, is HARMONIC ANALYSIS AND PREDICTION OF TIDES. 85 an approximate representation of one of the tidal components sought. These constants must, however, be modified and reduced in order to be adapted to practical use. 27. AUGMENTING FACTORS. In the usual summations with the primary stencils for all the short period components, except component 8, the hourly ordinates which are summed in any single group are scattered more or less uniformly over a period from one-half of a component hour before to one-half of a component hour after the exact component 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 component hour, as with component 5, and the amplitude obtained will be less than the true amplitude of the component. The factor necessary to take account of this fact is called the augmenting factor. Let any component be represented by the curve y=A cos (at+a) (326) in which A =the true amplitude of the component a=the speed of the component (degrees per solar hours) t=variable time (expressed in solar hours) a=any constant. The mean value of y for a group of consecutive ordinates from 7/2 hours before to 7/2 hours after any given time ¢, rt being the number of solar hours covered by the group, is t+7/2 : al cs (at + a) pute! al sin (ata) | t—7r, T AT t+7/2 t—7/2 ar sin (at+a+F)- sin (at+a-Z SG: SOU. ar . an CoS (at+ a) sin ge Sits: A cos (at+ a) (327) Since the true value of y at any time f, is equal to A cos (at+a) by (826), it is evident that the relation of this true value to the mean value (327) for the group 7 hours in length is A cos (at+a) he ona 360 (328). . ar ot . at ae SID A cos (at+a) 360 sin ) The quantity = the augmenting factor which is to be applied 360 sin | 2 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 approxi- mately realized in the summation for any component, but the larger the series of observations the more nearly to the truth it approaches. 86 U.S. COAST AND GEODETIC SURVEY. 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 component hour before to one-half of a component hour after the hour to be represented. In this case r 15 equals one component hour, or = solar hours. Substituting this in (328), the Tp 24 sin OP 320) 2 which is the formula generally adopted and is the one upon which the augmenting factor of Form 194 is based. If the second system of distribution of the hourly heights as described on page 65 is adopted, 7 equals one solar hour and formula (328) becomes augmenting factor= menting factor =——” — augmenting factor= a (330) 30 sin 5 It will be noted that formula (329) depends upon the value of p and therefore will be the same for all short period components (S excepted) with like subscripts. Formula (330) depends upon the speed a of the component and will therefore be different for each component. When the secondary stencils (described in sec. 25) 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 com- ponent A with speed @ and that the secondary stencils have been used for component B with speed 6. Then let p and p' represent the subscripts of components A and B, respectively. The equation for component B may be written y=B cos (bt +8) (331) In the primary summation for component A, the group of ordi- nates included in a single sum covers a period of one component A hour or — solar hours. Expressed in time t, midway of this interval and ene the exact integral component A hour to which the group applied, the average value of the B ordinates included in such a group may be written 15p wR ona dt 15p 115 COs a oeOise 15pb\_.. — bee) | ae r5p5 2] 5 (t+6+ 5 oe) sin (t+ 19a DA a! P5pb = (7S sin se) B nos, ABtOB) =F, B cos (bt +8) (332) HARMONIC ANALYSIS AND PREDICTION. OF TIDES. S7 In which F,, for brevity, is substituted for the coefficient 24 a . 15pb ;, Sin cluded in the A grouping to the true & 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 com- - ponent §, this coefficient may be taken as unity since the original S sums refer to the exact S hour. When the secondary stencils are applied to the component A group sums, the groups applying to an exact component A hour at any time ¢ and represented by that time, will be distributed over an inter- 1 val of a component B hour, or — solar hours. and gives the relation of the average B ordinate in- For an integral component B hour at any time ¢ within the middle day represented by a seven-day page of original tabulations the limits 1 : 1 of this interval will be ¢ — SP) and (: ahs oF) For the same page of tabulations, letting t represent the same time in the middle day, the limits of the group interval for the day following the middle one, are 1 1 (14 PSP and (14 2024). If we let n= —3, —2, —1,0, a 2b a 2b +1, +2, +8, EeSpecianely, for the seven 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 5p" 360pn , 15p" (t+ : — 3p) and (t+ pt oe ) Formula (332) gives the mean value of the B ordinate for grouping of the A summations. The mean value of (332) obtained by com- bining the groups falling in any particular day of page of tabulations in the limits indicated above is . ms FB ee (bt dt 15p C 4300p _15p! £08 + B) a 2b 130 1 ; 3606 15p' 1 a 1 —sin (r++ 20 at y| 2a, tp? 360pn =(= sin SP) FB cos (t+ B+ a) a =F FB cos (v¢ melo sen * (333) 1 if we put F 3 = sin = for brevity. 88 U. S. COAST AND GEODETIC SURVEY. Formula (333) 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 1 FFB 'S cos ( t+ p+ OOP) a -1F RB E (bi 20h 'es (-3 swe) ia peta) wi (-3 a) + cos (bt+ 8) cos (- 2 ee) —sin (bp +8) sin (- 9 =eoie) +cos (bt+ 8) cos (- i Ne —sin (6¢+ 8) sin (- 1 ae) + cos (bt +6) cos 0—sin (6¢+ B) sin 0 360bp a + cos (6t+ 8) cos go0bp —sin (6t+ 8) sin a 360bp a +c0s (bt +8) cos (3 2°02) — sin (or+0) sin (3 OP) | =4+ FFB E +2 cos OUP 4.9 cos oe 2 cos 3 OOP | cos (bt + B) 360bp a +cos (bt+ 8) cos (2 —sin (b¢+ 8) sin (2 09 360bp egars nee i) a 2 =4+F F,B] 2 > 3600n —1 [cos (bt+ 8) SIN = a a ae =4 FFB “sin 1S05P cos (bt +8). (334) a Replacing the equivalents of F, and F, in (334), the average value of the B ordinate as obtained by the secondary summations may be written 1260bp Dig. 15bp\ (PANU tapi \ on aie oat 8 oc 1 sin —5 ) ( ee 180bp B cos (bt +8) (335) a Since the true ordinate of component 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. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 89 This augmenting factor may be written bp Tp" 7 sin sae 1 ohn he heer 24a sin 2un 24 sin ESP . 1260bp 2 2a Me sin. ————— a The first factor of the above is to be omitted if the primary sum- mations are for component S. It will be noted that the middle factor is the same as the augmenting factor that would be used if component B had been subjected to the primary summations. 28. REDUCTION OF EPOCHS AND AMPLITUDES TO MEAN VALUES. In equation (321), page 84, y= A cos (p 9—$£) the quantity (— ¢) is the phase of the component at the time 0 equals zero—that is, at the beginning of the series—and p @ is equal to ¢ at the time y is a maximum; that is, at the time of ae component high water. The value of ¢ will therefore depend upon the time of the beginning of the series, which is more or less arbitrary, and the ¢’s of any component determined from different series of observations are not directly comparable. Expressions for the theoretical phases, or arguments, of the principal lunar components are represented in formula (100), page 35, and a general expression for this argument as modified by a constant « for a particular locality is given by formula (101), page 39. The last formula is an equivalent of the angle of (321) and may be written (at+ Vi tu—k«)=p 0—¢ (337) In the above the variable angle p 6=" @nz—0-a)+8—a-9 |=0 (371) At the time of this maximum, when 2n7—a—é ——\ ae Th a ar ie | a t the phase of component A will equal (2n r—a—?) +a and the phase of component B will equal y (Qn7—a—6) +8 Let ¢=phase of component B—phase of component A at this time. Then "=" (on t—a-0) +8—a (372) Substituting the above in (371) — Aa sin 6+ Bb sin (¢—8) = — Aa sin 6+ Bb sin ¢ cos 6— Bb cos ¢ sin 9 — —(Aa+ Bb cos ¢) sin 6+ Bb sin ¢ cos 0=0 (373) Then _ Bbsing ten oe a + Bbcos@ (374) For the resultant amplitude at the time of this maximum substitute the values of t from (370), in (366), and we have y=A cos (2n7—6) + B cos E (2n 7-9-2) +2 a =A cos#@+B cos | Qnr—8-a) +$-a-9| =A cos 0+B cos (¢—8@) (375) =A cos 6+B8B cos ¢ cos 0+8 sin ¢ sin 0 =(A+B cos ¢) cos 6+ B sin ¢ sin 9 B sin @ A+B cos @ — /A?+ B?+2AB cos ¢ cos (« —tan! 96 U. S. COAST AND GEODETIC SURVEY. From (374) §=tan™ ee jen eh (376) A7tTB cos Bp cos ¢ In the special cases under consideration the ratio 5 is near unity, Bsin @ and the difference between @ and tan is therefore very A+B cos ¢ small, so that the cosine may be taken as unity. The resultant amplitude may therefore be expressed by 2 _/ A? + B? +2AB cos é=Aq/1 oe a cos ¢ (377) The true amplitude of the component sought being A, the resultant amplitude must be divided by the factor PE B y i tipet2 7 os ¢ (378) in order to correct for the influence of the disturbing component. The corrections for acceleration and amplitudes as indicated by formulas (374) and (378) may to advantage be applied to the con- stants for component K, for an approximate elimination of the effects of component P, and to the constants for S, for an approximate elimination of the effects of components K, and T,. By taking the relations of the theoretical coefficients for the ratios q 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 3, the following values may be obtained. B Aa a Bb ss HTC Oh Ba onli scr Nall Ueno peasy mie a 0.33086 | 3.03904 | —2h-+»’—180°. NTacE Of, Kafr So occ cc te | Ne RR PU et 0.27213 | 3.66469 | 2h—2p’”. UATE CTO LHD a OI Gas or ene er eee ON SE CA Rat Gs 0. 05881 | 17. 02813 | —h+7. Substituting the above in (344) and (378) we have Effect of P, on K, : sin (2h —v’ Acceleration = tan-! ) (379) 3.0390 — cos (24 —v’) Resultant amplitude = 0.813 /1.6767—cos (2h—v’) (880) Effect of K, on 8, i Me Sh? Acceleration = tan sin (2h —2y’") 3.6647 + cos (2h — 2n’’) (381) Resultant amplitude =0.738 /1.9734-+ cos (2h—27’”) (382) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 97 Effect of T, on S, —sin (h—p,) 17.0281+ cos (h— p,) (383) Acceleration = tan Resultant amplitude = 0.343 8.5318 + cos(h— p,) (384) The above formulas give the accelerations and resulting amplitudes for any individual high water. For the correction of the constants derived from a series covering many high waters it is necessary to take averages covering the period of observations. ‘ables 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 = To take account of the longitude of the moon’s node, the factor of reduction from section 12 should be introduced. If the mean coefficients are indicated by the subscript 0, formulas (376) and (378) may be written sin ¢ Acceleration = tan TYRE + cos iu Resultant amplitude= V 1+ Crane +24 cos (386) f(A) In the cases under consideration the ratio ¥(B) will not differ ereatly from unity, the ratio ae 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 (385). Acceleration = Th Laas a) (387) Biber Bese Also because G_ im these cases is small compared with unity, the following may be taken as the approximate sd of (386): Resulting amplitude=1 eT +(42) J+23 7 > COS p- 1| (388) To allow for the effects of the longitude of oe 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 components P,, S,, and T,, is unity for each. Therefore, for the effect of P, on K,, the ratio Tay —— = F(K,), and for the effect of K, upon S,, this ratio is ORG) Wor the effect of ay upon 8, the ratio is ‘unity. 98 U. S. COAST AND GEODETIC SURVEY. 30. ELIMINATION. Because of the limited length of a series of observations analyzed the amplitudes and epochs of the components as obtained by the processes described in the preceeding sections are only approximately freed from the effects of each other. The separation of two compo- nents from each other might be satisfactorily accomplished by haying the length of series equal to a multiple of the synodic period of the two components. To completely effect the separation of all the com- ponents 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 component. The length of such a series would be too great to be given practical consideration. In general, it is therefore desir- able to apply certain corrections to the constants as directly obtained from the analysis in order to eliminate the residual effects of the com- ponents upon each other. Let A be the designation of a component for which the true con- stants are sought and let B be the general designation for each of the other components in the tide, the effects of which are to be eliminated from component A. Let the original tide curve which has been analyzed be represented by the formula y=A cos (at+a)+2B cos (6t+8) (389) in which y =the height of the tide above mean sea level at any time t. t=time reckoned in mean solar hours from the beginning of the series as the origin. A=R (A) =true amplitude of the component A for the time covered by series of observations. B=R(B) =true amplitude of component B for the time coy- ered by series of observations. a= —{(A) =true initial phase of component A at beginning of series. 8 = —¢(B) =true initial phase of component B at beginning of © “series. a=speeds of component A. b=speed of component B. Formula (389) may be written y=A cos a cos at+2 B cos {(b—a)t+f}cos at —A sin asin at—zD B sin {(b—a)t+}sin at =[A cos a+ B cos{(b—a)t+}] cos at —[A sin a+2 B sin{(b—a)t+$}] sin at (390) The mean values of the coefficients of cos at and sin at of formula (390) correspond to the coefficients (, and S, of formulas (316) and (317) which are obtained from the summations for component A. Let At and at=the uneliminated amplitude and initial phase, respectively; of component A, as obtained directly from the analysis. The equation of the uneliminated component A tide may be written y = A! cos (at+ a!) = A! cos a! cos at—A!' sin a! sin at (391) HARMONIC ANALYSIS AND PREDICTION OF TIDES. 99 Comparing (390) and (391), it will be found that A’ cos a! =mean value of [A cosa+ Zz Bcos {(b—a)t+8}] (892) A‘ sin et =mean value of [A sina+z B sin {(b—a)t+B}] (893) Let +=length of series in mean solar hours. Then the mean value of B cos{(b— ~a)t +6 \within the limits t=0 and (=r, is 2 {B cos {(b—a)t-+p}ai="° 7 roar lee ied an) sammie 180 sin $(6—a)r eg aa The mean value of B sin{(b—a)t+8}within the same limits is B cos { {4(b—a)r+ 8} (394) + {B sin (G—at+p}dr= 229 5 = [eos (ba) + +8} —cos 8 _ 180 sin} (b—a)r Fe PLEIN: Substituting (394) and (395) in (392) and (393), and for brevity letting B sin {4(6—a)r+8} (395) 180 sin 4(6—a)r be EG aee 0 we have A' cos at =A cosa+z F, cos {4(b—-a)r +8} (397) A' sin at=A sina+z F, sin {4(6—a)r+8} (398) Transposing, A cos a= FF, sin {4(b-—a)7+B—a!} (403) Multiplying (399) and (400) by cos a! and sin a!, respectively, _A cos a! cos a= A! cos? a!—= Fy, cos {4(b—a)r +8} cos a! (404) A sin a! sin a =A! sin? a@'—= Ff, sin {4(b—a@)r +8} sin a! (405) Taking the sum of (404) and (405) A cos (at—a) =A!—> F, cos {$(b-a)7r+B—-a!} (406) Dividing (403) by (406) (407) y F, sin {4(6—a)r+8-—a!} z= Fy cos {4(b—a)r+B—a'} tan (a!—a) = aes 100 U. S. COAST AND GEODETIC SURVEY. From (406) _Al-z Ff, cos {4(6—a)r+B8—a!} ae cos (a!—a) i te) Substituting the value /, from (396) and the equivalents R'(A), R(A), R(B),—¢(A) —¢(A), and—¢(B) for A', A, B, a’, a, and B, re- spectively, we have by (407) and (408) tan [¢(A) —¢1(A)] ~ 180 sin $(b—a)r Yar OO (Pay De in +(6—a RA) — eu a: a)T R(B) sin {4(6-a)7-¢(B) + 2 (A)} + R(B) cos {4(6—a)r—5(B) + (A)} 4(b—a)r (409) 180 sin $(6—a)r : R(A) = (A)— 2 1 (6 Zaye R(B) cos {4(—a)r7—5(B) + (A)} cos [¢(A) — ¢(A)] (410) Formula (409) gives an expression for obtaining the difference to be apphed to the uneliminated (A) in order to obtain the true ¢(A), and formula (410) gives an expression for obtaining the true amplitude R(A). These formulas can not, however, be rigorously applied, because the true values of R(B) and ¢(B8) of the disturbing compo- nents 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 eliminated values for the disturbing components, 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. Form 245 (fig. 32) provides for the computations necessary in applying formulas (409) and (410). 1 aft In these formulas the factors represented by a 6 foe and the angles represented by 4(6—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 (409) and (410). An examination of formulas (409) and (410) will show that the disturbing effect of one component upon another will depend largely sin $(6—a)r_ 4(b—a)r not equal to a, this fraction and the disturbing effect it represents will upon the magnitude of the fraction Assuming that 6 is approach zero as the length of series tr approaches in value ay or HARMONIC ANALYSIS AND PREDICTION OF TIDES. 101 any multiple thereof, or, in other words, as 7 approaches in length any multiple of the synodic period of components 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 components the less will be their disturbing effects upon each other. For this reason the effects upon each other of the diurnal and semidiurnal components or of any components of different subscripts is usually considered as negligible, and in the application of formulas (409) and (410) only components with like subscripts are taken into account. The quantities #(B) and ¢(B) of formulas (409) and (410) refer to the true amplitudes and epochs of the disturbing components. 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 cover 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 components 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 components should be used. 3l. LONG-PERIOD COMPONENTS. The preceding discussions have been especially applicable to the reduction of the short-period components—those having a period of a component day or less. They are the components that determine 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 components that are usually treated in works on harmonic analysis—the lunar fortnightly Mf, the lunisolar synodic fortnightly MSf, ¢he 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 meteorological conditions, often have an appreciable effect upon the mean daily level of the water. To obtain the long-period components, methods similar to those adopted for the short-period components with certain modifications may be used. For the fortnightly and monthly components the component month may be divided into 24 equal parts, analogous to the 24 component hours of the day. Similarly, for the semiannual and annual components the component 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. Instead of distributing the individual hourly heights, as for the short-period components, a considerable amount of labor can be saved by using the daily sums of these heights. The mean of each 102 U. S. COAST AND GEODETIC SURVEY. 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 component month or component year is divided into 24 equal parts, the stants separating the groups may be numbered consecutively, like the component hours, from 0 to 23, with the 0 instant of the first groups 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. Letting a=the hourly speed of any component, in degrees. p=1 when applied to a monthly or an annual component, and p=2 when applied to afortnightly orasemiannualcomponent. - d= day of series. s=solar hour of day, Then 5 360 p 1 component period=—_— solar hours (411) and 1 component month="—? solar hours (412) also 60 1 component ee — solar hours (413) Dividing the component month or component year into 24 equal parts, the length of wae 15 1 component division=—* solar hours (414) Therefore, to express the time of any solar hour in units of the com- ponent divisions to which the solar hourly heights are to be assigned, the solar hour should be multiplied by the factor a/15p. Thus, a Component division = 15p (solar hour of series) a = EE [24(d—1) +8] a =j5p PAd—-D +115] (415) 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 components from Table 3 the following numerical values are obtained for the coefficient re Mf. . . 0.036,601,10: MSf. . . 0.033,863,19; Mm. . . 0.036,291,65; Sa. amd oscar. 0002.07, 00. By using the appropriate coefficient and substituting successively the numerals corresponding to the day of series (d), the correspond- ing value of the component division to which each daily sum is to be HARMONIC ANALYSIS AND PREDICTION OF TIDES. 103 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. The distribution of the daily sums for the analysis of the long- period components may be conveniently accomplished by copying such sums in Form 142 (fig. 25), taking the component divisions as the equivalents of the component 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. 29) with such modifications as will now be described. In using the daily means as ordinates of a long-period component consideration must be given to the residual effects of any of the short- period components upon such means and steps taken to clear the means of these effects when necessary. Component S, with a period commensurate with the solar day, may be considered as being com- pletely eliminated from each daily mean. Components K, and K, are very nearly eliminated, because the component K day is very nearly equal to the solar day. Other short-period components may affect the daily means to a greater or less extent, depending largely upon their amplitudes. Of these the principal ones are components M,, N., and O,. In the distribution and grouping of the daily means for the analysis of the several long-period components the disturbing effects of the short-period components just enumerated, excepting the effect 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 S, there will always remain a residual effect of the component M, in the component MSf sums of the daily means, no matter how long the series, which must be removed by a special process. Let the equation of one of the short-period components be y=A cos (at+a) (416) Letting d= day of series, the values of ¢ for the hours 0 to 23 of d day will be - PA) OA Sas OA (P21) 00. a 94d =) 4 93. Substituting these values for t in (416) and designating the corre- sponding values of the ordinate y as ¥,, Y,, Y2 --- + Y23 the following are obtained: y, =A cos [24(d—1)a+a] ° y, =A cos [24(d—1)a+a+a] Yois3 Cos [24(d—1)a+a+2a] (417) Y2,=A cos [24(d—1)a+a+ 23a] 104 U. S. COAST AND GEODETIC SURVEY. Representing the mean of these 24 ordinates for d day by ya, we have yam aq A cos {24(d—1)a+a} [1+cosa+cos 2a+---+cos 23a] a sin {24(d—1)a+a} [sina+sin 2a+---+sin 23a] 1 , sin 12a 23 Bis 9) aang 54 4 saita | cos (24(¢ 1)a+a} cos 5 —sin (24(d—1)a+a} sin «| 1 sin 12a 247” sinda cos {24(d—1)a+a+11.5a} (418) Formula (418), representing the average value of the component 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 component A. It will be noted that if we let A represent any of the solar components, S,,8., 53, 5,, ete., the factor sin 12a, and consequently the entire formula, becomes zero for all values of d. ; By formula (418) clearances for each of the disturbing short-period components for each day of series may be computed and these clear- ances then applied individually to the daily means, or, if first multi- plied by the factor 24, to the daily sums. The labor involved in making independent calculations for the clearance of the effect of each short-period component 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 ¢= time reckoned in mean solar hours from the beginning of the series, then for any value of ya, which must apply to the 11.5 hour of d day, t=24(d—1)+11.5 and (419) at =24(d—1)a+11.5a | If the above equivalent is substituted in (418) and yg replaced by Ya, We have VLiy sum lag: Ya 94“ “sin 4a cos (at+ a) (420) which represents a continuous function of t; and for any value of t corresponding to the 11.5 hour of d day the corresponding value of ya Will be ya. This formula is the same as that for the short-period component A, except that it includes the factor a = — in the . . 2 coefficient. The speed a is, of course, a known constant, and the values of A and a@ are presumed to have already been determined from the harmonic analysis of the short-period components. Simi- HARMONIC ANALYSIS AND PREDICTION OF TIDES. 105 larly, the disturbing effects of other short-period components may be represented by Yo= & B = a cos (bt +B) lye sin’ 12¢ Yo 94% “sin 4¢ etc. cos (ct+7) (421) The combined disturbing effect of all the short-period components may, therefore, be represented by the equation sin 12a sin 4a Y =Ya+t Yo+ete. =57 A cos (at+ a) 1 _, sin 126 oe = 7p CoS (Bt +B) +ete. (422) This formula is adapted to use on the tide-computing machine. With the component cranks set mm accordance with the coefficients and initial epochs of the above formula, the machine will indicate the values of y corresponding to successive values of t. 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 coefficients of the terms of (422) should be multiplied by the factor 24 before being used in the tide-computing machine. Assuming that all the daily sums are used in the analysis, the augmenting factor represented by formula (329) which is used for the short-period component is also applicable to the long-period com- ponents, with p representing the number of component periods in a component month or a component year. Thus, for components Mm and Sa, p equals 1, and for Mf, MSf, and Ssa,p equals 2. For the long- period components 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. If we let formula (416) be the equation of the long-period component sought, formula (420) will give the mean value of the 24 ordinates of the day which, in the grouping for the analysis, is taken as represent- ing the 11.5 hour of the day or the ¢g hour of the series. Since the true component ordinate for this hour should be A cos (atg+a), it is 5 1 evident that an augmenting factor of 24 [o must be applied to the mean ordinates as derived from the sum of the 24 hourly heights of the day in order to reduce the means to the 11.5 hour of each day. The complete augmenting factor for the long-period components ma therefore be obtained by combining the above with (329) to obtain Tp 24 sin 3a oy ees aah sin 12a (423) Values obtained from formula (423) are given in Table 20. The following method of reducing the long-period tides, which conforms to the system outlined by Sir George H. Darwin, differs 106 U. S. COAST AND GEODETIC SURVEY. to some extent from that just described. In this discussion it is. assumed that a series of 365 days 1s used. Let the entire tide due to the five long-period components already named be represented by the equation y=A cos (at+a)+B cos (bt+8) + C cos (at+y) +D cos (dt+6)+E cos (e+e) 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’, C’, D’, and E’, equal A cos a, B cos 8, Ccos y, D cos 6, and E cos e, respectively, and AUS iB: COD andehy L equal —Asina, —BsinB, —Csiny, —D sin 6, and — Fsin e, respectively. (425) (424) Then formula (424) may be written y =A’ cos at+ B’ cos bt+ C’ cos t+D"' cos dt+ EH" cos et + A’’ sin at+ B’’ sin bt+ O”’ sin ct+ D"’ sin dt+ E"’ sin et (426) 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 components. The most probable values of these quantities may be found by the least square adjustment. Let ¥,, Yo, - - - + Yes 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 (426) the successive values of y and the values of t to which they correspond, 365 observational equations are formed as follows: Vin mea AG Se ENS (COR ORS yeti oe aA sia i) sin O2- Ae CSU LAG 1s COS 220 een. +A” sin 24a+B" sin 2464 2... (427) Yxes= A’ cos 24 X 364a+B’ cos 24X 3646+ .... + A’’'sin 24x 364a+ B”’ sin 2436404 .... 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 y, cos 0=A’ cos? 0+B’ cos0 cosO+ +--+: - + A’’ sin 0 cos 0+ 8B"’ sin 0 cosO+ +--+: y, cos 24a= A’ cos’ 24a+ B’ cos 246 cos 24a+ ---: - +A’’ sin 24a cos 24a+B’’ sin 246 cos 24a+---- KORRES Y 35 COS (24 X 364a) = A’ cos? (24 x 364a) + B’ cos (24x 364b) cos (24x 364a)+ ---:- +A’’ sin (24 X364a) cos (24X364a) +B" sin (24 x 3646) cos (24 364a)+ --+s > HARMONIC ANALYSIS AND PREDICTION OF TIDES. 107 Summing n =365 n=365 : SS yn cos 24(n—-1)a=A'SS cos? 24(n—1)a n=1 n=1 n=365 +A’’ SD sin 24(n—1)a cos 24(n—1)a n=1 n=365 +B’ > cos 24(n—1)b cos 24(n—1)a n=1 n=365 +B’ 3S} sin 24(n—1)b cos 24 (n—1)a n—) n=365 +O’ SD cos 24(n—1)e cos 24(n—1)a i ¢ n=365 +0” Sd sin 24(n—1)e cos 24(n—1)a n=365 +D’ D> cos 24(n—1)d cos 24(n—1)a n=! n=365 + D" D> sin 24(n—1)d cos 24(n—1)a =y n=365 + EH’ 3S cos 24(n—1)e cos 24(n—1)a nn n=365 + H’’ 3S sin 24(n—1)e cos 24(n—1)a (429) 1 n= which is the normal equation for the unknown quantity A’. In oe similar manner we have for the normal equation for the quan- tity . Z Yn sin 24(n—1)a =A’ 2» cos, 24(n—1)a sin 24(n—1)a + A”’ = 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 + (" Z cos 24(n—1)e sin 24(n—1)a+ OC” X sin 24(n—1)esin 24(n—1)a + D’> cos 24(n—1)dsin 24(n—1)a+ D’’> sin 24(n—1)d sin 24 (n—1)a + H’ Z cos 24(n —1)e sin 24(n—1)a+ FE” = sin 24(n— le sin 24 (n—1)a (4301 the limits of n being the same as before. Normal equations of forms similar to (429) and (430) are easily obtained for the other unknown quantities. 2934—24}—_8 108 U. S. COAST AND GEODETIC SURVEY. By changing the notation of formulas (286) to (288) the following relations may be derived: eS ie sin 24na cos 24(n—1)a 2 Lonel De ee ee PS LENE eal = cos? 24(n—1)a=4n+4 aa oe Ulery antl sin 8760a cos 87364 (431) ae sin 24a n=365 _ sin 24na cos 24(n—1)a SS sin? 24(n— 1)a=4n —-3-—__ = s ( ) 2 2 sin 24a ey sin 8760a cos 8736a (432) aS ere ce sin 24a n=365 = cos 24(n—1)6b cos 24(n—1)a asa sin 12n(b—a) cos 12(n—1)(b—a) me sin 12(6—a) es 12n(6b+a) cos 12(n—1)(6+a) sin 12(6+ a) sin 4380(b—a) cos 43868(b —a) Carr 4 sin 12(6—a) — 4380(b+a) cos 4368(6-+a) (433) sin 12(6+ yn sin 24(n—I1)a n=1 —2.14A’—4.15B’—4.90C’ +3.80D’ +.3.88.’ +181.95A’’+1.01B"’ +1.06C” +0.34D"’ +0.68E”’ n=365 DD yn cos 24(n—1)b as = pene 1 Saat, oO DOO — 50D — se ay Ay 0:83 0-92,61, — 0.09 )7" — OS n=365 A D> yo sin 24(n—1)b n=1 =4.29A’ +0.88B’ +0.920" +3.05D'+3.06E’ SRO FS Sab 08007 = 008)”. —O.10H n=365 DS yn cos 24(n—I1)e _ n=1 =) /6.A' + 0:56B" 4 183:19 C0), = 68D) =1 70H MA =) OD | O97 G =O 17a Oe n=365 - SS yn sin 24(n—1)e n=1 = 5.04A’ +0.92B’ + 0.970’ +3.24D' +3.25H’ +1.06A’’—0.80B’’ + 181.810" —0.10D’’ —0.20E" n=365 DS yn cos’24(n —1)d n—E —4 88A’—1.50B’ — 1.680’ +182.38D’ — 0.24’ Sea S05, 3.2406" = O000D” = O0ne (436) n=365 D yn sin 24(n—1)d i = —0.34A’ —0.09B’ —0.11C’ +0.00D’ + 0.00’ eye = (O08. =OL0O eS 262 07-00 4 n=365 ; DS yn cos 24(n—1)e n=1 =4.96A’ — 1.51B’ = 1-700’ —0.24D)' “8238 2% +3.88A’’+3.06B’’ + 3.250’ +0.00D’’ +0.00H’’ n==365 : SS yn sin 24(n—1)e n=1 = —0.70A’ —0.18B’ —0.21C’ +0.01D’ +0.00E’ +0.68A’’ —0.17B’’ —0.20C’’ + 0.00D"’ + 182.62 E’” 110 U. S. COAST AND GEODETIC SURVEY. 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 amplitudes A and a, B and 8, etc., of formula (424) are readily obtained, since yr A= 7(A’)?+ (A)? and a=tan— ae Tn calculating the corrected epoch, it must be kept in mind that the ¢ in this reduction is referred to the 11.5 hour of the first day of series instead of the preceding midnight. Before solving equations (436), if the daily means have not al- ready been cleared of the effects of the short-period components, 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 component is represented by equation (418). Intro- ducing the subscript s to distinguish the symbols pertaining to the short-period component, the disturbance in the daily mean of the mn‘ day of series due to the presence of the short-period component A, may be written wa pda sin 12a, Ysin= 94 “'s Sin 445 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 cos {24(n—1)a,+11.5 as+as} (437) sin 12a; sin sas [ysln COS 24(n—1a=57 A, cos {24(n—1)a,+11.5a,+as} dal sin 12a, 24 * sin ta, 4 [cos {24(n—1) (as +a) +11.54a;+a5} +cos {24(n—1) (a,—a) +11.5 ds tas}] (438) and [Yeln sin 24 (a= l)a cel sin 12a, 24> * sin! te. 4 [sin {24(n—1) (ag +a) +11.5a5+as} sin {24(n—1) (as—a) +11.5 as;+as}] (489) Then, referring to formulas (284) and (285) n=365 = [ysln cos 24(n—l)a= n= i=, sin 12a,[sin 12 x365(a4;+a) 48~ * sin sas sin 12(a,+a) sin 12x 365(a,—@) sin 12(a;—a) cos {12x 364(a,+a@) +11.5a,+ a5} cos {12x 364(a,—a) +1155 +4) | (440) HARMONIC ANALYSIS AND PREDICTION OF TIDES. Jit and n=365 D [ysln sin 24(n—Da= n=1 1 1 DS i 1 sin 12a, ke 12 x 365(as;+ 4a) Ein 1S e362 CH sees dee oe 3 48~ * sin 4a, sin 12(a,+a) sm 2 x 365(a,—a) sin 12(a,—a) sin {12 X364(a@,—a@) + 11.5a.+as} | (441) a Now let A’, =A, Cos as and (442) . Al’. =—A, sin as then (440) and (441) may be reduced as follows: n=365 > [ys], cos 24(n—1)a a= et sin 12a, [sin 12 x 365(a.+a) #48 sin 4a, sin 12(a,+a) sin 12 x 365(a,—a) Siig SOAS (3) cos {12 x 364(a,+a) +11.5a5} se cos {12 x 364(a,—a) + 11.545} | Ate = v 9 apr 1 sin 12a, = 12 X 365(a,+a) sin (19% SOK a) BBA 48 sin 4a, sin 12(a,+a) Reinet 2< 365 (dea): ut Sma). {12 x 364(a,;—a@) + 11.5 as} | A UIA AB) and n=36 5 So [Ysln sin 24 (n'— la meet 1 sin 12a, [= 12 x 365(a,+4a) ~ 48 sin da, ST GCa aaa {12 x 364(a, +4) +11.5a5} _ sin 12x 365(as— a) sin 12(a,—a) ooh sin 12a, [sin 12 x 365(a;+a) 48 sin 4a, sin 12(a@,; a) _ sin 12 x 365(a,—a) sin 12(a,—a) sin {12 x 364(a@,—a@) + 11.545} |’. cos {12 x 364(a,+a) +11.5a,} cos {12 x 364(a,—a@) + 11.5a,} jet (444) Formulas (443) and (444) represent the clearances for any long-period component A due to any short-period component As. The first must be subtracted from terms: corresponding to Lyn cos 24 (n—1)a and the latter from terms corresponding to Zyn sin 24 (n—1)a of formula (436) before solving the latter. In (443) and (444) 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 2 U. S. COAST AND GEODETIC SURVEY. be computed once for all. Separate sets of such coefficients must, however, be computed for the effect of each short-period component upon each long-period component. In the usual reductions in which the effects of 3 short-period components upon 5 long-period com- ponents 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 components. Mm. Mf. MSf. Sa. Ssa. MEOH CO ceed BSR poy otis Nal bs HRe Lee OME —0. 0556 +0. 0030 +5..739 —0. 1041 —0. 1046 (COA ERE ee GA ee ont ANE Ed RE —0. 1704 —0. 0377 —2. 923 —0. 0752 —0. 0755 SEVER SET, ARE NU LN Dalat She ea roan —0. 1708 +0. 0417 —2. 840 —0. 0018 —0. 0035 (SU) ioe pea Se ee ee erie BD Roe Pa aoa +0. 0441 +0. 0105 —5. 727 +0. 0048 +0. 0096 NINES GID) SiR Ras Ne and eo a UA —0. 0588 +0. 0368 +0. 0294 —0. 0176 —0. 0176 CCL OUR ERAT BRIBED PRES RE aad a NS —0. 0776 —0. 2236 —0. 1938 +0. 0025 +0. 0025 GSS) eco eR ON See A Terie Came itn —0. 0206 —(. 1526 —0. 1221 -++0. 0002 +0. 0004 SLES SIR iar a ate es) CCN TARE Ca +0. 1138 —0. 0854 —0. 0808 +0. 0001 +0. 0002 CON COB eer ah ees UIC RSP CONG AU BAe EN —0. 0648 +0. 0166 +0. 0157 —0. 1924 —0. 1934 COTM ahs Nie egy CUA cet ORSON OBE —0. 3476 —0. 0778 —0. 0816 —0. 1826 —0. 1831 SDP I CE BU eee Meat Ne) —0. 3452 +0. 0841 -++0. 0875 —0. 0046 —0. 0093 CSHEP Ne A SZ a +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. After the clearances have been applied and the normal equations (436) solved and the resulting amplitude and epoch obtained for each of the long-period components, 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 tis taken to corre- spond to 11.30 a. m. on the first day of series. In obtaining the numerical values of such quantities as Zyn cos 24(n—1)a and Ly, sin 24(m—1)a, m order to avoid the labor -of separate multiplication for each day, the following abbreviations have been proposed by the British authorities. The values of cos 24(1—1)a and of sin 24(n—1)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 distributed into 11 corresponding groups, so that all values in one group will be multi- plied by 0, another group by 0.1, etc. The cos 24(n—Il)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 cosines apply. 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, using for the purpose the hourly heights for 29 consecutive days, beginning with the first day day of each month. It is therefore desirable to have a method of using these means directly in the analysis for the annual and semi- annual components, thus avoiding any special summation for the purpose. The period of the annual component is Gear the length of the Julian year; that is, 365.25 days. If this period were divided 4From Scientific Papers by Sir George H. Darwin, Vol. I, p. 64. HARMONIC ANALYSIS AND PREDICTION OF TIDES. ELS into 12 equal groups and the mean of the hourly heights for each group taken, these means would represent the approximate height of the combined annual and semiannual components for the middle of each group, and the middle of the first group would be the initial oint from which the zeta (¢) as obtained by the usual process would e referred. As each group would represent 30° of motion for the annual component, or 60° for the semiannual component, to refer this ¢ to the actual beginning of the series of observations it would be necessary to apply a correction of 15° for the annual component or 30° for the semiannual component. In obtaining the monthly means by the usual process of including 29 days only, the year is divided only approximately into 12 equal groups. The following. table shows the comparison of the middle of each group actually used in the summation and the middle of each corresponding group obtained by dividing a Julian year beginning January 1, 0 hour, into 12 equal parts. In the summing for mean sea level each group extends from 0 hour on the 1st day of the month to the 23d hour on the 29th day, making the group from first to last observation exactly 28 days and 23 hours long. The middle of each group will therefore be 14.48 days later than the first observation of the group. = ' Middle of actual Middle of each group reck- group earlier than oned from beginning of middle of same year. division of Julian Month. year. Julian |Common| Leap |Common! Leap year year year year year Days. Days Days Days Days PRATIUIAT Veyee este scion ae a arate oie eine Poets emilee 15.22 14. 48 14.4 0.74 ED GHARY) Here se rena hee tact cree me bd oeck = creme 45. 66 45. 48 45.48 0.18 0.18 ar Clee ee eR Re oat nc Seto cratcin comes cecce ye ence 76.09 73.48 74. 48 2.61 1.61 PAG rat 59 SES aA Gees SE. BSI. gate Sak 106. 53 104. 48 105. 48 2.05 1.05 IVY neta ee eee ee wie etna Mee neice ota ee tele s och a sien 136. 97 134. 48 135. 48 2.49 1.49 ERT TT OS ee tee ee oe leer es ed otra Sek. ELL 167.41 165. 48 166. 48 1.938 0.93 TRUS Se th ats = SERIE A ey RA em ag en a 197. 84 195. 48 196. 48 2.36 1.36 JATIGAD RURAL a oe oe ee ES te a ee Oe 228. 28 226.48 227.48 1.80 0.80 SERED eI setee ee ee ae Seek a Pe te Sons sm ccs 258. 72 257. 48 258. 48 1.24 0.24 CCLODE ee aes eer MR Oe ihe STS aE ee 289.16 287.48 288. 48 1.68 0.68 INGOW ETH GE eae oe Meee wer ys st RE carn oi lote cate 319. 59 318.48 319. 48 1.11 0.11 ID yeoeias | OSes = 4 eh Noes a A ce aL oe 350. 03 348. 48 349. 48 1.55 0.55 SUSIE Joc 848 SSS Re Ee eeease- Seer en ene meee 2,191.50 | 2,171.76 | 2,181.76 19. 74 9.74 LGoTi Se |e pe LoL Roce DOs GCO BOE COTEE DE bEO Se eREE AS EEreet are Beemer area 65 Se onc ee 1,64 0.81 Speed of Sa component per day=0.9856°. 2 WernS reduced tOldesrces Ohisa. sana sete tate Ck ee os act ec ee cna sabe hae agee sicalals ole ci 1. 62 0. 80 Mame chon GON@Omsde pane sate wpe aajpncise nee se oaeees ce secs cnt cose ecient ae 13.38 14. 20 WOLEECLLONULOMGOROSA a: = See. sh cece cco cance ce ee ec eee Meme caeee ee citeteteesas aceeeats 26. 76 28.40 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 year is on an average 1.62° 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 0.80° 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 13.38° and 14.20°, for common and leap years, respectively, as a correction to be applied to the ¢ of Sa as 114 . U. S. COAST AND GEODETIC SURVEY. directly obtained, in order to refer the ¢ to the O hour of the 1st day of January, for the component Ssa the corrections will be twice as great. If the year commences on the first day of any month other than January, the corrections will differ a little from the above. Calculated im 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 j Common| Leap year. year. year. year. ° ° ° ° Danka eed tata s soa sees sect eee 13. 38 NEOPA eb yee eb aee sqaceeSueossoc 14. 86 15.19 ROS ae eRe E Sees eeeeee 12.80 13: 709) Aes Lees. os ae ees 14. 28 14.69 ares. Ls eye 15.19 TESTO SG) dials cecenesteneecrcr coches 13. 71 14. 20 MOT eres Seen neice he lee = = 14. 61 TASGOW OCG. he a2 san wisiiwetcins eieteteres ore 14.12 14. 69 OIE Te eet a Pela, ea a SA Si 15. 02 1G TMM) INO ll owaguesaqeooscesecssecce 13. 54 14. 20 WUTC El Sen eae eek sees cies 14. 45 145690||| Deen Seanad c)es cot eet 13.95 14. 69 As the group summed covers approximately one-twelfth of the period of the component Sa, or one-sixth of the period of Ssa, the augmenting factors will be as follows: Sa 1.0115, logarithm 0.00497. Ssa 1.0472, logarithm 0.02005. 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,+u) 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 year separately in order that the results may serve as a check on each other. 32. ANALYSIS OF HIGH AND LOW WATERS. The automatic tide gauge, 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 include all the high and low waters for a month or more may be utilized in determining approximate values of the principal harmonic constants, but the results are not as satisfactory as those obtained from an analysis of the hourly heights. | An elaborate mode of analysis of the high and low waters 1s con- tained in volume 1 of Scientific Papers, by Sir George H. Darwin. Other mathods are given by Dr. R. A. Harris in his Manuai ef Tides. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 115 The process outlined below follows to some extent one of the methods of Doctor Harris, extending his treatment for the components K and O to other components. The lengths of series may be taken the same as the lengths used as the analysis of the hourly heights, which are 29, 58, 87, 105, 134, 163, 192, 221, 250, 279, 297, 326, 355, and 369 days. It is sometimes convenient to divide a series, whatever its length, into periods of 29 days each. This permits a uniform method of procedure, and a com- parison of the results from different series affords a check on the reliableness of the work. 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. A convenient table for the correction of the lunitidal intervals, when the high and low waters have been given in standard time and the moon’s transits over the meridian of Greenwich are given in Greenwich mean civil time, will be found in Special Publication No. 26 of the U. S. Coast and Geodetic Survey. « If the tide is of the semidiurnal type, the approximate amplitude and epoch for component M, may be obtained directly from this high and low water reduction. On account of the presence of the other components 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 can not 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. The epoch of M, may be obtained from the corrected high and low water lunitidal intervals H WI, L WI by the following formula— M°, =4(H WI+L WI) x 28.984 + 90° (445) In the above formula H WI must be greater than L WI, 12.42 hours being added, if necessary, to the H WI as directly obtained from the high and low water reductions. 116 U. S. COAST AND GEODETIC SURVEY. The difference between the average duration of rise and fall of the tide at any place, where the tide is of the semidiurnal type, depends largely upon the component M,. It is possible to obtain from the seh and low waters a component 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 component 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 component M,. By assuming an M, com- ponent with epoch such as to make the component symmetrically situated in regard to the maxima and minima of the M, component, the amplitude necessary to account for the mean duration of rise of the tide may be readily calculated as follows: Let DR=duration of rise of tide in hours as obtained from the lunitidal intervals. a= Hourly speed of component M,. = 28.°984. M,=Amplitude of M, component. M,°= Epoch of M, component. M,=Amplitude of M, component. M,°=Epoch of M, component. Then, for component M, to be symmetrically situated with respect to the maxima and minima of component M, M,° =2 M,° +90° (446) 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 components M, and M, may be written yi, =M, cos (at+a) (447) Y,=M, cos (2at+ 8) (448) and the resultant curve y=M, cos (at+a)+M, cos (2at+8) (449) Values of ¢ which will render (447) a maximum must satisfy the derived equation. aM, sin (at+a) =0 (450) and for a maximum of (449) ¢ must satisfy the derived equation aM, sin (at+a)+2aM, sin (2at+8) =0 (451) For a maximum of (447) 2nt —a oS (452) in which n is any integer. HARMONIC ANALYSIS AND PREDICTION OF TIDES. IRIE Let “= the acceleration in the high waters of component M, due to the presence of component M,. With the M, wave symmetrically situated with respect to the M, wave, : will also equal the retarda- tion in the low water of component M,, due to the presence of com- ponent M,, 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 (449) ise ee t : (453) and this value of ¢ must ay equation (451). Substituting in (451), we have aM, sin (2n7—6) + 2aM, sin (4nr—2a+8 — 26) = (454) —aM, sin @—2aM, sin (20+ 2a—8) =0 But 2a—B8 = —2M,°+M,° (455) From (446), —2M,°+M,°= +90° according to whether the duration of rise is greater or less than ~ : or whether @ is negative or positive. _Then 2a—8= F90° (456) according to whether @ is positive or negative. Substituting this in (454) —aM, sin 6+2aM, cos 20=0 (457) and Mint. sino : M, =? cos 20 (458) the upper or lower sign being used according to whether @ is positive or negative. As under the assumed conditions @ must come within ‘the limits +45°, the ratio of - as derived from (458) will always be 2 positive. The duration of rise of tide due solely to the component M, is we . _ The duration of rise as modified by the presence of the assumed M, as i e02 28 Sena a DR (459) Therefore 6=4(180° —aDR) (460) 118 U. S. COAST AND GEODETIC SURVEY. Substituting the above in (458) we have Msn 0D) | cos 4aDR Mini 2 cos (180°—aDR) — ¥3 cos aDR (461) and is cos 44a M,= #3 Cac ane. M, (462) M, must be positive, and the sign of the above coefficient will depend upon whether aDAF is less or greater than 180°. The approximate constants for components S,, N,, K,, 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 components, using the same stencils as are used for the regular analysis of the hourly heights. Account should be taken of the number of items entering into each sum and the mean for each component hour obtained. The 24 hourly means for each component are then to be analyzed in the usual manner. The results obtained by this process are, of course, not as depend- able as those obtained from a continuous record of hourly heights. The approximate results first obtained can, however, be improved by the following treatment if a tide-computing machine is available. Using the approximate constants as determined above for the principal components and inferred values for smaller components, 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 com- ponent 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. In making the combinations the following formulas may be used: Let A’ and x’ represent the first approximate values of the constants of any component. A”’ and «’’, the constants as obtained from the residuals. A and x, the resultant constants sought. Then A=+(A’ cos x’ +A” cos x”)?+(A’ sin k’+ A” sin x”)? (463) and _ AY Sink oA Sine A’ cosx’ +A” cos x” (464) K=tan 33. HARMONIC PREDICTION OF TIDES. The methods for the prediction of the tides may be classified as harmonic and nonharmonic. By the harmonic methods the ele- mentary component tides, represented by harmonic constants, are combined into a composite tide. By the nonharmonic methods the HARMONIC ANALYSIS AND PREDICTION OF TIDES. 119 predictions are made by applying to the times of the moon’s transits and to the mean height of the tide systems of differences to take account of average conditions and various inequalities due to changes in the phase of the moon and in the declination and parallax of the moon and sun. Without the use of a predicting machine the harmonic method would involve too much labor to be of practical service, but with such a machine the harmonic method has many advantages over the nonharmonic systems and is now used exclusively by the U. S. Coast and Geodetic Survey in making predictions for the standard ports of this country. The height of the tide at any time may be represented harmonically by the formula Rite h=H,+2f H cos [at+(V.+u) —«] (465) in which h=height of tide at any time t. H,=mean height of water level above datum used for pre- diction. H=mean amplitude of any component A. f=tactor for reducing mean amplitude H to year of pre- diction. a=speed of component A. ¢=time reckoned from some initial epoch such as beginning of year of predictions. (V,+4u) =value of equilibrium argument of component A when t=0. xk=epoch of component A. In the above formula all quantities except A and t may be con- sidered as constants for any particular year and place, and when these constants are known the value of h, or the predicted height of the tide, may be computed for any value of t, or time. By comparing successive values of h the heights of the high and low waters, together with the times of their occurrence, may be approximately determined. The harmonic method of predicting tides, therefore, consists essen- tially of the application of the above formula. The exact value of ¢ for the times of high and low waters will be roots of the first derivative of formula (465), equated to zero, which may be written— ; dh dG 2A sm fat + (V,+u) —«]=0 (466) Although formula (466) can not, in general, be solved by rigorous methods, 1t may be mechanically solved by a tide-predicting machine of the type used in the office of the U. 8S. Coast and Geodetic Survey. The constant H, of formula (465) is the depression of the adopted datum below the mean level of the water at the place of prediction. For places on the open coast the mean water level is identical with mean sea level, but in the upper portions of tidal rivers that have an appreciable slope the mean water level may be somewhat higher than the mean sea level. The datum for the predictions may be more or less arbitrarily chosen, but it is customary to use the low-water plane that has been adopted as the reference for the soundings on the hydrographic charts of the locality. For all places on the Atlantic 120 U. S. COAST AND GEODETIC SURVEY. and Gulf coasts of the United States, including Porto Rico and the Atlantic coast of the Panama Canal Zone, this datum is mean low water. For the Pacific coast of the United States, Alaska, Hawaii, and the Philippines, the datum is in general mean lower low water. For the rest of the world, the datum is in general mean low water springs, although there are many localities where somewhat lower planes are used. After the datum for any particular place has been adopted its relation to the mean water level may be readily obtained from simple nonharmonic reductions of the tides as observed in the locality. The value of H, thus determined is a constant that is available for future predictions at the stations. The amplitude H and the epoch «x for each component tide to be included in the predictions are the harmonic constants determined by the analysis discussed in the preceding chapters. Each place will have its own set of harmonic constants, and when once determined will be available for all times, except as they may be slightly modified by a more accurate determination from a better series of observations, or by changes in the physical conditions at the locality such as may occur from dredging, by the depositing of sediment, or by other causes. The factor f is the reciprocal of factor F, discussed in section 12. It is introduced in order to reduce the mean amplitude to the true amplitude depending upon the longitude of the moon’s node. The factor f for any single component, therefore, passes through a cycle of values. The change being slow, it is customary to take the value as of the middle of the year for which the predictions are being made and assume this as a constant for the entire year. The error resulting from this assumption is practically negligible. Hach com- onent has its own set of values for f/, but these values are the same or all localities and have been compiled for convenient use in Table 14 for the middle of each year from 1850 to 1999. The quantity a represents the angular speed of any component per unit of time. In the application of formulas (465) and (466) to the prediction of tides this is usually given in degrees per mean solar hour, the unit of t being taken as the mean solar hour. The values of the speeds of the different components have been calculated from astronomical data by formulas derived from the development of the equilibrium theory which has already been discussed. ‘These speeds have been compiled in Table 3 and are essentially constant for all times and places. The quantity (V,+u), which was discussed in section 10, is the value of the equilibrium argument of a component at the initial instant from which the value of ¢ is reckoned; that is, when ¢ equals zero. In the prediction of tides this initial epoch is usually taken at the midnight beginning the year for which the pre- dictions are to be made. In strictness the V, or uniformily varying portion of the argument alone, refers to the initial epoch, while the wu, or slow variation due to changes in the longitude of the moon’s node, is taken as of the middle of the period of prediction and assumed to have this value as a constant for the entire period. The quantit (V.+u) is different for each component and is also different tor eac initial epoch and for different longitudes on the earth. In Table 15 there have been compiled the values of this quantity for the begin- ning of each year from 1850 to 2000 for the longitude of Greenwich. HARMONIC ANALYSIS AND PREDICTION OF TIDES. iat The values may be readily modified to adapt them to other initial epochs and other longitudes. Let L=west longitude in degrees of station for which predictions are desired. S=west longitude in degrees of time meridian used at this station. - For east longitude, Z and S will have negative values. Now let p=0 when referring to the long-period components. 1 when referring to the diurnal components. 2 when referring to the semidiurnal components, etc. then p will be the coefficient of the quantity T in the equilibrium arguments represented in (100), (215), and other formulas. Now, 7 is the hour angle of the mean sun and is the only quantity in these arguments that is a function of the longitude of the place of observa- tion or of prediction. At any given instant of time the difference between the values of the hour angle T at two stations will be equal to the difference in longitude of the stations. If, therefore, the value of the argument (V,+u) for any component at any given instant has been computed for the meridian of Greenwich, the correction to refer this argument for the same instant to a place in longitude L° west of Greenwich will be — pL, the negative sign being necessary as the value of 7 decreases as the west longitude increases. The instant of time to which each of the tabular values of the Greenwich (V,+w)’s of Table 15 refers is the 0 hour of the Green- wich mean civil time at the beginning of a calendar year. In the predictions of the tides at any station it is desirable to take as the initial epoch the 0 hour of the standard or local time customarily used at that station. If, therefore, the longitude of the time merid- ian used is S° west of Greenwich, the initial epoch of the predictions will usually be 8/15 mean solar hours later than the instant to which the tabular Greenwich (V,+4)’s are referred. In formulas (465) and (466) the symbol a is the general designation of the speed of any component; that is to say, it is the hourly rate of change in the argument. The difference in the argument due to a difference of S/15 hours in the initial epoch is therefore aS/15 degrees. The total correction to the tabular Greenwich (V,+u) of any year in order to obtain the local (V,+ wu) for a place in longitude L° west at an initial epoch of 0 hours of time meridian S° west at the beginning of the same calendar year is as The ceneral expression for the angles of (465) and (466) may now be written i [at-+(Vo+u)—x]—[at+Greenwich (Vo+u)+">—ph—x] (468) In order to avoid the necessity of applying the corrections for longi- tude and initial epoch to the Greenwich (V,+4u)’s for each year, these corrections may be applied once for all to the x’s 2 U. S. COAST AND GEODETIC SURVEY. S ; (G515—pL—-«)= —K : (469) Then (468) may be written at-+ (V,+u) —x=at+Greenwich (V,+u) —x’ (470) Thus, by applying the corrections indicated in (469) to the «’s for any station, a modified set of epochs is obtained. These will remain the same year after year and permit the direct use of the tabular Greenwich (V,-+)’s in determining the actual component phases at the beginning of each calendar year. Let Greenwich (V,+u) —k’ =a (471) then formulas (465) and (466) may be written h=H,+>3 fH cos (at+a) (472) for height of tide at any time, and DS afH sin (at+a) =0 (473) for times of high and low waters. Formula (472) may be easily solved for any single value of ¢, but for many values of ¢ as are necessary in the predictions of the tides for a year at any station the labor involved by an ordinary solution would be very great. Formula (473) can not, in general, be solved by rigorous methods. The invention of tide-predicting machines has rendered the solu- tion of both formulas a comparatively simple matter. 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 (472). It provided for the sum- mation of 10 of the principal components, 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 components were afterwards - constructed. 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 U. S. Coast and Geodetic Survey for the year 1883. The first machine designed to solve simultaneously formulas (472) and (473) is the U. S. Coast and Geodetic Survey tide-predicting HARMONIC ANALYSIS AND PREDICTION OF TIDES. LOS machine No. 2, which is described in the following section. A description of the machine is also given in Special Publication No. 32 of the U. S. Coast and Geodetic Survey. 34. U. S. COAST AND GEODETIC SURVEY TIDE-PREDICTING MACHINE NO. 2. The Coast and Geodetic Survey tide-predicting machine No. 2 was designed to sum simultaneously the terms of formulas (472) and (473) and to register the successive heights of the tide (h) by a dial - and pointer as well as graphically by a curve, and also to indicate the time or values of (¢) which satisfy equation (473) for the high and low waters. This machine was designed and constructed in the office of the U. S. Coast and Geodetic Survey. Designed in 1895, its con- struction was begun the following year, and after some interruptions it was completed in 1910. It was first used in making predictions for the U. S. Coast and Geodetic Survey Tide Tables for the year 1912 and has been used for all editions of the Tide Tables since that time. The general appearance of the machine is illustrated by Figures 12, 18, and 14. It is 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 components are made to rotate with angular speeds proportional to the actual speeds of the components; third, a system of cranks and sliding frames for obtaining harmonic motion; fourth,summation chains connecting the individual component elements, by means of which the sums of the harmonic terms of formulas (472) and (473) are transmitted to the recording devices; fifth, a system of dials and pointers for indicating in a convenient manner 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 representation of the tide automatically constructed by the machine. The machine is designed to take account of the 37 components listed in Table 3, including 32 short-period and 5 long- period components. 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. 12), 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. The front section, or dial case, rests upon the desk facing the oper- ator and contains the apparatus for indicating and registering the results obtained by the machine. ‘The middle section rests upon a depression in the base and contains the mechanism for the harmonic motions for the principal components M,, S,, K,, O,, N,, and M,. The rear section contains the mechanism for the harmonic motions for the remaining 31 components for which the machine provides. 72934—24}——_9 124 U. S. COAST AND GEODETIC SURVEY. The angular motions of the individual components, as indicated by the quantity at in formulas (472) and (473), are represented in the machine by the rotation of short horizontal shafts having their bear- ings in the parallel plates of the component frames. All of these component shafts are connected 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 component, and for any interval of time or increment in ¢t as indicated by the time dials the amount of angular motion.in any component shaft will be equal the increment in the product at corresponding to that component. Since the corresponding angles in formulas (472) and (473) are identical for all values of t, the motion provided by the gearing will be applicable alike to the solution of both the formulas. The mechanism for the summation of the terms of formula (472) 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. 12), and the mech- anism for the summation of the terms of formula (473) is on the right- hand side of the machine, which is designated as the ‘“‘time side” (fig. 13). In Table 37 are given the details of the general gearing from the hand-operating crank to the main vertical component shafts, together with the details of all the gearing in the front section or dial case. It will be noted that S—6 (fig. 16) 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 releas- able gears permit the adjustment of these indicators to any time desired. After an original adjustment is made so that the hour and minute hand will each read 0 at the same instant that the day dial indicates the beginning of a day, further adjustment will, in general, be unnecessary, as the gearing itself will cause the indicators to main- tain 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 (8, fig. 12). The minute-hand shaft, with a period of 1 dial hour, moves over a dial graduated into 60 minutes (2, fig. 12). 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 inscribed 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 arc-shaped open- ing through which the graduations representing nearly two months are visible at any one time (4, fig. 12). 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. 12). This pomter 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 the last day of December this pointer will move back one day to its original position. HARMONIC ANALYSIS AND PREDICTION OF TIDES. . 125 On the same center with the day pointer there is a smaller index (7, fig. 12) 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 im 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. 12) immediately above the large dial ring. A slower move- ment of the day dial is provided by a releasable gear on the vertical shaft S—6 (fig. 16). There are three main vertical component shafts S—15 (fig. 18), S-14 (fig. 19), and S—16 (fig. 14), to which are connected the gearing for the individual components. The period of rotation of each is 12 dial hours, and all move clockwise when viewed from above the machine. The connections between these main component shafts and the indi- vidual component crank shafts are, in general, made by two pair of bevel gears and an intermediate horizontal shaft, except that for the slow moving components Sa, Ssa, Mm, Mf, and Msf, a worm screw and wheel and a pair of spur gears are in each case substituted for a pair of bevel gears. In each case the gear on the main vertical shaft is. ecok e so that each component crank shaft can be set independ- ently. . Main component shaft S—13 in the front component section drives 9 individual component crank shafts representing 6 components, 3 of the components being provided with two crank shafts each. These 6 components are M,, S,, K,, N,, M,, and O,, the first three having the double crank shafts. Main component shaft S-/4 at the front of the rear component section drives 16 component crank shafts repre- senting one component each. These are M,, MK,S,, MN, »., S,, ue, and 2N in the upper range, and MS, M,, K,, 2MK, L,, M,, 25M, and P, in the lower range. Main component shaft S—16 at the back of the rear component section drives 15 component crank shafts. The components represented are OO, i,, S,, M,, J,, Mm, and Ssa, in the upper range, and 2Q, R., T,, Q,, p,, Mf, MSf, and Sa in the lower range. For each of the five long-period components motion is communi- cated from the intermediate shaft by a worm screw and wheel to a small shaft on which is mounted a sliding spur gear. The latter en- gages a spur gear on the component crank shaft, but may be easily disconnected by drawing out a pin on the time side of the machine, thus permitting the component crank shaft to be turned freely when setting the machine. Gear speeds.—The relative angular motion of each individual com- ponent shaft must correspond as near as possible to the theoretical speed of the component represented. ‘The period of rotation of each of the three main vertical shafts being 12 dial hours, the angular motion of each of these shafts is 30° per dial hour. Table 38 con- tains the details of the gearing from the main vertical component shafts to the individual component cranks, the number of teeth in the different gears for each component being given in columns I, II, III, and IV. In designing the predicting machine it was necessary 126 U. S. COAST AND GEODETIC SURVEY. to find such values for these columns as would give gear speeds approximating as closely as possible with the theoretical speeds of the components. By comparing the gear speeds as obtained with the corresponding theoretical speeds it will be noted that the accu- mulated errors of the gears for an entire dial year for all the compo- nents are negligible in the prediction of the tides. Releasable gears.—Releasable gears (52, fig. 19) on the main ver- tical shafts of the dial case and component frames permit the inde- pendent adjustment of the time indicators and individual component shafts. The details of these gears are illustrated in Figure 20. A collar C, with a thread at its upper end and a flange at the bottom, is fastened to the shaft by means of three steel screws. The gear wheel A fits closely upon this collar and rests upon the flange. It has sunk into its upper surface a recess a, which is filled by the flange of collar B. When in place, the latter is prevented from turning by a small steel screw reaching into a vertical groove c in the collie C. The lower surface of collar B is slightly dished, and the collar is split twice at right angles nearly to the top. When the milled nut D is screwed down with a small pin wrench, the edge of the collar B is pressed against the edge of the recess a with such force as to make slipping practically impossible. When the nut is loosened, the gear may be turned independently of the main driving shaft. A small wrench (56, fig. 19) is used for setting these gears. Each of the three main driving shafts is provided with a clamp (55, fig. 19) to secure the shaft from turning when the nut of the releasable gear is being loosened or tightened. Component cranks.—Secured to the ends of the individual compo- nent shafts, which project through the brass plates on both sides of the machine, are brass cranks (40, fig. 16) which are provided for the component amplitudes. Those on the left or height side of the machine are designated as the component height cranks and are used for the coefficients of the cosine terms of formula (472), and those on the right or time side of the machine are designated as the component time cranks and are used for the coefficients of the sine terms of formula (473). The time crank on each component shaft is attached 90° in advance (in the direction of rotation) of the height crank on the same shaft. For the components Sa and Ssa no time cranks are provided, as the coefficients of the sine terms correspond- ing to these components are too small to be taken into account. The direction of rotation of each component shaft with its compo- nent cranks is clockwise when viewed from the time side of the machine and counterclockwise when viewed from the height side. The details of a component crank are shown in Figure 21. The pointer a is rigidly attached to the crank as an index for reading its position on a dial. In each crank there is a longitudinal groove 6 with flanges in which a crank pin d may be clamped in any desired position. The crank pin has a small rectangular block as a base which is designed to fit the groove in the crank, and through the center of the crank pin there is a threaded hole for the clamp screw Attached to the under side of the crank-pin block is a small ‘spring c that presses the block outward against the flanges of the groove, keeping it from slipping out of place when unclamped and at the same time permitting it to be moved along the groove when setting the machine. The crank pin may be securely fastened in Special Publication No. 98. Fig. 12.—DIAL CASE AND HEIGHT SIDE, TIDE-PREDICTING MACHINE. ‘ANIHOVW ONILOIGSYd-SGIL ‘S0IS AWI1—'€1 ‘14 a oe "86 ON UO!}BOI|GNd |eIDedS Specia! Publication No. 98. Sie Sais D REAR END AND HEIGHT SIDE, TIDE-PREDICTING MACHINE. Fig. 14. Special Publication No. 98. 7 3 Fig. 15—MAIN DRIVING GEARS AND AUTOMATIC STOPPING DEVICE, TIDE- PREDICTING MACHINE. HARMONIC ANALYSIS AND PREDICTION OF TIDES, 1g any desired position by tightening up on the clamp screw, which, ressing against the small spring at the back, forces the crank-pin plisek outward against the flanges of the groove with sufficient pres- sure to prevent any slippmg. A milled head wrench B is used for tightening the clamp screw. A small rectangular block e of hard- ened steel is fitted to turn freely upon the finely polished axle of the erank pin. This block is designed to fit into and slide along the slot of the component frame. Positive and negatwe direction.—All the component shafts and cranks may be grouped into two ranges—those above the medial horizontal plane of the framework being in the upper range and those below this plane in the lower range. In the following discussion direction toward this medial plane is to be considered as negative and direction away from the plane as positive; that is to say, for all components in the upper range the positive direction will be upward and the negative direction downward, while for the components in the lower range the positive direction will be downward and the negative direction upward. Component dials.—To indicate the angular positions of the com- ponent shafts, the pointer (a, fig. 21) moves around a dial (4/, fig. 16) which is graduated in degrees. These dials are fastened to the frame of the machine back of the component cranks on both sides of the machine, those on the time side being graduated clockwise and those on the height side counterclockwise. These dials and pointers are so arranged that the angular position of a component shaft at any time will be the same whether read from the dial on the height side or from the dial on the time side of the machine, and at. the zero reading for any component the height crank will be in a peruive vertical position and the corresponding time crank in a orizontal position. At a reading of 90° the height crank will be horizontal and the time crank in a negative vertical position. With the face of the machine registering the initial epoch, such as January 1, 0 hour, of any year, the value of ¢ then being taken as zero, each component shaft may be set, by means of its releasable gear, so that the dial readings will be equal to the a of the corre- sponding component as represented in formulas (472) and (473). If the machine is then put in operation, the dial readings will, for successive values of ¢, continuously correspond to the angles (at+a) of the formulas, as the gearing already described will provide for the imcrement at. Component sliding frames.—For each component crank there is a light steel frame (42, fig. 16) fitted to slide vertically in grooves in a pair of angle pieces attached to the side plates of the machine. At the top of the component frame there is a horizontal slot in which the crank pin slides. As the machine is operated the rotation of the component shafts with their cranks cause each crank pin to move in the circumference of a circle, the radius of which depends upon the setting of the pin on the crank. This motion of the pin, acting in the horizontal slot of the component frame, imparts a vertical har- monic motion to that frame. The frame is in its zero position when the center horizontal line of the slot intersects the axis of the com- ae shaft; positive motion is the direction away from the medial orizontal plane of the machine and negative motion is toward the medial plane. The displacement of each component height frame 128 U. S. COAST AND GEODETIC SURVEY. from its zero position will always equal the product of the amplitude setting of the crank pin by the cosine of the component dial reading, and the displacement of each component time frame will always equal the product of the amplitude setting by minus the sine of the component dial reading. Component pulleys—EKach component frame is connected with a small movable pulley (43, fig. 16). For all components except Miao, aN. KOs 6. and Sa on the height side and components M,, S,, N,, and M, on the time side this connection is by a single steel strip, so that the pulley has the same vertical motion as the cor- responding component frame. Doubling gears.—Because of the very large amplitudes of some of the components two methods were used in order to keep the lengths of the cranks within practical limits. For the components M,,S,, and K, two sets of shafts and cranks were provided, so that the amplitudes of these components may be divided when necessary and a portion set on each. A further reduction in the length of the cranks for these and the other components named in the paragraph above is accomplished by the use of doubling gears between the component frame and movable pulley. Two spur gears with the ratio of 1:2 (48, fig. 16) are arranged to turn together on the same axis. The smaller gear engages a rack (46) attached to the component frame and the larger gear engages a rack (47) attached to the component pulley. Each rack is held against its gear by a flange roller (49), and counterpoise weights are provided to take up the backlash in the gears. Through the action of these doubling gears any motion in the component frame causes a motion twice as great in the com- ponent pulley. Doubling gears are provided on the height side of the machine for components M,, S,, N,, K,, O,, and Sa and on the time side for components M,, S,, N,, an es Scales for amplitude settings——The scales for setting the component amplitudes are attached to the frame of the machine and are, in general, graduated into units and tenths (44, fig. 16). The scales are arranged to read in a negative direction; that is, downward for the components of the upper range and upward for the components in the lower range. On a small adjustable plate (45) attached to each component pulley there is an index line which is set to read zero on the scale when the component frame is in its zero position. For setting the crank pins for the component amplitudes the cranks to be set are first turned to a negative vertical position. For the cranks on the height side of the machine this position corresponds to a dial reading of 180° and for the cranks on the time side to a reading of 90°. The scales on the height side of the machine, which are used in setting the coefficients of formula (472), are graduated uniformly one- half inch to the unit. On the time side of the machine the scales are modified in order to automatically take account of the additional factor involving the speed of the component which appears in each of the coefficients of formula (473). Dividing the members of this for- mula by m, the speed of component M,, it becomes Ly ail sin (at-+a)=0 (474) The modified scales are graduated 0.5 a/m inch to the unit. The use of the modified scales on the time side of the machine permits both a HARMONIC ANALYSIS AND PREDICTION OF TIDES. 129 the height and time crank for any component to be set in accord with the factor fH which is common to the coefficients of both formulas (472) and (473). There are also provided for special use on the time side of the machine unmodified scales graduated uniformly to read in a positive direction. Summation chains.—The summations of the several cosine terms in ' formula (472) and of the several sine terms in formula (473) are car- ried on simultaneously by two chains, one (27, fig. 16) on the height side and the other (28, fig. 17) on the time side of the machine. The chains are of the chronometer fuse type, of tempered steel, and have 125 links per foot. The total length of the height chain is 27.6 feet and of the time chain 30.6 feet. A platinum point is attached to one of the links of the time chain 3.5 feet from its free end for an index. Hach of these chains is fastened at one end near the back part of the machine by a pair of adjusting screws (53, fig. 14, and 54, fig. 13). From these adjusting screws each chain passes alternately downward under a component pulley of the lower range and upward over a com- ponent pulley of the upper range, spanning the space between the rear and front component frames by two idler pulleys and continuing until every component pulley on each side of the machine is included in the system. The movable pulleys are so arranged that the direc- tion of the chain in passing from one to another is always vertical a parallel to the direction of the motion of the component sliding rames. Summation wheels.—The free or movable end of each of the chains is attached to a threaded grooved wheel (29, 30, fig. 16), 12 inches in cir- cumference and threaded to hold more than seven turns of the chain, or about 90 inches in all. These are called the height and time sum- mation wheels. Each is mounted on a shaft that admits a small lateral motion, and by means of a fixed tooth attached to the frame- work of the machine and reaching into the threads of a screw fastened to the shaft the latter when rotating is forced into a screw motion with a pitch equal to that of the thread groove of the summation wheel; so that the path of the chain as it is wound or unwound from the summation wheel remains unchanged. The height summation wheel (29, fig. 16) is located near the front edge of the front component section, where it receives the height summation chain directly from the nearest component pulley. The time summation pulley (30) is located inside the dial case near the lower left side, and three fixed pulleys are used to carry the time chain ‘from the end component pulley to the summation wheel. Counter- poise weights are connected with the shafts containing the summa- tion wheels in order to keep the summation chains taut. When all of the component frames on either side of the machine are in their zero positions, the corresponding summation wheel is approximately half filled by turns of the summation chain. Any motion of a component frame in a positive direction will tend to un- wind the chain from the wheel, and any motion in the negative direction will tend to slacken the chain so that it will be wound up by the counterpoise weight. With several of the component frames on either side of the machine moving simultaneously, the resultant motion, which is the algebraic sum of all, will be communicated to the summation wheel. The motion of the component. frame being transmitted to the chain through a movable pulley, the motion of 130 U. S. COAST AND GEODETIC SURVEY. the free end of the chain must be twice as great as that in the pulley. The scale of the pulley motion is one-half mch to the unit, and there- fore the scale of the chain motion is 1 inch to the unit, and one com- plete rotation of the summation wheel represents a change of 12 units. The zero position of the height summation wheel is indicated by the conjunction of an index line (50, fig.16) on the arm attached to the wheel and an index line (41, fig. 16) on a bracket attached to the framework of the machine just below the summation wheel, the wheel itself being approximately one-half filled with the summation chain. The length of the chain is adjusted so that the summation wheel will be in its zero pésition when all the component frames are in their zero positions. It will be noted that the conjunction of the index lines will not alone determine the zero position of the wheel, since such conjunctions will occur at each turn of the wheel, while there is only one zero position, which is that taken when the com- ponent frames are set at zero. The zero position of the time summation wheel is indicated by the conjunction of an index point (//, fig. 12) attached to the time sum- mation chain and a fixed index (12, fig. 12) in the middle of the hori- zontal opening near the bottom of the dial case, and the length of the time summation chain is so adjusted that this conjunction will occur when all of time component frames are in their zero positions. Predicted heights of the tide.—When the machine is in operation, the sum of all the cosine terms of formula (472) included in the settmgs for a station will be transmitted through the height summation wheel to the face of the machine and there indicated in two ways—first by a pointer moving over a circular height scale (8, fig. 12) and second by the ordinates of a tide curve that is automatically traced on a roll of paper (14, fig. 12). The motion of the height summation wheel is transmitted by a gear ratio of 30:100 to a horizontal shaft which is located just back of the dial case. One complete rotation of this shaft represents 40 units in the height of the tide. From: this shaft the motion is carried by two separate systems of gearing to the height pointer on the face of the machine and to the pen that traces the tide curve. Height scale-—The height pointer is geared to make one complete revolution for a change of 40 units in the height of the tide. A height scale, with its circumference divided into 40 equal parts and ~ each of these unit parts subdivided into tenths, provides for the direct registering of the sum of the cosine terms of formula (472) as com- municated through the summation wheel. This scale has its zero” graduation at the top and is graduated positively to the right and negatively to the left. The height pointer can easily be adjusted to any position by means of a small milled nut (10,fig. 12) at the end of its shaft. If it should be desired to refer the predicted heights to mean sea level, this pointer must be adjusted to read zero at the same time that the summation wheel is in its zero position; but if it is desired to refer to some other datum, the pointer will be adjusted according to the elevation of mean sea level above this datum. For the value of h in formula (472) the pointer will be adjusted to a reading corre- sponding to the adopted value of H, at the time the summation wheel is in its zero position, then this value of H, will be automatically included with the sum of the cosine terms of that formula. As the Special Publication No. 98. Fig. 16—DIAL CASE, TIDE-PREDICTING MACHINE. — Special Publication No. 98. 08 ewes DIAL CASE AND FRONT COMPONENT SECTION, TIME SIDE, TIDE- Figs U7: PREDICTING MACHINE. Special Publication No. 98. Fig. 18.—VERTICAL DRIVING SHAFT, FRONT COMPONENT SECTION, TIDE- PREDICTING MACHINE. Special Publication No. 98. REAR COMPONENT SECTION, TIDE- PREDICTING MACHINE. 19.-FORWARD DRIVING SHAFT, Fig. HARMONIC ANALYSIS AND PREDICTION OF TIDES. eel l machine is operated the height pointer will indicate the predicted height of the tide corresponding to the time shown on the time dials. In order to increase the working scale of the machine when predict- ing tides with smaller ranges, two additional circular height scales. are provided, one with the circle divided into 20 units and the other into 10 units, with the units subdivided into tenths. These scales may be easily removed or replaced on the machine, the scale in use being secured in place by a small button at the top (9, fig. 12). The 20-unit scale may be conveniently used when the extreme range of the predicted tide at any place is between 10 and 20 feet, and the 10-unit scale when the extreme range is less than 10 feet. If the 20- unit scale is to be used, the value of each coefficient of both the cosine and the sine terms must be doubled before setting the component cranks, and if the 10-unit scale is used these original coefficients must first be multiplied by 4 before setting the values in the machine. If the extreme tide is less than 4 feet, the 40-unit dial may be readily used as a 4-unit scale by considering the original unit graduations as. tenths of units in the larger scale. In this case the coefficients of the cosine and sine terms of the formula must be multiplied by 10 before entering in the machine. The factor used for multiplying the coeffi- cients to adapt them to the different height scales is called the work- ing scale of the machine. Working scales of 1, 2, 4, and 10 are now in general use to take account of the different ranges of tide at the places for which predictions are made. Predicted tumes of the tide.—Simultaneously with the summation of the cosine terms of formula (472) on the height side of the machine the summation of the sine terms of formula (474), which was derived from formula (473), is being effected on the time side. Being con- cerned only with the time at which the sum of the sine terms is zero, no provision is made for registering the sum except at this time, which is indicated on the machine by the conjunction of the index point on the time chain and the fixed platinum index in the dial case. Near the time of a high water the index on the chain moves from right to left and near the time of a low water from left to right. The con- junction of the movable and fixed index is visible to the operator of the machine and he may note the corresponding dial readings for the time and height of the high or low water. Automatic stopping device.—This device provides for automatically stopping the machine at each high and low water. Secured to the hand-crank shaft is a ratchet wheel and just above the ratchet wheel is a steel pawl (25, fig. 15) operated by an electromagnet (26) mounted under the desk top. - The electric circuit for the electromagnet is closed by a contact spring that rests upon a hard-rubber cylinder (31, fig. 16) on the rear end of the shaft on which the time summation wheel is mounted. A small platinum plug in this rubber cylinder comes in contact with the spring, which is fitted with a fine motion adjustment, when the time summation chain registers zero. This closes the circuit and draws the pawl against the ratchet wheel, thereby automatically stopping the machine. The lateral screw motion of the shaft on which the rubber cylinder is mounted prevents the platinum plug from coming in contact with the spring on any revolution other than the one which brings the time chain to its zero position. The circuit is led through an insulated ring on the hub of the hand crank where a contact is kept closed by a spring. After 182 U. S. COAST AND GEODETIC SURVEY. the operator has noted the time and height readings of the high or low water he may easily break the circuit at the crank hub by a slight inward pressure against the crank handle, thus releasing the arma- ture and pawl and permitting the machine to be turned forward to the next stop. By means of a small switch (23, fig. 15) just below the crank the circuit may be held open to prevent the automatic device from operating when so desired. Nonreversing ratchet.—Upon the crank shaft, close to the bearing in the desk frame, there is a small ratchet wheel and above this there is a pawl (24, fig. 15) that is lifted away from the wheel by friction springs when the machine is being turned forward but which is instantly thrown into engagement when the crank is accidentally turned backward. By pushing in one of the small buttons (22, fig. 15) just above the crank the pawl is locked so that it can not engage the ratchet, thus permitting the machine to be turned backward when desired. Pressure on another button releases the pawl. Tide curve.—TVhe tide curve which graphically represents the rise and fall of the predicted tide is automatically traced on a roll of paper by the machine at the same time that the results are being indicated on the dials. The curve is the resultant of a horizontal movement of the paper, corresponding to the passing of time, and a vertical movement of a fountain pen (13, fig .12), corresponding to the rise and fall of the tide. The paper is 6 inches wide with about 380 feet to the roll, which is sufficient to include a little more than a full year of record of the predicted tides at a station. The paper should be about 0.0024 inch thick in order that the complete roll may be of a suitable size for use in the machine. Within the dial case, near the upper right-hand corner, is a mandrel (33, fig. 16), which can be quickly removed and replaced. It is designed to hold the blank roll of paper, the latter being wound upon a wooden core especially designed to fit.on the mandrel. At the bottom of the mandrel is an adjustable friction device to provide tension on the paper. From the blank roll the paper is led over an idler roller (34, fig. 16), mounted in the front plate of the dial case, then across the face of the machine for a distance of about 13 inches to a feed roller (35, fig. 16), then over the feed roller to the receiving roller (36, fig. 16), upon which it is wound. The feed roller governs the motion of the paper across the face of the machine and is provided near each end with 12 fine needle points to prevent the paper from slipping. The feed roller is controlled by the main vertical shaft of the dial case through gearing of such ratio that the feed roller will turn at the same rate as the main vertical shaft; that is to say, one complete turn of the feed roller will represent 12 dial hours in time. The feed roller being 6 inches in circumference the paper will be moved forward at the rate of one-half inch to the dial Sous A ratchet and pawl (37, fig. 16) are so placed as to leave the paper at rest when the machine is turned backward. If desired, the paper feed can be thrown out of action - altogether by turning a small milled head on the ratchet gear. To provide for the winding up of the paper on the receiving roller there is a sprocket wheel (38, fig. 16) held by adjustable friction to the upper end of the feed roller. Fitted to the top of the receiving roller is a smaller sprocket which is driven by a chain from the feed- roller sprocket. The ratio of the sprockets is such as to force the HARMONIC ANALYSIS AND PREDICTION OF TIDES. decoys receiving roller to wind up all the paper delivered by the feed roller, the tension on the paper being kept uniform by the friction device. To remove a completed roll of record the smaller sprocket is lifted from the receiving roller and a pin (39, fig. 16) at the back of the dial case is drawn out, releasing the upper bearing bracket. The bracket can then be raised and the receiving roller with its record removed. A similar bracket secured by a pin is provided for the removal of the mandrel on which the blank roll of paper is placed. Marigram gears.—The pen that traces the tide curve is mounted in a carriage which is arranged to slide vertically on a pair of guidin rods and is controlled from a horizontal shaft at the back of the dia case. On this shaft there is mounted a set of three sliding change gears (18, fig. 17), which are designed to mesh, respectively, with three fixed gears mounted on a shaft just above. By sliding the change gears in different positions any one of them may be brought into mesh with its corresponding fixed gear. These gears provide for ratios of 1 : 1, 2:1, and 3 : 2, according to whether the innermost, the middle, or the outer gears are in mesh. At the outer end of the shaft containing the fixed gears is a thread-grooved wheel 4 inches in circumference (19, fig. 17), to which is attached one end of the pen-carriage chain (20, fig. 17). The chain is partly wound upon the wheel and from it passes through the dial case to the front of the machine, then upward over a pulley near the top to a counterpoise weight within the dial case. The pen carriage is secured to this chain by means of a clamp and can be adjusted to any desired position. Scale of tide curve.—With a working scale of unity the arrange- ment is such that the motion of the height summation wheel as transmitted to the curve-line pen through the marigram gears with ratio of 1 : 1 causes the pen to move vertically 0.1 inch for each unit change in the predicted height of the tide. If the marigram gears with ratio 3 :2.or 2:1 are used the unit of height will be represented by a vertical movement of the pen of 0.15 or 0.2 inch, respectively. For any working scale other than unity the above unit equivalents must be multiplied by the number representing that scale. The scale ratio of the tide curve will depend upon the unit of height used for the predictions. Taking the foot as the unit, the following scale ratios are obtained: Marigram | Marigram | Marigram gears 1:1.] gears3:2. | gears2:1. VORA CS CAle pepper ia hncuinmet en, SOR PO ERR ERNPTAN. 5 UIP SC Oe 1 1 1 PVOnksn oscar meres rere RGAE oe i) hs cy AIRY Veg lai Leona ive Le 1 VOL ines calersaeeme ccs Muar Wmmonpeee oa CT ON Soa reM per eve Sh 1:30 1: 20 1:16 MWienking ScalenlOusea (ets VEAP er iy Ce ee ATER hl re tes De 1 9 Pens.—The curve-line pen (1/3, fig. 12) and the datum-line pen (14) are each of the ordinary fountain type. Each is fitted with a | metal lock joint, so that it may be quickly removed and replaced in the same position, and is pressed against the paper by a light coil spring when in use. The curve-line pen is mounted in a swivel arm on a light carriage which slides vertically along two rods. The datum- line pen is mounted in a swivel arm that may be adjusted so that the mean sea-level line will be traced midway between the upper and lower edges of the paper. 134 U. S. COAST AND GEODETIC SURVEY. Hour-marking device—The arm for the datum-line pen is secured to the outer end of a shaft which carries two armatures, one for the upper and the other for the lower of two electromagnets (17, fig. 17). A spring keeps the armatures at equal distances from their respective electromagnets. The upper electromagnet is designed for fasieettae the hours on the datum line and is in a circuit that is opened and closed by a platinum-tipped contact spring resting upon the edge of an ivory disk in which are embedded, equally spaced, 24 narrow strips of platinum (32, fig. 16). The ivory disk is mounted on the shaft of the hour pointer, and as this rotates the platinum strips successively make an electric contact that throws the datum-line pon downward for an instant, making a corresponding jog in the datum ine, the downward stroke of the pen indicating the exact hour. An extra strip of platinum placed close to the one representing the midnight hour causes a aauble jog for the beginning of each day, the - downward stroke of the second jog indicating the zero hour. High and low water marking device-—The lower electromagnet is in a circuit that is closed when the platinum index on the time chain (11, fig. 12) is in contact with the fixed platinum index (12); that is to say, at the times of high and low waters. When this contact is made, the electromagnet attracts the armature, which throws the datum-line pen upward, causing a corresponding upward jog in the datum line, and thus automatically marking the time of the high or low water. A small switch (21, fig. 15) just above the hand-crank shaft permits the cutting out of the current from the two electromagnets. Adjustment of machine-—The adjustment of the machine should be tested at least once each year and at any other time when there is any reason for believing that a change may have taken place. The following adjustments are required. Height-chain adjustment.—All amplitudes should be set at zero, so that the turning of each component crank shaft will produce no motion in the height chain. This should bring the summation wheel to its zero position, but on account of a certain amount of backlash and flexures in the machine this wheel may not be in an exact zero position even when the chain is in adjustment. Now, set a single component with a very small amplitude and operating the machine with the hand crank, note whether the index of the summation wheel oscillates equal distances on both sides of its zero position. If not, the chain should be adjusted by the adjusting nut at its fixed end at the back part of the machine. Time-chain adjustment—The adjustment of the time chain is similar to that of the height chain. The zero position is indicated by the conjunction of a small triangular-shaped index on the chain and a fixed platinum index in the middle of the horizontal opening in the dial face. A small amplitude being set on one of the component time cranks and the machine operated by the hand crank, the chain index should oscillate equal distances on both sides of the platinum point. If it does not, the necessary, adjustment may be made at the fixed end of the chain. Hour-hand adjustment—This must be so adjusted that it will register the exact hour at the same instant the circuit for the electro- magnet is closed for the hour mark on the marigram, which is indi- cated by a downward stroke of the datum-line pen. It is also neces- Special Publication No. 98. Fig. 20.—DETAILS OF RELEASABLE GEARS, TIDE-PREDICTING MACHINE. Pa Fig. 21.-DETAILS OF COMPONENT CRANKS, TIDE-PREDICTING MACHINE. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 135 sary that thezero hour or beginning of the day shall correspond to the double hour mark on the marigram. This adjustment may be accom- plished by moving the hour hand on its shaft after releasing its set screw. A finer adjustment may be effected by changing the position of the contact spring back of the dial face. Minute-hand adjustment.—This is to be adjusted to read zero on the exact hour indicated by the hour hand and the closing of the electric circuit for the hour mark. The adjustment may be accom- plished either by moving the minute hand on its shaft after releasing its set screw or by means of the releasable gears on the main vertical shaft of the dialcase. The adjustments just described are those which need be made only occasionally. Other adjustments are taken into account each time the machine is set for a station. Setting predicting machine.—The time indicators on the face of the machine are first set to represent the exact beginning of the period for which predictions are to be made, which will ususally be 0 hour of January 1 of some year. The hour and minute hands should always be brought into place by the turning of the operating crank in order that the adjustment of these hands relative to the electromagnet circuit may not be affected. The date dial may, however, if desired, be set independently, using the binding nut just above the large dial ring for releasing and clamping. If only a small motion of the date dial is necessary, it is generally preferable to set it by the operating erank. The year index should be set to indicate the kind of year. In the usual operation of the machine a ratchet prevents the operating crank from being turned backwards, but this ratchet may be released when desired by pressing on a button in the side of the machine just above the orate After the face of the machine has been thus set to register the beginning of the predictions the three main vertical shafts of the two component frames should be clamped to prevent them from turning. To set the height amplitudes.—All the component cranks on the left or height side of the machine are first turned, by means of the releas- able gears on the main vertical shafts, to a vertical position, the cranks of the upper range of components pointing downward and those in the lower range upward, in which position all angles will read 180°. For the long-period components the cranks can be more quickly brought to the vertical position by drawing out small knobs on the time side of the machine, thus disconnecting the gearing. The cranks are then turned by hand to the desired position and the knobs pushed back into place. The amplitudes may now be set according to the scales attached to the sides of the machine. The crank pin is unclamped by a small milled head wrench and is then moved along its groove until the index at the scale registers the amplitude setting given in Form 445, when it is clamped in this position. If no amplitude is given for any component, the corre- sponding crank must be set at zero. To set time amplitudes.—The process is similar to that for the height amplitudes, the cranks on the time side of the machine being first turned to a vertical position with all angles reading 90°. The cranks are to be set with the same amplitudes as were used for the height side, the modified scales automatically taking account of the true differences in the amplitudes. For the components Sa and Ssa the amplitudes are set on the height side only. 136 U. S. COAST AND GEODETIC SURVEY. To set component angles.—After the amplitudes have been set and checked on both sides of the machine the angles are set for the beginning of the period of predictions, these settings being given in Form 445. The angles may be set from either side of the machine, except for components Sa and Ssa, for which there are no dials on the time side, as the readings are the same for both sides. As each component angle is set its releasable gear is clamped to the main vertical shaft. After all the angles have been thus set the three main vertical shafts must be unclamped to permit them to turn. Changing height scale——There are three interchangeable height scales, known as the 40-foot, the 20-foot, and the 10-foot scale: ‘The 40-foot rmg may also be conveniently used as a 4-foot scale. The scale to be used for any station is indicated in Form 445. In remoy- ing a scale from the machine a small button at the top is turned to release the ring, which is then lifted slightly as it is beng removed. The desired scale is then placed on the machine and secured in place by a button. Before removing or replacing the height scale it is desirable that the height pomter be set approximately 45° to the left of its zero position in order to interfere least with the removal or replacement of the scale. The datum or plane of reference.—The hand-operating crank should be turned forward or backward until the index of the summation wheel on the height side of the machine indicates mean sea level. It must be kept in mind, however, that as the index lines may come in conjunction at each complete rotation of the summation wheel there is a possibility of being misled in regard to the mean sea-level osition. When in doubt, the operating crank should be turned orward to obtain a number of conjunctions, the corresponding height dial reading for each being noted. The conjunction that corre- sponds most closely with the average of such height readings will be the one that applies to the true zero position. Each complete turn of the height summation wheel will cause a change to the height reading of 12 units, 6 units, or 3 units, respectively, according to whether the 40-unit, 20-unit, or 10-unit dial is used. The height hand, which can be released by the milled nut on the face of the machine, may now be set to the scale reading that cor- responds to the height of mean sea level above the datum which has been adopted for the predictions, this value being given in Form 445. The marigram gear.—There are three gear combinations, designated as the 1:1, 3:2, and 2:1 ratios. The gear ratio to be used for any station is indicated in Form 445. When it is necessary to change the gear ratio, the machine should be first turned to its mean sea- level position. The change is then effected by sliding the lower set of gears horizontally, being careful to hold the upper set with one hand to prevent it from turning when the gears are released. Before engaging the gears in their new ratios the counterpoise for the pen carriage should be brought to a position approximately midway between the limits of its range of motion. The 1:1 ratio is obtained by sliding the lower set of gears as far as possible toward the height side of the machine, thus engaging the innermost gears; the 3:2 ratio by moving these gears toward the time side until the outer gears are engaged, and the 2:1 ratio by engaging the middle gear of each set. In setting up the machine for successive stations there is a me- chanical advantage in making the necessary gear changes before HARMONIC ANALYSIS AND PREDICTION OF TIDES. LST setting the new amplitudes if the gear changes are in the order of 2:1, 3:2, 1:1, and after setting the amplitudes if the gear changes are in the reverse order. This precaution will lessen the chances of jambing the curve pen carriage and throwing the height chain off its pulleys when setting the amplitudes. Inserting paper roll.—To place the paper on the machine, remove the mandril that is mounted within the dial case near the upper right-hand corner and slip the roll of paper over the mandril, the roll being so placed that the winding is clockwise when viewed from above and when on the machine the paper unwinds from the outer side of the roll. In placing the roil on the mandril care should be taken to see that the small projection on the base of the latter enters the cavity in the wooden core, so that the roll will fit flat against the base. After the mandril with the roll of paper has been returned to the machine and secured in place the end of the paper is passed around a roller to the face of the machine, across the face, and over the feed roller at the left of the machine. The end is then inserted into the slit in the receiving roller, which is given a few turns to take up the slack paper and make it secure. Before passing the paper over the feeding roller and on the receiving roller these rollers should be released to permit them to turn independently, the release being effected by turning the small milled head on a ratchet stud gear near the base of the feeding roller and by lifting off from the top of the receiving roller the small knob holding the connecting ehain. After the paper has been secured to the receiving roller these connections should be restored. Curve pen adjustment—With the machine in its mean sea-level position, the curve pen must be adjusted to bring the pen point on _ the mean sea-level line as drawn by the base-line pen. This adjust- ment may be effected by releasing the pen carriage from the oper- ating chain and moving it to the desired position, where it is clamped in place by the binding screw. Verification of machine settings.—Each step in the adjustment and setting of the machine should be carefully checked before proceeding with the next step. After the setting of the machine for any station has been completed an excellent check on the work is afforded, if the eto as for the same station for the preceding year are available, y turning the machine backward several days and then comparing the predicted tides with those previously obtained. Predicting.—The datum and curve fountain pens are filled and put in place, the electric cut-out switch under the base of the machine closed, and the ratchet of the operating crank set to prevent the machine from being turned backward. If the predicted height of the tide for any given time is desired, the machine may be turned forward until the required time is regis- tered on the time dials and the corresponding height read off of the height dial. If the predicted high and low waters for the year are desired, the operating crank is turned forward until the machine is automatically stopped by the brake at a high or low water. To avoid the strain on the machine due to sudden stops, the operator should watch the small index on the time chain, and as this approaches the fixed index in the center of the opening on the face of the machine, turn the crank more slowly until the machine is stopped as the indexes come 138 U. S. COAST AND GEODETIC SURVEY. in contact with each other. The time and height may then be read directly from the dials on the face of the machine. The movement of the height pointer before the stopping of the machine and also the tide curve will clearly indicate whether the tide is a high or low water. After the tide has been recorded an inward pressure on the crank handle will release the brake and the machine can be turned forward to the next tide, the process being repeated until all the tides of the year have been predicted and recorded. 30. TIDAL CURRENTS. 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. Components 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. In the reversing type the current flows alternately in opposite directions, the velocity increasing from zero at the time of turning to a maximum about three hours later and then diminishes to zero again, when it begins to flow in the opposite direction. By consid- ering the velocities as positive in one direction and negative in the opposite direction, such a current may be expressed by a single har- monic series, such as V=A cos (at+a)+B cos (at+6)+C cos (ct+y) +etc.. (475) in which V=velocity of the current in the positive direction at any time t. A, B, C, etc. =maximum velocities of current c)»mponents. a, b, c, etc. =speeds of components. a, B, y, etc. =initial phases of components. In the rotary type the direction of the current changes through all points 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 representation of such currents it is therefore necessary to have two series—one for the north and the other for the east component. For the analysis of either type of current the original hourly veloci- ties or the resolved hourly velocities are tabulated in the same form used for the hourly heights of the tide. To avoid the inconvenience 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 ~ HARMONIC ANALYSIS AND PREDICTION OF TIDES. 139 set of constants, one for the north components and the other for the east components. Although the predicting machine was designed, primarily, for the prediction of the tides, it is adapted also to the prediction of tidal currents. The currents involve both direction and velocity, while the tides involve height only. The predicting machine can not be used directly for the determination of direction, but it is used for the summation of component velocities in the same manner as for the summation of component heights in the prediction of the tides. It is therefore directly applicable for the reversing type of current, in which only a single direction need be considered, the velocities bemg taken as positive or negative according to the direction in which it is flowing. For the rotary type of current all velocities might be resolved into two directions at right angles to each other, such as the north and east, and velocity predictions made for each of these directions inde- pendently. The labor, however, of recombining the north and east components into the resultant velocities would be practically pro- hibitive without a machine especially designed for this purpose. For the predictions of the reversing current two methods are employed. The first is of general application and requires that the harmonic constants of the current components be obtained from an analysis of the current velocities. The machine settings are then computed in the same manner as for the prediction of the tides and using the same forms with slight modifications in the headings, the amplitudes being expressed in knots instead of feet. The approxi- mate extreme range will be taken as twice the maximum current in one direction. The height dial unit will be taken as the knot instead of the foot, and zero velocity will be taken to correspond to mean sea level. If the machine is now set up and operated in the same manner as for the prediction-of tides, the current velocities may be read directly from the face of the machine for any desired time, the positive values being for the velocities in the direction originally adopted as positive, preferably the flood current, and the negative values for the velocities in the opposite direction. The machine will be automatically stopped at each maximum flood and ebb current, and slack water will be indicated by the zero position of the recording hand. The velocity of the current for any desired time and the times of maximum veloci- ties and of slack water may be also obtained from the predicted curve. A second method of predicting the reversing current, which is more indirect than that just described, is applicable to a hydraulic current in a strait. Such a current is caused by the difference in the head of the tidal waters at the two ends of the strait. Except for the lag due to the inertia or momentum of the water, slack would occur at the time the water is at the same level at both ends and the Maximum velocities at the times of greatest difference in the head. For this method of predicting it is necessary that tidal harmonic constants should be available for both ends of the strait. Let these ends be designated by M and JN, and let the single sub- script refer to the tidal constants at J/ and the double subscript to those at N, and for convenience call the direction of flow from M to N as flood or positive and the reverse direction as ebb or negative. 72934—24}——10 140 U. S. COAST AND GEODETIC SURVEY. Excepting for the lag, therefore, the flow will be positive when¥the elevation of the water at Mis higher than at N and negative when the water at N is the higher. Let H,=amplitude of any component A for station M. H,,=amplitude of same component A for station N. x,=epoch of same component A for station M. x,,=epoch of same component A for station N. L,=\ongitude of station M (positive if west, negative if east). L,,=longitude of station N (positive if west, negative if east). _ p=subseript of component A. (V,+4u),=local V,+wu for component A at station M. (V,+u),,=local V,+u for component A at station N. Then (V,+%),=(Vo+u),+p (L,—L,,) (476) The equations of the heights of the tide due to component A at stations M and N, respectively, may be written y, =f, cos [at + (Vo+u),—k)] =fH, cos [at+(V.+),] cosx,+ fH, sin [at+(V.+u),] sink, (477) Uli. H, cos [at + (Vo+ u) Ky] Tre dy, cos [at a ( Vo = u), rs p(L, a Ly) TF Ky] = il, cos [at + (V,+4u) y\ cos-[k;,= p(L, 74 Ly) | +f. d by sin [at + (V,—u) Al sin on Pp (L,— L,,)| (478) | The difference in the height due to component A, positive when the tide at M is the higher and negative when the tide at N is the higher, may now be written Y= Yi =} {H, COS K, — Joby, COS [k,, aes p(L, ay by hip cos [at + ( Vo “it u),] +f {H,sin x,—H,,sin [x,,— p(L,—L,,)]} sin [at + (V.+u),] =fVH? + H,? — 2H, H,,.cos [k,=«,,+ p(L, —L,)|X ay} [op Mid Geel STUN ys pL, —L,,)] Cos E + ( ie ae u), —tan H, cosx, — H,, cos |x, —p(L,—L,)] Dp GE aE Te 1] (479) If we let : H=JH?+H,?=2H) H, cos |k,—x, + pL)=L,)] (480) an Os Uh H, sin Kyi H,, sin lias p(L, —L,,)] ah H,cos «,—H, cos ky, pL SEM ee and substitute in (479), we have y =fFH cos [at+ (V.—u),—k] (482) In (481) the quadrant of « is determined by the signs of the numer- ator and denominator, which correspond, respectively, to the signs. of the sine and cosine of the angle. ee HARMONIC ANALYSIS AND PREDICTION OF TIDES. 141 Sunilar formulas will represent the height difference due to the other components, and the sum of all will give the resulting difference in the head of water at station M and station N. This sum for successive values of ¢ is readily obtained by use of the tide-predicting machine, which will give the times of the maximum and minimum and zero differences and also the difference in the head of the water for any desired time. In general, the current will flow from M to N when the value of y is positive and in the reverse direction when y is negative, but on _ account of the inertia of the water there will be a lag which will cause the maximum strength of flood and ebb to occur some minutes after the time of greatest head, and also the slack water to be some time later than the time of zero difference in head. The amount of this lag may be determined from actual observations. th the prediction of the slack waters by the use of the predicting machine the necessity of taking account of the lag for each individual slack is avoided by modifying once for all the epochs determined from Formula (481). Let t,=lag or average difference between time of zero difference in head and time of following slack water, and let ft ior bt 1, (483) Then, when ¢ represents the time of zero head, t’ will represent the time of the corresponding slack water. Substituting in (482) we have y =fFHH cos [at'+ (V,+4u),— (x +at,)] (484) in which t’ will represent the time of slack water when y equals zero. To adapt the above to the use of the Greenwich (V,+4), we have from (467) (V,+u),=Greenwich (V,+u) +a 8/15— pL, (485) Substituting in (484) and letting Kk’ =x+at,—a@ S/15+pL,=x+pL,—a (8/15 —t,) (486) we have y =fFH cos [at’ + Greenwich (V,+ 4) —x’] (487) In formulas (481) to (486) it will be noted that the x has for con- venience been taken as referring to the longitude of station M and the corresponding values for the local (V,+wu) and ZL are therefore used. The S refers to the meridian of the standard time used in the calculation. The x’ is adapted to the meridian of Greenwich and also takes account of the lag in the current. The elements such as represented by formula (487) may be readily summed by the tide-predicting machine. While the resulting differ- ences in head will refer to time t, the face of the machine will indicate time t’, and when the difference in head registers zero, ¢’ will indicate the time of the corresponding slack water. _ Formula (487) may also be used for the prediction of the strength of the current, but if the lag in the strength differs from the lag in the slack waters a separate set of «’s must be computed. The strength of flood current will correspond to the maximum positive differences in the head and the strength of ebb to the greatest negative differences. 142 U. S. COAST AND GEODETIC SURVEY. 36. FORMS USED FOR ANALYSIS AND PREDICTIONS. The forms used by the U. S. Coast and Geodetic Survey to facilitate the work of the harmonic analysis and predictions of the tides, together with an example of their use, are shown in Figures 22 to 34. A series of tidal observations at Morro, Calif., for February 13 to July 25, 1919, is taken as the example to illustrate the analysis and the computation of the settings for the predicting machine. Form 362, hourly heights (fig. 22).—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. 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 red ink for the 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 columns are to be numbered consecutively, beginning with one (1) for the first day of the series. The series analyzed should be one of the following lengths: 15 (14 days for diurnal components), 29, 58, 87, 105, 134, 163, 192, 221, 250, 279, 297, 326, 355, or 369 days. 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. 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. Stencils (figs. 23 and 24).—The first figure is a copy of the com- ponent WM stencil for the even hours of the first seven days of the series, and the latter illustrates the use of the same. ‘This stencil being laid over the page of hourly heights shown in Figure 22, the heights applying to each of the even component hours for this page show through the openings in the al where they appear con- nected by diagonal lines, thus indicating each group to be summed. For each component summation, excepting for S, there are pro- vided two stencils for each page of tabulated hourly heights, one for the even component hours and the other for the odd component hours. 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 component S no stencils are necessary, as the component hours correspond to the solar hours of the tabulations, and the horizontal sums from Form 3862 may be taken directly as the component hour sums. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 143 Form 142, stencil sums (figs. 25 and 26).—The sums for each com- ponent hour are entered in Form 142, one line of the form being used tor each page of the original tabulations. The total of the hour sums in each line of the form must equal the corresponding page sum of | Dersnmuenporcomerce TIDES: HOURLY HEIGHTS | | Statio:;___Morro, Californias Year; — 1919 | Chief of Party: ___Be Be Latham. Pat, 55° 22" We Tone. 1209 51" We Time Meridian; ___120 We. _ Tide Gauge No. 107~ Scale 1:9 Reduced to Staff. Month and Day. 11—792 Day of Series. | Hour. 0 1 2 ui 26 3. 3.6 ERERE 24.9 lad fed [ed [es] [ad [o) [od | os i ha 2.0 2.2 2.2 2.6 16.6 || peer ed | 18 L 1 6 1.5 JERERERE 20 | a : 18.1 21 | 40 aa 23.0 22 4. 27.6 23 | 45 30.6 | Sum for 29 days, 1 to 29 of Divisor=696; mean for 29 days= Tabulated by MeLeaSe _________ Date_Apr.22,'19, Summed by _#,4,K%——____—— DateNove 22, 1920, Fig. 22. 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 component the totals for each component hour are obtained, the divisors from Table 32 entered, and the component hourly means computed (fig. 26). These means should be carefully 144 U. S. COAST AND GEODETIC SURVEY. checked before proceding with the analysis. Large errors can usually be detected by plotting the means. Form 244, computation of (V,+4u) (fig. 27) —This form provides for the computation of the equilibrium arguments for the beginning of Derantuenror commence TIDES; HOURLY HEIGHTS Station: Year: Chie Of EP srt ys oe ie loa ts ee ee eee OTE. Time Meridian: ___________. Tide Gauge No. Scale 1: Reduced to Staff. Sum for 29 days, 1 to 29 of = .Divisor=696; mean for 29 days= BT x 1 beach eee sean) ee Se ene eeeS SLANTED 0 ER LN Fie. 23. the series of observations, the computation being in accordance with formulas given in Table 3. For the most part the form is self- explanatory. The values of 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), may be obtained from Table 4 for . HARMONIC ANALYSIS AND PREDICTION OF TIDES. 145 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 2. The corrections necessary in order to refer the elements to any desired Derantuenroncommencs § TIDES; HOURLY HEIGHTS Station: __ Stencil for component M. Year: CTE eo Ba iy EES SS ES OO Tg fi act 2 Lone: Time Meridian: ____________ Tide Gauge No. Scale 1: Reduced to Staff. P| | Rex | Y 27 Pea | al gs Cae = es ea 23 igesh che bree eel | a =r al —_— ( 4.5) 1p | CED) Wy ie (37) | Geadee| | ede Enel eases e i) Se ave a Pe aa Tebulated by ae Date Summed by Ce Date Fic. 24. 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 146 U. S. COAST AND GEODETIC SURVEY. 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 Form 142 DEPARTMENT OF COMMERCE TIDES: STENCIL SUMS, U.S. COAST AND GEODETIC SURVEY REntions LetorrouuCal I forn dag awe slconh MRI neat Mob O ree Component: __"M't Length of series>..163,. Series begins:.1919=Feb e=15=0 Long.: 120° 51" We Da; 8 5 ¥r Ho. Deo. co Kind of time used: ........._.120° We_......... Computed by Brede As Kummall,Deo.2,1920. 1l—8i7 le, , Page. o> 1 2 3 + 5 8 7 8 9 10 11 24035 2006 1769 1609 21.0 2500 2620 31.9 39.2 D408 3109 2704 21.8 17.5 14.4 13.6 11.2 12.3 1466 20065 2167 230.9 2407 24.5 19.7 16.9 11.0 9c6 1262 17e% 28el 2409 2704 2706 29.8 2lel 2604 1860 1705 1765 2207 2206 2600 S405 3605 41.2 3505 2825 2 3 4 5 2165 2164 1769 1862 1665 19.9 2469 2909 S707 BHeB 35e2 2906 6 7 8 2003 1608 1508 1201 1265 15.1 21.0 2104 2506 2408 25.0 28el 23el 1601 13.3 1301 1566 2307 2806 3509 BOol 3409 2726 2306 2505 2500 2104 21.6 2008 2505 2702 2907 4505 560% 3206 2705 9 2009 18.5 165.2 12.1 11.3. 13.8 18.1 26.5 26-8 28-6 28.0 2906 10 16.9 13.2 10.2 8.7 11.5 1565 18235 21.5 24.4 2565 2807 2405 Tl 1826 15.0 1205 1567 17%2 23.2 29.5 41el 36,7 S404 27.8 24.0 12 24eB5 2505 20.4 20.7 21.0 2406 S21 S3le7 S207 3605 Sle6 2506 13 25-7 173 13.2 1060 11.9 12.5 16.2 2005 2403 3002 3005 2404 14 1607 1266 865 827 965 1463 1900 2504 3002 2704 2702 2602 15 19.0 1661 1665 156% 2001 2665 3707 S7e7 4002 39.05 S506 2906 16 29.6 22.8 21066 22.5 2507 Sle? 31le9 34.9 3508 SBel 2805 2705 17 22.9 18.6 14.5 111 10.5 12.65 15.3 19.0 2504 24.5 2409 2306 18 1564 10.0 662 32° 409 1062 1660 2462 2405 2501 2504 2706 19° 16.7 15.4 13.2 15.5 19.8 29.4 S18 3508 3805 3709 3807 28035 20 27.6 2100 1908 2004 2802 2909 31.5 3601 3604 39.9 285.9 22.8 tom. 43721 35605 300.5 28605 32528 401.7 495.8 57805 63504 64526 593.5 523.4 Page. 12 13 14 16 16 17 18 19 20 21 22 23 1 22.7 2lel 1725 13.5 1465 1705 17.9 2602 2602 2767 26.9 28.0 576.6 2 2508 1766 1704 186.9 2105 2808 3201 35el 3605 3509 3508 2666 55205 3 17.5 1765 1407 15.4 21.1 2365 2902 3305 35.5 39.5 30.0 24.8 547.8 4 23.2 2008 12.9 900 71 709 1400 20.8 22.2 24.9 25.8 25.3 538.0 5 2705 2002 1669 15.5 16.4 2508 2565 3003 27.9 29.0 34.0 25.4 5897.5 6 2308 2300 2000 24.2 23.1 25.2 27.5 28.8 394 38.2 28.7 24.4 562.8 7 19.5 1505 1708 1466 1565 2002 298 31.4 35.7 S33el 3006 29.0 574.5 8&8 22.4 19.4 1265 808 704 1007 14.9 1965 2401 2606 32.0 27.3 55802 9 22.1 18.1 1504 1400 1%el 19el 2463 2903 38.0 2920 28.0 24.6 52802 10 23.2 2206 2605 22.0 24e4 2609 2807 3308 28cl 30c? +2604 2405 535601 11 19.4 18.2 11.8 12.5 1501 1668 2523 26.6 29e1 2900 3004 22.9 55008 12 21.5 1355 84 4.6 4.4 867 15.0 21.0 25.5 31.9 28.1 27.0 53603 13-2107 1868 1665 17066 1766 2106 2609 36.1 35.5 3604 3062 27.0 54202 14 2866 02306 2704 2604 26.0 2808 3608 32.5 31.5 28.2 2820 25.0 56603 15 2003 1664 12.9 10.5 14.6 2165 2200 2507 27.8 Slel 2504 2206 58406 16 2009 14.9 100.9 6061 408 909 16.9 2502 S21 3008 30.6 28.9 580.2 1¥ 2163 22.2 17%el 1608 1709 2607 2706 3600 3305 3605 3504 Sel 54507 18 2202 2lel 21.5 23.6 31e2 S18 30.4 31.7 31635 28.2 2706 2101 51404 19 19.6). 15.5.9 12.5) 12.3 20.0 12.2 19.9 26.1 2866 24.1 22.5 19.7 542.6 20 20.4 15.2 7.68 3.6 3.0 Jel Wiel 2202 2let RIB Deed 270% S58eL “445.6 37500 51800 28609 31007 388.8 481.6 56926 61507 62006 59007 515.9 11095.0 Fig. 25. is used in Table 5, the soe tabular value must be taken with its sign reversed. For the middle of the series the nearest inte- eral hour is sufficient. The values of J, », &, 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 J is always positive. Although Table 6 is computed for the epoch, HARMONIC ANALYSIS AND PREDICTION OF TIDES. 147 January !, 1900, it is applicable without material error for any series of observations. 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 NV Form 142 ag pete ta al TIDES: STENCIL SUMS. Station: Morro, California, po LE Patt eaesgatnry Component: /"""___ Length of eeries: .263._ Series begins: 1919 - Febe—13-0 fiong.:120° 51' We Days. Yr fo. De. Er. Kind of time used: AAT Pea _. Computed by SS Page. Oo 1 2 3 + 6 6 7 8 i=) 10 11 21 2569 18-1 14.8 1465 10-6 11.1 17.3 2308 23.1 2404 2401 2226 22 16.68 14.8 7.7 De? 666 liel 1965 23502 2665 ° 2706 3008 24.9 23 1768 1567 15el 20e1 2106 S007 3505 S7%e5 39.0 42.8 335.9 28.4 2A 7.2 6.8 642 Gel 66e 8.0 _ 9e7 10.9 .18.5 12.1 11.0 9.4 Sums-21-24 6707 5504 4308 4604 4565 6009 7908 9502 10609 106.9 9908 855 m= 1=20 43701 356035 300.3 25 325.8 401.7 495.8 578 4 645.6 593.5 5235.4 Swims ~ 504.8 411.7 34401 33209-56921 46206 57506 67507 74203 75205 69505 60807 Divisors.- 164 163 162 #165 -164 4165 4165 165 164 165 163 162 Meanse= 3.08 2.53 Bele 2002 2e25 2.684 3.55 4613 4.53 4.56 4.25 3.76 tums, _———————————— SS ——E—E—E——EE—————————————————————————————eee Page. 12> 13 14 16 16 17 is 19 20 21 22 23 21 23563 18.2 17.0 1705. 25.5 24.0 2807 32.9 3509 42.1 34.7 Slel 558.8 22 225 2007 2002 2620 2609 Sle? 36.2 34.0 40.0 31.3 26.2 20.5 551.4 25 2301 1665 1565 1166 1169 1507 1906 2501 2666 26.2 24.0 24.5 5738 Fi Bini la OG ATA LOO OPE I? AU I) on ey 7.8 159.8 Sums 21-24 72.2 5909 5507 5666 6500 7065 86.2 9504 10800 10666 92.6 83.9 1843.8 nm: 1220 243.6 375.0 3518.0 285.9 310.7 388.8 481.6 569.6 613-7 620.6 590.7 513.9 11093.0 Sums. 51508 43469 37307 345.5 S75e7 45901 56768 66500 721.7 72702 683.5 59768 1295608 Divisors.j 162 163 163 ‘163 162 162 163 163 162 162 165 .163 Means.= SelB 2.67 2029 2012 2031 2.85 348 4.08 4.45 4.49 4.19 3.67 Fic. 26. for interpolation. If the series falls beyond the limits of this table, the following formulas may be used: u of L,=2&—-2v—R (p. 48) (488) uot M,=é—7+Q@ . (p. 51) (489) The values of £ and v may be taken from Form 244, the values of R and Q from Tables 8 and 10, respectively, using the arguments [ and P for the middle of the series. 148 U. S. COAST AND GEODETIC SURVEY. | In finding the difference between the longitude of the time meri- dian (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 DEPARTMENT OF COMMERCE, TIDES: Computation of V,+u. U. & COAST AND GEODETIC SURVEY. sc 120%51'w iBtation --Movern es Gall Sern ait eos ot a eal ee = Lat. ...92_ 22 Ne Long. ..120285 Won yr. mo. a. hr. (3 Greenwich wry d. 2 Beginning of series 191.9. Rebe.13. | [Obeyinrne saa dill (S|iens Length of series 163... Time mer. 120.00W._ =S ' yt. ™. da. hr. t Greenwich 4) Middle of series ___1919 May __E Boe Ceo r) Compute all values to tivo decimal places. Tables in Harmonic Analysis and Prediction of the Tide. Table 4, for January 1 of year_...--.—-|-------GOO "| el ie eats POUR Table 5, correction to Ist of month_-__--|_---__-_-' 48-47 sid besos _35 Table 5, correction to day of month Table 5, correction to Greenwich hr----| ee nai) chit yeni Zh y= 119.02 |@- 32. 24! 322 32! 281.55 |e f: M=r(Tables) =----- 2h: eee 25 a i @ms(taune) UE | Tg 4-68 | Meena oie 10282) TTS 701| sg 2 O00! (9)—E (Tab!o6) = ee a a 150-93. M, ot Tugcgtowl 1 =__ 60.00 | COE 16 563) | === we ee ore ne ee V.4u=2(M)= 187. 36. Bisson me ee sce fen Sen 5 ee Ah Ae ee (11)= 20" (TableOj= --- OO K, N; ae RCS Oa (MK), 53-03 | t0=---------Bk-47_! +Mp—_....... 46084 | yy we (asye180° +e 93.68 | (12)=P=(5)—(9) met ae) || ay =86.78} AP: ay Eyam) |e -60. 00 Cea lgt a og | Vott= -60200.| voew—_... 288-98 s Vetu= BS: 68 UA Tane aa ene amen Wie ee Co =... 520206) visu -_ 398-30 | K, (2N), Ss (MN), (15)=S%—E* - =-------- = 0 (2) COR Ie PN 320 .06 _ 356.60 | + Mi =.--------- £6: 84 (16)=(3)+(15) =~. Sel Cast al —apn=-___ 16-63 | _as)- —86 .78 eae +N: =__820-06 | i aneagsar = DEFT | ypu 299-57 | veewm_. 288-20 : Vetu=.......366-90., 86.78 |= mec sess ance || ee me OURS BS 2 APE V,+ua3(33)—_ 204-90 --|_------ ee 6-90 i (18)=(1)= (2) = me mnnnne T Ba 557 L, 0; 1 (MS), | (19)= (17) + (18) = ~-----==-== ae o (2) eee Aeplolsls Wey) at 334-15 | + ay—_ 208-350 | 4 m= 46-84 +3y—__— 270.6 | _ gy = 51-46 | — ary 40-77 |45,.- _ 358. 30 | OE a ree Vi+u = Hlivaru eee 405.14 (2)=(1)+Q) =| fi i End EAR (22)=(20)—(21) = Mickie So. Nonna 202.45 | +@)= = ......1:598: 60, COA Oe ca y o7 + (14) = Sis -46. 84 | (28m (9)— (8) — oe nneennnnnnseeneee ES 309-76 | (25)=(23)+ (24) = : (28)=(1)—(9) -- (29)=2 (28)-.---=-. (30)= (29) +90° CSN eI a A AS AO | eee S| An hes: casceaese nee ind (BL ur (16) = (8) ime aa coe cesses eee cel lly agtet=e omy a gee ly ey Art| ohne se = ee CoN Oa A IR SS Ope ited lal oe | (CIO Oya ee OT RR WAG Oa SO of ag) Sots) | (33)=2 (15)---.-== -_.-. S| se Se arce -..) Vetum+cs) = 96.7 8. 2 . (2Q), AL Sa i (34)=(2)=(1) -2=----- 299 290 (35)=2 (34)----- =. + Q-= 2222 2 55.91 +Q.-= 32 | (36)= (26)—(35) = ------- —(1s)—__ 86 -78 | + (a3) = (1)=(3)—(2) 29 Votun......+99 13 | viu- (Ge) CT) a ee EO LOIN INU Soler SAM veal Rea a ----2--2---- = --=- | =2--- DSS aS ERE TEESE + Greenwich hour= original hour+(S*+15). * Positive for West longitude; negative for East longitude. Computed by-Le. Ae Alpert, .......--- Feb.25,1921. Duplicated by...be.P> Disney... Febs 25,1921e, eermonast rape orice 11—7956 (Date.) (Date.) - Fig. 27. 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 andthe 12th hour the following midnight, Form 244 will still be apRuCanle if the longitude of the time meridian (S) is taken equal to the civil time meridian plus 180°. For example, HARMONIC ANALYSIS AND PREDICTION OF TIDES. 149 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°. Form 244a, log F and arguments for elimination (fig. 28).—Items (1) to (11) are compiled here for convenience of reference for 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 Form 2442 - TIDES: Log F and Arguzents for Elimination. Station Morro. California. Length of series _162 days. Series begins 1818,” Feb: 15, 0. Component : Log F. : aioe : Log F. : Conronent Log F. ages o.q201 ns! tty : 9.9726 + ME : 0.0092 sone at 2° e160 Yo, 2 9.9982 «: a 0.0083 AMR. 24,0.0472 | 10, > 0.0264: MM : 9.9863 a Ta 409.9569 -.-2,-.00 : 0.0929 : MS, 2M : 9.9922 : ty 9.8853: P : 0.0000 : Ke 2 0.0896 : Mg: (9.9982 =: @, 2Q + 00264 : USE : 9.9932 2, aUMs : 9. 9897 : Bos Sr, Se, 0.0000 Ma 9.9772 My: 909863: AAs: 9.9932 +: Sa, Sea : 0.0000 My 99794 Pa ; 9.0264 (1) = N= item (6) from Form 244 = 245911 (2) = E = item (7) from Form 244= 21.76 (3) = P= item(12) from Form 244 = 53.03 (4) = (n4v") = item (3) - 3 item (10) ,from Form 244 = 327° (5) = (bv" ) = item (3) = $ item (11),from Form 244 = 331 (6) = (nP, } = item (3) = item (4),from Fom 44= 41 (7) = Log R, from Table 7 = 9.9657 (8)-= Log Q, from Table 9 = 9.8592 (9) = Natural mumber from Log F(K,) = 1.038 (10) = Log £(Kz) = 10 - icg F(K2) = 9.9528 (II) = naturel number f(K2) from (10) = 0.897 Fic. 28. argument. Items (9) to (11)are obtained after the rest of the form has been filled out. The log F for each of the listed components, 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 components L, and Log F(L,) =log F(M,) +item (7) (490) Log F(M,) =log F(O,) +item (8) (491) If the tidal series analyzed was observed between the years 1900 and 2000, the log F(L,) and log /(M,) may be taken directly from 150 U. S. COAST AND GEODETIC SURVEY. Table 13, using the year of observations, together with item (1), as argument. Form 194, harmonic analysis (fig. 29).—This form is based, pri- marily, upon formulas (316), (317), (324), and (325) and is designed for the computations of the first approximate values of the epochs (x) and the amplitudes (H) of the harmonic constants. Form 194 TID DEPARTMENT OF COMMERCE COAST AND GEODETIC SURVEY Station.___Morro, Californiae ™. vw he Component.....bL_. Series begins. 2919... ROD yee Sent (Oe ES: HARMONIC ANALYSIS. (1) Hours 0 to 11-.-.-- oe _2.25|_ 2.84 | (2) Hours 12 to 23...-. 2 .67 2 29) 2.12 2.3]| 2.83 | 3.48] 4.08 | 4.45) 4.49| 4.19) 3.67 (tee =0 14 | -0.17| -0.10/-0.06| 0:02 0-05| 0.05! 0.08! 0.07] 0.06/ 0.09 (alas ovat | 0.00] 0.09] 0G 06) 0.071 0.08] 0.051 o-00 | ooo Ron | 6.26| 5:20) 4.41) 4-14 |. er 7.01| 8.21 | 8.98] 9.05] 8.44] 7.43 (6)=Last half of (5).-- 7.01| 8.21] 8.9819 05! 8. 44| 7.43 “ay (is) (16) (19): (18) (12) @=@) |_anxan_} gi, |_@xU3)_| eps, |_GDX05)_| gin, |_@=@_|_aDxas) nxage Da axa 1245 nx20° an 12c, -0-10 | -Q-100) 000} 0.000) 1.000|_-Q:100) . : =0-23.|.-0-222! .107|.=0:035) 707 “20-163 000 | -=0.-23_|.-0-199} 1.00.) -O:110) 00) 0.000; . 0-17 | -0-120} 707) .-0-021)— .707|. =O. 120) 1. 500 |.=O-24 | -0+070] .000/ 0. 000|-1.000) -O- 140) . 39 |. =0::04 | =0.:010)- .77}.-0-042)- .707|_=0:.028)_. 500 | Ree _=0-72)| va 0: 258]120- .-0-025 0.000\-1.00| —O0.050| ooo! 0.000 sin. |(18)X(@2) | ggg, | _8)x(24) |_(5)+00) (37) | (28) | 9) (30) (+6) _| 2H=CD | gin, | CDXAD| os, (aspx cai) | (2e)+(27 (s7) (32) | (33) | so sin. | 9X) | gp, |_Gax(G8) Pe EI exon es nX120° 12 Wee j x60" 12% 1255 nx120° 0} 0.000 +1) =2.010 0} 0.000 -1|.. 4,910) 0| 0.000 +1] =1.760 ee 0140, Letom 9089 f/ Is in Ist quadrant when we have +s and +c. Pot is in 2d quadrant when we bave +4 5nd —c. “ Is In 3d quadrant when we have —s and —e. ’ {s in 4th quadrant when we have —sand +e. eked a (a ee ae (41)= ¢’, for beginnmg of the series__-_.. (42)= local V+ (From Form 244) _.... CN eR en - (45)=log. cos. {/-----.---.- (46)=(38)—(44), or (39)—(45)-_. (47)=log. (augmenting factor+12)_-_...____. (48)= log. (reciprocal of 12) for component 8 -.. (49)= (46)+-(47), or 48}4+-448)= log. R’_-_.-... — (50)=og. factor F (From Form 244'a)__ (51)= (49)+-(50)=log. HW’... ...---__._.. (52)= natural number from (49}= RB’... Computed by._Le Pe Disney, 8-5 5630 )1:16474 |9-41162 9.85794 /0-335806 18-39 7949. 8-6 98.36 )0-8 26681-01368! 7714. Ree 326-05. 143.19| | 8.92206 8.92082/8.9208 2/8920 8 2/8. . . ‘ 0: 09550/8- 34569 897. 66.14) 0.060 (63)=naturel number from (51)=_H'....--------------------- 0.046 1 .226 Febe25,19216_. Verified by ._Le As Alpert __, Fabe25,19214 (Date) 7 Wate.) Fig. 29. Provisions are made for obtaining the diurnal, semidurnal, terdi- urnal, quarter-diurnal, sixth-diurnal and eighth-diurnal components, but only such items need be computed as are necessary for the par- ticular components sought. For the principal lunar series M,, M,, M;, M,, M,, and M,, com- pute all items of the form. For the principal solar series S,, S,, 5,, and S,, items (14), (16), (33), (35), and (37) may be omitted. Re HARMONIC ANALYSIS AND PREDICTION OF TIDES. Pe For the lunal solar components K, and K,, items (14), (16), and (23) to (37) may be omitted. *For the diurnal components J,,0,, OO, P,,Q,,2Q, and p,,items (5), (6), and (14) to (34) may be omitted. For the semidurnal components L,, N., 2N, R,, T., X3, us, v2, and 2SM, items (3), (4), (8) to (16), and (23) to (37) may be omitted. For terdiurnal components MK and 2MK, items (5), (6) ,(9), (12), and (18) to (37) may be omitted. For quarter-diurnal components MN and MS, items (3), (4), (8) to (25), and (35) to (37) may be omitted. In the bottom portion of the form the symbol of the component is to be entered at the head of the column or columns indicated by the subscript corresponding to the number of component periods in a component day, the remaining columns being left blank. The hourly means from Form 142 (fig. 26) are entered as items (1) and (2) in regular order, beginning with the mean for 0 hour. Item (4) consists of the last five values of item (3) arranged in reverse order. Item (6) consists of the last six values of item (5) in their original order. For the computations of this form the following tables will be found convenient: Table 19 of this publication for natural products, Vega’s Logarithmic Tables for logarithms of linear quantities, and Bremiker’s Funfstellige Logarithmen for logarithms of the trigonometrical functions. In the last table the angular argu- ments are given in degrees and decimals. In choosing between items (44) and (45) the former should be used if the tabular value of (41) in the first quadrant is greater than 45° and the latter if this angle is less than 45°. In referring (41) to the proper quadrant it must be kept in mind that the signs of the natural numbers corresponding to (38) and (39) are respectively the signs of the sine and cosine of the required angles. Thaberore (41) will be in the first quadrant if both s and ¢ are positive, in the second quadrant if s is positive and c negative, in the third quadrant if both s and c are negative, and in the fourth quadrant if s is negative and ¢ positive. In obtaining (49) use (46) + (47) for all components except S, and (46) + (48) for component S. The log factor F' for item (50) may be obtained from Form 244a. Form 194 is designed for use when 24 component hourly means have been obtained and all the original hourly heights have been used in the summation. If in the summation for a component each component hour of the observation period received one and only one of the hourly heights, it will be necessary to take the log-augmenting factor from Table 20 and add this to the sum of items (46) and (48) to obtain item (49), striking out item (47). This form is also adapted for use with the long-period components. Assuming that the daily means have been cleared of the effects of the short-period components in accordance with section 31, and that these means have been assorted into 24 groups to cover the component period, the 24 group means may then be entered in Form 194 in place of the 24 hourly means used for the short-period components. Then, treating the components Mm and Sa the same as the diurnal tides and the components Mf, Msf, and Ssa as the semidiurnal tides, the form may be followed except that the log- augmenting factor must be taken from Table 20 and then combined with items (46) and (48) to obtain item (49), striking out item (47). 152 U. S. COAST AND GEODETIC SURVEY. To obtain Sa and Ssa from the monthly means of sea level, or tide level, determined from the first 29 days of each calendar month, the following process may be used: Enter the monthly means beginning with that for January in alternate spaces provided for the hourly means on the front of Form 194, placing the value for January in the space for the 0 hour. For convenience consider all the intermediate blank spaces as being filled with zero values and make the computations indicated by (3) to (12) and (18) to (21). Correct the coefficients of s, and c, from 12 to 6, at top and foot of columns (9), (12), (19), and (21). In bottom of form enter Sa in column having subscript 2 and Ssa in column with subscript 4 in order to obtain correct augmenting factors and strike out numerals indicating subscripts. For (38) and (39) take the logarithm of twice the values of 6s and 6c as obtained above. The ¢’s as obtained from (40) must have the following corrections applied in order to refer them to 0 hour of the first day of January—common years, Sa correction = — 13.38°, Ssa correction = — 26.76°; leap years, Sa cor- rection = — 14.20°, Ssa correction= —28.40°. For convenience in recording the results it is suggested that the ¢ as directly obtained from. (40) be entered (in its proper quadrant) in the space just below the logarithm from which it is obtamed, and that the ¢ corrected to ‘the first day of January be entered in the same line in the vacant column just to the right. The V+4u, computed to the first day of January, may then be entered immediately under the corrected ¢’s and the x’ of (43) readily obtained. For (49) the combination (46) + (47) will be used Form 452, R, «, and § from analysis and inference (figs. 30 and 31)— This form provides for certain computations preliminary to the regular elimination process. The constants for components K, and S, as obtained directly from Form 194 may be improved by the application of corrections from Tables 21 to 26; and constants for some of the smaller components, which have been poorly determined or not determined at all by the analysis, may be obtained by infer- ence. If the series of observations is very short, the inferred values for the constants of some of the components may be better than the uneliminated values from Form 194. Form 452 is based upon section 29. It is designed to take account of the diurnal component on one side (fig. 30) and the semidiurnal components on the other side (fig. 31). The amplitudes and epochs indicated by the accent (’) are to be taken from Form 194 and the quantities indicated by the asterisk (*) from Form 244 or 244a. If the series is less than 355 days, values for S, and 25M may be omitted. For all short series the values in columns (4) and (8) are to be computed in accordance with the equivalents and factors in columns (3) and (7), respectively. If the series is 192 days or more in length, the « of M,, P,, and K, for column (4), and the log R of M,, P,, and K, for column (8) may be taken directly from Form 194, and if the series is 355 days or more in length the « and log F of all the com- ponents for which analyses have been made may be taken directly from the same form. When a value is thus taken directly from the analysis, the corresponding equivalent in column (3) and factors in column (7) are to be crossed out. The tabular values of items (12) and (13) for the diurnal com- ponents and items (14) to (18) for the semidiurnal components may HARMONIC ANALYSIS AND PREDICTION OF TIDES. 153: be obtained from Tables 21 to 26 or from plotted curves representing these tables, but for a series of 355 days or more in length the accel- erations may be taken as zero and the resultant amplitude factors as unity. Form 452 DEPARTMENT OF COMMERCE UW. S, COAST AND GEODETIC SURVEY TIDES: f, «, AND s, FROM ANALYSIS AND INFERENCE. Station _.......! MOLTO SMCALSTOLTIE smitty 1s WhOye MNES ALLEN EAU EME ENR AS ASE Length of Series ........... 283.0. DIURNAL COMPONENTS. days. Series begins ......... 1919, February 13... log. F(J,)* 2 From Anatyeis AND INFERENCE. S| reow anatvats anv Drrezence. oO °° a =(4)— a = tus [=O] 2 E pm Ged (6) g @ 5 5 © (0 dec.) o Factors. (4 dec.) log. 0.079” log. H/(Q,) log. R(J,) +8.8976 +9.7814 aoe etl RG) Kj+(14).......--..|_ 12805.) 328556, _3 log. R/(K,) kee ---AhOD | 30608.) Jog. (03) 0128 |. 93.5 | 255.9 | 198 _ log. R(K,) 9.9724 R(K,) Saale ras| er were epee ‘Tog.0.071 | + 8.8513) ag | ey log. R(O,) | + 9.7550 | ----|-----= 9-4 --} -- 2 FS --|------' log. Q* = gs agee te og. ROM, e | (9)=P*=___ 93095 (2 dec.); (10)=F(K,)*= 22988 (3 dec.); (11)=(h-4/)*= S2L(0 dec.) i APA (12)=aeceleration in K, due to P,=F(K,)xTable 21=...22}._____(1 dec.) “0, | log R’(0,) | 9.7550 (18)=resultant amplitude, XK, and P,=1+4[F(K,) x (Table 22 ]=.....22.03__(2 dec.) " IR(0,)=R10,) (14) =(Kj}-0)=..11.6__(1 dee.) 00 | tog. 0,043 — 0.0929 8.3220 Tae +9.5198 $+ 9.9724 log. F(K,)* | +0.0160 log. R(P,) 9.5082 RE) log. 0.194 +9.2878 jog. RCO) _| + 967550 | log. R(Q;) 9.0428 R(Q,) log. 0.026 +8.4150 log. R(Q,)__|_+ 97550. log. R(2Q) 8.1700 R(2Q) log. R/(S;) ee R(S,)=R(S,) Jog.0.038 | +8.5798 log. R(O;) +9.7550 log. R(p:) 8.3348 R(o;) Date Computed by.1:A.Al pert, Fab.28,1921' Duplicated by La P. Disney, teb.28,1921 Fig. 30. The «’s of K, and 8, are to be corrected by the accelerations as indicated before entering in column (4), and in computing item (14) for the diurnal components and (21) for the semidiurnal com- ponents the corrected x’s are to be used. If the two angles in item (14) for the diurnal components, or in items (20) or (21) for the semidiurnal components, differ by more than 180°, the smaller angle 154 U. S. COAST AND GEODETIC SURVEY. should be increased by 360° before taking the difference, which may be either positive or negative. In computing column (8) it will be noted that the corrected log R’s of K, and 8, are to be used in inferring other components depend- ing upon these. DEPARTMENT OF COMMERCE U. S. COAST AND GEODETIC SURVEY TIDES: #, «, AND s, FROM ANALYSIS AND INFERENCE, Station _.MOrro, California week 8 8 SE IT nn nn nn nn nnn nn en nnn nnn nnn nee nn nn eer rce cane wene non ene nne ens ceemnenewennnn--cccwwees own menesseseccesense “Length of Series -...........--- GS in ees days. Series begins. 1919, February 13,000 : SEMIDIURNAL COMPONENTS. From ANALYSIS. From ANALYSIS AND INFERENCE: FRoM ANALYSIS AND INFERENCE. : 4 R vv K ig =(4) wen (6) g (1) (2) (3) (4) (7) a | Factores log. 0.272 log. R(S.) log. F (K,)* log. R(K,) __ RK) _| +9.4346 +9 .5017 —0 .0472 8-881 2 a A 2 Ft. (8 dec.) | °(2dee.) || 3 Equivalent. ° (1 dec.) K, TS + 9.1553 | +9 .4201 =9 .9657_ 8.6097 log. R/(M;) ROL)=R/(M,) log. R’(Nas wy |..O0O18. |. 99,2 (9)H1*=... 2h 76._(2 dec.); (10)=P*= 53-93. (2 dec.); (11) =F(K,)*_-.2. 897._(8 dec.) log. R(2N) (2)=(h—9 = BOD (0 dec.); (03)=(h=p,)*= CS aay dec.) |_—_|__2@N) (14)=acceleration in S, due to K,=Table23- Xf(K,)=.....ba4____.°(1 dec.) Ry | log. 0.008 (15)=acceleration in S, due to T,=Table 25, -.....--- =. 820 ._....° (1 dec.) recon HG@6)== (sea LB) ate iertoewe ee tO =..20.6 (1 dec.) "RR, _(17)=resultant amplitude, S, and K,=1+[Table 2X F(K)J= be 92. edee.) [FR Tog. R(S,) | (18)=resultant amplitude, S, and T,= — Mable veoh an ee AO NOB sie (2 dec.) 5 ~ (19) 5 (19)=og. (17)-Hlog, (18) sesso =_929955____(4 dee.) Tog. R(S;) (20)=(M3—N;)=.....240.5_ (1 dec.); (21)=(S?=Mi)=...=422___(1 dec.) R(S:) SSS SSS 555454555... gg = SE Tee Fae log. R(S3) log. R(T) R(@,) log. 0.007 log. R(M,) log. R(X3) Ra) log. 0.024 log. R(M,) log. R(t) Rus) log. 0.194 log. R(N,) log. R(vq) R(v2) 25M | log. R’(2SM) RSM) =F’ (A8M) Ha "2 Date Computed by _1+A.M vert, Fab.28,1921 Duplicated by LP. Disnay.,Fat.28,1952 Fie. 31. Form 245, elimination of component effects (fig. 32).—This form is based upon formulas (409) and (410). One side of the form is designed for the elimination of the effects of the diurnal components upon each other and the other side for use with the semidiurnal components, the two sides being similar except for the listing of the components. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 155 The symbol A represents the component to be cleared, and the symbol B is the general designation for the disturbing components. The symbol applying to component 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. The ‘‘table’’ in the headings of columns (2), (3), and (4) refers to Table 29 in this publication. For column (2) it will be found con- venient to copy the logarithms of the R’s of B from column (8) of . Form 245 DEPARTMENT OF COMMERCE ceeee meee TIDES: ELIMINATION OF COMPONENT EFFECTS HT LA Re Ort Pe eee a. a yr. m. L Length of series __ 163__.___. days: Series begins 1915, Feb. 13)... 0. | 5) 6) 8) Table t(Ba) fas @xsin (5) (ayxcas (5) RESULTS *(no dec.)} (2) (3) R(B;)XTablo| R(B;)x Table Sess |) = (9)=R’(A,) from analysis mae | (10)=(9)—(7) | (11) =Hog. (6) O00 | (12)=log. (10) (13)=(11) —(12)=Iog. tan 6 ¢ | #(14)=3 5 -| (15)=log. cos 6 ¢ 004 } (16)=(12)—(15)=Iog. R(A,) (17)=(16)-+-log. F(A)=log. H(A2) 6.2207 =| (19)=/(A,) from analysis oan (20)=(14)+-(19)=¢ (A) (21)=(20)+(V.+0)=«(A,) oho tidbit ad ed do Component 4,=..L (ay go rete) 2) eed ete S| Ue I (9)=R’(4,) from analysis SSS SSS ===) =a-===-| (10)=(9)—(7) {| (1)=log. (6) 1 _| (12)=log. (10) _| (13)=(11) —(12)=lHog. tan 3 ¢ _| *44)=3 --| (15)=log. cos 6 1 Fa aia Hodebonoonno? i OQ). | 185 1 151 | 000. = : (16)=(12)—(15)=Iog. R(A,) 8.2397... 2.3292 | 002. (17)=(16)-+Hlog. F(A3)=log. H(A) S086] 5.8! I (18)=H(A,) amex 2OZ056! $2592 | ---== |n--an-| ---=-| ---== == | (19)=1(A,) from analysis ET Sey [ora jaa | = — | (20) =(14)+(19)=F (AQ) eat Dre, (21)=(20)+(Vo-+u)=x(A2) = 307.5 Component 4,=_M3_ (9)=R’ (A,) from analysis = _1.-245 (10)=(9)—(7) = 1,246 (11)=lIog. (6) = 7.6021. - | (12)=log. (10) = 90,0955 =| (13)=(11)—(12)=log. tan 85 = 7,5066 |: wionan—.| *(14)=8 5 = mai fe | (15)=Iog. cos 8 5 = O01 | (16)=(42)—(15)=Iog. R (A,) : _| (7)=(46)-+Hlog. F(A,)=log. H(A) = 0.0887 ! | (18)=H(A2) = 227 | | (19) =2(4,) from analysis ZS: = | (20)=(14)+(19)=r (A,) “ Sums =| +0. 006 !=0, 003, | (21)=(20)+(Vatu)=«(4s) 3 f or (14) is in the Ist quadrant when (6) is + and (10) is +. Stor (14)isin tho 2d qusdrant when (6) 1s + and (10) is =. Computed by __1-_ A. Alpert. mice Qn iia diest quadrant when (6) 1s — and (10) is —. ¢ For (14) is in tho 4th quadrant when (6) 1s — and (10)is +. Ag allie - um Verified by-_-_U2_Fs Disney sys Fig, 32, Form 452 on a horizontal strip of paper spaced the same as Table 29. Applying this strip successively to the upper line of the tabular values for each component the logarithms of the resulting products for column (2) may be readily obtained. Similarly, for column (4), the ¢’s of 3Bgfrom 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 eon and the differences obtained. k« The natural numbers for column (3) corresponding to the loga- rithms in column (2) can usually be obtained most expeditiously 72934—24}—_11 156 U. S. COAST AND GEODETIC SURVEY. 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 com- DEPARTMENT OF COMMERCE Sy spr nor Bsa Lf) TIDES: STANDARD HARMONIC CONSTANTS FOR PREDICTION Approximate extreme range...._8¢8 “tt. Height dial Qe e eer tes 2 = =: Marigram ecale_-........%28¥ _ The DATUM is . @-plene ceren =f. fe =bow FWater i Mean {aetes Low Water which is 2040 __ ft. below mean sea level, and is upproximatety-the datum of Coast and Geodetic Survey the { Aopen i: German }eharte. 9.908 | 304.3). ||+. 0,.016.|...87.2). 0.274 | 107.7. +. 0.103 | 289,8 |i+ 2.0 1 0.42....|+.69.2) 290.8 0.007 |105.6 |+39,3 | 0.08 | 4225.1] 144.9. +. = Compiled by ...usPeDa...March 28,19255.._._ Verified by _PaJ.aHa__ March 29,1923, Dew Fic. 33. pohent 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. HARMONIC ANALYSIS AND PREDICTION. OF TIDES. Loa In the use of this form it will be noted that the R’s and ¢’s refer- ring to component B are to be the best known values whether derived from the analysis or by inference, but the R’ and ¢’ of component A, entered as items (9) and (19), respectively, must be the unmodified values as obtained directly by Form 194. Form 444, standard harmonic constants for predictions (fig. 33).— This form provides for the compilation of the harmonic constants for use in the prediction of the tides and also for certain permanent preliminary computations to adapt the constants for use with the U. S. Coast and Geodetic Survey tide-predicting machine No. 2. The form is used in a loose-leaf binder. The components are listed in an order that conforms to the arrange- ment of the corresponding component shafts and cranks on the pre- dicting machine. The accepted amplitudes and epochs are to be given in the columns provided for the purpose. At the bottom of the page a space is provided for indicating the source from which the con- stants were derived. The column of Remarks provides for miscellaneous information pertaining to the predictions. This includes the kind of time in which the predictions are to be given—the approximate extreme range of tide at the place for determining the proper scale to be used, the height dial, the marigram gear, the marigram scale, and the datum to which the predicted heights are to be referred. The extreme range may be estimated from the predictions for a preceding year or may be taken approximately as twice the sum of the amplitudes of the harmonic constants. The height dial, mari- gram gear, and marigram scale which are recommended for use with different extreme ranges are given in the table below. Working scale, height dial, marigram gear, and scale. f ! | (1) (2) (3) (4) (5) (6) (7) (8) F . Mari- Mari- Tide f 3 : Mari- Mari- Extreme range (feet). Worng aeigit gram gram ideh ‘ar rae Seats scale. tide. tide. gram. Feet. Feet. Inches. HD Oa = ol or a a 10 4 24eM| 6 0.5 3.0 2.0 GATS TGVET LS ey See Ee pe es a 10 4 Sine 1:8 0. 67 4.0 15 Bra CLG. ue ea ae 10 4 eat The 14 1.00 6.0 1.0: CARS TADS SS eee ere 4 10 2 1:15 1.25 050) 0.8 (ick la ELS tes yh pa Mek et a Pa 4 10 274 1:20 1.67 10.0 0. 6: OLG 1 Oso wa are es fee ek iva ie 4 10 eal 1:30 2. 50 15.0 0.4 TO:GS grasa ser eee eh eS yiee 2a 2 20 er! 1:30 2.50 15.0 0.4 I TE eis a 5 ee Oe 2 20 oe 1:40 3.33 20.0 0.3 LO 6— ZO" Sena eal Os lee 1 Ry 20 eh 1:60 5. 00 30.0 0.2 P20 1G 29. tetera mea ay lo 1 40 221 1:60 5.00 30.0 0.2 296-39! bu eae a ET ea ey Oe Ne 1 40 32% 1:80 6. 67 40.0 0.15 DONG aise meena emir waberS Co cn 1 40 Vy 1: 120 10. 00 60.0 0.1 Pars (1), or extreme range, is the argument for determining the working scale and marigram gear for each station. Column (2) contains the factor to reduce the amplitudes of the harmonic constants to the working scale. Columns (3) and (4) contain the height dial and marigram gear that should be used for ranges of tide within the limits indicated in column (1). Column (5) gives the height scale of the tide curve. Column (6) shows the amount of rise or fall of the tide represented by each inch, vertical measure, on the marigram. 4 Column (7) gives the extreme range of tide that can be represented on the marigram, which is 6 inches wide, with the different marigram scales. Column (8) shows the amount of rise and fall on the marigram for each foot of actual tide. 158 U. S. COAST AND GEODETIC SURVEY. The principal hydrographic datums in general use are as follows: Mean low water for the Atlantic and Gulf coasts of the United States and Porto Rico. Mean lower low water for the Pacific coast of the United States, Canada, and Alaska, and the Hawaiian and Philip- pine Islands. Approximate low water springs for the rest of is world, with a few exceptions. For use on the predicting machine the datum must be defined by its relation to the mean sea level, and this relation is usually deter- mined from a reduction of the high and low waters. Column A of Form 444 is designed for the differences by which the epochs of the components are adapted once for all for use with the unmodified Greenwich (V,+4)’s of each year. These differences take account of the longitude of the station and also of the time meridian used for the predictions, and are computed by the formula = pL—-% (492) in which x’ —x=adapted epoch—true epoch. p=subscript of component, which indicates number of periods m one component day. For the long-period components Mm, Ssa, Sa, Msf, and Mf, » should be taken as zero. L=\longitude of station in degrees;+if west,—if east. a=speed of component in degrees per solar hours. S=longitude of time meridian in degrees;+if west,—if east. The values of the products & for the principal time meridians may be taken from Table 35. For any time meridian not given in the table the products may be obtained by direct multiplication, taking the values for the component speeds (a) from Table 3. Column B is designed for the reduction of the amplitudes to the working scale of the machine. The scale is unity when the 40-foot height dial is used, 2 for the 20-foot height dial, 4 for the 10-foot height dial, and 10 for a 4-foot height dial. The working scale should be entered at the head of the column and used as a factor with the amplitudes in order to obtain the values for this column. Columns C and D are designed to contain the adapted epochs in positive and negative forms which may be used additively with the Greenwich (V,+4u)’s. It will be found most convenient to compute column D, first, by applying the difference in column A to the « in the preceding column and entering the result with the negative sign. If the direct application of the difference should give a negative result, this must be subtracted from 360° before entering in column D. The values for column C’ may then be obtained by applying 360° to the negative values in column D. Form 445, settings for tude-computing machine (fig. 34).—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 or convenience the order of arrangement of the components is identical in the two forms. The name of the station, the time merid- HARMONIC ANALYSIS AND PREDICTION OF TIDES. 159 jan, the height dial, marigram gear, marigram scale, and datum plane are copied directly from Form 444. 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 DEPARTMENT OF COMMERCE ee Sorm No. 445 TIDES; ‘SETTINGS FOR TIDE COMPUTING MACHINE Sration _...Morro, California. Compo- AMPLITUDE DIAL SETTING NENT || SETTING | Jan. 1, 04| FEB. 1, 0" [Dzc. 31,24" Resoer Vln: lea lead Lie Gl Agana Samer Lilhe ye MRE A The datum is WpHME nena eee = Lose Vater. 5 : et HES fae a } which is... 2240... .. feet below mean sea level, and is appresimatel the datum of ‘Coast and Geodetic Survey . the {RoiSivadey Soaks ncorman} charts. ane) Osco aap Nest ce -- t= 2th .,{ SE ._.° SPITE _---}) -- aie 4 : abe | Machine settings computed by ..._.. LePeDe March 28, 1925 Verified by___FadiesHa_Marah 29, Why AGG) PIGUICLEG! DY gerne ioa sm snes ice LE Sete og i ele ae a ee aan eee ae Sone nee enerce eee aot! i} | Datel seers cere EL AS a TB EN Te = Fic. 34. 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 160 U. S. COAST AND GEODETIC SURVEY. 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. For the dial settings for January 1, 0 hour, the Greenwich equilib- rium arguments of (V,+w)’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,+4u)’s a strip similar to that used for the factors f should be pre- ared. Thesame strip will serve for all stations for the given year. For the dial settings it is customary to use whole degrees, except for component M,, for which the setting is carried to the first decimal of a degree. The settings for February 1 and December 31 are used for checking purposes to ascertain whether there have been any slipping of the gears during the operation of the machine.. To obtain the dial set- tings for February 1, 0®, and December 31, 245, prepare strips similar to those for the f’s and (V,+4)’s. On one enter the angular motion of the components from January 1, 0" to February 1, 05; on a second and a third strip, the angular motion for February 1, 0" to December 31, 245, 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 our 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 angular changes for computing the settings for any day of the year may be obtained from Tables 16 and 17. PART II.—TABLES. EXPLANATION OF TABLES. Tasie 1. Astronomical constants and formulas.—There are given in this table some fundamental astronomical constants and formulas, which are used in the computation of other tables which follow, with references to the authorities from which they were obtained. The form and degree of precision is for the most part identical with that of the original source. It will be noted that T is the time expressed in Julian centuries reckoned forward from Greenwich mean noon on December 31, _ 1899 (Gregorian calendar), which corresponds to December 19, 1899, by the Julian calendar. (See p. 11 for an explanation of these cal- endars.) By the Julian calendar the date corresponding to an inte- gral value of T will always be December 19 of a year ending with the figures 99 A. D. or 02 B. C.; for example, by the Julian calendar, T=—1 on December 19, 1799; —2 on December 19, 1699, etc. The Gregorian calendar was first introduced in 1582. By this calendar, T=—3 on December 29, 1599; —2 on December 29, 1699; —1 on December 30, 1799; 0 on December 31, 1899; +1 on January 1, 2000; +2 on January 1, 2100; +3 on January 2, 2200; etc. In the formulas for the true longitude and distance of the moon the notation has been changed in order to be in accord with the notation of the present volume, and the terms not used here have been omitted. The terms containing k? were for the reduction to the ecliptic and have been omitted here because it is desired to repre- sent the position of the moon in its orbit rather than in the ecliptic. The longitude in the orbit is referred to an origin (’ of fig. 6) selected so that the longitude of the moon’s nodes will be identical in either the ecliptic or the moon’s orbit. TABLE 2. Astronomical quantities and relations—The values com- piled in this table for convenience of reference are based upon Table 1. The mean longitudes of the lunar and solar elements and also the rate of change in these elements are derived from formulas of Table 1. The rate of change, although computed for the epoch January 1, 1900, will apply without material error to all modern times, since the variations in the rates for all the elements are very small. The inclination of the earth’s orbit to the ecliptic changes about 0.013 of a degree in a century. The value computed for epoch January 1, 1900, may therefore be used without material error for all modern times. The inclination of the moon’s orbit to the ecliptic is regarded as an absolute constant. The eccentricity of the earth’s orbit changes about 0.000042 per century. The value as computed for the epoch January 1, 1900, may therefore be used as a constant. The mean value of the eccentricity of the moon’s orbit is also used as a constant. The. mean radius of the earth is taken as the cube root of the product of the polar radius and the square of the equatorial radius, this being the radius of a sphere having the same volume as that of 161 162 U. S. COAST AND GEODETIC SURVEY. the earth. In expressing this radius in feet the result is rounded off to the nearest hundred, since a greater precision is not warranted by the data from which it is obtained. The numerical values of several other important quantities that appear in the text are also included in Table 2 for convenience of reference. TaBLE 8. Principal harmonic components.—This table gives a list of the principal harmonic components used in the prediction of the tides. The symbol by which each component is generally designated and a brief description suggesting the derivation of the component are given in the first and second columns, respectively. A general discussion of the argument (V+) will be found in section 10. The formulas for these arguments are derived from formulas (100), (176), (190), (194), (208), (215), (228), and (230) in the text. References to the overtides, compound tides, and meteorologi- cal tides will be found in sections 18, 19, and 20. The speed, or average rate of change in the argument, in general, depends entirely upon that part of the argument designated as V, the wu being an inequality that does not affect the average rate of change. The speed formulas are readily derived from formulas for V by substituting for the variable elements T, h, s, p, and p, the corresponding hourly rates of change in these elements, represented by 6, n, o,@, and a@,, respectively. The value for @ is 15°, this being the hourly rate of change in the hour angle of the mean sun. The values for the other elements may be obtained from Table 2, and by substituting in the formulas the corresponding numerical values for the speeds of the components are readily obtained. An explanation of the double expression for the argument for component M, will be found in section 14. The coefficients are discussed in section 11. The coefficient for- mulas of Table 3 are derived from formulas (100), (176), (190), (194), (208), (215), (228), and (230) of the text. In the coefficients of the solar components the factor G has been introduced in order that the 3 general lunar coefficient, a (¢) ax (function of \) may be used as a common coefficient factor for both the lunar and solar components. The mean values of the coefficients are obtained by mutliplying the constant factors by the mean values of the variable factors. The numerical values of the constants are given in Table 2, and the mean values of the variables, which depend upon some function of J, are given in the formulas indicated by the references. The mean value of the coefficient does not include the general coefficient. For the evectional and variational components »,, A,, M2, and p,, two mean values are given for each coefficient. The first is that derived from the given formula and the second is a value obtained by Prof. J. C. Adams, who was associated with Sir George H. Darwin in the investigation of the Harmonic Analysis of Tidal Observations, and who in his computations carried the development of the lunar theory to a higher order of precision than is provided for in this work. (See pp. 60-61 of Report of British Association for the Advancement of Sciences for the year 1883.) The second value may therefore be resumed to be a more precise determination of the mean coefficient or each of these components. HARMONIC ANALYSIS AND PREDICTION OF TIDES. 163 The factors F for reduction have been compiled from the formulas indicated in the column of references. The table includes the cor- rected factor for the component M,. The factor in general use for this component is represented by formula (204) on page 52. TaBLeE 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 (A), 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 2, 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 theselimits. 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. 11 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 givenin Table 2, 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 pean 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 ut of time the equivalent Greenwich hour must be used for the table. TaBLE 6. Values of I, v, é, v', and 2v'’ for each degree of N.—This table has been computed for epoch January 1, 1900, using the con- stants of Table 2, but the tabular values are applicable without material error to any series of observations within modern times. mae following formulas were used in the computations for this table: Cos I=cos 7 cos w—sin 7 sin w cos N = 0.91369 — 0.03569 cos N (103) of p. 41. cos 4(w—2) Wane — Ey) = — tan $N (104) of p. 41 4(w+7) MOI AS Tan H(W-g-1) =5E eee tan 4N (108) of p. 41 cos $(w—7%) | : sin $(w—1) _ in vy sin 27 eae ae sin y sin : 2+38e7/e\r 8. cos v sin 2+ 53a (¢) 77 sin 20 sin v sin 27 Gos v sin 27-£0.3357 (229) of p- 57 164 U. S. COAST AND GEODETIC SURVEY. sin 2y sin? J (Mies ea ii fee 2p sin? Tae etna (2) s sin? w 2+3e \c,/ M as sin 2v sin? J cos 2v sin? J+ 0.0728 231) 01 (p.. Of TaBLe 7. Values of log Ra for amplitude of component L,.—This table has been computed by means of the following formula: R,=[1—12 tan? 47 cos 2P +36 tant $/]-? (177) of p. 48. The argument P=p—é. The value of p and WN for any date between 1800 and 2000 may be taken directly from Tables 4 and 5 ae values of € and J as functions of N may be obtained from able 6. TaBLe 8. Values of R for argument of component L,.—This table has been computed by the following formula: 6 sin 2P oe Ho ae 47—6 cos 2P’ (178) of p. 48. The argument of P=p—é. The values of p and N for any date between 1800 and 2000 may be taken directly from Tables 4 and 5, ea values of € and J as functions of N may be obtained from able 6. TaBLe 9. Values of log Q. for amplitude of component M,.—The formula upon which this table is based is Q,=([5/2 —9/2 tan? 47+ 9/4 tan‘ 47+ 3/2(1 —tan? $2) cos 2P]-}, (191) of p. 50. Attributing to J its mean value (w=23.°452) from Table 2, the above may be written Q.=[2.310 + 1.435 cos 2 P]-. By means of the latter formula a single argument table of the log- arithms of Q, has been prepared, which is applicable without serious error to any series without regard to the exact value of J. TaBLE 10. Values of Q for argument of component M,.—The formula for which this table is based is 2—3 tan? 4/ 4—3 tan? 4] Attributing to J its mean value (w=23.°452) from Table 2, the above formula may be written Tan Q=0.483 tan P. an — tan P (195) of p.51. By means of the latter formula a single argument table has been prepared which is applicable without serious error to any series with- out regard to the exact value of J. TaBLE 11. Values of wu for equilibrium argument.—The values for the table have been computed for each degree of NV by the formulas HARMONIC ANALYSIS AND PREDICTION OF TIDES. HD for this argument given in Table 3, the elements of the formulas being functions of N, which are given in Table 6. The w’s of com- ponents L, and M, are functions of both N and P. The uw of L,= u of M,—(R from Table 8) and the wu of M,=4 (uw of M,)+(@ from Table 10). For any time between the years 1900 and 2000 the w’s of L, and M, may be conveniently obtained from Table 13. TaBLE 12. Log factor F' for each tenth degree of I.—This table is based upon the formulas for factor F'in Table 3. The factors for L, and M, being functions of both J and P are not included directly in the table. These may be obtained from the following formulas: log F (L,) =log F (M,)+ (log R, from Table 7) [See (186) p. 49.] log F (M,) =log F (O,)+ (og Q, from Table 9) [See (206) p. 53.] Yor any time between the years 1900 and 2000 the factors for L, and M, may be conveniently obtained from Table 13. It will be noted that the above formula for log F’ (M,) accords with the factor gen- erally used in practice rather than the theoretically correct factor, as indicated in Table 3 (see p. 52). TABLE 13. Values of u and log factor F for components L, and M,.—This table includes the values of wu and log F for components L, and M,, for each 5° of N for the years 1900 to 2000, inclusive. The values are based upon the following formulas: u of L,=2&-2v—-R=(u of M,) -—R p. 48. u of M,=&—-v+Q=4(u of M,)+Q prale Ff (L,) =F’ (M,) x Ba (186) of p. 49. F (M,) =f (O,) xX Q. (206) of p- Doe The values of u of M, may be taken from Table 11, F (M,) and F (O,) from Table 12, R from Table 8, Q from Table 10, R, from Table 7, and Q, from Table 9. The factors F of M, represented in this table are those which have heretofore been in general use (see p. 52). For the corrected F (M,) the tabular logarithms should be increased by 0.1523, which is the logarithm of the correcting factor 1.42. TABLE 14. Factor f for middle of each year 1850 to 1999.—The factor f is the reciprocal of factor Ff. 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. TaBLeE 15. Equilibrium argument (V.+u) for beginning of each year 1850 to 2000.—The equilibrium argument is discussed in section 10. The tabular values are computed by the formulas for the argument in Table 3, 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 T 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 N 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 montb, day, and hour, and are computed from the 166 U. S. COAST AND GEODETIC SURVEY. hourly speeds of the components as given in Table 3. The differ- ences refer to the uniformly varying portion V of the argument, it being assumed that for practical purposes the portion w 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, asgivenin 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 component, which represents the number of periods in a com- ponent day. West longitude is to be considered as positive and east longitude as negative, and the subscripts of the long-period components are to be taken as zero. This correction is to be subtracted. The (V,+4u) 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 w 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 the augmenting factors is given in section 27 of the text. The tabular values for the short-period components are obtained by formulas (329) and (330). -For the long-period components the augmenting factors were com- puted by formula (428). TABLES 21 To 26.—These tables represent perturbations in K, and S, due to other components. They are based upon formulas (379) to (384), inclusive. TABLE 27. Critical logarithms for Form 245.—This table was 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. Component speed differences.—The component speeds as given in Table 3 were used in the computation of this table. TaBLE 29. Elimination factors.—These tables provide for certain constant factors in formulas (409) and (410). 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: 180 sin 4(b—a)r a 4(b—a)r Second value=natural number 180 chee = positive. x 4(b—a)r First value =logarithm of always taken as HARMONIC ANALYSIS AND PREDICTION OF TIDES. 167 sin $(b—a)r Third value =4()—a)r, if is positive, 3(b—a)r or 4(6—a)r +180, if a is negative. 16 = TaB_LE 30. Products for Form 245.—This table is designed for obtaining 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 component 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 component speed is greater or less than 15p, when in the summation it is desired to assign a single solar hour to each successive component hour. For the usual summations in which each solar hour height is assigned to the nearest component hour no attention need be given to the asterisk. The table is computed by substituting successive integral values for d in formula (264) and reducing the resulting solar hour series (shs) to the corresponding day and hour. The solar hour to be tabulated is the integral hour that immediately follows the value the (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 (264), are to be considered as negative in the application of the table when the speed of the com- ponent is less than 15p. When the component 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. The following example will illustrate the use of the table: To find component 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: . Solrhaume (2° 13 14" 45. 16e 717) 18; ‘19, 20, 21%, 'doreag Difference___.—-5 -5 -5 -6 -6 -6 -6 -6 -6 -7 -7 —-7 ‘Component PO tom ame S&S. OF. G10 1 Tae 1 wa ae eS In the results it will be noted that the component 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 component OO hours corresponding to solar hours O to 18 on the 22d day of series. The O 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 168 U. S. COAST AND GEODETIC SURVEY. 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: Solarthours! 08) 20720) 3¥. a5 6h eg 8h) GLY vaON site 12h 13° 0 tI As aan seer Gc een Differences. +14, +14, +14, +15, +15, +15, +15,+15,+15, —9, —9, —9, —9, —9, —9, —9, —9, —8, —8 Component OO hours. 14/415, (116,118, )a9, , 20) ton “22 237 UN iO! Ow, te, euighh) at) 5 Ogee ee In the results it will be noted that component 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 components have been arranged in accordance with the length of the component days. TABLE 32.—Dvivisors for premary stencil sums.—This table contains the number of solar hourly heights included in each component 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—Component A is the component for which the original primary summations have been made, and component B is the component 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 (273), and the corresponding ‘‘Component A hours” from formula (271), 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 components K and R and the negative sign for components P and T. For brevity all the 24 component 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 than the difference for the given hours on that page. For an exam- ple, take the hours for page 2 for component OO as derived from component J. According to the table the difference for the compo- nent 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 component 2Q as derived from component O the three differences usually required for each page are given in full. : oe use of the table may be illustrated from the example above, as ollows: Page 2— Component J HOU Esme ee MeO cette 2 3, 4, 5 6, G 8, 9! Ova Differences___ +9, 9, 9, QO HOR AOS NOM TO ee aa e 9, 9 Component OOhours_2. 9, 40; 901, 02) 6 D4 5, 1G, ay? 825 Oe eo Component J hours) 4). Men! 12,0 V8 TRO TSS Ea VR PO N20) 2s nae Difference ____ +9, 9 9 9 Component OOMoursl seat a2 eos 0, 1, 2, ah 4, 3; 6, 7%, See ee eS ee ee ee HARMONIC ANALYSIS AND PREDICTION OF TIDES. 169 The period 24 hours should be added or subtracted when necessary in order that the resulting component hours may be between 0 and 23. TaBLE 34. For summation of long-period components.—This table is designed to show the assignment of the daily page sums of the hourly heights to the component divisions to which they most nearly correspond. The table is based upon formula (415). The compo- nent division to which each day of series is assigned is given in the left- hand column. For components Mf, MSf, and Mm there will fre- quently occur two consecutive days which are to be assigned to the same component division. In such cases the day which most nearly corresponds to the component 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 component division. For component Sa a number of consecutive days of series are assigned to each component division. In the table there are given the first and last days of each group. i a TABLE 35. Products +5 or Form 444.—This table contains the ponds of the speed of each component as given in Table 3 and the ongitude of each of the standard time meridians. These products are for use in formula (492), which gives the value 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. U. S. 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 component 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 convenience of reference, the gears being designated by the letter G and the shafts by the letter 8. 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 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. TaB_e 38. U. 8. Coast and Geodetic Survey tide-predicting machine No. 2—Component gears.—This table contains ‘ie details of the gearing from the main vertical component shafts to the individual component cranks. Column I gives the number of teeth in the bevel gear on the main vertical component shaft; column II, the number of teeth in the gear on the intermediate shaft that meshes with the gear on the vertical component shaft; column III, the number of teeth in the gear on the intermediate shaft that meshes with the gear on the component crank shaft; and column IV, the number of teeth in the gear on the component crank shaft. 170 - U. S. COAST AND GEODETIC SURVEY. For the long-period components the worm gear is taken as the equivalent of one tooth. For each of these components 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 component crank shafts except that for component Sa in which case a ratio of 1:2 is introduced. The component crank-shaft speed per dial hour for each com- | column I _ column III column II column IV Sa the product of both values appearing in each of the columns IT 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 components and the accumulated error per'year due to the difference between the theoretical and the gear speeds. For covenience of reference the table includes also the maximum amplitude settings of the component cranks. TABLE 39. Synodic periods of components.—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. ponent is equal to 30° x For the component a ee ee TABLES FOR THE ANALYSIS AND PREDICTION OF TIDES 729384—247. 12 171 TaBLE 1.—Astronomical constants and formulas. [From original sources.] Wiean distance, ieanun to, suns e ae eer = 92,897,416 statute miles.* Mean distance, earth to moon________________ = 238,857 statute miles.* Radius of earth, CQuaAtOrian teem Cee ae = 3,963.34 statute miles.* Radius of earth, 610) EN gape wk URE ear a ape eer = 3,949.99 statute miles.* Ratio of mass of moon to that of earth________ = 181454 Ratio of mass of sun to that of earth__________ = 333,432: 1.T Eccentricity of earth’s orbit =0.016,751,04 — 0. 000,041,80 7 —0.000,000,126 72.7 Eccentricity of moon’ B Orbithc nee oe Ue eae = 0.054,899,720.f Inclination of earth’s orbit to plane of ecliptic: = 23° 27’ 8. 26" —46.845’" T —0.005, 9! T?+-0.001,81'" T3.7 Inclination of moon’s orbit to plane of ecliptic =5° 08" 43. 304, 6k a Mean longitude of sun = 279° 41’ 48.04’ +129,602,768.13’’ T+1.089’’ T?.+ Longitude of solar perigee = 281° 13’ 15.0’ +6,189.03’’ T+1.63’’ T?+0.012”" 73.7 Mean longitude of moon = 270° 26’ 14.72’ + (1336 rev. +1,108,411.20’’) T+9.09’’ T?+0.006,8’’ 73. : Longitude of lunar perigee =334° 19’ 40. 87" (11 rev. +392,515.94'’) T —37.24'’ T?—0.045”" T3.f Longitude of moon’s node = 259° 10’ 57.12’’ —(5 rev. + 482,912.63’’) T7+7.58’’ T2+0.008’’ 73. True longitude of moon in its orbit (in radians) =mean longitude (in radians) +2e Sin |(s =) 2 62 sun 2 (sp) 252 eae (elliptic inequality). ee me sin (s— Qh St ciD)) See a wa ee (evectional inequality). im? Sin 2 (Si 11) see esa. Sn (variational inequality). Reciprocal of true distance of moon from earth =reciprocal of mean distance ae € cos (s—p) +a’e? cos 2(s—p)__-------- (elliptic inequality). +42a’me cos (s —2h+p)_-------__--------- (evectional inequality). --a’ amt COS; 2 (S'--h) eee 2 ele (variational inequality).|| T=Number of Julian centuries (36525 days) from Greenwich mean noon on December 31, 1899. e = Eccentricity of moon’s orbit. s = Mean longitude of moon. p = Mean longitude of lunar perigee. h = Mean longitude of sun. ™m = Ratio of mean motion of the sun to that of the moon. a’ = Reciprocal of product of moon’s mean distance by (1—e?). * American Ephemeris for year 1923, p. xvi. + Astronomical Papers for the American Ephemeris, by Simon Newcomb: Vol. VI, pp. 9-11, and Vol. ae pt. 1, p. 224. ¢ The Solar Parallax and Related Constants, by William Harkness, p. 140. i An Elementary Treatise on the Lunar Theory, by Hugh Godfrey, 4th ed., p. 53. 172 HARMONIC ANALYSIS AND PREDICTION OF TIDES. _TaBLe 2.—Astronomical quantities and relations. [Derived from Table 1.] 173 Mean longitude of lunar and solar elements. Epoc (Gregorian calendar, Greenwich mean civil time). Lunar Solar ; Moon. | perigee. | SU2- | perigee. s Pp Pi ° °o °o ° SOO Sane MO NOUN seca satan case ue eee Woe cise so oelotels 342.313 | 225.453 280.407 | 279.502 HOO Waele OMOUT Sete ones oak oceans cece Se becuse see 277.026 | 334.384 | 280.190 | 281.221 ZOOM anegi OMMOUIere es. Noel. kee See la 211. 744 83.294 | 279.973 | 282.940 Moon’s Rate of change in mean longitude (epoch, Jan. 1, 1900). node. N ° 33. 248 259. 156 125. 069 Juli. Per é Per Per Per an century common year (86525 days). (365 days). solar day. solar hour. ° ° ° ° ING TA, egal Oy eget ae 1336 7.+307.892 | 13 r.+129.384,82 | 13.176,396,8 | c=0. 549, 016,53 Hunan perizes -./2.2.52 5.25 ...25-2- 11 r.+109. 032 40.662, 47 0. 111, 404,0 | w=0. 004, 641,83 AS] one se PU Bak Oe ee 1007.+ 0.769 359. 761, 28 0. 985, 647,3 | n=0. 041, 068,64 Solar perigees 0). Bi 1.719 0. 017, 18 0. 000, 047,1 |m1=0.000, 001,96 Moon’s nodes se fe sos2s. oe ee —(5 7.+134. 142) —19.328,19 | —0.052,953,9 | —0.002, 206,41 NotTE.—r=1 revolution or 360°. u Inclination of earth’s equator to the ecliptic (w), epoch Jan. 1, 1900....................-2....-.- =23. 452 iuchnationvonmoon7s orbit tothe ecliptie:(@)..-- tase: ee eee ee ee = 5.145 HECEHLIC Ay OteAtLAySIOL bit, (1) epoch gant), 1900) 22 5526-2 a sa e8 An Semel eae eis see = 0.01675 PIECE TIERICINVAOle MOONS] OUDIt (6) seo eccminae Mas tela antes cee eae eeenic eee ecco an oe enmon eames = 0.05490 Mennmiradiasyoneart il (@) ths eho sass con sacccces ose ooeksls ween os% =3,958.89 statute miles= 20,902,900 feet mean radius of earth (a) Mass of moon (M),. ( Mass of earth (£) mean distance of moon (c) 3 ) =0.000,000,055,900 Mass ofsun___(S) mean radius of earth (@) \3_ Mass of earth (#)’*\ mean distance of sun (¢) =0.000, 000,025,806 Mass ofsun (3S) ee distance of moon (c) Si a Mass of moon an* mean distance of sun (¢;) gv lbs ! 4 4 =x(¢) =0.000,000,000,926,51; 5x( =) =0.000,000,000,001,10 rate of change in longitude of sum (7) rate of change in longitude of moon (c) e=0.003014 é*;= 0.000281 sin w=0.39798 COs w=0.91739 sin 7=0.08968 cos i= 0.99597 i Rr : Let witein sin? w= 0.15839 cos? w=0.84161 sin? 7= 0.00804 cos?i= 0.99196 m?s=0.005,596 sin 40=0.20323 COS 4w= 0.97913 sin 2w=0.73021 cos 2w= 0.68322 =0.074, 804=m me=0.004,107 sin? 4w=0.04130 cos? 4m= 0.95870 sin‘ 40=0.00171 cos! $0=0.91910 4 wise ot cf 19d (68%) gg d (291) (291) (691) (£02) (L91) (891) (L191) G57) 9¢ d gg ‘d gg d (G91) (¢9T) (G9T) (G91) (981) (G9T) (S9T) U. S. COAST AND GEODETIC SURVEY. 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S. COAST AND GEODETIC SURVEY. Taste 4.—Mean longitude of lunar and solar elements af January 1, O 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; pj=mean'longi- tude solar perigee; V= longitude of moon’s node.] Year. s p h pi N Year. S Sire gun Di N ° ° ° ° ° ° ° | ° ° ° 1800..... 342.31 | 225.45 | 280.41 | 279.50 Sah Ae dete y= = 5 = 28. 44 | 181.24 | 279.82 | 280. 40 107. 55 iSO as 111.70 | 266.12 | 280.17 | 279.52 IBEC PEM eae ee 171.00 | 222.02 | 280.57 | 280. 41 88. 16 TSO2s2- 241.08 | 306.78 | 279.93 | 279. 54 354. 59 || 1854..... 330. 38 | 262.68 | 280.33 | 280.43 68. 84 TOS eee 10. 47 | 347. 44 | 279.69 | 279.55 38020) Leooeene = 69.77 | 303.34 | 280.09 | 280.45 49. 51 1804..... 139. 85 28.10 | 279.45 | 279.57 Bl5293e| saben 199.15 | 344.00 | 279.85 | 280. 46 30.18 TRO See 282. 41 68. 88 | 280.20 | 279. 59 296. 55 |), 1857... .. 341. 72 24.78 | 280.60 | 280. 48 10. 80 1806..... 51.80 | 109.54 | 279.96 | 279.61 277.23 || 1858... . 111.19 | 65.44 | 280.36 | 280.50 351. 47 USOT 181.18 | 150.20 | 279.72 | 279. 62 257.90 || 1859.._.- 240,49 | 106. 10 | 280.12 | 280.52 332.14 1808... -- 310.57 | 190.86 | 279.48 | 279.64 238. 57 || 1860....- °9. 87 | 146.77 | 279.88 | 280.53 312. 81 1809..... 93.13 | 231.64 | 280.23 | 279.66 219.19 || 1861.._.- 152.43 | 187.54 | 280.63 | 280.55 293. 43 . PSION: 292.51 | 272.30 | 279.99 | 279.67 199. 86 || 1862..... 281. 82 | 228.20 | 280.39 | 280.57 274. 10 Teil 351.90 | 312.96 | 279.75 | 279.69 180.53 || 1863..... 51.20 | 268.87 | 280.15 | 280. 58 254. 78 1S 121.28 | 353.63 | 279.51 | 279.71 161. 20 || 1864..... 180.59 | 309.53 | 279.91 | 280.60 235. 45 STS see 2 263.84 | 34.40 | 280.26 | 279.73 141.82 || 1865.._.. 323.15 | 350.30 | 280.66 | 280.62 216. 07 TV BBE OR: 75.06 | 280.02 | 279.74 122. 49. || 1866..... 92.53 30. 96 H 280. 42 | 280.64 196. 74 Te Sees 162.61 | 115.73 | 279.78 | 279.76 103.17 ,}) 1867....- | 221.92 71.63 | 280.18 | 280.65 177. 41 181622 5.2 292.00 | 156.39 | 279.54 | 279.78 83. 84. || 1868... - 351.30 | 112.29 | 279.94 | 280.67 158. 08 TSU =e Be 74.56 | 197.16 | 280.29 | 279.79 64. 46 || 1869..... 133. 86 | 153.06 | 280.69 | 280.69 138. 70 TSU hese 203. 94 | 237.82 | 280.05 | 279. 81 45.13 |} 1870_.... 263. 25 | 193.73 | 280.45 | 280.71 119, 37 1819..... 333. 33 | 278.49 | 279.81 | 279.83 25. 80 || 1871...:.| 32.63 | 234.39 | 280.21 | 280.72 100. 04 » ARP eee 102.71 | 319.15 | 279.57 | 279. 85 6547 7) 1872... 162.02 | 275.05 | 279.97 | 280.74 80. 72 ihe PA eee 245.28 | 359.92 | 280.32 | 279.86 347.09: || 1873... - 304. 58 | 315.83 | 280.72 | 280.76 61.34 PR22e ues 14. 66 40.59 | 280.08 | 279.88 327.76 || 1874..... 73.96 | 356.49 | 280.48 | 280.77 42.01 SPRY a 144. 04 81.25 | 279.84 | 279.90 308. 43 || 1875..... 208. 35 37.15 | 280.24 | 280.79 22.68 1824: 25% 273.43 | 121.91 | 279.61 | 279.91 289.11 || 1876.4... 332. 73 77.81 | 280.01 | 280. 81 Sa 1825.2 256 55.99 | 162.69 | 280.35 | 279.93 269. 72: || 1877-.... 115. 29 | 118.59 | 280.75 | 280.83 343. 97 TR26525. 2 185.38 | 203.35 | 280.11 | 279.95 250. 40 || 1878...-- 244.68 | 159.25 | 280.51 | 280.84 324. 64 ake Rae 314.76 | 244.01 | 279.87 | 279.97 231. 07.}| 1879..... 14.06 | 199.91 | 280.27 | 280.86 305. 31 1282-525 84.15 | 284.67 | 279.64 | 279.98 211.74 || 1880..... 143. 45 | 240.58 | 280.04 | 280.88 |} 285.98 L829 sees 226.71 | 325.45 | 280.38 | 280. 00 192. 36 || 1881..... 286.01 | 281.35 | 280.78 | 280.89 266. 60 eSO Ree 356. 09 6.11 | 280.14 | 280. 02 173.03 || 1882..... 55.39 | 322.01 | 280.54 | 280.91 247. 28 ESTA 125. 48 46.77 | 279.91 | 280.03 153.70. || 1883... - 184. 78 2.67 | 280.31 | 280.93 227. 95 TERRA ees 254. 86 87. 43 | 279.67 | 280.05 134. 37 || 1884... .] 314.16 | 43.34 | 280.07 | 280.95 | 208.62 TS33 oe ce 37.42 | 128.21 | 280.41 | 280.07 114.99 || 1885..... 96. 72 84.11 | 280.81 | 280.96 189. 24 ibchy. eee. 166.81 | 168.87 | 280.18 | 280.09 95. 66. || 1886..._-. 226.11 | 124.77 | 280.57 | 280.98 169. 91 1R35L 2 296.19 | 209.53 | 279.94 | 280.10 ONS A WI SsT ese 355. 49 | 165.44 | 280.34 | 281.00 150. 58 183622222 65.58 | 250.20 | 279.70 | 280.12 DG Ole |] L888 S55 02 124.88 | 206.10 | 280.10 | 281.01 131. 25 TSS (ae ee 208. 14 | 290.97 | 280.44 | 280.14 37.63 || 1889...--. 267. 44 | 246.87 | 280.84 | 281.03 111, 87 1838s eee 4 337.52 | 331.63 | 280.21 | 280.16 18. 30 |} 1890..... 36.82 | 287.54 | 280.61 | 281.05 92. 54 1839... 2: 106. 91 12.30 | 279.97 | 280.17 358.97 || 1891..... 166. 21 | 328.20 | 280.37 | 281.07 73.22 W840 22. 236. 29 52.96 | 279.73 | 280.19 339. 64 || 1892..... 295. 59 8.86 | 280.13 | 281.08 53. 89 184 ee 18. 85 93.73 | 280.48 | 280.21 320. 26 || 1893..... 78. 16 49.63 | 280.87 | 281.10 34, 51 18427 ee 148. 24 | 134.39 | 280.24 | 280.22 300.93 || 1894.._.. 207. 54 90.30 | 280.64 | 281.12 15.18 1843.....- 277.62 | 175.06 | 280.00 | 280.24 | 281.61 |} 1895#.._.| 336.93 | 130.96 | 280.40 | 281.13 355. 85 US44 ea 47.01 | 215,72 | 279.76 | 280.26 262. 28 |} 1896..... 106.31 | 171.62 | 280.16 | 281.15 336. 52 1845..... 189.57 | 256.49 | 280.51 | 280. 28 242,90 || 1897..... 248. 87 | 212.40 | 280.91 | 281.17 317.14 1846..... 318.95 | 297.16 | 280.27 | 280. 29 223. 57 || 1898..... 18. 26 | 253.06 | 280.67 | 281.19 297. 81 W847 52. 2% 88. 34 | 337.82 | 280.03 | 280.31 204. 24 |} 1899..... 147. 64 | 293.72 | 280.43 | 281.20 278. 48 1848..... 217. 72 18.48 | 279.79 | 280.33 184, 91 1849..... 0. 28 59. 26 | 280.54 | 280. 34 165. 53 Teo0sseee 129.67 | 99.92 | 280.30 | 280. 36 146. 20 aie eee 259.05 | 140.58 | 280.06 | 280.38 126. 87 i . HARMONIC ANALYSIS AND PREDICTION OF TIDES. Kit TaBLE 4.—Mean longitude of lunar and solar elements at January 1, 0 hour, Greenwich mean civil time, of each year from 1800 to 2000—Continued. rie Yeas. s p h Di N Year. Ss Dp h Pi N ° ° ° ° ° c) ° ° ° ° 1900... .. 277.03 | 334.38 | 280.19 | 281.22 | 259.16 |} 1952..... 323.15 | 290.16 | 279.60 | 282.12 | 333.45 1901..... 46.41 | 15.05 | 279.95 | 281.24 | 239.83 || 1953..... 105. 72 | 330.94 | 280.35 | 282.13 | 314.07 1902..... 175.80 | 55.71 | 279.71 | 281.26 220.50 || 1954..... 235.10 11.60 | 280.11 | 282.15 294. 75 1903~.... 305.18 | 96.37 | 279.47 | 281.27 | 201.17 |} 1955..... 4.49 | 52.26 | 279.87 | 282.17 | 275.42 1904. ..8 74.57 | 137.03 | 279.23 | 281.29] 181.84 || 1956..... 133.87 | 92.92 | 279.63 | 282.18 | 256.09 1905... 217.13 | 177.81 | 279.98 | 281.31 162. 46 || 1957... 276.43 | 133.70 | 280.38 | 282.20 236. 71 1906.... | 346.51 | 218.47 | 279.74 | 281.32 | 143.13 || 1958..... 45.82 | 174.36 | 280.14 | 282.22 | 217.38 ae Dy eee 115.90 | 259.13 | 279.50 | 281.34 | 123.81 || 1959_.... 175.20 | 215.02 | 279.90 | 282.24 | 198.05 ie) BOOS Ese o% 245, 28 | 299.79 | 279.27 | 281.36} 104.48 |} 1960..... 304. 59 | 255.69 | 279.67 | 282.25 | 178.72 19092 7 27. 84 | 340.57 | 280.01 | 281.38 85.10 || 1961.....| 87.15 | 296.46 | 280,41 | 282.27 | 159.34 1910..... 157.23 | 21.23 | 279.77 | 281.39 65..77-|| 19622422 216. 53 | 337.12 | 280.17 | 282.29 | 140.01 POU os 286.61 | 61.89 | 279.53 | 281, 41 46.44 || 1963..... 345. 92 17.78 | 279.93 | 282.30 | 120.69 A912 56,00 | 102.55 | 279.30 | 281. 43 27.11 |} 1964..... 115.30 | 58.45 | 279.70 | 282.32 101.36 1913.....| 198.56 | 148.33 | 280.04 | 281.44 into | 1965-2. 257.86 | 99.22 | 280.44 | 282.34 81. 98 1914..... 327.94 | 183.99 | 279.80 | 281.46} 348.40 || 1966..... 27.25 | 139.88 | 280.20 | 282.36 62. 65 1915 223 97.33 | 224.65 | 279.57 | 281.48 | 329.07 || 1967..... 156.63 | 180.54 | 279.97 | 282.37 43.32 1916... 226.71 | 265.32 | 279.33 | 281.50 | 309.75 || 1968..... 286. 02 221, 21 | 279.73 | 282.39 23. 99 AON Tie): 9.27 | 306.09 | 280.07 | 281.51 290. 36 1918..... 138.66 | 346.75 | 279.84 | 281.53 | 271.04 1919.>...| 268.04 | 27.41 | 279.60 | 281.55.) 251.71 1920.....| 37.43 | 68.08 | 279.36 | 281.56 | 232.38 ODN ce 179.99 | 108.85 | 280.10 | 281.58 | 213.00 1922, 309.37 | 149.51 | 279.87 | 281.60 193. 67 T9232. 78.76 | 190.18 | 279.63 | 281.62] 174.34 1 208.14 | 230.84 | 279.39 | 281.63 | 155.01 $925. 3 350, 71 | 271.61 | 280.14 | 281.65 135. 63 1926225).2 120.09 | 312.27 | 279.90 | 281.67 116. 31 Com )27 (Se 249. 47 | 352.94 | 279.66 | 281.69 96. 98 1928... .. 18.86 | 33.60 | 279.42 | 281.70 77.65 W929E = 22 161.42 | 74.37 | 280.17 | 281.72 58. 27 1930... .- 290. 81 | 115.03 | 279.93 | 281.74 38. 94 19312252. 60.19 | 155.70 | 279.69 | 281. 75 19. 61 Sy ‘1932 es 189.58 | 196.36 | 279.45 | 281.77 0. 28 asp yewes 332.14-| 237.13 | 280.20 | 281.79 | 340.90 1934..... 101.52 | 277.80 | 279.96 | 281.81 | 321.57 aR eas -| 230.91 | 318.46 | 279.72 | 281.82 | 302.25 1936-.... 0.29 | 359.12 | 279.48 | 281. 84 282. 92 Bane 142.85 | 39.89 | 280.23 | 281. 86 263. 54 AQS8ee 272.24 | 80.56 | 279.99 | 281.87 | 244. 21 19392... - 41.62 | 121.22 | 279.75 | 281.89 | 224.88 1940..... 171.01 | 161.88 | 279.51 | 281.91 | 205.55 1941..... 313. 57 | 202.65 | 280.26 | 281.93 | 186.17 GFT es 82.95 | 243.32 | 280.02 | 281.94 | 166.84 1943..... 212, 34 | 283.98 | 279.78 | 281.96 | 147.51 1944..... 341.72 | 324.64 | 279.54 | 281.98 | 128.19 19452: ==. 124, 28 5.42 | 280.29 | 281.99 | 108.80 1946..... 253.67 | 46.08 | 280.05 | 282.01 89. 48 1947..... 23.05 | 86.74 | 279.81 | 282.03 70.15 1948..... 152.44 | 127.40 | 279.57 | 282.05 50. 82 1949..... 295.00 | 168.18 | 280.32 | 282.06 31. 44 ™, 19505... .- 64.38 | 208.04 | 280.08 | 282.08 1 DAaih aa 9 ee 193.77 | 249.50 | 279.84 | 282.10 | 352.78 178 U. S. 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. Ss 9) h pi N Month. S p h Pi 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] 18.37 | 118.28 0. 01 —6.35 || Nov, 1} 45.62] 33.87 | 299.64 0.01 | —16. 10 June 1 | 189.64} 16.82 | 148.83 0.01 | —8.00 |} Dec. 1} 80.92 | 37.21 | 329.21 0.02 | —17. 69 DIFFERENCES TO BEGINNING OF EACH DAY OF MONTH FOR COMMON YEARS.! Day. Ss p h Pi N Day. s p h Pi N o° ° ° ° ° ° ° ° ° ° ese 0.00 0. 00 0. 00 0. 00 ONOOH | Persea 210. 82 1.78 | 15.77 0.00 | —0.85 Pat A a 13. 18 0.11 0. 99 0.00 | —0.05 || 18....... 224. 00 1.89 | 16.76 0.00 | —0.90 Ssoosasc 26. 35 0. 22 1.97 0.00 } —0.11 |] 19....... 237.18 2.01 | 17.74 0.00 | —0.95 AU aE 39. 53 0. 33 2. 96 0.00 | —0.16 |) 20.--.--. 250. 35 2.12 | 18.73 0.00 | —1.01 Doseeene 52. 71 0. 45 3. 94 OO |) Obra) Paleceeces 263. 53 2.23 | 19.71 0.00 | —1.06: GEuanee 65. 88 0. 56 4.93 0.00 |. —0.26 |) 22....... 276. 70 2.34 | 20.70 0.00 | —1.11 Westeecs 79. 06 0. 67 5.91 0:00 | —0:32 || 23-.2.--.. 289. 88 2.45 | 21.68 0.00 | —1.16 ebeaacne 92. 23 0.78 6. 90 0.00 | —0.37 || 24....... 303. 06 2.56 | 22.67 0.00 | —1.22 Osdcsads 105. 41 0. 89 7. 89 0.00 | —0.42 |} 25....... 316. 23 2.67 | 23.66 0.00 | —1.27 LOR 118. 59 1.00 8. 87 0.00 | —0.48 |] 26....... 329. 41 2.79 | 24.64 0.00 | —1.32 Le 131.76 1.11 9. 86 O%00)) Or Sai 272 seee—: 42, 59 2.90 | 25.63 0.00 | —1.38. 1S ee 144.94 1.23 | 10.84 0:00 |. —0158 |) 28.2... .. 355. 76 3.01 | 26.61 0.00 | —1.43 Te aueuHe 158. 12 1.34} 11.83 0.00 | —0.64 || 29,...... 8. 94 3.12 | 27.60 0.00 | —1.48 D4ecses 171. 29 1.45 | 12.81 0.00 |- —0.69 |} 30....... 22.12 3.23 | 28.58 0.00 | —1.54 15s S ee. 184. 47 1.56 | 13.80 0.00 |} —0.74 || 31..-...- 35. 29 3.34 | 29.57 0.00 | —1.59 Geese 197. 65 1.67 | 14.78 0:00 |. —0:79 |) 325.22... 48. 47 3.45 | 30. 56 0.00 | —1. 64 | DIFFERENCES TO BEGINNING OF EACH HOUR OF DAY, GREENWICH CIVIL TIME. | Hour. s Pp h Pi N Hour s i) h Pi N | . ° o ° ° ° ° ° ° es ° Oe seaBar 0.00 0. 00 0.00 0.00 OhOO) ||) Wescencse 6.59 0. 06 0. 49 0.00 | —0.03 . ae ae 0. 55 0. 00 0.04 0. 00 0300) || Ssaaeas= 7.14 0. 06 0. 53 0.00 | —0.03: Pal ees 1.10 0.01 0.08 0. 00 0.00 |} 14....... 7.69 0. 06 0. 57 0.00 | —0.03. i Saat 1.65 0. 01 0.12 0:00 | OF 017)| 152.2222. 8. 24 0.07 0. 62 0.00 | —0. 03 Hocottds 2.20 0. 02 0. 16 0.00 | —0.01 |} 16....... 8.78 0.07 0. 66 0.00 | —0.04 Beoaeods 2.75 0. 02 0. 21 OF00) 5 OLO1 | Pie 9. 33 0. 08 0.70 0.00 | —0.04 Beesasbe 3. 29 0. 03 0. 25 0.00 | —0.01 || 18...-.... 9. 88 0. 08 0. 74 0.00 | —0.04 Usa cane 3. 84 0.03 0. 29 0.00 | —0.02 || 19......- 10. 43 0. 09 0.78 0.00 | —0.04 Sra 4.39 0. 04 0. 33 0.00 | —0.02 || 20....... 10. 98 0.09 0. 82 0.00 | —0.04 Quieres 4.94 0. 04 0. 37 0.00 | —0.02 || 21....... | 11.58 0.10| 0.86 0.00 | —0.05- TEAR RSS 5. 49 0. 05 0. 41 O200)))) —0. 02 )//2225220 2. 12.08 0.10 0. 90 0.00 | —0.05 WWeebeeds 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, ofleap 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. al s >] 3 é, b) Positive Positive when Wis b etween 0 a °. 5 N always. between 180 and See ae WES is I v é 2 a c ° Di 2H POPE oni Saioe 28. 60 if. 9 Diff. ° 0 0. 00 Diff 2 Di Hage oc Re 28. 60 19 0. 00 17 0.00 T- " Diff. a 2 Sa i | 28. 60 0 0.19 13 0. 00 36 2° ea ey 3 Pere Wd TRO Pe ene 17 0.13 2 J 0 0.56 | 28 4) 47 0.97; 24 0.28 359 i aia Beal aa 19 0.51] 46 0.40; 28 (ON i| ae eA liens Aaa aa 98.5g| 2 O23 yy 19 0.67 a 0.85 | 59 357 ve eteeeere es 98.58| © 5 onset! 17 eee 1.12 18 . 84 13 1,14 Sep OV «| ee 4 19 1.01 u aed 13 1,42 | "28 ae 7) el ah Spada se 1 1.31 , : 14 Tea es a Cy" a i el 28. 2 1 1.50 19 -18 17 0. 94 29 Ma Baits) 5 1.68 18 1.35 16 1.07 13 1.99 3 ORR ete 2 'T 28.53 | 19 1,51 17 1.20 13 2.27 28 aa8 ze >: Benes oaee 98. 52 il nee |. 18 1.68 i Bate 58 251 SR) Bd a = 2 c j if i BevsD |e) Tile ose be Te ae Wilbee £ 7 2 2.83 | og | 350 Beets! : 1. : in Be eae | es re 2 ze 4 | 19 2,18 ” ek me 58 348 eee oS 2 61 | . ile 28.45 | 2 2 ENiist 18 Dest ee ee 13 3.67 g | ate eg aot (ei anos, |e ie ona as 28 | 346 ee : 2.98 i 13 4,23 18 opp asa 2 Been | aks 2. 68 oy 2 CMB OSes one 98. 39 2 16 2, 84 16 2.12 |. 3.94) 18 : 13 4.51 ae ica al: 1) go] | aaa] Bl Kos 27 | as fet Bu) 2) $8) we) Re ee eae — SE Be eae 3 ; i Se sees ee eee 98. 29 | | 18 3.50 16 2. 77 13 5. 60 a a 1 Cn cd Sa : 4. 06 ; 13 5. 87 24 il je was.96 |° 3 18 3. 66 27 | 339 oH ogose esac. 98. 23 3 4.24 1 3.82 16 2.90 3 4.42 | 18 ; 3.03 | 18 6.14 A TE an Sse ee Bal ee ase Ye eh eee ‘ 3. 12 4 oe 2840/4) 460) yg/ 414 i ee Se Sa ek | 3 o : se P12), S Lose 12 B30) ae 340| 22 6. 94 935 28. 09 iS 2 AG lat 16 FiGe |) aS 7.2) 1 Beil ae 28.06 | 3 one 4. 62 ue 747} 3g | 338 ae Bape ae aie 16 3.65 a 4 5.48 |! 18 78) 46 3.73 | 18 713 332 27.98 arf #98) ig] 890) 95 7.99} 35) 331 moe) 4) Bp 5.1 ents |p sale eo 4 Si soiie ae Eee: 1 4. 02 a ; 4 5.99 | 22 -25] 76 414| 12 8. 50 309 27.86 ue BAe ye Naat iy ae BuWOi 4 goss | 38 ree a 4 6. 16 : 12 9.00 |-..29.1 32% 2B es horde as Walley pererS 15) ey geen 7. om Byes meso}! Wiles Sap] 1 hic BOG 72 Ge a cede BBee eee ec eo Prosit ®\ sees |) 2 lve sg 10 eee aly ean ae 524 AL ae eae 5 B ‘ 4, 27, 5) eg} 16 Bidet 22 lat acl Zee wollen eee 393 AG. eet yh 97.58 16 6. 31 15 4.97 12 10. 22 Be 322) Aye Ae 8 753 5 7,15 ; 11 10: 464 ||). 25 |e 4328 7G 5 ah eR Bee 5 7.31|", 16 6.46 | 15 5. 08 ri ee err 27. 43 E at GOH 15 Paci te lak aecie aa 319 SSE aa 63 1 16 Atel] 288 |G 763) ye] 8:90] yy| 5.4 : 23 | 318 : 7.94| 15 7.04) 4 5.52 | it 11. 38 31 5. 63 11 - 60 3] use| 22). 315 180 U. S. COAST AND GEODETIC SURVEY. TABLE 6.—Values of I, v, &, v’, and 2v'’ for each degree of N—Continued. tt Positive when Nis between 0 and 180°; negative when WV is Positive between 180 and 360°. N always. N I v & py’ 2p!’ ° 2 Diff. © Diff. e Diff. ° Diff. : Dif ae 7 a OE 2 D7e32) is Togs He TO) a) 5.634) jy) (th) Aa. Sees aee maealen ZG eae te en ate 27.27 8.10 7. 32 5. 74 12. 04 314 aa a PoP hi) RObiltie ae 7 AGN) 1/4, 584) |\ 1) |a ainiae ea Beer ee ee Bis IM B.40/|5) 42 7760)|) 15 B95 ja tpi ae Aa emer mente Hoes she 27.09 8.55 7.73 6.05 12. 68 311 Bo ees e DiOallae, ce TSTH gee 6.15] 10\iae ai eat ye eneeenn Sie Poe DOTNET Ri MB. 84 (yao 8.00} 7% 6.25]; 40) | ABs 08) a2 aero 7 aa 26.91 a+) 8.98 8.14 6.35 13.28 308 meee tO) oe 26.85| & OID iat BIOTA aS 6.45.|, 1O'lo¢ mie.as a 307 1 Lee ni aaa pone 9.26| 4 g.49| 13 6.54 [o 19.1 08 Gili elie g06 FGMES idee 26.72 9. 40 8. 52 6. 64 | SERB E86 305 SEL eee 26.65)|) 2 Ones 8.65] 33 6.73)|., g | ae «Ald. 05H pe lea 204 Bape scab aaP 26.59} 8 BPCTL | eas S77 lac 45 6: 82,|.¢ Gi| Ohio 23 tenes ponmeie | see 26. 52 9.81 8.90 6.91 14. 40 302 is eae 26.45 i oo4| 13 g.02| 12 7.00 8 14.58} 18| 301 7 Ea Be: 38) CUM glo 07 Huy 18 9114 |) 12 709 |% 2 Gk a1. 7d Gea oa ee 26.31 10.19 9. 25 7.17 14.92 299 Gop yee Te Dé PAN ela 0632 |e aye ig 7,26'|5 8 |e 5,08] pte tees Gaye cca men Te Da TTI ila: 44) 12 gis liag te 7.34 | 8 iui 5, ate aa Ga eG 26. 10 10.56 9. 59 7.42 15.39|. 22 |) ge pee Gc ia 26.03/10) fide 10,6810) 17 Jy 9.70) a tt Uni, 401) 2 eae le lla ae Gene Sie BSvOnte Sune lOn7ola a Clsidian 7.574], 8 4) \ ae 6B ar ees Gree obs eat 25. 88 10. 90 9.92 7.64 15. 83 293 GSee ner D5: SO) ethan OUN|) =2l ea eLON02 a 7.72 |) 3 )oe Ae 96) (apy lene 7 gee B52 \Bh on MY. 12)) 0 lam lola ete 7.791), 210s a6, 10) ire lean (ie ee 25. 65 11.23 10. 22 7. 86 16. 23 290 ise 25:57 |(tr ee hip e138 10} 10.32 19 7.92 | 8 | 03 aes. 252 teal aes ple a 254900) heal a atten PFA ay etOLeet I 7.900 2 i a6. al rae wee TA RULE ASUS rs ee Slee) py lee cts) nas 8.05 ||. ¢ |) 6.58] anal eRe Thee i Mes 2533) 10h 03 i iauatt: 63 2 aloe ue B11}, 6): 5 apheol) Aaions i hy Sam D525) Wey Opieltev2i| 21 7a i MOLeSI| ee 8.17}. 6 | cos. 80) ot gees Gc te eee DbAT HN ve] Qa Blt > 5 lia gtOo ze lg 8.23 1.0 |) Ae, 86: 90M al ame Tee a se 25.09/80 oi ei 01480 1028 jee lad. Bo nee 8.28 |") 2 | ca Miri OO uce lees TB ee! 2501) 05/5 || imi ost 20 3 ie 10. 93 ||.0i)8 8.34) bs. 8 | os 708 ease (Cn a 24921) || we 206)! Vel a aT OL || ae 8.30 |" 5 | a geet eet See. Dara ||P) linn, eta) We |) ry ai.io8 |) Aa 44]! 2} ca ais | le ean Slee: Ret BA76)| Big | iap 02d 2N| og do gD Mae 3.48] 0 5 | (oles beahoneag Soma ee PANSTIN ie liam ul2 2 aes) oy 22 le eee Ma ete) (0 Sl rT: Sones. ee Deo Oh Si ae zis Ee ie aah ooN Tae 8.57) 4 | do mao pean S4ee oc eee | DE SOW ot 2 ny 2 4ad abe | ip Vit) 36N Ola 8.61 |) 2 | sie) 52h aera aege cis eee ea 2a42 | || ay A2.48)) | je il. 42 | ne 8.64] 4 | a A SB a ou pee Sieh eS DEBS RE 8 ay 2 all M8 ny ehtlear aa 8.63 |.) 2] ca ges Haye aaa Shin te 73 A211) 2 Nig A260 Hie 81) se ies gene S71 OS ae Ae age eee Sia. ails BATCH iro Il tp gi2465i|. sce Heh cLi58)| ane B.74 |... | op Oe 7 Nh pet page ee a AY palor | 129 )) watz: 7004 2 | canted | Ane B76 002 | ae aera ent eaeeal ee 23, 98 12.75 11. 68 8,79 17.77 270 181 TIDES. ANALYSIS AND PREDICTION OF IC HARMON te ? f if wey Positive Ww 1s between 0 and 180 ne; tive when Nis hen. N 9 Za h : ° between 180 and 360°. a Positive aS eae as always. Pero Ree oh wae v! N FF g FT eee ® - Snare OC aero Berane y Dif. 8.79 2 69 ti Diff. : 11. 68 4 17.79 2 ae { om bia ey Se oo eee T |e 9 70] 4| m72) | 881 2 (20 ata, 82)l «+b! = -26 i E 4 85 3. : . 86 “33 23.72 9 11.83 3 : , ee i ab : 8.87 1 7.82 1 12.90 3 11.86 3 8.88 ne SEAT) aera 2 aaa ga ; sie A f a 81 ae 2 12.95 2 ie 4 ow : 1773 : He BE 38o peg it a2 09 5 Tose 2 8.90; |W. : 260 , : ie 1 «74 = Dots ae ah ee ae ; Kj 1 17.6 5 13.02 0 OT a 8.88 5 He dali 23.09 9 veils Oly age 0 " Oe Wee iar AO ST ae 13. 0 8.87 1 17.57 6 | O55 Te ee i 13.02] 4 ut 8 a 2) Wal) 5 fe a eae ee 9 po Oa i) ara 2 1 : ce Ieee Me 104..-2........ 22.64) 9 p : SE ae 17.38 | 252 =I an . bos 11.96 1 8.80 2 17.30 8 sais A 12. 2 2 11.95 2 8.78 3 251 saa) Rl aoe 5 2 a 2 11.93 2 aes 17. 22 8 950 ai ae ‘ 15 3 4 ft Saas ass} gel | al ee ee 22.28 : 12.89 4 11.86 3 : ' 16.95 | 44 ae ! E 4 8.65 : aa a 3 4 16.84 11 246 11.83 4 &.61 4 73 Rs : at 4 11.79 4 8.57 5 ne _ 245 zee 12. f 5 11.75 5 16.62 12 244 aig 2 1h af 5 ae P S Be 4 ee 13 243 5 B Fy 16.3 13 n 12.67 6 11.65 am) i] Beye) aw) a ts) ame } 12. 7 8h “10 21.58 8 11. 54 6 : 2 oe 15| 240 8.31 6 6 Se f 12.48 7 11.48 7 8, 25 6 Sie e 239 Bi arsine | eas g| 1.4 g 5 15.81) 45 238 : ; ait 6 5.66 21.32 8 11,34 8 : a ie aay : 8.13 7 15.50 7 12. 25 8 11.26 8 8.06 ; : gids 9 ue H 11.18 8 é 15.33 | 47 38 ul 2. .99 8 16 7 21.07 8 11.10 9 ve g | 234 5 7.91 8 14,99 8 oe ao 9:|/aawiee ses ; ; d 8 el m8] §) RE) B) ae) eal eneet wl 3 kee ; : Ta 8 62 a Bh se 20.82 8 ee iS wee a f07 9 i643 5p a A : 5 9 es mes] 8] ts He listo. 6) | itt j 14.23) op | 280 ae Be 8 4: 7.4 9 14.03 | 59 298 BE 0.50} 49 7.40 8 gn ee ae 1 Q 10 13. 83 21 F 10. 38 12 7.30 1 wae a2) 3] HB) B) BS) 3) | Bel mw Tei i 20, 35 7 . 7.2 10 13. 40 22 225 is 0.13) 43 7.10 132..-.-.... eT Rela. beni : of) 838 5 10.00 13 7.00 20. 28 8 10.77 i 133....2...... 20. 20 7 fo 62) 3° ee ce eae 20. 13 e 182 U. S. COAST AND GEODETIC SURVEY. TABLE 6.—Values of I, v, £, v’, and 2v'’ for each degree of N—Continued. Positi Positive when W is between 0 and 180°; negative when WN is ae ty between 180 and 360°. N : MAAN RP TAVIS La iL HN I os t afi 2p!" ° ° Dif. © Diff. ° | Dig. ° | Dig. ° Dig.| Te Sat 0,13 1,7 splay a0. 62 ta oys7lre, 7-00|!(:" 44 || sebsats fene Haals eh 20. 05 10. 47 9.73 6.89 12.96 |. oa | 224 187.2 to98 | ZN iy olan e 36 0) gee 6.78 ||) 44 || (0 MZ a veel eae "i Aa foi91)|© 2 ize ao. 15||% 18 9.44) 19 6.66 ||), 42 || Xt Aap ae yl ee liauees FIG why bE hea 19. 84 9. 98 9.29 6.55 12. 25 221 40.2 9.77! 2 hve cage 8 F2 Hoe. 13:16 48 toe 6.43] 22 | a ee eae 7 ae 9 SEL gaz! 8 ales a as B97], 40 6.31]! 12 4c Grgell ee aceate | SVD 19. 64 9. 45 8. 80 6.18 11.51 218 M32 pet Pine v9.07 1% TSieelpesll; 17 oq Ub. 06 ||, 12 | canned anette i AR i oR ip. 51 | 2 agai 23 gaeys 12 5.93 |) 75 | Mi Me OMii ole wale stele has 19. 45 8. 89 8.28 5. 80 10.74 215 M4602 19.39| 8 8.69| 30 euro || 418 5.66 | fa | (tv M048 ho aes ee EE abi 19.33! 6 B49 || 6 20 got | 0 22 6.52) Ut i ato. 20 | 2 waa TAR ens N 19.27 8.28 7.72 5. 38 9. 94 212 M490 9.22 | Blac es07) © 3) ye age |) 20) eo 5 o4 |) 21 Be ioe ie aed Sika i9.16|- 8 7.85), 2 g. 82) 1/20 B09) pap 9.38} 38] 210 TAL es 19.11 7.63 7.12 Oct as 9.10 209 152. i005 |.°) 8 mia hy 28 Got aye 4.80] 72 | (i OB ait eee ll ames am hi 19.00] 3 7.18% 38 ei70lia a es ee 8.6210, 2a). aa Lae me es 18. 95 6.95 6.49 4.50 8.23 206 155220000 ABOU] F [ce G72 18 oe | ge 1627 |) 22 lie S434 10 10 lee Breet anal amete es Gee #286 |" 2 6.48% 2) E05) 419) 13 7.64) 30} 204 Lerpe OR 18, 82 6.24 3 5. 82 4. 03 7.34 203 Teer 9 Aaa 1878 #7 599 |b 2) EEO I ee BigT pac 7.08 |.1 130.) «BOR Tiedt Ne ag 74 YF 5 74)| b 22 5.36 | ( 23 370 || 12 6.74] 39) 201 eS ree ea My 18.70 5.49 ; 5.13 3. 54 6.43 200 161.2 1g66| 4 Be gee 4.sg| 28 2.87 | ad G12}. 3h |. dos eons | AURIS 12.62) ° 3 4/08 | 6 23 Bes Oe, 220) ae 5. 81 |.) Se vheelie Tegra att a 18.59 4.72 4.41 3.03 5. 50 197 its eae Sa 13.56) 0 2) io pe 4e | 028) my peace | 122 | popes) 1 | te oe aes Tege GES Ae53 Me a a9 |) h 22 201 |) ¢ 22 Beg) tre 2.87 1 Seale TC nn a sh 18. 50 3. 92 3. 66 2.52 4.55 194 ee Wa.47 | 9.3 | or pes | US! | ou ag. at| 022 | cy eo ga | BUS) oe ae eet Cs a as 18.45 | 2 3:38 Bae a7) ice aia ete 3.01 |). 32), 198 1609 Was 18. 43 3.10 2.90 1.99 3. 59 191 170.2. Te | #2 2. 88 (|) 8 2 2.64 | 26 iat | 2 18 8:27 | 32 haga Wit 1c AN | 13:30 4 2 2155 | 28 p.38,|| ES.) pan 63) 8 T 2.94 1133) aes Trae ee: 18. 37 2.27 2.12 1.45 2. 62 188 yee. eS 1936") 82 1,99 | (28 1.86 | SES BOT tae 20] tytn, COE | 1a94 | 0)? 1.771 Aes a. 60 | S22 09 | 48 97 |}. 32 i. 51e6 LTaee Pence 18. 33 1.42 1.33 0.91 1.64 185 aves) 50) 18.32) 0 0h fy. ct 14) S28 | a (a 07 | BE oy ee zs || 18 | ok ene Fit ea ig.2 | MY 0:86 | $28 0.80} 3% Tee as 99") Se | aie T75oue Mea 18.31 0.57 0. 54 0.37 0.66 182 Ton ec ig. 31 | 01? 0.29 || 628 0 27 ||. bee pts |e ee 033 |i coe lee BE TRHSo LEMS, 5 18, 31 0.00 0.00 0. 00 0.00 180 183 HARMONIC ANALYSIS AND PREDICTION OF TIDES. 0LZ 06 GES8 6 £698 °6 OLL8 °6 G6L8 6 8188 °6 GLG G6 898 6 9898 6 €618 6 L088 °6 6888 °6 086 oot 6898 °6 9198 °6 TOLB *6 PP88 6 468 6 GBS SOT LG98 °6 GPL8 “6 F288 6 F068 *6 6868 *6 062 OIL 6918 "6 FEBS “6 £168 °6 0668 °6 £906 °6 966 SIL L188 "6 $968 °6 6206 "6 0016 6 6916 6 00€ O21 1806 °6 GOT6 6 OLI6 ‘6 GEC6 6 8626 °6 SOE Gol G16 6 8166 6 8886 6 [686 6 6F6 6 org Ost O&8F6 6 €8F6 ‘6 §£96 6 0896 6 £696 °6 STE GET 9196 °6 L1L6°6 $916 "6 6816 6 6186 °6 0GE OFT 8966 °6 6166 ‘6 1000 ‘0 6100 0 ¥£00 0 CGE SFI 6920 ‘0 8920 °0 0220 0 6920 0 ¢9c0 0 0&8€ OST 2690 0 8290 0 Lgg0 0 £890 ‘0 9020 ‘0 Gee GST 6960 ‘0 $060 0 £580 0 6080 °0 0S20 ‘0 Ore Oot €Tél ‘0 O&ct 0 LIT ‘0 G90T '0 9860 ‘0 GPE gor 8991 0 Sect 0 LIFT 0 cost *0 LOTTO ose OLT €S61 ‘0 G6LT ‘0 TF9T 0 00ST ‘0 L981 0 GGé GLI GSTe 0 9961 ‘0 O6ZT 0 869T 0 6LFT 0 09¢ 08T 8226 0 L202 ‘0 €F8T ‘0 PLOT ‘0 LIST 0 ° ° d : ' 066 086 oLG 09 0G “rT quauodusoo fo apnzydun sof “y boy—) atavy, 8968 °6 6968 °6 £006 *6 8506 *6 9816 6 9€26 "6 LS86 °6 0086 6 £996 6 9F86 °6 ¢#00 0 8920 0 8240 ‘0 8690 0 8060 ‘0 C601 ‘0 FCT 0 OFet 0 SLEL ‘0 oVG 1806 °6 £116 6 9816 6 L926 °6 96£6 6 1686 ‘6 OLG 06 L¥06 °6 £616 °6 9616 6 9926 °6 FEE6 6 66£6 6 G9G 98 6106 °6 6S16 6 4666 6 6666 6 8956 6 TGF6 6 096 08 Te16 6 6026 *6 0166 '6 98&6 “6 8686 6 8SF6 6 gcc GL [066 6 T1266 SEE6 ‘6 96£6 6 TSF6 ‘6 O16 6 0G¢ OL 6626 6 0986 °6 816 6 £156 °6 9656 ‘6 GLS6 ‘6 SFG G9 F166 L9t6 °6 8196 6 9996 °6 TL96 °6 £996 “6 OFG 09 8hS6 6 G696 °6 FE96 6 £196 6 OTL6 6 £116 6 GEG gg 0026 ‘6 FEL6 6 [926 ‘6 7616 6 0286 ‘6 £86 6 0&6 0S 0186 6 1686 ‘6 0166 °6 9266 "6 OF66 °6 1966 *6 GGG oy FS00 ‘0 0900 °0 +900 0 9900 0 9900 °0 ¢900 0 066 OF 6420 ‘0 8820 0 G6C0 ‘0 GIZ0 ‘0 L610 ‘0 G810 0 GIG ce 6FF0 ‘0 810 '0 88¢0 0 LS¢0 0 16800 L660 0 OIG 0€ L900 9690 0 9790 °0 8640 0 6S¥0 0 L0¢0°O S06 GG F£80 ‘0 6920 0 £690 0 8290 0 G90 0 9090 0 006 06 8660 ‘0 9060 ‘0 0280 ‘0 620 0 6990 ‘0 0690 0 S61 oT 8cI1 0 6101 0 8160 0 #280 0 98200 $990 ‘0 06T Or TIZr ‘0 G60 0 1860 ‘0 8280 0 £820 '0 [690 0 S8T G OFCT 0 LITT ‘0 GOOT “0 1680 '0 6620 °0 8020 °0 O8T 0 ° ° d 0&6 066 olG 006 ob I o8L I 184 U. S. COAST AND GEODETIC SURVEY. S2Ss 6wom ono » eno rs) San 4S Sana ses SSS BSE SL a, e DODD MOODCD CACDtD GMM DANN ANN A Sno mOM ow MOM SHO 6wOW © See OOH BAT BEAN Aaa SSeS hy eminrr Sree meee ae Son hon oe) rere Sot FAHD MOO AMM AEN ©0rR © Barareh See i Br : ee A é 2 oS HAA «Sad cna San eees tS e nN SCOn HOG rAt WOM NOW OtK © Ss SAS Bes SOK wHS OA ee °, se SSSI) GES) SEM ae ee ~_ e q N s _ us) 5 a SNP 2WOOD ANM HON MNO OHO O SHG HSH BSS GES TNS rind oS a 2 See ANS Ss Sas Se Ta ~ se =) nN S) — im 8 . pie sete es hed ACS ON WEL Rae el Let AIC ORG SES eI a AN PIE cna ; & z 3 SOAR HAR NAOT NON Wt BOW Oo fa) 3° SHO ASK 0 roo SHO KAHN © S e rere Saree rae > A co e ° N eo & < Se a & CHa ANDO PHO HHH 0100 rRiInM O = 4g Str sos Ser sist add cwa o S a o5 i seers eee rere =e oS S a Serr has i=) de Le S oo Crs Ctr ARK AMR FON MAH © = =] So~ SAS iftdid wt BSH SHN OS S ° Soon oe ee | Serre Sere Bo Be | xt = e n > + OY ie =| S08 oe : S 8 SM OND OM ANH OMD WMO CO > SS! SHS SHAHN St BON SOK WGN S ° Leela ba Ss ~ = a SS o ~ a SCOoo COM MOH DHN COM HthRO CO 3s a SAS BSH ASS ANS SOK 6H S S qd Bo ‘ ase eee Ser re -_ N | n _ o & aq 8 SOS HOM MOR OHM AMM SCHr oO Ss = Sais NOS HHH FHS SKS BH S fea} ° Soo bos he he | are oO = ° <_< = ix] —_ sl = on wn =) 5 Smt HON HOH Ont HMSO OHO SO = SAx SHO SSS SSS Ors HM S Os] ° ee onl wn —) ° © aq I = 3 > 3 COG OFAN O-OrY WMD OOM NOt © 3 SANs 6s BIS SAH ros HAG SO e [>] for} ° i=] ce Ea Coot OND C-1re ONE BOS MOM SO i telaes| aie ANS b sy READE Stee) | Meebo ast (MICRONS ais Z Sr SSK Oe CO SSH GAR S ° ee) ; 0° ae | S6o »men OHWSo WOM nS 9OMm SO La) DOS ss BeniAN Aae Aas woo & ome FAN AAN ANN Aad AAA A Sno mon OfS MOM SHS W a, c) LS TNN OO = in us oOwor Oe 8 oon Ont WHF © iB HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLE 9.—Log Qa for amplitude of component My. Log Q:. Diff. IP o 9.7133 0 180 9. 7133 2 179 9.7135 2 178 9. 7137 4 177 9.7141 4 176 9.7145 6 175 9.7151 7 174 9.7158 7 173 9.7165 9 172 9.7174 10 171 9. 7184 10 170 9. 7194 12 169 9. 7206 168 13 9. 7219 13 167 9. 7232 15 166 9. 7247 16 165 9. 7263 17 164 9. 7280 18 163 9. 7298 19 162 9: 7317 20 161 9. 7337 1 160 9.7358 22 159 9. 7380 98 158 9. 7403 oA 157 9. 7427 35 156 9. 7452 7 155 9.7479 o7 154 9. 7506 28 153 9.7534 | 30 152 9. 7564 31 151 9.7595 150 31 9. 7626 33 149 9. 7659 34 148 9. 7693 | 35 147 9. 7728 | 36 146 9. 7764 37 145 9. 7801 38 144 9. 7839 | 39 143 9.7878 | 40 142 9.7918 42 141 9. 7960 42 140 9. 8002 23 139 9. 8045 138 45 9. 8090 46 137 9. 8136 46 136 9. 8182 135 185 Log Qa. Diff. ° ° ° . 45 | | 295 9. 8182 ne 135 315 46 | 226 9. 8229 4g | 301384 314 47| 297 9, 8278 ay Di.) 138 313 48 | 298 9, 8328 a 132 312 49 | 229 9. 8379 A 131 311 50 0 9. 8430 : 130 310 31 231 9, 8482 a 129 309 52 | 232 9, 8536 128 308 53 233 9. 8590 ee 127 307 Baal 4p 9, 8645 pf 28128 306 55 | 285 9. 8701 ng 192123 305 a6 | e236 9. 8757 oe iad 304 57 | 237 9, 8814 ea ies 303 58 | 238 9. 8872 Py) | Wee 302 59 | 239 9, 8931 2 Le v2 301 60 | 240 9, 8990 rg) 120 300 61| 241 9.9049 eo} 119 299 62| 249 9.9109 aia fea as 298 63 | 243 9. 9169 GO ani? 297 64| 244 9. 9229 ay | stale 296 65 | 245 9, 9289 oO 115 295 66 | 246 9, 9349 ey ee 294 67 | 247 9. 9408 aoa lis 293 68 | 248 9, 9468 pay hese) 292 69 | 249 9. 9527 ea vag 291 70| 250 9. 9585 pay fg sty 290 71 | 251 9. 9642 2 109- 289 72 | 259 9, 9698 8] 108 288 73| 253 9.9753 pg fd 2107 287 74| 254 9. 9807 oF] 106 286 75 255 9. 9859 50 105 285 76| 256 9. 9909 ag | 104 284 77 | 257 9. 9957 ei e038 283 73 | 258 0. 0002 re 102 282 79| 259 0.0045 4g Ve sot 281 80! 260 0. 0085 a 100 280 si | 261 0.0122 By 99 279 s2| 262 0.0156 98 278 83 263 0. 0186 EY 97 277 84} 264 0, 0213 a 96 276 85 | 265 0. 0236 Ls 95 275 86 | 266 0, 0255 3 94 274 87 | 267 0.0271 a 93 273 88 | 268 0. 0282 5 92 272 s9 | 269 0. 0288 y 91 271 90| 270 0.0290 90 270 186 U. S. COAST AND GEODETIC SURVEY. TaBLeE 10.—Values of Q for argument of component M,. PM @ | Die |) uP @,, |SDit. ||) P Q | Dig. ||. Pp | @ | Dit. ° ° io} ° ° ° ° ° 0 |fieek0.0F 45% 45\| 12588 | pit. 90] /90.0!], 9 1 ee mias | fiz] 0 5 tl Sbo.5 46| 26.6 g1| 92.1 136 | 155.0 Biel. Of. 02 AT!) Pome || Vue 8 92 | joa], 2 Oil keiyae7 |" fieies nag oe 31) Sei1.5 4 ve oe 43 | fase | | 8 93 | 7096.2}, 2 Uhersias.) ieaes | bearers 4| Wae1.9 49| 29.1 o4| 98.2 139 | 157.2 Siem he 0.2 50 | jag | | 08 95| 100.3) 2h |}: 140] t57.9| = 07 Gul pam olen, O° sas!) (3088 ||" Pa 96 |: 002.3i]" 2:9 Vic jcaaa | fasts || 5 02 7 || ci3.4 m2 | panez 97 | 104.3 142 | 150.3 Sal Hers. oa) 0-2 a3 |, paeat | hae 9g | <206.2i|' 3:9 ll aiaaa | jiemo| | 4 OF of era a pene 2 oa | gaa |) 99 | imogaj), 2-0 rbcraas | freo7 pg 2 1o| 4.9 55| 34.6 100 | 110.1 145 | 161.3 Ta 1 0) Pca 56 | ass} q°9 ll ostot | oanioq|)) 7-8 horas | fies | agtd 12 |) 665.94 S208 a7 | see] 2 ll veao2 | easel 7? lhoovaa7 |) bearer 13| 6.4 58 | 37.7 103 | 115.5 14g | 163.2 14] 1816.9 | Oe so: Pasig |) ge |araoe) vainz.3t|) 18 liceoeaao: | tease images 15 |) opt7.4y] Sees a0) j3a9] ath il pao | ameoll | Pe iecmeo| fleaa |) ace 16 || eti7.9 gl) 41.1 106 | 120.7 151 | 165.0 17 |) 2418.4 Oe ee ge} f4aag| | de? | wso7 | «122.3i/)! 7-Oiinertaal | teare | haeece 13,|) TOP AE a3 | f4a5 |) de2|) cntos | caas.9]|"* 7-6 Hecovtes | Bigere pt aieee 19; 9.4 64] 44.7 109 | 125.5 | 154] 166.7 20 |/ aito.o9 (2 08 a5 | j4aco| ar8 ll) iidio | paaz.oj| 18 lveetas | Meme iimedse BL RHO Sae Oe a6] fara] 4:8 || wait | capeist 13 lheertee |) Beman | ime 22| 11.0 67| 48.7 112} 129.9 157| 168.4 23 |) erie) leo 6 68] t5qa |.) 454 ||| wat8 | caal3h > 7-4 lboeies | Memo faee Ba |) fried eee eo | i5ns | de4 |} oaya | pase.zi|'* $3 lsicteo|) fees | 9 gee 25| 12.7 70| 53.0 115 | 134.0 160 | 170.0 26 |iraqta.aip 32 0-6 | jes] 2:3 |] care | 23530 73 lhcactel |) Elles ime 27 |} agra ik oe 0-8 m2} 56.1) 76) m7) 1865) 75) Jez] Ima} O8 2g| 14.4 73 | (ory 1g | 137.7 163 | 171.6 29 |) wits.) MOB Wonca | Paoa | Fae S ll) ciate | aes:on\0l 1-2 hace) | iia mre 307] | Hans. 61 Be a | j6b0| a2 lj: a20 | (ago! 1? lhectes || fame nau 31| 16.2 76 | 62.7 121} 141.2 166 | 173.1 a2 | Hanng-arpet 0:6 | teas | | 2 ll seae | gaan aiite 17. lleactey | igaaelme cae a3 |b gol7.4 poe Oe 781 Peso | dr! Nl) S93 | caus. ai 10 lh cies | eigaci eee 34| 18.0 79| 68.1 104 | 144.4 169 | 174.6 35 |) ots.7 |! OF tis 80 | jena | | ac8 i 20025 | alas, 2g) 72 Ih yoago)| | Bien ne yee 36 |; gaa | 4) 0-8 ai | | ms | | 8 | ee | nite aye FO lutte | Poa ee 37 | 20.0 s2| 73.8 127 | 147.3 172| 176.1 33 |b qe. | Wee a3 | bisa? | 8 il coaae | oplas. 3/00 00 i cattas | pigene emer Sie kee Roane ga) famr | | 22 |] -a20 | (algo. 21/0) 8-81, coca | Siagen aeee 40| 22.1 35| 79.7 130 | 150.1 175 | 177.6 TES? sont telat 96 | jabs | | 22 i craan | gaso.og|(® O83 hvgige | tiveet meee PO et aaah hd a7 | p8es8 |) SO Ul .1d82 fade? 22 i cela7 |) een ea ee 43| 24.2 38 | 85.9 133 | 152.6 | 178 | 179.0 44 || jv25.0}) 28 go] iano] | 29 ll ja34 | pes. at) 0-8 igo | Aiea ae 2 45| 258| °% 90} 90.0; 7 135 | 154.2] 180 | 180.0 i HARMONIC ANALYSIS AND PREDICTION OF TIDES. 187 TaBLe 10.—Values of Q for argument of component M,—Continued. q | pis. || P q | vis. || 2 q | pig. || P Q | Dit. ° ° ° ° a) ° ° 180:0] + gg] 225] 25.8] .°9.5|f 270) 270.0) 54 |/ 315] 338.2 a 180.5 226 | 206.6 m1 | 272.1 316 | 335.0 ROP ee ey en 2074 20m art Sona era a tee lel an7l | aise es Istsy|) } a |) 228 |) 208.8) © oie m3 | 276.2 31 || 318], 9336.5 ae 181.9 229| 209.1 | 274| 278.2 319 |. 337.2 ce |) 280 ood Cis Ns Bradl). Baore |p) aay || 820 | paar i 1826) | gical | yw232 | @210.8 baie Gretta, 246 In 28221 pai | ag B2l le 838.6 y Pee 183, 4 232} 211.7 | 277 | 284.3] 4 322 | 339.3 HER arey || t. 233 Ate ale lin) Beale BSB | ae | 323 3400] 8 Pee ahaa] | pe8t | O88 6 isch 1 igo) PO) PBA | pointy: be B24 lhe 840. 06 14:9] | | 235 | 214.6 |... 280} 290.1| - 325 |’ 341.3 nea) Oe ||| 6 | ate.| EOS | ast!) ger. | 4°83 || 396 | a4. ae 185.9) 9:2) = 237 | 286.8 | TT, 282] 293-8) 3:9 |] 327) 3426 ee 186. 4 938 | 217.7| , 283 | 295.'5 328 | 343.2 SSRN fee eh \|'1 4, 200 || 2tses | ey Aelia | Bae Bora | ae || Saaeh| aaais ir 7 eB 87-4 | 9:3 |) 240 219.9) >|, 285] 2990] 3:7] 330) atts 0.8 187.9 | - 241 | 221.1 286 |. 300.7 331 | 345.0 em WE eyHi) . eae asia LM) Ai May | Bozls |b | aaa) Basle ue : ee : 188.9) 93 |) 243) 225.5]. 1:5 |] 988) 308-9) JB] 333) 346.2 06 189. 4 244] 204.7). 289 | 305.5 334| 346.7 ce Ta Mn ee 22s 2250 1 200] 307.0]. 73 335 | 347.3 ae 190:5 | 2 || 246] 227.3) Ff) BoE} 308.5) ffi 336] 347.9 ae 191.0 247 | 228.7 292 | 309.9 337 | 348.4 mem ON esl 1,3 /248\\) 23008 oh, 203 | 3i1.3| 14 | 338 | 349.0 ue Wzy) Gees 4240 | wa Sipe t ile 2eele: 317 | stra || en 839 [i> 340.8 a 192.7 250 | 233.0 295 |) 314.0] ~ 340 | 350.0 Eee (hse 951,|, 1 234)5)|, ie 172 296 | 315.3| Sil 341] 350.6 ae 193.8) fg] 2] BEL) Tg] 2]. B65) TF 842] Bota ae 194.4 253 | 237.7 298 |, 317.7 343 | 351.6 BOSON tee cull ss 254 |e 5 239.3 | at 4Cul\i 200 318.9 V2) 2 (aaah |e 3521 ae MOG PSs!) i255 |! HAAG how be Sap.|: 3200 | ite? || p7B45) (ee-252.6 ae 196. 2 256 | 242.7 301 | 321.2 346 | 353.1 . 96.8). 98 || | 257) 2445] el) 302] 3223] Ti gar) aise | Oe 1974) | yl 25a! 296.2 bo.9)-Fe 0 Boal) Basa | 2ded |] os 34s) e854 9 ne 198.0 259} 248.1 foal 2304/2 3244 349] 354.6 198.7| G2 || 260) 249.9] Tg], 305] 325.4) 7p ]| 350] 355.1 tee Pasa) ootea|)) b26il || 2251.8 bord OMe SOBs 326-4 | | they || aeaod |eya55.a re 200.0 | 262} 258.8 307 | 327.3 352 | 356.1 PAP ee 265 | 25507 | pole) 08 1828-8 |i aen | casa iste IL) ae 214) QP | BBA] 257.7 | 3h] 809) 8.2] Og | 3H) 357.1 ae 292.1 | 265:| 259.7 310 | 330.1 355 | 357.6 Pea a6 |) 2608 | Bll iain |. 80.9 | )y05 || ie aoe |, soy, Oe 23.5) 97 || 27} 263.8) FT| 312) B38) gi] 357 | 358.5 ae 204.2 | 268) 285.9 313 | 332.6 358 | 359.0 BOM Gray. 259) garg SON. aia | saad | eek I a5Ol ecole oe 25.8 270 | 270.0) > 315 | 334.2 360 | 360.0 188 U. S. COAST AND GEODETIC SURVEY. TABLE 11.—Values of u for equilibrium arguments. [Use sign at head of column when WN is between 0 Bas 180°, reverse sign when W is between 180 and Mo2,No c (@) Q Ni nh Ky K, |2N,MS| M3 |MiMN| Me Mg 2G a OO | MK | 2MK} Mf | WV d, 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} 0.388 | 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] 0.23] 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] 131} 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] 1.371! 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 11 | 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 | 349 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} 4.23] 0.56) 0.84) 1.12] 1.67] 2.23] 2.23] 7.82] 2.55] 0.88] 5.03 | 345 16 | 2.98} 2.12} 4.51] 0.60] 089) 1.19] 1.79] 2.38] 2.388 | 8.34] 2.72] 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 19} 3.52) 2.51 | 5.33] 0.70) 1.06) 1.41] 2.11 | 2.81] 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 | 295} 2.97 | 10.38) 3.38} 1.17] 6.67 | 340 21} 3.89) 2.77) 5.87) 0.77) 1.16) 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] 11.389 | 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] 1.34] 7.65 | 337 24] 4.42) 3.15] 6.68! 0.88) 1.31] 1.75] 2.63 | 3.51 | 3.55 | 12.39] 4.03] 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 | 4.19] 1.46] 8.29 | 335 26 | 4.78 | 3.40) 7.21} 0.94] 1.42} 1.89] 2.83 | 3.78] 3.83 | 13.39] 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] 1.63] 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} 1.11 | 1.67 | 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 33 | 5.99) 4.26) 9.00] 1.17 | 1.76] 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} 2537 |) 13522) (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 | 21:43 | 6.88 | 2.48 | 13.80 | 317 44} 7.79| 5.52] 11.60} 1.50} 2.24] 2.99} 4.49| 5.93) 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 Notre.—For Lz and M; see Table 13; for 25M and MSf, take u of M2 with sign reversed; for P,, Re, Si, S2, Sz, Ss, T2, Mm, Sa, and Ssa, take u=0. HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLEe 11.—Values of u for equilibrium arguments—Continued. 1&9 [Use sign at head of column when J is Rebweens oar 180°, reverse sign when NW is between 180 and an Sa OLD (IOM~ GGG YORK I Ane NEO Ws SB NOR NNN ONAN ANP PAR AMD BAM ann x oon iz) 09 _ woos IID ope SSe2 8a Baa Raa Ske B Sey OND CONF NASH BON O SATS PPO WN HHO OOD CRO wow AE 233 900050 900000 G00000 G050G0 c00000 Go 90G0 III ABH OD Com Ky 11.82 12. 04 12. 26 12.47 12.68 12. 88 13. 08 13. 28 13. 48 13. 67 13. 86 14.05 14. 23 14. 40 14. 58 14.75 14. 92 15. 08 15. 24 15.39 15. 54 15. 69 15. 83 15. 96 16. 10 16. 23 16.35 16. 47 16. 58 16. 69 16. 80 16. 90 17.00 17.09 17.17 17. 25 17.33 17. 40 17. 46 17. 52 17. 58 17. 63 17. 67 17.71 17.74 17.77 M2,No 2N,MS A, Hy Y | On r<) NID APD HMw RPNOO NOt ON bo Oo NN PNY NYY PHN PNY NYN PNP NEP Per Pee poe pee pee pee eee oe oe Ot OOO esse 228 sg He He Bmw Whmht Heo ooo ss M3 SS8 8 QF FO, Be Hh NYS NNN NNN W AD Onn rp ] (=) woes caggo woNsD gGED oord owsD wwe wm NE —_ oo M,,MN SSS Ses FSR SAS F “1 tO 00 00-352 NOD SES SRR BES et tt OoD RNO Se SSS 8 Shh FAP PR RR RR RR RRR pO ogOge gogo GErEs poDDG WerE goto gErES oe 7 ww hy Nw AI~i 4 PHD PRR DRDH PHA AHA ABH BAH Dan nan son gin wore \ailasptany Pie WWW WOW Pee FOO COND Nr Mg ie OO Ts ERR BSS 283 £38 28S Dee hh SRE 00 Or 01,Q; Mg 2Q, ai (6X@) MK No MNO COS 3S eon bore Cont Moc Be Oe RSS Sue & 228 R82 See coon ARO ar C= > on S56 or 51 90.99.00. 90:G0,30 90.9000 909000 909000 G000G0 go00G0 SONI NNN NNN NNN NNN PHD PHP Sn & + = =: 6.42 | 22.31 | 7.15 6.55 | 22.75 | 7.28 6.68 | 23.18 | 7.41 6.80 | 23.60 | 7.54 6.92 | 24.02 | 7.67 7.05 | 24.44 | 7.80 7.17 | 24.85 | 7.92 7.29 | 25.26 | 8.04 7.41 | 25.66 | 8.17 7.53 | 26.06 | 8.28 7.65 | 26.46 | 8.40 7.76 | 26.85 | 8.51 7.88 | 27.23 | 8.62 7.99 | 27.61 | 8.73 8.10 | 27.98 | 8.84 8.21 | 28.34 | 8.95 8.32 | 28.70 | 9.05 8.42 | 29.06 | 9.15 8.53 | 29.41 | 9.25 8.63 | 29.75 | 9.35 8.73 | 30.09 | 9.44 8.83 | 30.42 | 9.53 8.93 | 30.74 | 9.62 9.03 |,31.06 | 9.71 9.12 | 31.37 | 9.79 9.22 | 31.68 | 9.87 9.31 | 31.98 | 9.95 9.40 | 32.27 | 10.03 9.48 | 32.55 | 10.10 9.57 | 32.82 | 10.17 9.65 | 33.09 | 10.24 9.73 | 33.35 | 10.31 9.81 | 33.60 | 10.37. 9. 88 -85 | 10. 43 9.96 | 34.09 | 10.49 10.03 | 34.31 | 10.54 10.10 | 34.53 | 10.60 10.17 | 34.74 | 10.65 10. 23 | 34.95 | 10.69 10.30 | 35.14 | 10.73 10.26 | 35.33 | 10.77 10.41 | 35.50 | 10. 81 10.47 | 35.67 | 10.84 10.52 | 35.83 | 10. 87 10.57 | 35.98 | 10.90 | 10.62 | 36.12 | 10.93 2MK or © Nee [—) © 30 0 31D D tomb Ses BID CO [CS) It} (SOTEN) Orn He HOO mp aS 1 SPP FSP PPR RR RRR Rg ogg wget BND gowgD gow wot WON NNN NNN Wb CS S D wo ~I co Mi Note.—For Le and M; see Table 13.. For 25M and MSf, take u of Mz with sign reversed, for Pi, Ro, Sz Se, Ss, Ss, Tz, Mm, Sa, and Ssa, take u=0. ——_—-—- 190 U. S. COAST AND GEODETIC SURVEY.» Ch TaBLe 11.—Values of u for equilibrium arguments—Continued. [Use sign at head of column when J is Ber ae 180°, reverse sign when’ N is between 180. and | Mo,No 01,9 N| J, | Ky | Ke |2N,MS} M3..|MyMNI Me Ms 20 oO-| MK |2MK} Mf |W ie - ear.| WV uofLs | Ditt oe Nh a wofM, | Diff. | Log F es ; ne (In) Diff. | Log F(Mi)| Di 191 ; (ee Soy iff. N 4) 330) —18.7 a -— 4. 208.5 1915 | | 325 | | p:14.3 P| 10.9 9.9730 : Bae ssg| 3 219.4 447 9.7651 ae ~29| 39 AA) as Gy) | RET 643 330 Be mee | 15g | |} aes 19.5 g3os3 | 298 9. 8294 1916 ‘ 5.6 978.3 | 22-8 9. 8835 153 9. 8993 699 325 305 | +8.6 21.6 9, 8815 20 9.9567 | ou4 320 300 +13.6 5.0 297.9 ae 108 9.9741 174 315 995 | . +-17.4 3.8 314.9 17.0 9. 8923 i 316 310 1917 2.2 307.4 | 12.5 9.9153 | 230 9. 9425 290| - 419.6] | 23.) } oudioa:| 1 300 Bigees | jgost | {nee 285 419.4 FOND 336.8 | ~ ee 449 9.8244 600 300 Bao | | fied | oe 344.3] | 25 9.9948 | - 506 295 275 9.6 | | &3 Pere | an Oo Ooiss | 310 9.7738 191 7.9 356.2| , 28 0.0954 | . 296 9.7368 | 300 290 8| 270 44.7 5.2 0.1305 |, 322 peeres | 9838 |. eee 265 een Teall 1.4 ? 63 9.7018 ill 280 260 Bigzs 4.4 6.6 Bag 0.1368 | * 15 275 259 eee | eS 12.1 5.5 0.1141 297 |' —-9- 7033 4 1919 ! 0.4 18,1 6.0 0. 0740 401 9.7167 134 270 j Eee =a1yo)| | rhe! 0.0300 | | 240 9.7419 252 265 245 Bog, |. t ia8 25.2 : 398 9.7789 370 250 240 6.8 2.6 33.8 8.6 9.9902}. 486 255 235 233.7 3.1 etre | othe 9, 9582 320 9. 8275 1920 : 3.2 eet) | aa | peat 234 BiR22 | | gs. | yaad 230 —0.5 18.2 9.9200 | . 148 919539 | 670 245 225 +2.6 3.1 | 77.7 ' 67 0.0203 664 240 220 55d 258 97.9 | 20-2 9. 9133 493 235 Bis!) Penaeras |. oe leg | oaee 9. 9139 eg | 100626 1921 : 1.6 Big | 15-1 9.9210 71 | 0.0830 134 230 210 49.2 11.5 9.9338 | | 128 0.0580} 200 225 oe +10.4 0.9 143.3 174 | 0.0113 467 220 200 410.1 0.0 152.2 8.9 9.9512 | 510 215 795 | | ::9:0| 7 At wee | Sie | | beer 209 99603 g 1922] 1 2.0 165.1 5.9 9.9948 227 9.9141 462 210 190 S70 | 5.0 0. 0172 224 | 9. 8760 381 205 185 user | 4 ee 170.1 196 Peet ezas | eyed 180 tog| 32 4g| 47 0. 0368 209 195 175 —2.9 3.5 179.2 4.4 0. 0509 141 9, 8257 192 ; tet 13.6 475 9. 7060 71 85 85 —18.4 1.6 20.7 real 0.0620 200 80 Gp isco | |) ae Baal eae O.o105 | 525 9. 7260 i 1929 5 4.7 gage | (iiea’|) loaiiencs O23 | (Monrsag. |") 6822 75 55} —10.3 15.5 9.9045 | 383 9.3053 | 466 70 ae ay a ee 56.3 | B88 | a) gb37' || | 8a88 65 Nina SZ Me 76.7 | 20-4 9.8977 609 60 40 ite | | B40 Baty ee 9.8331 | 146 9.9246 ; 190.6 | 20-9 9. 8811 20 9. 9677 431 55 9. 8923 112 9. 9664 13 50 9.9134 | 480 45 2 4) . e td. 00 to 2000—Con if u and log F of L2 and M, for years 19 —Values of u TaBLe 13. i N i LogF (M;).| Diff. | wu of My Diff. | Log F(L:2).| Diff. g w of Le Diff. Year. N °o 40 . 9184 i ° 9. 8923 252 ‘ a 35 ° “ 120.6 16.1 9. 8498 670 30 1929 40 +7 5.7 9.9175 401 9. 7828 558. Die 136.7]. 41.9 AO | 9.7270 420° 20 ei 35 +12.8 4.8 148.6 9.0 0.0134 avi len 9. 6850 281 193 30 +17.6 3.3 17.6) bag, 0. 0840 774 15 25} $20.9.) . 4. 165.0 6.3 9. 6569 141 10 20 +21.3 4.2 0. 1614 588 9. 6428 7 ae 171.3 5.7 0. 2202 16 9. 6421 129 15 +17.1 10.4 177.0 5.6 0. 2218 572 0 1931 10 +6.7 13.9 182.6 5.8 9. 6550 267 355 5 —7.2 10.6 0. 1646 781 9. 6817 408 350 i 188.4 6.3 0. 0865 720 9.7225 546 345 By 0 has 4.5 194.7 waa 0. 0145 576 9.7771 662 19. 355 Pe 3 0.3 202. 0 9.1 9. 9569 412 340 350 =29! $433 211.1 12.0 9. 8433 677 335 345 —18.7 4.9 9. 9157 262 9.9110 468 330 s 223.1 16.2 9. 8895 119 9. 9578 6 325 aM 340 mie 5.9 239.3 21.0 9. 8776 18 9. 9572 449 1 335 =U 6.3 260.3} 93 9 9. 8794 151 320 330 —1. p 6.2 283.5 20. 4 9. 9123 633 315 305 +4) 5.9 9. 8945 282 9. 8490 594 310 , 303.9 15.6 9.9227 410 9.7896 478 305 a 320 +10. 4 fi 319.5 11.4 9. 9637 523 9. 7418 340 19’ 315 +15. 5 3.9 330.9 8.7 0. 0160 592 300 310 +19. 6 1.9 339.6 Fil 9. 7078 204 295 305 +21.4 fe 0.0752 | + 59 9. 6874 71 290 346.7 6.1 0. 1302 316 9.6803 55 285 1935, 300 +20.3 4.9 352.8 5.5 0.1618 70 9.6858 181 295 +15. 4 8.5 358. 3 5.4 0. 1548 393 280 290 +6.9 9.4 B67 5.4 9. 7039 310 275 285 —2.5 7.1 0.1155 530 9. 7349 428 | 270 i 9.1 6.0 0. 0625 496 9.7777 554 265 36 280 —9. 3.6 15.1 6.6 0.0129 418 9. 8331 658 19 975 —13.2 0.3 onl & 8.1 9.9711 313 260 270 8h 1.6 29.8] 10.4 9.8989 701 255 265 —11.9 2.9 9. 9398 208 9. 9690 595 250 ‘ 40.2 13.6 9.9190 103 0. 0285 270 245 1937 260 —9. 3.6 53.8 177 9. 9087 24 0. 0555 160 255 —5. 3.8 GAR 20.7 9. 9063 63 240 250 =1.@ 3.8 92.2 20.3 0. 0395 446 255 245 +2.2 3.4 9. 9126 136 9. 9949 522 20 112.5 16.8 9. 9262 196 9. 9427 475 225 F 240 +5.6 2.9 129.3 19,7 9.9458 240 9. 8952 383 1935 235 +8.5 2.1 142.0 9.6 9. 9698 262 220 230 +10.6 152 151.6 7.4 9. 8569 284 215 225 +11.8 0.0 9. 9960 256 9. 8285 186 210 159. 0 6.2 0.0216 | 946 9. 8099 97 39 220 +11.8 ileal 165. 2 5.3 0, 0432 141 205 19 mis | | juan? | ek OE | Ate 9. 8002 Ee iiss 210 +8.3 3.3 0. 0573 44 9. 7989 68 195 ; 175.2 4.5 0. 0617 56 9. 8057 145 190 he 205 +5, 3.8 179.7 4.5 0. 0561 139 9.8202 223 19 200 +1.2 3.6 184.2 4.5 0. 0422 194 185 195 —2.4 3.1 188. 7 4.9 9. 8425 301 180 190 —5.5 ONT 0. 0228 218 9. 8726 381 175 193.6 5.5 0. 0010 219 9. 9107 457 | 170 He 185 —7..6 12 199.1 6.5 9. 9791 200 9. 9564 513 19 180 ons 0.3 205.6 7.9 9. 9591 171 165 175 —9.1 0.6 213.5 10.1 0. 0077 509 | 160 170 —8.5 iL, 83 9. 9420 130 0. 0586 380 155 223.6 13.0 9. 9290 83 0. 0966 78 150 1942 165 —7.2 1.8 236.61 16 6 9.9207 31 0. 1044 299 160 —5.4 9) 5) 253.2 19.2 9, 9176 28 145 155 —3.2 2.6 272.4 18.9 0. 0745 563 140 150 —0.6 oi 9. 9204 91 0. 0182 652 135 291.3 16.0 9. 9295 160 9. 9530 624 130 43 145 +2.1 2.6 3807.3 | Jo 5 9. 9455 230 9. 8906 542 19 140 +4.7 2.4 319.8 9.8 9. 9685 297 ; 125 135 ove 1.9 329.6 eu 9. 8364 441 120 130 +9.0 1.0 9. 9982 349 9. 7923 334 115 ; 337.3 6.5 0. 0331 365 9.7589 225 110 1914 125 +10. 0.3 343. 8 5.6 0. 0696 306 9. 7364 G44 120 +9.7 2.1 349. 4 5.2 0. 1002 115 +7.6 4.4 354.6 110 +3.2 i HARMONIC ANALYSIS AND PREDICTION OF TIDES. 201 TABLE 13.—Values of u and log F of Lz, and M, ‘for years 1900 to 2000—Contd. Year.) W u of Lz Diff. u of Mi Diff. | Log. F(L2)} Diff. |Log #(Mi)| Diff. N ° ° ° ° 1944| 110 ee) | aie 46 | gl | onmeN2 | | ye 9.7364| 110 1945| 105 _3.0 359.7 0.1149 9.7249 105 300), | J-9%6,| 8:8. es: Ben | 0.1061 dre |: ease]: | 08 100 em | S|) tage! (G2 | ger | 128) | owen | ee) | ae : 0.2 . 1.5 ; 446 . 370 1946 85| 17.9 24.0 9. 9885 9.7973 85 an) Sea) | BS dass | ore | ouaaag!| - | S92 9.8467 | £98 80 my | mpais|! 32 9.9163 | 328 9.9050 | 383 75 1947 70 8.2 63.1 9.8963 - 9.9603 70 0 5.2 21.5 99 279 65 340)| - ee gife)| 203 9. 8864 aa 9.9382 | 270 65 60 Poe | eee sazi2,| - 22.8 9.8884 | 2 9.9677 | 205 60 55 $81) 5 fale | 138 9.9030 | 336 9.9088 | 289 55 1948 50| +13.2 140.7 9.9312 9.8382 550 Bayi cee |) | Bee 151.4 | 10.7 9.9740; 428 9.7734| 848) ~ 45 40; +198) 228 mule | | fe 0.0319 | 372 9.7208 | 328 | | -40 35, +190, 58 166.7 | 88 0.1025 | | 238 9.6820 | 388 35 1949 30| +13.4 172.7 0.1749 9.6570 30 25 42.0| 14 178.4| 22 0.2196 | 447 9.6455 | 115 25 al | ewe, 188 18a | | 6 0.2046 | 158 9.6475 | 420 20 i) | gost | Se isaig | 38 0.1407] $38 9.6629 | }es 15 1950 10| —23.1 964 | 2 0.0643 9.6924 10 11 27 680 437 0] 0.0660 | 24 gual |i) 38 230 1976 | 295 —8.0 193.8 0.0374 9. 8033 295 2 | aol | | ES dogs |.) 281) Bongorar | a) Sen) | wokwent |. GHee | | 290 215 ee) | oes paged |) 1 (B88 9.9301 | 2/3 gigeza | | Peel 215 210 Pe le BEST ore 9.9573 | 228 9.9436 | 224 210 1977| 205 5.9 | 225.2 F 9.9399 0.0042 205 Baa | | avetaun.| | 22 maby |, dens |, seliest'|| © 3184) 1 Motoest |) i aes\| 1% 200 195 Spa pee g5ntp)|) ere 9, 9218 4 o.1070| #33 195 1978 | 190 41.0 274.2 9.9208 0.1199 190 tes |" Ooteat2 |) 24 ports | 182 |, | worse | a0! | pobog7i|. eS | | ass 180 os ae N76. Stace gygs7.| )gse oves02 || ‘eee 180 175 eas] || Hrs aig} 193 9. 947 ine 9.9949 | 233 175 1979| 170 +8.1 328, 8 9.9649 9.9413 170 165 Haig | | Oe BREED iain 2 9.9862 | 233 9.3936 | 344 165 160 esta || 8 maga! "| bee 0.0099 | 287 98538 | a8 160 155 Hes) il og BETaB iI ah 0.0340 | 313 91 8208 ieee 155 1980} 150 44.3 352.7 0.0553 9.7989 150 145 wore | 32 sis | eae 0.0696 | 183 9.7342 | 1? 145 140 Beig | a8 nga |e are 0.0729 e 9.7783 8 140 135 230) a CaM beats 0.0630 | ov 917816 |) ae 135 ags1. |. "130 |. 011.4 11.9 0.0416 9.7947 130 125| 13.4; 29 mao |) |e'8 oyoiza | 2° oyeis4 |) | vee 125 | 4| 32tzrs | OA Bs || a obesity | 1) oa 9,a5a3.| | (829 120 C9 EONS I se au ditearete 9.9524 | 283 9.3990 | 457 115 1982| 110] —10.5 46.0 9.9278 9, 9523 110 105 7) | 302 Galo | ene go1o1 | 127 oraoai | | 508 105 100 save) 38 any |) e208 9, 9006 2 0.0307 | 2/6 100 95 HOI hig SS rae 9.9004] 494 o.o14s| 16 95 1983 90 45.3 122.9 9.9104 9.9597 90 ees +9.5| 32 18m |, 4a" 99315,| | 255 9.3394 | 708 85 30) +130) 33 Daeey ee 9.9645 | 330 ofgais,| | pete 80 ape e is. 3)| | 288 TTS, | ayers 0.0099 | $34 9.7646 | 312 7 1984 70 |). sea5l4 164.3 0.0659 9.7201 70 Ga jh snatao' |. 228 mona |) OF onazes | | fee 9.6889 | 312 65 60 SL ee 176.0| 38 0.1741 ae 9. 6706 # 60 1985 55 25636 181.4 0. 1839 9. 6652 55 50) 162) 28 PRTAO |) Die 0.1474 | 383 9.6729 9/4 50 45 —21.5 0.9 193.2 71 0. 0849 650 9. 6941 | 350 | - 45 aay ty onks,| 10.8 Boos | pee 010198) ee gu72g1 |) 230 40 1986 35 | 90,1 209.0 9.9639 9.7781 35 Sie Sts. 9 a2 aaoyay| aoe quant) [oy aes 9.8395 | 614 30 9 ee (i a Payee ae 9.3920 | 200 9.9035 | 840 25 20 —4.2 6.5 256. 6 23. 4 9. 8768 14 9. 9506 18 20 1987 15 42.3 280.0 9. 8754 9. 9524 15 10 ate, bie BO1,0\K, Sayre 9; 8880) lf) dae 9.9068 | 438 10 Plies eidap yl) oes Bua3) | hae g.ai5o| 220 9.3400 | 868 5 Cilia Gere tie Sea Gh 9572 | EH? 917750). Gap 0 1988 | 355| +22.6 338.3 0.0151 9.7215 E 355 350 Hoa7 |) OT aie 3 0.0868 id 9.6820 | 325 350 See fe sete UN) eee 351.9] 575 0.1629 | fag 9.6565 | 374 345 340 saad Oe aara7 Wace 0.2165 a g.e451 | 1 340 i989 | 335 —6.0 3.3 0.2126 i oyeaga! ee 335 SSO is oe Oy hea ae ee ee Okeb23 hess 330 325 | ) —to.8 | +0 ee ee 0.0309 | 244 giegta) |, 28? 325 320 | | —19.4 . 22.5 . 0.0132 9.7341; 2 32) 72934—24+——14 904 U. S. COAST AND GEODETIC SURVEY. TaBLE 13.—Values of u and log F of Lz: and M; for years 1900 to 2000—Contd Year.| N | wofLs | Diff. | wof Mi. | Diff. |Log F(Z2).| Diff. [Log F(M).| Diff. | W o °o ° ° 1989] 320] 19.4] 54 225A Nan oer) ||) ee 9: 7841 | tiene 320 18901) |) 845 | 1 16.4 31.5 | 9. 9595 9.7906 315 4.6 11.7 386 676 aio | ) yeas |) 58 aon | ee 9.9209 | 386 9.3582 | 676 310 305 PEGS |) | eee sano | ope oxsoe7 |) (28 9.9976 | 694 305 300 lo | | 28 ais |’ joe 9, 8859 ne 9.9776 a 300 1991} 295 2ANG 102.2 9.8876 9. 9827 295 290 +97) 32 az2i6,| Poe gioont | | > 9, 9433 a 290 235 | +140) 3:3 fag2| 138 giaas7 | | 28 9.8848 | 285 285 280] +17.0| 29 149. 7 te 9.9601 | 34 gigaes |) Gpee 280 logo | 275] 418.3 158.6 0.0029 9.7819 275 270.) gearzis’) | 2 des] ieee 0.0479 | 43 9.7493 | 326 270 265| +1361 34 wes |) oe quossy,| 9) Sc, 9.7290 iS 265 1993 | 260 77 176.9 0. 1137 9. 7211 260 6.9 Bil 10 38 255 SHB |) pele wa7i0)| /} 2A onreay || 1 9.7249 | 438 255 250 =o] 38 187.0 ? 0.0933 | 214 9.7400 | 22 250 245 —a6| 36 agg ls ee axes | 7 32 9.7662 | 262 245 1994] 240| —10.2 198.3 0.0218 9. 8033 240 0.3 6.9 340 477 235 99| 8 a 9.9878 | 30 9.g510 | 327 235 230)| | geeases || Sea Bisis | ee oxanay | | 280 9.9086 | 348 230 295 eed | (eee got) Ab8 9.9389 | 209 g:9731 | | leas 225 1995 | 220 —3.8 237.9 9. 9252 0.0372 220 2.7 7 17.2 69 496 oo | (wesw 260 |) (apes | 20k | eee |e) | SO ee 205 Aud) ee 293,1 | 186 9, 9228 51 0.0839 | 202 205 Shae 2.1 ; 15.4 : 102 ; 444 1996 | 200 46.2 308. 5 9.9330 0.0395 200 1.7 11.9 147 519 195 SECON appa!) | nt8 9.9477 |. a7 9.9876 | 329 195 90 +89) §8 goog | (ies o.g6c1 | 38 qvagep || ages 190 185 +92) 03 29780)| Ge 9.9870 | 209 918055 | | (eae 185 1997 | 180 48.6 343.0 | | © 0.0091 9. 8607 180 1.6 5.3 211 268 175 70) | 5c gapia)| | 22 0.0302] 211 93339 | | «es 175 170 fas | ake zaaio | 9 okoem7)| a3 O:8148 | | tas 170 165 ebis2) |), [eee a57u5 | | oe 0.0586 fs 9. 8035 ae |) 16s | 1998 | 160 aii 1.9 0.0605 9.7999 160 6 3.6 A 81 45 155 =61} 26 apa) bare 0.0524 | 484 9.8044) 445 155 150 opt | (ise g1g3) |’ Wigs avaass | | 26° o.g174 | 330 150 M5 | aden | bee #69)| Here o,oiza | | 238 9118305) |. Gara il arses 1999/ 140| —11.8 23.4 9. 9866 : 9. 8714 140 135 | | pestle | eC BUS)| aoe 9.9613 | 393 9.9130] 45° 135 130 —99/ 28 H2n0) | aya 9.9389 | 228 9.9627 | 97 130 2000 | 125 25 55.9 9.9215 0.0141 125 120 ||| gaia) ot 7309 | 280) | .9;9103| 2 | hososte 1) ean ee 115 —09) 38 ats | 208 9.9066 | 32 0.0540 | 428 115 110 42.8 114.7 . 9, 9111 0.0142 110 ‘2 HARMONIC ANALYSIS AND PREDICTION OF TIDES. 205 iad | TaBLE 14.—Factor f for middle of each calendar year, 1850 to 1999. Component. 1850 | 1851 | 1852 | 1853 | 1854 | 1855 | 1856 | 1857 | 1858 | 1859 Tio: sh Ree Are Sree 0.892 | 0.948 | 1.007 | 1.061 | 1.105 | 1.138] 1.158} 1.165 | 1.160] 1.141 LE oe eS ae 0.922 | 0.959 | 0.999 1.037 | 1.069] 1.092] 1.107] 1.113 | 1.108} 1.095 Ep eee eae 0.816 | 0.887 | 0.977] 1.075| 1.168] 1.246] 1.298] 1.317} 1.302] 1.254 Tae eel eenip es aor ae 1.163 | 0.905 | 0.725 | 1.055 | 1.263 | 0.944] 0.469} 0.962 | 1.283} 1.001 Mond anes. te 1.023 | 1.675, 1.974 | 1.559 | 1.118 | 1.860} 2.348] 1.872) 1.177] 1.776 Mo*, No, 2N, Jo, u2, v2-| 1.027 | 1.017] 1.005] 0.993] 0.981 | 0.972] 0.966 | 0.963 | 0.965} 0.971 cee Ss ag 1.042 | 1.026 1.008] 0.989] 0.972 | 0.958] 0.949 | 0.945 | 0.948 | 0.957 PEZINON os oe Pe. f 1.056 | 1.035 | 1.011] 0.986 | 0.963 | 0.944] 0.932 | 0.928 | 0.931 | 0.942 Mie Seek | 1.085 | 1.053 | 1.016 | 0.978 | 0.944 | 0.918} 0.900 | 0.894 | 0.899 0.915 He Nd Seer Se et 1.114 | 1.071 | 1.021] 0.971 | 0.927] 0.892] 0.869] 0.861 | 0.867} 0.888 HOLA nee oe 0.874 | 0.933 | 0.998 | 1.059! 1.110] 1.150] 1.174 | 1.183! 1.176| 1.153 O10 ME Ee aan 0.631 |. 0.786 | 0.983 | 1.204 | 1.422] 1.608] 1.735 | 1.783 | 1.745 | 1.627 AC) Se a 0.948 | 0.976 | 1.004] 1.029] 1.048 | 1.062] 1.069] 1.072] 1.070] 1.063 TLE oe See 0.974 | 0.993 | 1.010 | 1.022] 1.029] 1.032] 1.032] 1.032} 1.032| 1.032 it eae Gone eae 0.743 | 0.856} 0.990 | 1.129] 1.257| 1.360] 1.427| 1.452| 1.432) 1.370 Wiens GR os 1.094 | 1.059 | 1.016] 0.973 | 0.933 | 0.900] 0.879 | 0.871 | 0.878 | 0.897 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1.110 | 1.066 | 1.013 | 0.955 | 0.898 | 0.852] 0.829 | 0.832 | 0.863 0. 912 1.072 | 1.041 | 1.004] 0.964 | 0.926 | 0.898 | 0.883 | 0.885 | 0.904 0. 936 1.179 | 1.086 | 0.988 | 0.897} 0.823 | 0.773 | 0.749 | 0.753) 0.784 0. 840 0 1 1 8 7 0.568 | 0.924 | 1.225 1.117 | 0.865 L att 1. 082 - 091 0. 840 il 2.227 | 1.792; 1.046] 1.260} 1.680 -609 | 1.164] 0.812] 1.189 1.731 0.980 | 0.991; 1.004 | 1.016 | 1.026] 1.084] 1.038] 1.037 | 1.032 1. 024 0.970 | 0.987 |} 1.006 | 1.024] 1.040) 1.051} 1.057 | 1.056 | 1.049 1. 036 0.966 | 0.983 ; 1.008 | 1.032 | 1.054 | 1.069 1.076) 1.075 | 1.036 1. 048 1 1 0 0.941 | 0.974 | 1.011} 1.049 | 1.081} 1.105 | 1.117 -115 | 1.100 1.073 0.922 | 0.966 | 1.015] 1.065} 1.110] 1.143) 1.159 - 156 | 1.135 1.099 1.116 | 1.065 | 1.005 | 0.941 | 0.880 | 0.832 | 0.808 . 812 | 0. 843 0. 896 1.447 | 1.230 | 1.008} 0.807 | 0.647 | 0.540 |) 0.489) 0.497 | 0.563 0. 6&5 1.050 | 1.032 | 1.007 | 0.979 | 0.951 | 0.928 | 0.916 | 0.918 | 0.933 0. 958 1.029 | 1.023 | 1.011] 0.995} 0.976 | 0.960 | 0.950 | 0.952 | 0.964 0. 981 1.270 | 1.145 | 1.007 | 0.872] 0.755 | 0.670 | 0.629 | 0.635 | 0.689 0. 783 0.928 | 0.968 | 1.011 | 1.054} 1.091 | 1.117) 1.180] 1.128] 1.111 1. 082 Component. 1870 1871 1872 1873 1874 1875 | 1876 1877 1878 1879 1.028 | 1.079 | 1.120) 1.147] 1.162] 1.164] 1.154] 1.180 1. 094 1.014 | 1.050 | 1.079 | 1.099] 1.111 | 1.112] 1.104] 1.087 1.061 1.014 | 1.112 1.201 | 1.269 1.309 | 1.315 | 1.287} 1.227 1.144 1.148 | 1.224 | 0.816 | 0.545 | 1.087 | 1.270 | 0.858 | 0.543 1. 036 1.300 | 1.185 | 2.004 | 2.286) 1.656 | 1.227] 1.998 | 2.269 1. 645 1.000 } 0.988 | 0.977 | 0.969 | 0.964 | 0.963 | 0.967 | 0.974 0. 984 1.001 | 0.982} 0.966 | 0.954 | 0.947 | 0.946) 0.951 | 0.961 0. 976 1.001 | 0.976 | 0.955 | 0.939 | 0.980! 0.928} 0.935 | 0.949 0. 968 1.001 | 0.965 | 0.933 | 0.910 | 0.896 | 0.894} 0.904] 0.924 0. 953 1.002 | 0.953 | 0.912 | 0.881 | 0.864 | 0.862) 0.874 | 0.900 0. 938 1.022 | 1.080 | 1.127] 1.161] 1.179; 1.182] 1.169] 1.140 1.098 1.067 | 1.290 | 1.500 | 1.666} 1.764 | 1.779) 1.708 | 1.563 1. 366 1.014 | 1.037 | 1.054 | 1.065 |) 1.071 | 1.072] 1.068 | 1.059 1. 044 1.015 | 1.025 | 1.030] 1.032] 1.032) 1.032) 1.032 | 1.031 1.027 1.044 | 1.181 | 1.300 | 1.390 | 1.442 | 1.450] 1.413] 1.335 1,224 1.000 | 0.957} 0.919 | 0.891 | 0.874 | 0.872 | 0.884] 0.908 0. 943 *Factor f of MS, 25M, and MSfare each equal to factor f of Mo. Factor f of Pi, Re, Si, S2, Ss, Se, T2, Sa, and Ssa are each unity. 206 _ U. S. COAST AND GEODETIC SURVEY. TaBLe 14.—Factor f for middle of each calendar year, 1850 to 1999—Continued. Component. 1830 1881 1882 1883 1884 1885 1886 1887 1888 1889 Anispi saudeio. o. 8 faeer 1.047 | 0.991 | 0.932 0.878} 0.840 | 0.827 | 0.841] 0.880] 0.934 0. 994 TAS open 2 aie eee 1.027 | 0.988} 0.949 | 0.914] 0.890 | 0.882] 0.891} 0.915 | 0.950 0. 990 Kg i aGidumn set ateaeee 1.048 | 0.951} 0.866 | 0.800} 6.760 | 0.748 | 0.762} 0.803 |) 0.869 0. 955 Tig soc) le os See cists 1.246 | 1.020] 0.786 | 0.944] 1.152) 1.171 | 1.006 | 0.824] 0.945 1. 205 My. eeiders eres shares 1.046 | 1.528 | 1.824] 1.529] 0.970 | 0.877 | 1.364] 1.721 | 1.593 1.075 Mo*, No, 2N,Ao,u2, ve-| 0.996 | 1.009} 1.020} 1.030 | 1.036 | 1.038 | 1.036 | 1.029} 1.020 1. 008 Mapcc. 42-38 see deee 0.994 | 1.013 | 1.031 | 1.045 | 1.054] 1.057} 1.054 | 1.044) 1.030] 1.012 1.017 | 1.041; 1.060 | 1.073 1.077 | 1.072] 1.060} 1.040 1. 016 1.026 | 1.062 | 1.092) 1.111} 1.118) 1.111] 1.091] 1.061 1.024 1.0385.) 1.084 | 1.124] 1.151 | 1.160] 1.150} 1.123) 1.082 1. 033 ‘Oy,.Qi,2Q) p1 --.-.--- 1.043 | 0.980 | 0.916 | 0.860 | 0.820 | 0.806 | 0.821 | 0.862 | 0.919 0. 9&3 Oca ere) ts oe 1.144 | 0.926 | 0.739 | 0.599 | 0.513 | 0.486 | 0.516 | 0.604 | 0.746 0. 936 MICS heces se cees 1.023 | 0.997} 0.968 | 0.941 0.922] 0.915 | 0.922] 0.942 | 0.969 0. 998 VB EES Ae ape eee 1.019 | 1.005 | 0.988 | 0.969 | 0.955 | 0.950 | 0.955 | 0.970 | 0.988 1. 006 Migs soos ee eS 1.092 | 0.953 | 0.823 | 0.717 | 0.649 | 0.626 | 0.651 | 0.721 | 0.828 0. 959 Mim os ieatigz sian Se 0.984 | 1.028} 1.069 | 1.102] 1.124 |) 1.131 | 1.123 | 1.101 | 1.067 1. 026 | Component. 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 A jssssssteceiscsesee 1.049 | 1.096 | 1.132) 1.155 | 1.165 | 1.162] 1.146] 1.118] 1.077 1. 026 1S PO eae a Re she 1.028 | 1,062 | 1.088 | 1.105 | 1.112] 1.110] 1.099] 1.078] 1.048 1.012 DENG Sate uJ eee 1.052 | 1.148 | 1.230] 1.289] 1.316 | 1.308 | 1.267; 1.197} 1.108 1.010 Moo = cis tomes oe ee 1.153 | 0.709 | 0.683 | 1.185 | 1.219 | 0.704 | 0.607 | 1.141] 1.229 0. 897 2 SBE RE Sead Ampase die 1.323 | 2.091 | 2.158 | 1.434 | 1.369 | 2.176 | 2.240] 1.471 | 1.166 1.781 Mo*, No, 2N, A2,u2,¥5-| 0.996 | 0.984 0.974 | 0.967 | 0.963 | 0.964} 0.969 | 0.978 | 0.989 1.001 IMR Sater be seth 0.993 | 0.976 | 0.961 | 0.950] 0.946 | 0.947 | 0.954 | 0.967 | 0.983 1.002 Mas MING? fossesenece 0.991 | 0.968 | 0.948 | 0.934} 0.928 | 0.930] 0.939 | 0.956 | 0.977 1. 002 | (Mg Sted ee 0.987 | 0.952 | 0.923 | 0.903} 0.894 | 0.896} 0.910 | 0.934} 0.966 1. 003 RE Sie eS ce 0.982 | 0.936 | 0.898 | 0.873 | 0.861 | 0.864] 0.882] 0.913 | 0.958 1. 004 01, Qi2@ ype nee. 1.046 | 1.100 | 1.142] 1.170 | 1.182] 1.179 | 1.160} 1.125} 1.078 1.020 OOPS eee ase eae 1153] L375) desl) stense |) E7800 1 cGls |e T6600 ih tole reales G 1. 058 INU eae oe De 1.024 | 1.045 | 1.059 | 1.068] 1.072] 1.071 | 1.065} 1.054 | 1.036 1.013 MERE oe cea cos ages 1.019 | 1.028 | 1.031} 1.032} 1.032 | 1.032) 1.032) 1. 030 | 1.025 1.014 UN es a a aes 1.098 | 1.230 | 1.339 | 1.416 | 1.451 | 1.441 | 1.387 | 1.296 | 1.175 1.038 NIN) 2s er see ae 0.983 | 0.941 | 0.907} 0.883 | 0.872 | 0.875 | 0.892 | 0.921 | 0.958 1.001 Component. 1900 1901 1902 1903 1904 1905 1906 1907 1908 | 1909 ANSE See SAE Renee nice 0.968 | 0.910 | 0.861 | 0.832} 0.829 | 0.854} 0.900] 0.957 | 1.016 1. 069 Ee a niche Peyote oh elas 0.973 | 0.934} 0.903 | 0.885] 0.883 | 0.899 | 0.928 | 0.965 | 1.005 1.042 KG cisco sisia sists oes 0.916 | 0.838 | 0.782 | 0.752 | 0.750 | 0.774} 0.825] 0.900 |* 0.992 1.090 1 DS Ses SOREL 4 98 0.753 | 1.030} 1.193 | 1.117] 0.925] 0.858] 1.051 | 1.221} 1.062 0. 653 IMyee ese s tae aeasasn 1.902 | 1.399} 0.858] 1.069 | 1.507] 1.643 | 1.340] 0.946 | 1.479 2.112 Mo*, No, 2N, Ao,us, v2.| 1.013 | 1.024] 1.032] 1.037] 1.038 | 1.034) 1.026 | 1.016} 1.003 0. 991 Mia rye ee ce ee 1.020} 1.036} 1.049} 1.056] 1.057 | 1.051 | 1.039 | 1.023 | 1.005 0. 986 M4 MING Sess 1.027} 1.049] 1.066} 1.076} 1.076] 1.068 | 1.053] 1.031 | 1.007 0. 982 IMG 22 ] oO & 4 S oD Seaa sonst mn pal = gs ws aoncd oor eo Ea = Das sli all Saal Sood oON = oO x mel N NS ASH gS te) oR To) N rico at co = NN en N é S a ca 8 BEE € 09T 6 OFS L G08 8 SIG ¥ S6r 8°L 6 TL ° OO as BASS s tr a eae) soHn R ene 4 6961 | 896T 6 661 | F°002 | 6002 | F GET | 666 | F002 | 8 '00G | E GBI | 8 GET | €°00G | 8002 | E66 | 8 66T | 002 0086 | GO82 | F O82 | L612 | 6 OLE | G 082 | F'O8S | L 6LE | 0'O8S | TL 08S | FO8S | 9622 | 6612 | 1 08z T'988 | PLS | 9°8ST | 69S | L'8cE | F'6ES | LOST | 68h | 2 'OGE | GIES | LOFT | OTP | 2 °STE | Gees GFIL | 8°STS | G'OLE | ESE | OFET | EPS | GHEE | 60S | F OST | F'0Se | 9°0SE | $°99 | T LOT | 67292 € COE | OBE | SPELT | T°L0G} L808 | LCS | GS 6ST | F'EFS |) 8 PSE | ESOT | Tele | €°16e | 2s | 9°TET GPIL | 8°SIS | 6 OTE | SSE | OPEL | EHS | GS FEE | 6°0S | E OST | F'0G% | 9°0GE | G99 | T“LOT | 6°L9z C'Ghe | CPHL | Teh | L908 | 0'9%2 | LSet | SSG | 8 60E | L°60G | 9 GOT | F'6 G "866 | 6 CET | 1°26 8 FST | OTF | 9182 | FOES | OFZ | GIT | 609% | 9'°OIS | 266 | 8LFE | G'9ES | OUST | 9°EL | L°0%E SSL | 9°F8S | BES | 26S | 8°06 | 98 | L'Sh | 8 1SG | O'8F | F'POG | ET L°806 | T°L. 6 SOT TTS | OST | 9°SH | SLE | LES | F-8CL | LOE | FLITE | GC 12s | 9'PAT | OLS | LITE | 911Z | FOIT FFOL | POLIT | 6°LFS | F'SGE | TSO | P°EET | 600% | SOE | OL GL | 8°8ET | 9°EhS | LSE | 8° G 801 | SLL | 6'9F% | ESSE | 629 | FSET |) LCOS | 8'80E | FEL | 8°68 | T°O9T | 0°99% | B°SEEs | 9°GF T ‘ZEIT | 1°06 | § 88 | b S62 | T'F6 | 085% | OCS | 6'69C | L6G | ERIS | HLT | O'S2% | Lee | TZ8T L*G0Z | 6064 | €°6L | OTT | 6602 | 0°662 | 286 | LOT | TOE | $'G0E | 8 BE | OTF | 6°62 | 9°81 ¥G o% 6'T 9°S 4G 1% 6'T 9% £°% 14 8°T 9% £3 0% 00 0°0 0°0 0°0 00 0°0 00 00 00 0°0 0°0 00 0°0 0°0 O-O8T | O'O8T | O'O8T | O'OST | O'O8T | 0'O8T | O'OST | O'OT | 0'ORT | O'OST | O'O8T | O'O8T | OORT | OORT QLLT | S°LLT | T°SLT | FLLT | Q°LLT | G°LLT | T'SLE | H°LLT | 2°22 | 6*LLT | @set | FZL0 | L*L2T | O'8zT F682 | F'8 8°98 | T'9TG | L°G6S | 6°L 9°18 | P°S0G | T°LL0 | G'6KE | LOO | G'68T | L°99% | E'ShE G°G9Z | 8°SSS | GSH | 6 GLE | 8'09G | F LFS | & CES | ESS | SLES | L'06G | F°S0G | GOES | GB LTS | 8°90 0 OSE | 8 GFE | 9°6FE | F°OGE | LOSE | 8 Gre | 9°GPE | E OSE | T'OGSE | 6 GFE | 9 OFE | F'OSE | LOSE | 6 GFE T1908 | 0'6E | 8°EEL | O'F0G | G°S0E} TTS | 0'COT | O°0Ge | T°L 8 ZaT y ‘VES | O°ETS | PSST | FEST 9 The | L'ShL | Thr | 8 6C8 | 0622 | 89GT | O'S | S°SOE | FLEL | 66 | S'8FE | TILe | TOLL} 8 °OL G £66 | $6 8°G8 | 0 E12 | 8686 | 8°9 LS | 0'GIG | F680 | L°9 OFS | 9 11S | $886: 1° F 69% | 8°96 | GFZ | 8692 | B°LGS | G9FS | 8 'HES | 609 | GPS | Z'BES | L9G | GSES | L°OFS , 9°82S 6 °19Z | 8°9TS | S°SLT | 9°96 | S HST | O'SHL | SLOT | O'GST | 6 SIT | 9°8L | B°LE | 6°86 | 91S | F8 POL | 9OL | F6CT} 0'09G | T'8IE} OLT | H'9L | 60S | 2692 | 6 '8GE | F'8e | FOOL | L°8IS § 910 6 O€T | F'88S | G98 | S66] 16 | € 190] 60S | $'6SC | F'6S | F6IG | 6 8L | O'262 | 8'Ge | GFT GSst | €°91G | L Fee | O'OTE | TG | G8 G'8E | 9 POT | 9'PEL | GS FOT | CG FET | 2092 | F682 1 '8TE G'ChS | CPHL | Leh | L908 | 0-900 | L°SeL | S$ Ge | 8 60E | L°60Z | 9 °60T | 76 G "866 | 6 Z6I | 1°26 €'OF | 9°LFL | 9°9E@ | 9°30E | G'9E | SOFT | G'OSe | E THE | FOS | L°SST | 6'990 |] 612 | €°S8 | O°6ST 9°96 | $08 | FOE | bP PEL] FS 0 T6E | #°S G6LT | P'GhE | BEST | BREE | F'GOT | 8OCe . GIT 8 O61 | 6OXT | FFT | 9 TST | 8 °@8T | GOST | F'IGT | 8 S6T | 90% | 0°60 | F FIZ | O'9TS | 9°LIZ | O'LTS gts 8 & La 60 oT L’% '¢ Lik FU | 6 FL | 9LT | SST | LBL | Sst OOPS | F'ShS | OLST | LES | GCE | EES | FEST | GSS | GS SEE | O'6FS | O'FOT | E'E9 | O'GES | SHS ° ° ° ° ° ° ° ° ° ° ° ° ° ° LO6I | 996T | S96T | P96BT | E96T LG6T | 9S6T | SS6T | #S6T 6961. | TO6T | O9GE | 6961 | 86T €S6T @ 661 | L661 | 2002 sg “-"Usg 9°66 | 8612 | 1 086 See Oss | EPO | 9ST? 32 “WY, 8°G8 | @L8T | L886 Saimin eae SIN TEB6Gs |G GGs | OS8Cie |e pies t otenetete es eeee cre aN 8°98 | S181 | L886 | See aa ea G PLE | 8 CLT | TL nee FSCT | © Th | O'L86 a TPL Ge ECG: le RaGeleS aer cient ee ee FaS8G eGarele |POeS n= eeeen sec onan terre es oe, Gea GROLG. 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COAST AND GEODETIC SURVEY. 216 £102 | 8 ‘66T 9082 | 6°6L2 $902 | 9°F0T 6°10 | 0°622 gg | FEST 610z | 0'622 T'8cr | 0°18 8°60T | $29 TOE | $ ‘0ST 8CLT. | 9°26 O'ShE | 0°92 TOT | 2 '992 o'sTe | 9°T9T sree | 6°SL (fecha 00 | 00 0081 | 0“O3T 6LLT | Z°LLT 6°88 | 9°6IZ £°c6 | 28 bere | Tose Z‘IL | 9°ser 9'Tht | 8°89 240r | 6 "18% Lite | ¥'968 f21e | 1'¥ee ZHIL | "eke Zore | 1°Z9r Tae Daian ‘Scr | 0°18 8°6LT | 1662 0-991 | 9:9 001% |€"802 ST | 9‘IT 8 °20G | 8“9IT ° o 6861 | 886r L861 986T Z't0z | L°66t | '00z | 9'00z | 1102 | 9°66r | 1°00z | 97002 | I *10z | 9'G6T | 07002 | ¢*00Z | 0°10 | $°66I | 0'00G | G°00G ©7777-7777 TT BSE 9082 | 8616 | T'08G | £°08@ | 9°08G | 8 °6LG | TO8G | £°O8Z | OSG | 87620 | OONE | E70RE | S08 | 8°6Le | O-08G | G'O8Z wg ser | 9:96 | 62 | 27622 | Soot | 2°88 | 0°0 | Ze | Ger] 2°08 | O-~e | E:e9e | OAT | 8° | OPE | E'9cs TTT Zee | 0°00 | 6'OF | O'TFL | T'2Fs [0 ‘BTe | T'89 | Z'sct | Z'scz | OFS | ZL | LPT | Gceze | O'VGe | S68 | OFOT © SW W'S) | HRIL| GIs | Gore | O° | BRET | O'R | Ose | F OTT | O-EO | BRE | ETH | THE | FCIe | L'sok | OH Z'eez | 0-008 | 6'0F | 91H | Tavs | O'sTe | T'89 | B'scr | Z'8cz | OFS | SPL | LPLT | $'92% | OWE | Be6 | OOT ~ aeons (13 8-9eT | 0°09 | Tore | F'8IZ | B°LIT | O'S | G'TOE | 8°10Z | STOL | 0°92 | 8°98Z | ES81 | S78 | 08 | 899% | F'sOT --- 7 see gic BOC Oc T'o2, | Fee | Zee | Suet | ech | Esse | Sere | F'ZET | 1°12 | ecTee | G-6Te | °2OL | GcPse | G'e0e | S°6ST | OGL | NW 9°892 | 9°L1T | 9°928 | eez | cess | F°08 | O-LeZ | Zee | FORT | GE | S"eGT | TSE | OST | 8°8SE | T'8ST | "ZTE |---- 7 Ss Bese BESS HC LHL | #°29 | S*0ce| Sere | #OsE | L°cF | L°808 | ETS | GST | Beh | O'FOE | FOZ | 9°EOT ee PRE 11 6 ‘OIL | F°61Z | $062 | 2:0 | 9°69 | O°EAT | Fees | FOE | ZG | G'IIT| F'6LT | FBG | L'8TE “1d Gert | 9°81 | Z'88z | O-gge | 0°89 | O'FAT | PFHZ | 6'FIE | S'S%_ | HIE | 2°10% | L°1Le | S'TFE pepe Siti 1°G1@ | 6IZE | 2082 | 8°82 | 9728s | E°S8 | HZ] 98h | 8's | 2°09 | 2602 | FS | O-LOT Serial OOF | F'T8 | O°OLT | 8°8G% | 8°LFE | 0°06 | F°6LT | 8'89Z | Z'8GE | 9°OOT | BERT | 8°82 | O°L “Oy eee Re Gees Ome ape HO Cee ER Cel Gm ero mteOecr a \edg Iau all GaGa 0G Ge GR RS Rae eae SOF Sel 00 {00 |00 |o00 |00 |00 |00 |00 |00 |00 |00 |00 | 0° OGag| eo eee es 9 4G 0081 | O'OST | O°OST | OORT | OST | O'OST | O“OST | O°OST | O°OST | O°O8T | O'ORT | OORT | 0°OST Os | ore eee pogo GTLLT | 2°21 | FLLT | L221 | O'RLT | G°LLT | G*L2T | L722 | O'SAT | GLA | Gc2LT | 8 “LLT | O°SLT 8 Pee e ese OREES See ar 16 | $202 | 9°c0g | Z:2z | G86 | €'F2Z | 0-262] 88 | 2°08 | 1°08 | S622 | F'Ece | 92d I eae ee SESE af 8466 | FUE | Sele | 6-10E | 068% | G'sTE | 0°26 | 1°08% | Z7e9z | 8°S8Z | G°TLZ | L'8Gs | B“L¥e i Se ea Serer a 1G ¥ GPE | Z0se | 6 "GFE | LZ GPE | FGFS | S "OSE | 6 ErE | L-OFE | Gere | Z'OGe | O-OGE | LEFE | G-6FE Cpae pase ahi O'1S | 9211 | 6'eIZ | o'eTe | Sug | Greet | O'eSt | 8°6 | G*9t | 9'ZIZ | T1ze | 9°79 | 6'eOr | "ES | F'9zE | L'8¢ | Be OW Fer | $09 | F'Ice] IIe | FOIL | OTF | 6°960| ETT | L°¢8 | G°9 | Gce9t | O'VOT | LT9 | BLE | O'S | 2 “OST |” eee) 0001 | £:92e | "80 | 0702 | 0°26 | L722 | 0-0 | FT | 8°96 | 9°22 | 8 TOE] L°8T | H'6 | F 2 | 2'860 | SFT | "NZ #860 | F-eze | 2 11e | 2662 | F282 | e'eTe | 6 "TOE | 9°06z | E°62z | E°S0E | 8°62 | O'ZBS | O-OL | "SEZ | L°G8S | TOE |-- tN T‘Z8t | 0-0Fe | 2°96 | Frect | O'TTT | O'89T | S°22T | S48 | 12h | FOL] O'89 | O'S | B'see| STE | L:2bE | L-1OE |oc ee ey £703 | O-Ost | z-zez | 1962 | Z-ece | O'9aT | L°S8T | S°SF | EcsvE | T'8L | ZzET | B°G6E | 97es2 | Ges | ¥:08 | G°9eT |--7777 7 9822 | O-O2T | T'92% | 2°92 | 8°982 | OF8 | Behe | Leh | 90% | T:2¢ | STIs | SOL | 169 | 6ST Se ne &°G0z | 0°022 | 9'86z | 9'2ze | 8°9GE | O'E9 | 86 | LSI | L°CST | O°6IS | 9'8PZ | 6°22 | B"90E | GIT | SOF | T's9 |---77 77777 8'98T | 0°09 | T'6Ie| F'8Iz | B'LIT | OP | 6'TOE| $'10Z | 8TOT | 0°92 | 898% | SS | G8 | O'S | 8996 | F'G9r | Se TW LILT | 9't6z | 0702 | 6728 | 8:02 | G'16e | 8°TF | L°SeT | O'RET | $°2Oe | BTS | L°8ST | BFE | 8-LOE | ORF | Sigst jw 7 eee eee ony 8‘cer | ease | z-6gl | Fete | 9°9IE | SOT | BOSE | 6°262 | 2°66 | F°9LZ | 0°66 | FSV] SIS | ZS | £09 | B79 | 7-7 eee i! 26ST | ZHST | O'S | T-esT | 9°S8E | S°8sT | ZFET | 6°00Z | 9°L0G | S"TTZ | BST | GSTS | F'8TS | BFIS | STIS | S720 |= - 777 6 ee ceed Cay al 0 eam Cie Lie al Bae SOs yl | CO beg CgB loc Glen | Oleg! Folie! GGT yiilen| os ae 0102 | 8'c6 | 9°¢ | G:o2z| 6'R8T| T'68 | 6% | 818 | O86 | FOOT | EFT | 9°98G| F'LET | T'S6 | BT | OL | é “$e ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° ° Gger | F861 | es6r | Z86r | Ts6r | Ost | GL6L | SLT | L261 | OLEL | GLOE | FLOE | LEE | ZLEL | TL6T | OL6T “quouodur0;, “‘panuryu0g—Qo0g 97 O98] ‘val wopuaypo yova fo Buywurbaq yw yornuaassy fo wprprsau wof (n-+-° A) quaunbip wnwquinby— GT Gav J, 217 HARMONIC ANALYSIS AND PREDICTION OF TIDES. oS a gon eo x5 a “_ 19 al 6 1S o¢ a ima’ G"9SG G "066 Lys 0 GE 08 T 96 9 COT 1 G9L "89 0 "826 a) oF 00 As ite) v=) xt 1D HOD Ga) OA BONN COMO COO o Ker Sosr on) oo g (sr) © GA a oD S369 05 AS srs es 9 iss) aS Oo BENG & ior) ie) a io} ise) ie) — cor ie] HARMONIC ANALYSIS AND PREDICTION OF TIDES, 283 TABLE 19.—Products for Form 194—Continued. [Multiplier=sin 75°=0.966.] 1 2 3 449 | 2.415 | 3.381 .459 ; 2.425 | 3.391 468 |» 2.434 |. 3.400 2.444 | 3.410 488 | 2.454 | 3.420 497 | 2.463 | 3.429 507 | 2.473 | 3.439 fat ee fe tt | Le hes me L; rs aI Qo 1.517 | 2.483 | 3.449 | 4. 6.347 | 7.313 | 8.279 9. 245 1.526 | 2.492] 3.458] 4. 6.356 | 7.322 | 8.288 9, 254 1.536 | 2.502 | 3.468] 4. 6.366 | 7.332 | 8.298 9. 264 1.546 | 2.512 Pish73 | « 6.376 | 7.342 | 8.308 9. 274 1.555 | 2.521 | 3.487] 4. 6.385 | 7.351 | 8.317 9. 283 1.565 | 2.531 | 3.497 | 4. 6.395 | 7.361 | 8.327 9, 293 1.575 | 2.541 | 3.507] 4. 6.405 | 7.371 | 8.337 9. 303 1.584 | 2.550] 3.516) 4. 6.414 | 7.380 | 8.346 9.312 1.594 | 2.560] 3.526) 4. 6.424 | 7.390 | 8.356 9. 322 1.604 | 2.570 | 3.536 | 4. 6 434 | 7.400] 8.366 9. 332: 1.613 | 2,579) 3.545] 4. 6.443 | 7.409 | 8.375 9. 341 1.623 | 2.589 | 3.555 | 4. 6.453 | 7.419 | 8.385 9.351 1.633 | 2.599 | 3.565 | 4. 6.463 | 7.429 | 8.395 9. 361 1.642 | 2.608 | 3.574 | 4. 6.472 | 7.438 | 8.404 9. 370 1.652 | 2.618 | 3.584) 4. 6.482 | 7.448 | 8.414 9.380 1.662 | 2.628} 3.594] 4. 6.492 | 7.458 | 8.424 9.390 1.671 | 2.637 | 3.603 | 4. 6.501 | 7.467 | 8.433 9. 399 1.681 | 2.647} 3.613 | 4. 6.511 | 7.477 | 8.443 9. 409 1.690 | 2.656 | 3.622) 4. 6.520 | 7.486 | 8.452 9. 418 1.700 | 2.666] 3.632 | 4. 6.530 | 7.496) 8.462 9. 428 1.710 | 2.676 | 3.642 | 4. 6.540 | 7.506 | 8.472 9. 438 1.719 | 2.685 | 3.651 | 4. 6.549 | 7.515 | 8.481 9, 447 1.729 | 2.695 | 3.661 | 4. 6.559 | 7.525 | 8.491 9. 457 1.739 | 2.705} 3.671) 4. 6.569 | 7.535 | 8.501 9. 467 1.748 | 2.714 | 3.680 | 4. 6.578.| 7.544 | 8.510 9. 476 1.758 | 2.724 | 3.690 | 4. 6.588 | 7.554 | 8.520 9. 486 1.768 | 2.7384 | 3.700| 4. 6.598 | 7.564 | 8.530 9. 496 Riva en ParASet ese 0OH | onde 6.607 | 7.573 | 8.539 9. 505 1.787 | 2.753 | 3.719 | 4. 6.617 | 7.583 | 8.549 9. 515 1.797 | 2.763 | 3.729!) 4. 6.627 | 7.593.1 8.559 9. 525 -840 | 1.806 | 2.772 | 3.738 | 4.704} 5.670 | 6.636 | 7.602 | 8.568 9. 534 -850 | 1.816] 2.782 | 3.748) 4.714] 5.680 | 6.646 | 7.612 | 8.578 9. 544 -860 | 1.826 | 2.792 | 3.758 | 4.724] 5.690 | 6.656 | 7.622] 8.588 9. 554 - 869 | 1.835 | 2.801 | 3.767 | 4.733 } 5.699 | 6.665 | 7.631 | 8.597 9. 563 -879 | 1.845 | 2.811 | 3.777 | 4.743) 5.709 | 6.675 | 7.641 | 8.607 9. 573 -889 | 1.855 | 2.821 | 3.787 | 4.753] 5.719 | 6.685 | 7.651 | 8.617 9. 583 -898 | 1.864 | 2.830) 3.796 | 4.762] 5.728 |) 6.694 | 7.660 | 8.626 9. 592 -908 | 1.874 | 2.840! 3.806] 4.772% 5.738] 6.704 | 7.670 | 8.636 9. 602 -918 | 1.884 | 2.850 | 3.816) 4.782} 5.748 | 6.714 | 7.680 | 8 646 9. 612 -927 | 1.893) 2.859 | 3.825 | 4.791 § 5.757 | 6.723 | 7.689 | 8.655 9. 621 UGE Ra SAAB oroe some -937 | 1.903 | 2.869 | 3.835 | 4.801 } 5.767 6.733 | 7.699 | 8.665 9. 631 Create a a iN -947 | 1.913} 2.879 | 3.845 | 4.811 § 5.777) 6.743 | 7.709 | 8.675 9. 641 99S Sst a Sed -956 | 1.922] 2.888) 3.854 | 4.820} 5.786) 6.752 | 7.718 | 8.684 9. 650 W00 tose 5 ese seee ses 966 | 1.932) 2.898 | 3.864] 4. 5.796 | 6.762 | 7.728 | 8.694 9. 660 0 1 2 3 5 6 7 8 9 } 234 U. S. COAST AND GEODETIC SURVEY. TaBLE 20.—Augmenting factors: SHORT-PERIOD COMPONENTS.* / Formula, augmenting factor =P _— : 24 sin 22 : 2 Augment- Logarithm. Remarks. ing factor. Diurnal Ji, Ki, Mi, O1, OO, Pr, Qi, 2Q, p1 | 1.0029 | 0. 001241 : Semidurnal Ke, Lz, Mz, No, 2N, Re, To, 1.0115 0. 004972 ||Each tabulated solar hourly height used Ja, w2, v2, 28M. once and once only in summation; Terdiurnal Ms, MK, 2MK........-.....- 1. 0262 0. 011220 group covers one component hour; Quarter-diurnal M;, MN, MS:....-....- 1. 0472 0. 020029 component ‘day represented by 24 Sixth-dittrnall Mg ks -S. paeeae: Posie 1.1107 0. 045605 means. ‘ . oda Righth-diurnal Mgoe oo. cent oe= 1. 2092 0. 082498 SHORT-PERIOD COMPONENTS.* (seam Heep eres “ae @ 360 sin oy Augment- Tipeeeieiica Augment- 7 osarithm R baat ing factor.| 795 ing factor.| 708 x cn ; Teese 1.0031 0.00134 || Pi.....- 1. 0028 0. 00123 Gy 3-8 1. 0029 0. 00125 |} Qi-:...- 1. 0023 0. 00099 OT 8 1.0116 0.00500 |} 2Q--...- 1. 0021 0. 00091 gies 1.0112 0. 00482 pe Sosa 1.0115 0. 00499 Ma... 1. 0027 0. 00116 Pode sas 1.0115 0. 00496 Hacholcout . : ‘ ponent hour of observation Viger 1.0107 0. 00464 |} A2.----- 1.0111 0. 00479 period receives one and only eneaielar ourly heights in the summation; = . ie i Wits p: Net aS Wh no ny aioe p: aaa group covers one solar hour; each solar M. pa. 1. 1028 0.0 495] rea 1.0023 0.00100 day represented by 24 hourly heights a ee P aier ire Pls yo ao : j and component day represented by 24 Mg....- 1. 1934 0.07680 || MK....| 1. 0250 0. 01074.'}| Teens: No.... 1.0103 0. 00447 || 2MK... 1. 0238 0. 01021 Phe eee 1. 0099 0.00430 || MN.... 1.04381 | =. 0.01833 Qj... 1. 0025 0.00107 || MS....-. 1. 0456 0. 01935 0O.. 1. 0033 0.00144 || 2SM.... 1. 0123 0. 00532 LONG-PERIOD COMPONENTS. ais augmenting factor=—Ys> ye sin ) sin —* “* -—_ sin 12¢ Augment- ing factor. Logarithm. Remarks. 1. 0050 0. 00218 1.0205 0. 00880 || Daily sums used as units in the summation for the divisional means, and 1.0192 0. 00825 all daily sums used; component month for Mm, Mf, MSf, and compo- 1. ous 0. pore nent year for Sa and Ssa represented by 24 means. 1. 011 0. 00499 *For component Sj, S:, S3, etc., the augmenting factor is unity. ,. HARMONIC ANALYSIS AND PREDICTION OF TIDES, TABLE 21.—Acceleration in epoch of K, due to P. [Argument h—3y’ refers to beginning of series.] \ Series. ~N ~ h—}’ \ 58 days. +14.6 87 days. +12.6 105 days. 134 days. 163 days. 192 days. ° +10.1 MGuieeancea tae rw mo WO NHR CNW WOW CHO DROP + Oo _ | rNN ONO FPOM HRO NOM Oe SS Free NS NS a eC 0 Sh +++ +++ ++ ° +0. 2 +0. 7 +0.9 +1.0 | 2S Sr S99 5 bo CO aI ooo CO D2 > _ taal 235: 221 days. ° +2. 4 NES PNW CONN ros bo CrOie reCOtD DOO ++ | 250 days. +2.0 +0.8 | Se eR Pb he POSES | OO MeN NOS amo +++ +1 279 days. IES SS Co oe REY 28) OOM mMOow D=~I1c1 Ooo +++ +41 297 days. I S Wire WRT PW OWR DOC YEH NES Spy yew HN +++ +++ 22.—Ratio of increase in amplitude of K, due to P;. [Argument h—4y’ refers to beginning of series.] 326 days. ° + + PND Per SOS9 fee NE HOS OOrF OFF MOM FOO KFPOT COO Om +++ +++ 58 days. 87 days. ° +0. 01 +0. 09 +0. 16 +0. 20 +0. 23 +0. 24 +0. 22 +0.18 +0.12 +0. 04 —0. 03 —0. 10 —0.15 —0.18 —0219 =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. 192 days. ° +0. 01 +0. 01 +0. 02 +0. 02 +0. 03 +0. 03 +0. 04 221 days. 250 days. 279 days. +0. 03 +0. 06 +0. 08 +0. 09 +0. 09 |-+0. 09 +0. 08 +0. 07 +0. 04 +0. 02 0. 00 —0.02 —0. 03 —0.04 —0. 04 —0.03 —0.01 +0.01 +0. 03 297 days. ° +0. 05 +0. 07 +0. 08 +0. 09 +0. 09 +0. 08 +0. 06 +0.04 +0.02 0.00 —0. 01" —0. 02 —0. 03 —0. 03 —0. 02 —0.01 +0. 01 +0. 03 +0. 05 236 U. S. COAST AND GEODETIC SURVEY. TABLE 23.—Acceleration in epoch of S, due to Kz. [Argument h—v’ refers to beginning of series.] \Series. E { ~ 15 29 58 87 105 134 163 192 221 250 279 297 326 - rn days. | days. | days. | days. | days. | days. | days. | days. | days. | days. | days. | days. | days. —y! | ° ° i-} ° ° ° ° ° ° ° ° | ° ° 0 | 180 | +3,2 | +5.9 |410.1 |+10.4 | +8.0 | +3.2 | +0.4 | +0.1 | 41.3 | +2.9 | +3.2 | +2.4] +0.9 10 | 190 | +7.2 | +9.6 |+12.3 410.0 | +6.7 | +2.0 0.0 | +0.3 | +1.9 | +3.3 | +2.9-|.4+1.9 | +0.5 20 | 200 }+10.8 |4+12.6 |+13.2 | +8.4 | +4.7 | —0.6 | —0.5 | +0.5 | +2.4 | +3.3 | +2.2 | +1.1 0.0 30 | 210 |+13.7 |+14.6 |412.5 ; +5.7 | +2.3 | —0.9 | —0.9 | +0.7 | +2.6 | +2.9 | +1.3 | +0.3 | —0.5 40 | 220 |+15.4 )4+15.0 | +9.9 +2.5 | —0.4 | —2.2 | —1.3 | +0.8 | +2.5 | +2.0 | +0.3 | —0.6 | —1.0 50 | 230 |+15.4 |4+13.5 | +5.8 |'—1.1 | —3.0 | —3.4 | —1.6 | +0.8 | +2.0 | +0.9 | —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 | —1.6 70 | 250 | +8.6 | +3.7 | —4.4 | —7.4 | —7.2 | —4.9 | —1.7 | +0.5 | +0.1 | —1.6 | —2.6 | —2.6 | —1.7 80 | 260 | +1.9 | —3.0 |} —8.8 | —9.5 | —&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 | +0.8 130 | 310 |—14.7 |—12.9 |.—8.0 | —2.4 | +0.5 | +3.1 | 41.7 | —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 | +1.7 | —0.7 | —1.5 | —0.7 | +0.8 | +1.6 | +1.7 150 | 330 | —9.1 | —6.4 | —0.6 | +4.4 | +5.9 | +4.9 | 41.6 | —0.5 | —0.8 | +0.3 | 41.9 | +2.4 | +1.7 160 | 340 | —5.3 | —2.3 | +3.3 | 47.3 | +7.7 | +4.8 | +1.3 | —0.3 | —0.1 | +1.3 | +2.7 | +2.8 | +1.6 170 | 350 | —1.1 | 41.9 | +7.0 | +9.4 | +8.4 | +4.2 | +0.9 | —0.1 | +0.7 | +2.2 | +3.2 | +2.8 | +1.3 180 | 360 | +3.2 | +5.9 |+10,1 |410.4 | +8.0 | +3.2 | +0.4 | +0.1 | +1.3 | +2.9 | +3.2 | 42.4 | +0.9 | TABLE 24.—Ratio of increase in amplitude of S, due to Ky. [Argument h—»’’ refers to beginning of series.] \ Series. NS 15 29 58 87 105 134 163 192 221 250 279 297 326 : NN days. | days. | days. | days. | days. | days. | days. | days. | days. | days. | days. | days. | days. —y > ° ° ° ° ° ° ° ° ° ° ° ° ° 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 | 0.00 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 |+4-0.04 |+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 |+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 |+C.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 |+0.05 |+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 |+0.02 |—0.02 | 0.00 |+0.03 |+0.06 |/+0.06 |+0.03 |+0.02 | 0.00 180 | 360: |+0. 26 |+0. 24 |+0.15 |4+0.03 |—0.02 |—0.04 |—0.01 |+0.03 |+0.05 |+0.04 |+0.01 | 0.00} 0.00 237 HARMONIC ANALYSIS AND PREDICTION OF TIDES. TABLE 25.—Acceleration in epoch of S, due to To. [Argument h—pi refers to beginning of series.] Series. —0.1 —0.4 0 —0.5 —0.6 —1.1 SUL —1.3 | 1.2 —1.5 —2.0 srl be) || stn! —0.8 =0.7 1.5 | —1.1 | —0.6 | —0.4 | eet ng: —2.0 —2.0 —2.0 —2.0 | —2.2 | —2.4 | —2.3 —1.5 +0. 6 —0.3 —2.0 —2.2 —2.3 —1.9 —2.2 —2,4 +2.9 | +2.5 | +2.0 | +1.5 41.4 —1.5 —1.9 —2.2 +3.0 | +2.7'| +2.1 | +1.5 | +0.9 | +0.3 —0.2 —0.7 +3.1 +3.0 | +2.9 | 42.4 | +1.8 | +1.2 | +0.6 | +0.2 ib —1.6 —2.0 —0.5 —1.0 —1.5 —0.8 —0.4 0 —0.2 —0.8 290 | +2.9 | +2.7 | +2.1 | +1.3 | +0.9 | +0.2 300 | +2.6 | +2.3 | +1.6 | +0.9 | +0.4 280 | +3.2 | +3.0 | +2.5 | +1.8 310 | +2.2 | +1.9 | +1.1 | +0.4 220 | +2.6 | +2.8 | +3.1 230 | +2.9 | +3.1 | +3.2 240 | +3.2 | +3.3 | +3.2 320 | +1.7 | +1.4 | +0.6 330 | +1.2 | +0.8 | +0.1 340 | +0.7 | +0.3 190 | +1.1 200 | +1.6 210 } +2.1 260 | +3.4 270 | +3.3 350 | +0.1 360 =—0:4 a, li 4 238 U. 8. COAST AND GEODETIC SURVEY. TABLE 26.—Resultant amplitude of S, due to T;. [Argument h—p) refers to beginning of series.] 0. 99 Series. ne - HARMONIC ANALYSIS AND PREDICTION OF TIDES. 239 TABLE 27.—Critical logarithms for Form 245. Natural | lLoga- Natural | Loga- Natural | Loga- Natural | Loga- Natural Loga- number. | rithm. |/number.| rithm. |]|number.| rithm. |/number.} rithm. || number.| r.thm, OXO00) oes ose 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 . 0@2 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 . 1604 9.0150 154 9. 1862 - 204 9. 3086 - 005 7. 6533 055: 8. 7365 - 105 9, 6192. 155. 9.1880 - 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 -1ll 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 117 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. 0774 -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 2 ray 9. 2419 -225 |}; 9.3513 . 026 8. 4066 -076 | 8.8780 -126 | 9.0987 .176 9. 2443 . 226 9. 35382 . 027 8. 4233 -077 8. 8837 -127 | 9.1021 oir 9. 2468 ; 2er |? “FO, 3551) . 028 8. 4394 . 078 8. 8894 - 128 9. 1056 .178 9. 2493 . 228 9.3570 - 029 8. 4549 - 079 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. 4844 . O81 8. 9059 131 9.1157 -181 9. 2565 . 231 9. 3627 032 8. 4984 . 082 8.9112 ~ 132 9. 1190 . 182 9. 2589 - 23. 9. 3646 033 8.5119 . 083 8.9165 | - 133 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 - 186 9. 1320 . 186 9. 2684 . 236 9.3720 . 03 8. 5623 . 087 8.9371 - 137 9. 1352 187 9. 2707 o200 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 &. 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 &.9891 - 148 9. 1688 .198 9. 2956 - 248 9.3936 . 049 &. 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 j | | [ical U. 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HARMONIC ANALYSIS AND PREDICTION OF TIDES. 0 0 GTS868 °0 ELEE0S “S+- OL6BERP 0 L89LF0 “E+ OOZE6T °0 OLZ09S “T+ 6S0%20'0 696990 "I+ 678900 0 968910 “T+ 826886 6 628416 ‘0+ _ O8ZF6F ‘0 TS0z1 “¢+ PLOOTF 0 Q9TOLS ‘S++ 628208 ‘0 C6L1E0 S+- EEPZLT “0 LIPL8P “T+ GECOL6 '6 6SLE86 “OT WSZ GTS86E “0 ETE80S “o—. 0 0 R68C8L "6 GLEFFS ‘0+ TESFL6 6 ZPhOERE ‘O— €LZ09T °0 OSE9FF “I— EEPZLT ‘0 LIPL8P I — T9GPST 0 PSP8SS “T— 9FP06L ‘6 8ZSL19 ‘0+ EShC98 "8 #98610 ‘0+ TOSEZ9 6 TOST ZF ‘O— 678900 ‘0 968910 “T— 9LLS6T “0 PSS69S “T— ta OL6E8P 0 L89L40 “§— 868SE2 6 SLEPPS “O— 0 0 SEPCLT 0 LIPL8P T— T0662 ‘0 $@L066 “T— 6LEL0E "0 & G6LTE0 “S— OLSOTE “0 898220 — ESPZ98 °8 #9820 '0+ TOSEL9 °6 T2ST LP “O— 678900 0 968910 “T— OOZEGT 0 OLZ09¢ “I — 060SZE 0 626EIT “S— tnt oozesT 0 | 6¢0Pz0'0 | 6F8900°0 | gzease'6 -| ogzrer-0 | FZE0IF ‘0 022092 I— | z9e9c0'1— | 96scr0'1— | 6zsF26°0— | TFS0zt-e— | 99T9Z9-2— 1eck6°6 | 22090 | gerezt-o | T9zFRT-O | OFFOGL6 | egpzoR's ZPORPG “O+ | OSEOFE T+ | LIFLSFT-+ | FRERzGT+ | SzezT9“0—-| FeRzZ0‘0— eerelt'0 | 11066z"0 | 6zg208°0 | ozc9te-0 | ecrzog‘s | 10Se19°6 LIPLSP I+ | ozz066'T+ | zezte20°Z+ | £68220°2-+ | #S9z20-0— | TzgTZP-O+ 0 pesto.'6 | sescez’6 | esrz02°6 | oOzesT"0 | 6FR900°0 0 s0ee0S ‘0-+ | ezerrs ‘0+ | TFFGSG°0-+ | OZz09¢-T— | 9B8STO-T— FERTOL “0 0 e6he19'8 | SISFI6’S | Iz9rTe’0 | OTOTRE-0 g0ee0¢ “0— 0 19010 "0+ | ee1zs0°0+ | 6Z¢e90°2— | FOES ‘T— g68ce2'6 | e6re19°8 0 e6pero's | ezTeze-o | ooze6r’0 GLeFPs‘0— | L90TFO‘0— 0 1901F0°0+ | sr9ror-%— | 02¢099°T— 8h49L'6 | SISFI6’S | g6re19's 0 TisTee 0 | esrroz ‘0 TFFS8S 0— | £E1Z80'0— | L90TFO“0— 0 ZILGPIZ— | LESto9 I— oozest'o | 1zopte'0 | ezteze'0 | rzsTee0 0 868981 6 022099 T+ | ezge90°z+ | sroFOT z+ | ZILSbI ‘2+ 0 CLERES “0+ 6F8900°0 | 919T81°0 | OOzE6T“0 | EsFFOz‘0 | ReRceL’6 0 968910 T+ | FOZEIS ‘T+ | 02Z09¢ I+ | Leeto9 “T+ | ezerFs 0— 0 10sez9'6 | szese6‘6 | 6P8900°0 | esorzo'o | szegeo0 | sescez"6 Tee LF 0+ | 6z8rL6 ‘0+ | 96810'T+ | 296990°T-+ | GFL880"T— | SLePFs-0— eep298'8 | Lzeeeo’e | 1osez9°6 | 89260L'6 | eT0eIZ‘0 | sz69e0°0 FE8CLO‘O— | FSFORF “0+ | T2eTLF-O-+ | sBezIs-0-+ | FeTEEDT— | 6PL880"T— 6269626 | cz9060°6 | 6echI6e's | seser9’s | goseee-o- | I8FsIz-0 z19929 0— | FOzezT“O— | LeTzs0-0— | TZ01FO‘0— | zez9RT‘z— | 80FZFEOT— By aL ay si NZ tN 6L820E 0 C6LTEO “3 — TOSEL9 6 IZSTLP 0+ 678900 ‘0 968910 “T+ TOSEL9 6 TeStLp “O— 826886 °6 628FL6 O— 678900 °0 968510 “T— 690620 ‘0 696990 “T— 826980 ‘0 662880 T+ 868SEL 6 GLEFPS “O+ 0 0 868982 6 CLEFPS ‘O— CT90F0 ‘0 £80860 “I— WH EEPcZt 0 LIPL8b “I — 678900 0 968S10 “E+ 00ZE6T °0 022095 "I++ £9F298 °8 FE8620 0+ LZ6EE9 “6 PSPOEP “O— TOSEL9 6 TEST LP ‘O— 8926026 . 88SZIS ‘0— 6LOETZ ‘0 POTEED “T+ 826980 ‘0 6FL880 T+ B68SEL 6. GLERES ‘O+ 0 0 SPZEFL 6 Bggess "O— oT GEc026 ‘0 6SLEE6 “0O— 9LLS6T °0 PSS69S “T+ 060928 0 626E1T “3+ 666962 ‘6 G1S9Z9 “O-+ £29060 6 POZEZT 0+ : 6SSF16 “8 LE1680 0+ Gese19 8 [LOT#O ‘0+ 908688 ‘0 BBLO8T “S+ ISFSIZ “0 80FZF9 “I+ GT90F0"'0 £60860 1+ GPCEFL 6 segess ‘O+ BO EO OY | Oe ee ee eee = oe ‘SENANOdWOO TYNYOIGINGS *ponuryu0g—(p—4q) bo, puwn (vn—4Q) saouasaffip paads quauodwoj—'8z AIAV, 242 U. S. COAST AND GEODETIC SURVEY, TaBLe 29.—Elimination factors. ‘ [Upper line for each component gives the logarithms of the factors; middle line, corresponding natural numbers; lower line, angles in degrees.] SERIES 14 DAYS. DIURNAL COMPONENTS. _ Disturbing components (B, C, etc.). Component sought (A). - ir Mocososossensccse 61 9.7607 576 98 8. 1857 014 175 Plevecceccoceccccescs 9, 3150 - 207 8. 6530 - 045 172 Or 00 P, 9.3150 | 9.7890 | 9.7203 - 207 -615 - 525 é 264 93 255 8.3839 | 8.3839 | 9.9958 - 024 - 024 - 990 356 4 346 9.7890 | 9.3150 | 9. 8578 -615 - 207 721 267 96 7 wee ce 8.3826 | &. 7358 Ase -| 024 . 054 ts 9 171 SaaS 26 ule ee 8. 9571 O24 le as ee - 091 351 es 342 8. 7358°} "8. 95714). 2S. - 054 SOG Ah ere sae 189° 18 ee 9.7968 | 9.0878 | 9.3355 - 626 122 . 217 91 100 82 8.2015 | &. 3320 | 8. 2581 -016 021 -018 3 12 174 8.1361 | 8.7710 | 9.9990 -014 . 059 - 998 182 11 353 + 9. 8516 | 9.1065 | 9.3331 e711 - 128 215 79 88 70 9. 0878 - 122 9. 3355 see c news 9, 7298 . 537 256 Si Pl 9.7607 | 8.1357 .576 014 262 185 9.9990 | 9.3344 .998 . 216, 353 276 9, 8290 | 8.6530 675 045 85 188 8.1361 | 9.8516 014 711 178 281 8.7710 | 9.1065 - 059 .128 349 272 9.9990 | 9.3331 . 998 215 7 9.3283 | 9.9967 213 . 992 89 12 7.1244 | 9.7298 .001 | ~.537 0 104 ee 9. 3369 aE i] = 21% 9. 3369 .217 77 HARMONIC ANALYSIS AND PREDICTION OF TIDES, 243 TaBLE 29.—Elimination factors—Continued. SERIES 15 DAYS. SEMIDIURNAL COMPONENTS, Disturbing components (B, C, etc.). Component sought (A). Ke Le Mz Ne 2N Rs S2 Te dg 770) v2 2SM. 1 MER Re Reo ee 9, 7534. 8. 9437. 9. 2494 8. 9063 |9..9986 9, 9950 |9. 9892 |9. 6707. |8. 7223. |9. 2966 || 8. 8476 : weeee | D672. »| . 088) | .175 .|,.081. | .997-:|..989. || .975.|}..468~ | 053° | 198 | .070 f bie 260 342 244 326 353 345 338 247 339 257 168 Teg See 8 Sc 9. 7534 |.....- 9, 7627 |8.9055 |9. 2507 ./9..7927 |9.8276 |9. 8585 |9.9961 |9.3018 |8. 1941 | 9.3301 SOMME Ee ares -579 | .080 | .178 -620 | .672 | .722 | .991 | .200 | .016 .214 100 aes 262 344 246 92 85 77 347 | . 259 357 88 BO. ION: Sane Rane 8. 9437 |9. 7627 |...... ,9.7627 |8.9055 |8. 7291 |8&. 1941 |8.4114 |9. 8276 |8. 1941 |9.8276 |8 . 1935 ; SUSRA a Oe aay | =e scts -579 | .080 | .054 | .016 | .026 | .672 | .016 | .672 - 016 18° OS 2 cas 262 344 10 3 175 85 357 275 6 Ng meysee cals x 9, 2424 |8.9055 |9. 7627 |....-.- 9.7627 |9. 2760 |9.3018 (9.3204 |8.1941 |9. 8276 |9.9961 | 9.0793) Led) -O80p 1 Oto etc -579 | .189 | .200 |.209 | .016 |.672 | .991 - 120 ‘ F 116 16 bl A 262 108 101 93 3 275 13 104 PANS) IS aa 8.9063 |9.2507 |8.9055 |9.7627 |...... 8. 8167 |8.6888 |8. 4856 9.3018 |9.9961 |9.6823 18.5765 , ; O81, | .178, | .080_| .579 |. 2... 066 | .049 | .031 200. | .991 | .481 - 038 34 114 16 98 maha 26 19 11 101 13 111 22 noe: Rie re Sa 9.9986 |9. 7927 |8.7291 |9.2760 |8. 8167 |...... 9.9987 |9.9950 |9.7195 |8. 5420 |9.3168 | 8. 4114 ; .997 | .620 | .054 |.189 | .066 | ..... -997 | .989 | .524 | .0385 | .207 - 026 if 268 350 252 334 eee 353 345 255 347 265 175 ore eae eee eee 9.9950 |9. 8276 |8. 1941 |9.3018 |8.6888 |9.9987 |....-- \9. 9987 ,9. 7627 |8.1935 |9.3301 | 8.1941 989 | .672 -016 | .200 | .049 | .997 ! ..... . 997 | -979, | .016: | .214 - 016 15 275 357 259 341 ri Hoey 353 262 | 354 272 3 Dareiseia tens ck 9. 9892 |9. 8585 |8. 4114 |9. ia 8. 4856 |9. 9950 |9.9987 |...... 9. 8010 |7.6684 |9.3364 | 8.7291: O71 || ht2enal - O20erh -031 | .989 | .997 | ..... -632. | .005 | .217 - 054 22 283 185 287 349 15 ai nee 269 182 280 10 Ne stiecoce ccc swei 9.6707 |9.9961 |9. 8276 |8..1941 |9.3018 |9.7195 |9. 7627 |9.8010 |.....-. 9.3301 |8.7786 | 9.3018 -468 | .991 | .672 | .016 | .200 | .524 |.579 | .682 | ..... -214 | .060 - 200 ; 113 13 275 357 259 105 98 91 [Xe os 272 190 101 HB occcccceccee+ |G @229 |9.3018 |8. 9141, |9. 8276 |9.9961 |8. 5420 |8. 1935 |7.6684 |9.3301 |...... 9.7627 | 8. 1926. ; -053 | .200 | .016 | .672 | .991 | .085 | .016 | .005 |.214 | ..... O79 -016 21 101 3 85 347 13 6 178 to} I aa 98 9 V9eececsccceccee (Ue 2966 |8. 1941 |9. 8276 |9.9961 !9.6823 |9.3168 |9.3301 |9.3364 |8. 7786 |9. 7627 |....-- 9.1043 -198 | .016 | .672 | .991 | .481 | .207 | .214 | .217 | .060 | .579 | ..... 127 103 3 85 347 249 95 88 80 170 262 Ae 91 QSM. cccccccccce |S 8476 (9.3301 [8.1935 |9.9793 |8.5765 |8. 4114 |8. 1941 |8.7291 |9.3018 |8. 1926 9.1043 | ...... -070 |.214 | .016 | .120 | .038 | .026 | .016 | .054 |.200 |.016 |.127 | ..... 192 272 354 256 338 185 357 350 259 351 269 soos 244 U. S. COAST AND GEODETIC SURVEY. TABLE 29.—Elimination factors—Continued. SERIES 29 DAYS. DIURNAL COMPONENTS. Disturbing components (B, C, etc.). Coen sought A). ; : agi Ri Mi OQ; OO - Py Qi 2Q Si Pl ae eel) ear Sore rere 8. 6955 | 8.6896 | 8.7199 | 8.8144 | 9. 2092 | 8. 6937 8. 6672 | 9.0538 | 8.3224 PE cooing . 050 . 049 - 052 - 065 . 162 . 049 . 046 113 . 021 eet 351 341 328 13 322 319 310 336 344 1G oui ae petting mete 8.6955 | -....- 8.6955 | 8.7517 |.8.7517 | 9.9818 | 8.7199 | 8.6937 | 9.9954 | 8.0542 OG0'* |)" eee - 050 . 056 . 056 - 959 - 052 - 049 . 990 - 011 9 iS 351 338 22 331 328 319 346 354 MESES Rabe SPV NO) A 8.6896 | 8.6955 | 22... 8.8144 | 8.7199 | 9.0674 | 8.7517 | 8.7199 | 8.4418 | 7.9579 049 G50) })\ a 2 065 052, 117 - 056 052 028 009 19 9 Eee 347 32 161 338 328 175 183 OP) Hie, eee 8.7199 | 8.7517 | 8.8144 | 222... 8.7185 | 8. 2616 | 8.6955 | 8.6896°| 8.3262 | 8.9810 052 056 (G5) | eee 052 -018 050 049 021 096 32 22 13 4 44 174 351 341 196 OO er Ae 8. 8144 | 8.7517 | 8.7199 | 8.7185 | .....- 9.0332 | 8.6848 | 8.6504 | 8.9334 | 8. 4666 065 056 052 OBZ pes sae - 108 048 045 086 029 347 338 328 316 309 306 297 324 332 Pas Oy Sa eee 9, 2092 | 9.9818 | 9.0674 | 8.2616 | 9.0332 | ...... 7.7378 | 8.2260 | 9.9954 | 8.6248 162 959 117 018 COS ah olen 005 017 990 . 042 38 29 199 186 51 be 357 348 14 202 Qe 4. Peer cans 8.6937 | 8.7199 | 8.7517 | 8.6955 | 8.6848 | 7.7378 | ...... 8.6955 | 8.4846 | 9.9857 049 052 056 050 . 048 005) 1 ..... 050 031 968 41 32 22 9 54 3 se 351 17 25 PE SIND SS ald 8.6672 | 8.6937 | 8.7199 | 8.6896 | 8.6504 | 8.2260 | 8.6955.) ...... 8.5377 | 9.1825 046 049 052 049 . 045 -017 O50; |" ees 034 152 50 41 32 19 63 12 Sete 27 35 SiLOR Ble OG 3 er teen 9.0538 | 9.9954 | 8.4418 | 8.3262 | 8.9334 | 9.9954 | 8.4846 | 8.5377 | ..22.. 8. 1807 - 113 990 . 028 -021 .| .086 . 990 - 031 OSE iy fees -015 24 14 185 352 36 346 343 333 ade 188 prt 2: Wee ahs se 8.3224 | 8.0542 | 7.9579 | 8.9810 | 8.4666 | 8.6248 | 9.9857 | 9.1825 | 8.18077} > 27... - 021 - O11 . 009 - 096 . 029 - 042 . 968 - 152 SOLS Gi yh eee 16 6 177 164 28 158 335 325 172 Bor lee HARMONIC ANALYSIS AND PREDICTION OF TIDES, 245 R TasLe 29.—Elimination factors—Continued. SERIES 29 DAYS. SEMIDIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component oN et ROO SO OO sought (A). | Ke Ie M2 Ne 2N Re S2 To de 2 v2 28M We. ES a AL AA 8.8144 |8.7517 |8.7199 |8.6937 |9. 9954 |9.9818 |9.9587 |9. 2092 |8.3224 |8.0542 | 9.0054 PAL aH .065 | .056 | .052 -049 | .990 -959 | .909 | .162 021 - O11 101 Roe 347 338 328 319 346 331 317 322 344 354 145 S08 ee 8.8144. |...... 8. 6955. |8. 6896 ig 6798 |7.9581 |8.9810 |9. 2842 |9.9857 |7.7378 |8. 2616 | 8.6248 - 065 add 050 | .049 048 00' -096 | .192 968, 005 018 042 13 u 351 341 332 178 164 150 335 357 186 158 Migss..3..5 SAN 4 8.7517 |!8.6955 |...... 8. 6955. 8.6896 |8.3262 18.2616 |8.7772 |8.9810 |8.2616 |8.9810 | 8.2588 . 056 050 | ...-.. 050 049 021 018 060 096 018 0 018 22 9 351 341 8 174 159 164 186 196 167 Naee.2. 8. ROCK 8 7199 8.6896 |8. 6955 |.-.... 18.6955 |8.4846 |7. 7378 |8.3278 |8.2616 |8.9810 |9.9857 | 7.5900 - 052 304963); 05020 .0 . 050 - 031 - 005 - 021 . 018 -096 | .968 . 004 32 19 9 oh 351 17 3 169 174 196 25 177 PA. (9 oe Re Peet PB 8. 6937 |8. 6798 |8.6896 |8.6955 |.....- 8.5377 |8.2260 |7. 4179 |7.7378 19.9857 |9.1825 | 7.7379 : -049 | .048 | .049 | .050 | soos - 034 - 017 - 063 - 005 968 152 - 005 41 28 19 9) kab 27 12 178 3 25 35 6 ye S.s ee 9.9954 |7.9581 |8.3262 |8.4846 |8.5377 |...-.- 9. 9954 |9.9818 |9.0538 |7.2754 |8.1807 | 8.7772 990 009 021 031 -} .084 | J.... . 990 959 113 002 015 060 14 182 352 343 333 346 331 336 359 188 | 159 Base 2). enue 8 9.9818 |8.9810 |8. 2616 |7.7378 |8. 2260 |9. 9954 |...... 9.9954 |8.6955 |8.2588 |8.6248 | 8.2616 959 096 018 005 017 2990; Seek... 990 050 018 042 018 | 29 196 186 357 348 14 346 shal 193 202 174 Baty se | ORR |g 9587 |9. 2842 |8.7772 |8.3278 |7.4179 19.9818 |9.9954 |...... 8.4418 18.5780 |8.8324 | 8.3262 909 192 060 | .021 | .003 959 990) Fea. 028 038 068 021 43 210 201 191.' 182 29 14 185 207 217 8 NOISE crsiciein ict 9. 2092 |9. 9857 |8.9810 |8.2616 |7. 7378 |9.0538 |8.6955 |8.4418 |...... 8.6248 8.9640 | 7.7378 162 968 096 018 005 113 050 O28) hh ids, 042 - 092 005 38 25 196 186 357 24 9 175 Boos 202 212 [dis tnmjoinie voids 85/3 8.3224 |7. 7378 |8. 2616 18.9810 |9. 9857 |7. 2754 |8. 2588 18.5780 |8.6248 |...-.. 3. 6955 | 8.2539 - 021 - 005 - 018 . 096 . 963 - 002 - 018 - 038 TORQ Wiech | .050 - 018 16 3 174 164 335 1 167 153 158 ese] 9 161 Dimas Becca Sbee 8.0542 |8.2616 |8.9810 |9.9857 |9. 1825 18.1807 |8. 6248 |8.8324 |8. 9640 |8. 6955 | Su elie 8.5015 -O11 -018 - 096 . 968 ~L52 -015 . 042 . 068 . 092 OO ul ete area . 032 6 174 164 335 325 172 158 148 148 sible ese 151 OSIM ucla) svorcare 9. 0054 |8.6248 |8. 2588 |7.5900 |7.7379 |8. 7772 |8 -101 - 042 018 . 004 - 005 . 060 215 202 193 183 354 201 246 U. S. COAST AND GEODETIC SURVEY. TaBLE 29.—Elimination factors—Continued. 3 SERIES 58 DAYS. DIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought A). yh Ky Mj QO; (exe) P) Qi 2Q Si. Pl Tipe Ot SY SE PERE Ue Ce 8.6896 | 8.6657 | 8.6504 | 8.8039 | 9. 1056 | 8.5715 | 8.4713 | 9.0154 | 8.3059 aR ee 049.5] 6 oG46e8 Fe O45 1 2. 064 fs 128 | 08% +]. OBO |i J 104 *]~ ~.020 ae 341 322 297 25 284 278 259 313 329 Kee. Be RE BERR 8. 6896 | ....-- 8.6896 | 8.7185 | 8.7185 | 9.9254 | 8.6504 | 8.5715 | 9.9818 8. 0520 ~ O40 SOEs. . 049 . 052 . 052 - 842 . 045 . 0387 . 959 O11 19 Sos 341 316 44 303 297 278 331 348 Mipec S. 2 Papeete SO REGO 8.6657 | 8.6896 | ...... &. 8039 | 8.6504 | 9.0427 | 8.7185 | 8.6504 | 8. 4403 7. 9572 046 1049.) eieslt . 064 . 045 -110 . 052 . 045 . 028 . 009 38 19 a 335 63 142 316 297 170 186 Oise Ses SSR SU ee 8.6504 | 8.7185 | 8.8039 | ...... 8.5737. | 8.2588 | 8.6896 | 8.6657 | 8.3224 8. 9640 . 045 . 052 NOGA) | SS . 037 . 018 . 049 . 046 . 021 . 092 63 44 25 = es 88 167 341 322 16 212 COPS. 2 ERE Me rsae- 8. 8039 | 8.7185 | & 6504 | 8.5737 | .....- 8. 8349 | 8. 4575 | 8.3057 | 8. 8391 8. 4112 . 064 . 052 045 BOS TN euketce . 068 . 029 . 020 . 069 . 026 335 316 297 272 ajk 259 253 234 287 303 Pye Se ERE RE 9.1056 | 9.9254 | 9.0427 | 8.2588 | 8.8349 | ...... 7.7379 | 8.2155 | 9.9818 8. 5907 . 128 . 842 -110 . 018 SOG8) Tih ie b.2k . 005 . 016 . 959 . 039 76 57 218 193 101 2 354 335 29 225 Qaee Sh 2 See ee BRE R. 8.5715 | 8.6504 | & 7185 | 8. 6896 | 8.4575 | 7.7379 | .....- 8.6896 | 8. 4645 9, 9418 . 037 . 045 . 052 . 049 . 029 QO57 a 25 8 . 049 . 029 - 875 &2 63 44 19 107 6 Te | Rea 35 51 2@QveL See PRa ek He sy. 8.4713 | 8.5715 | 8.6504 | 8.6657 | 8.3057 | 8.2155 | 8.6896 | ...... 8. 4887 9.0969 . 030 . 037 . 045 . 046 . 020 . 016 wO49e b) hE - 031 - 125 101 &2 63 38 126 25 19 a 53 70 Sie nae ee Bb 9.0154 | 9.9818 | &. 4403 | 8.3224 | & 8391 | 9.9818 | 8. 4645 | 8.4887 | ...... 8.1761 . 104 959 . 028 . 021 . 069 . 959 . 029 OBB ae oe . 015 47 29 180 344 73 331 325 307 we 196 YOST E NSA AEM Lane cs oa a | &. 3059 | 8.0520 | 7.9572 | 8.9640 | 8.4112 | 8.5907 | 9.9418 | 9. 0969 8.1761 . 020 - O11 . 009 . 0$2 . 026 . 039 . 875 .125 . 015 34 12 174 148 57 135 309 290 164 HARMONIC ANALYSIS AND PREDICTION OF TIDES. SERIES 58 DAYS. TaBLE 29.—Elimination factors—Continued. SEMIDIURNAL COMPONENTS. 247 Component sought (A). 1 ee 8.7185 - 052 44 Ngee ok Belo 8. 6504 045 63 8.5715 - 037 82 4. AO 9. 9818 . 959 2SM..........--|/8. 9185, - 083 250 iby, Mp 8.8039 |8. 7185 .064 | .052 335 | 316 fab 8. 6896 ee 049 WS | sae 8.6896 |.....- £0490| £9), 2 eS 8. 6657 |8. 6896 046 | .049 38 19 8. 6244 |8. 6657 042 | .046 57 38 7. 9579 |8.3224 -009 | .021 183 | 344 8. 9640 is 2588 .092 | .018 212 | 193 9, 2209 18.7480 .166 | .056 240 | 222 9. 9418 |8. 9640 875. | .092 51] 212 7. 7379 |8. 2588 -005 | .018 6 167 8. 2588 |8. 9640 -018 | .092 167 148 8.5907 |8. 2475 -039 | .018 225 206 Disturbing components (B, C, etc.). No 2N Re S2 T: de 8.6504 |8.5715 |9.9818 |9.9254 |9.8237 |9. 1056 . 045 - 037 - 959 - 842 . 666 -128 297 278 331 303 274 284 8. 6657 -|8. 6244 17.9579 |8. 9640 |9.2209 |9. 9418 -046 ‘| .042 -009 | .092 | .166 | .875 322 303 177 148 120 309 8.6896 |8.6657 |8.3224 |8.2588 8.7480 |8. 9640 - 049 . 046 - 021 . 018 . 056 - 092 341 322 16 167 138 148 Se eEe 8.6896 |8. 4645 17.7379 |8.3193 |8. 2588 ek ae - 049 - 029 - 005 - 021 - 018 okie 341 35 6 157 167 8.6896 |...-..- 8. 4887 |8.2155 |7. 4165 |7. 7379 O49) ies ok 2 - 031 - 016 - 003 - 005 19 ae ae 53 25 176 6 8. 4645 |8. 4887 |.....- 9.9818 |9.9254 |9.0154 - 029 Oo, Heese . 959 - 842 . 104 325 307 4 331 303 313 7.7379 |8. 2155 |9.9818 |...... 9.9818 |8. 6896 - 005 - 016 959) WAP. 2 . 959 - 049 354 335 29 Sige 331 341 8.3193 |7. 4165 |9.9254 |9.9818 |...-. ~ |8.4402 - 021 - 008 - 842 3959. | PSs. - 028 203 184 57 29 Dems 190 8. 2588 |7.7379 |9.0154 |8.6896 |8.4402 |.....- - 018 - 005 - 104 - 049 4028) |) 22s 193 354 47 19 170 e 8.9640 |9. 9418 |7. 2736 |8.2475 |8.5270 |8.5907 . 092 . 875 - 002 -018 . 034 - 039 148 309 3 154 126 | © 135 9.9418 |9.0969 |8.1761 |8.5907 |8. 7366 |8. 8933 - 875 - 125 015 - 039 - 055 - 078 309 290 164 135 107 116 7. 5898 |7.7356 |8.7480 |8. 2585 |8.3224 |7.7379 - 004 - 005 - 056 -018 -021 - 005 187 348 222 193 344 354 v2 248 U. S. COAST AND GEODETIC SURVEY. TaBLE 29.—Elimination factors—Continued. SERIES 87 DAYS. DIURNAL COMPONENTS. Component sought (A). Disturbing components (B, C,etc.). Jy Ki Mi Qi (6X6) Tse ee ol us a Bae | Bae 8.6798 | &.6244 | 8.5225 | 8. 7857 see. . 048 . 042 . 033 . O61 e 332 303 265 38 aac Be 5 AS Fhe 8.6798 | .....- 8.6798 | 8.6607 | 8.6607 HOSS yi) Veto . 048 . 046 . 046 28 A 332 294 66 Mas: 3. sausee ae sae- 8.6244 | 8.6798 | ...... 8. 7857 | 8.5225 . 042 0483/1 pial! . 061 . 033 57 28 322 95 Opes Fo Bye Gee. 8. 5225 | 8.6607 | & 7857 | .....- 8. 2641 . 033 . 046 BOOTS {eso . 018 95 66 38 133 OOs 2) aon esis: 8. 7857 | 8.6607 | 8.5225 | 8.2641 | ...... . 061 046 . 033 OUSW OT oes 322 294 265 227 a pS 8:.: te oe. ORS &.9030 | 9. 8237 | 9.0002 | &. 2539 | 8.3377 . O80 . 666 . 100 . 018 . 022 114 86 237 199 152 Oya. SRE se BoReS &. 3232 | 8.5225 ! &. 6607 | 8.6798 | 7.8138 . 021 . 033 . 046 048 . 007 123 95 66 28 161 2S 8.1 ee a 7.9841 | 8.3232 | 8.5225 | 8.6244 | 8. 4337 . 010 . 021 - 033 . 042 . 027 151 123 95 57 Brera. een Be 8.9481 | 9.9587 | 8.4376 | 8.3155 | 8.6579 - O89 . 909 . 027 . 021 . 045 71 43 195 337 109 Pros. UE eS cc cic'e 8.2780 | 8.0476 | 7.9556 | 8.9351 | 8.3116 .019 - O11 - 009 . O86 ..020 47 19 170 132 85 Qi 2Q Si 8, 3232 | 7.9841 | 8.9481 021 | .010 | .089 237 209 989 8, 5225 | 8.3232 | 9, 9587 033 | .021 909 265 237 317 8, 6607 | 8.5225 | 8.4376 i 033. | 027 294 265 165 8, 6798 | 8.6244 | 8.3155 048 | .042 | 021 332 303 23 7.8138 | 7.4337 | 8.6579 007 | .027 | .045 199 351 251 7.7367 | 8.1982 | 9.9587 .005 | .016 . 909 351 323 43 ch ana 8.6798 | 8. 4303 ee 048 | .027 332 52 8.6798 | 22... 8, 4014 Voasie | | cae 025 ps)|} wel 80 8, 4303 | 8.4014 | .....- hoot | | sOar Ione 308 BRO Lhe 9, 8640 | 8.9351 | 8, 1689 1731 | 086 | .015 284 | 256 156 HARMONIC ANALYSIS. AND PREDICTION OF TIDES. 249 TaBLE 29.—Elimination factors—Continued. SERIES 87 DAYS. SEMIDIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought (A). Ko Le Mo No 2N Re Se To do 2 v2 2SM Meigen ois semeeesy |e Sces 8. 7857 |8.6607 |8. 5225 |8.3232 |9. 9587 |9. 8287 |9. 5416 |8.9030 |8. 2780 |8. 0476 | 8. 7538 Mapes -061 | .046. | .033 | .021 | .909 | .666 | .348 | .080 | .019 | .O11 . 057 Bist» 322 294 265 237 317 274 231 246 313 341 75 Brats ati fe sepyates e Sr Soe ibse See e 8.6798 8.6244 |8. 5247 |7.9576 |8.9351 |9. 1055 |9. 8640 |7. 7367 |8. 2539 | 8.5315 SOBER) bee 048 | .042 | .033 | .009 | .086 | .127 | .731 | .005 | .018 . 034 Or ia siese 332 303 275 175 | 1382 89 284 351 199 113 Mg de oh sae) os 8.6607 |8.6798 |...... 8.6798 |8. 6244 |8.3155 |8. 2539 |8.6976 |8.9351 |8. 2539 | 8.9351) 8. 2286 O46, 1) S048 e 0 | oe. a 048 | .042 |.021 | .018 | .050 | .086 -018 | .086 - 017 66 PaSuN te one 332 303 23 161 LSS Reko2 199 228 141 Noarep ceca ceisites ois 8.5225 |8.6244 (8.6798 |.....- 8. 6798 |8. 4803 |7. 7367 |8. 3068 |8. 2539 |8.9351 |9. 8640 | 7. 5883 033 | .042 | .048 | ee -048 | .027 | .005 | .020 |: 018 | .086 -731 | .004 95 57 28: | obs 332 52 9 146 161 228 76 170 ON gos passe 8.3232 |8.5247 |8.6244 8.6798 |... 8.4014 |8.1982 |7.4165 |7. 7367 |9. 8640 | 8.9351) 7.7314 -021 | .033 | .042 | .048 | ..... .025 | .016 | .003 | .005 | -731 | .086 - 005 123 85 57 287 sees 80 37 174 9 76 104 18 Lice tae eae 9.9587 |7.9576 |8.3155 |8. 4303 |8. 4014 |...... 9.9587 |9.8237 |8. 9481 |7. 2740 |8. 1689 | 8.6976 ONIN. | HOOT ue ~O21 | O27 Bln 025. bese. 2s -909 | .666 | .089 | .002 | .015 . 050 43 185 337 308 280 a5e8 317 274 289 356 204 118 toys an pone 9. 8237 |8.9351 |8. 2539 |7. 7367 |8. 1982 |9.9587 |...... 9. 9587 |8.6798 |&. 2286 |8.5315 | 8. 2539 -666 | .086 !.018 | .005 !.016 | .909 ! _.... -909 | .048 | .017 ! .034 - 018 86 228 199 351 323 | 43 oes 317 332 219 247 161 Toren) Saale (9. 5416 |9.1055 |8.6976 |8. 3068 |7. 4165 |9.8237 |9. 9587 |... . $ 4376 18.4358 |8. 5521 | & 3155 -348 | .127 | .050 | .020 | .003 | .666 | .909 | ..... 027 | .027 | .036 02 i 129 271 242 214 186 86 43 eee 195 262 290 23 Dan ae A ee 8.9030 |9. 8640 |& 9351 |8. 2539 |7. 7367 |8.9481 |8.6798 |8.4376 |...... |8.5315 |8. 7629 | 7.7367 -080 | .731 | .086 | .018 | .005 | .089 | .048 | .027 | ..... .034 | .058 - 005 114 76 228 199 351 7 28 1G5G eres 247 275 9 Baseacices spew eee \8. 2780 |7. 7367. |8. 2539 |8.9351 |9. 8640 |7. 2740 |8. 2286 |8. 4358 |8.5315 |...... |8.6798 | 8.1849 -019, | 005, | .018 | .086 |.731 | .002 | .017 | .027 | .034 | ..... - 048 - 015 47 9 161 132 284 4 141 98 UTR Bema 28 122 Pg ee eeeeres cet 8, 0476 |8. 2539 |8. 9351 |9. 8640 |8. 9351 {8.1689 |8. 5315 {8.5521 |8. 7629 |8.6798 |...... 8. 3401 -O11 | .018 | .086 | .731 | .086 -015 | .034 | .036 | .058 | .048 | ..... . 022 19 161 132 284 256 156 113 70 85 S52) eee 93 SAS CORE ai 8. 7538 |8. 5315 |8.2286 |7.5883 |7.7314 |8.6976 |8. 2539 |8.3155 |7. 7367 |8. 1849 |8.3401 | ...... -057 |. 084 | .017 | .004 | .005 | .050 | .018 | .021 | .005 | .015 |.022 | ..... 285 247 219 190 342 242 199 337 351 238 267 Braet 250 U. S. COAST AND GEODETIC SURVEY. TABLE 29.—Elimination factors—Continued. ‘SERIES 105 DAYS. DIURNAL COMPONENTS. Disturbing components (B, C, etc.). es (OE sought )- Ji Ki Mi QO; (exe) Pi Qi Sirtete ieee te ee rcyes eh renee «| ELIReMSS 8.6704 | 8.5885 | 8. 4422 | 8.4953 | 8. 8322 | 8. 2332 pete . 047 . 039 . 028 - 031 068 | ~.017 214 248 | 271 158 291 305 1 oh eae eel Ne a SHAG (Oi Sal Nahata) 8. 6704 | 8.5381 | 8.5381 | 9.7311 | 8. 4422 ROS fi Plime does fe . 047 . 035 . 035 . 038 . 028 146 | ae 214 236 124 257 271 Wales a is eo Mh aie 8. 5885 8.6704 eerie 8. 4953 | 8. 4492 | 8. &219 | 8. 5381 . 039 BOAT Plager e - 031 - 028 . 066 - 035 112 146 202 89 42 236 OEE eee ae ae 8, 4422 | 8.5381 | 8.4953 | ...... 8. 2803 | 8.1856 | 8.6704 - 028 . 035 OSU iliwereciee - O19 - 015 . 047 89 124 158 2 67 20 214 LOKO VARIES Als cee means 8. 4953 | 8.5381 | 8.4422 | 8.2803 | ....-- 8. 4500 | 7. 9556 . 031 - 035 . 028 POLST Shenae - 028 . 009 202 236 271 293 313 327 LE peers stelle mane 8. 8322 | 9.7311 | 8.8219 | 8.1856 | 8.4500 | ...... 7. 8500 - 068 - 538 . 066 015 NOZSua le Meee . 007 69 103 318 340 47 ies 194 hs ar nee Serr eemrens 8. 2332 | 8.4422 ! 8.5381 | 8.6704 | 7.9556 | 7.8500 |! ....-- - O17 . 028 . 035 - 04% - 009 SOO eres 55 89 124 146 33 166 P PAD Been ede ane Race 7. 7808 | 8.2382 | 8. 4422 | 8.5885 | 6. 4362 | 8.2067 | 8.6704 - 006 - 017 - 028 - 039 - 000 . 016 . 047 21 55 89 112 179 132 146 Sieeacye teen Se ea i 8.3722 | 9.9393 | 8.9548 | 8.6113 | 7.5174 | 9.9393 | 8. 2396 . 024 . 870 -090 |, . 041 . 003 . 870 . O17 18 52 266 288 175 308 322 ico asa shed ayeeoaeose 8.1065 | 7.0766 | 8.3679 | 8.8929 | 8.1640 | 8.4685 | 9.7951 - 013 - OOL . 023 . 078 - 015 - 029 . 624 144 178 32 54 121 74 268 eaetce HARMONIC ANALYSIS AND PREDICTION OF TIDES. 251 TasLe 29.—Elimination factors—Continued. SERIES 105 DAYS. SEMIDIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component : sought (A). | Kg Le Moe Ne 2N Re Se Te de u2 v2 2SM | Kip ise asd oh paey aia Ya =e seays 8. 4953 |8.5381 |8. 4422 |8. 2332 |9. 9392 |9.7311 |9.1892 |8.8322 |8.1065 |7.0766 | 8.6847 sores -031 | .035 | .028 | .017 | .069 | .5388 | .155 | .068 | .013 | .001 . 048 esi le 202 236 271 305 308 257 205 291 216 182 97 Moose oes Vesa 8.4953 |--.--. 8. 6704 |8.5885 |8. 4347 |8.9311 |8.8929 |7.6403 |9.7951 |7.8500 |8.1856 | 8.4685 SOs ter | eee -047 | .039 | .027 | .085 | .078 | .004 | .624 | .007 | .015 - 029 TEMS ae 214 248 282 106 54 2| 268 194 340 74 Moi) oat di erenils he 8.5381 |8.6704 |.-.--- 8. 6704 |8.5885 |8. 6113 |8. 1856 !8.3896 |8. 8929 |8.1856 |8.8929 | 8.1585 alOssedy 1s Niel OL Tin ee erey= -047 | .0389 | .041 | .015 | .025 | .078 | .015 | .078 -014 124 WAG hi dee 214 248 72 20 148 54 340 306 40 Neste: cmon et 8. 4422 |8.5885 |8. 6704 |....-- 8. 6704 |8. 2395 |7.8500 |8. 4362 |8.1856 |8. 8929 |9.7951 | 7.2638 O28 ee OSOM ls OFM [iia ccce -047 | .017 | .007 | .027 | .015 | .078 | .624 - 002 89 112 146 | .... 214 38 166 114 20 306 92 6 QIN os avo diaeaee == 8. 2332 |8.4347 |8. 5885 |8. 6704 |.....- 7.1241 |8.2067 |8.3366 |7.8500 |9.7951 |8.7943 | 7.8368 BOLZer ne O2A pile Odum ei OA (ae jean ae -001 | .016 | .022 | .007 | .624 | .062 - 007 55 78 112 HAG 7 oe 4 132 80 166 92 58 152 Beare re ene 9. 9392 |8. 9311 |8. 6113 |8. 2395 |7.1241 |...... 9.9392 |9.7311 |8.3722 |8. 3410 |8.3820 | 8.3896 BOOS, | OSann | 048 | 0L7 |) 00L e252 . 869 538 | .024 | .022 | .024 - 025 52 204 288 322 356 Aad 308 257 342 268 234 148 ems Sets 9.7311 |8.8929 |8.1856 |7.8500 |8. 2067 |9. 9392 |....-. 9. 9392 |8. 6704 |8.1585 |8. 4685 | 8.1856 538 | .078 015 007 O1G 869), eee 869 047 014 029 015 103 306 340 194 228 52) teers 308 214 320 286 20 Cae eee ete poe = 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 | .538 OE) MN Bees 090 005 012 041 155 358 212 246 280 103 52 Z 266 192 338 72 DET SS Se eee oe ee 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 OSD Ra Saas 029 | .046 007 69 92 306 340 194 18 146 94 286 252 166 ) Toe aes eae 8.1065 |7.8500 |8.1856 |8.8929 |9.7951 |8.3410 |8. 1585 |7. 6654 |8.4685 |...... \8. 6704 | 8.1117 013 007 015 078 624 | .022 014 005 O29 )7F ae aeee 047 013 144 166 20 54 268 92 40 168 74 2 146 60 Fi a ea 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 eA a aesere 018 178 20 54 268 302 126 74 22 108 PAY NS eee 94 2SMe ei. becca: 8. 6847 |8.4685 |8.1585 |7. 2638 |7. 8368 |8.3896 |8. 1856 |8. 6113 |7.8500 |8.1117 |8.2581 | ...... 048 029 014 002 069 | .025 015 041 007 013 CSS NT ae 263 286 320 354 208 212 340 288 194 300 266 BA 72934—24}——_17 say i 952 U. S. COAST AND GEODETIC SURVEY. TaBLe 29.—Elimination factors—Continued SERIES 134 DAYS. DIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought Nh ky M QO; (0x) Py Qi 2Q Si p Pfr eS I Re et eh el Na RS 8. 4360 | 83946 | 8.2695 | 8.0361 | 8.7345 | 8.2094 | 8.0980 | 8.6047 | 7.7771 eee 027 025 019 O11 . 054 016 012 0 006 205 229 239 170 253 264 288 319 201 TRG ye sia .g cote eyeccn los tela 8.4360) |) oo s-e- 8. 4360 | 8.2628 | 8.2628 | 9.5078 | 8.2695 | 8.2094 | 9.8992 | 7.1819 OZ TPn assests 027 018 018 . 322 019 016 793 - 002 155 205 214 146 228 239 264 294 356 Mi ere -faidg asteiee coe dace 8.3946 | 8.4360 | ...._. 8. 0361 | 8.2695 | 8. 4838 | 8.2628 | 8.2695 | 8.8500 | 8.2196 025 OZ Tih ars ere O11 019 . 030 018 019 071 O17 130! 155 190 121 23 214 239 89 332 Open cs ecko. ce cae 8. 2695 | 8.2628 | 8.03861 | ...... 8.1796 | 7.9151 | 8.4360 | 8.3946 | 8.5206 | 8.6697 019 018 (OMI TF epee a 015 . 008 027 025 033 0474 121 146 170 u 1 14 205 229 80 322 OORe se ere ce eae 8. 0361 | 8.2628 | 8.2695 | 8.1796 | ...... 8. 4760 | 8.1133 | 7.9812 | 8.2156 | 7.8315 O11 - 018 . 019 OLS ease . 030 . 013 . 010 - 016 - 007 190 214 23 249 i Poe 262 273 298 328 211 1 ER ene Cea ig Seaphe tk 8.7345 | 9.5078 | 8.4838 | 7.9151 | 8.4760 | ...-.. 7.6424 | 7.9951 | 9. 8992 8. 2746 . 054 a2 - 030 . 008 SOSOKB sees . 004 . 010 . 193 - 019 107 132 337 346 OR Nee sees 191 216 66 308 Qithinec 2st. ss. sess 8. 2094 | 8.2695 | 8.2628 | 8.4360 | 8.1133 | 7.6424 | ___... 8. 4360 | 8.2605 9. 6387 . O16 . 019 . O18 - 027 . 013 5 O04 ia bee . 027 . 018 - 435 96 121 146 155 87 169 tas 205 55 117 ZQise sods pee ees awe 8. 0930 | 8.2094 | 8.2695 | 8.3946 | 7.9812 | 7.9951 | 8.4360 | ...... 7. 9233 8. 7610 - 012 . 016 . 019 . 025 . 010 . O10 AO QZ ies | tee . 008 - 058 72 96 121 131 62 144 155 wees 30 92 Syeee se oe ahs. oes aeae 8. 6047 | 9.8992 | 8.8500 | 8.5206 | 8.2156 | 9.8992 | 8.2605 | 7.92383 | ...... 8.3143 . 040 . 793 . 071 . 033 . 016 . 793 . 018 = OOS So Ere Sole Hogi 41 66 271 280 32 294 305 330 aks 242 DURE Sieisa Me ce kaise eo 7.7771 |.7.1819 | 8.2196 | 8.6697 | 7.8315 | 8.2746 | 9.6387 | 8.7610 | 8.3143 | .....-. . 006 - 002 . 017 . 047 . 007 . 019 . 435 . 058 ODT ele Sits 159 4 28 38 149 52 243 268 118 ase HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLE 29.—Elimination factors—Continued. SERIES 134 DAYS. SEMIDIURNAL COMPONENTS. 253 Component sought (A). Disturbing components (B, C, etc.). Ke Le M2 No IN 2 popes 8.0361 |8. 2628 |8. 2695 |&. 2094 i pNRS - 011 - 018 - 019 . 016 eae 190 214 239 264 850368 |). 55 8. 4360 |8.3946 18.3215 OME |) site - 027 025 | .021 170 Bess 205 229 254 8. 2628 |8. 4360 |.-.-.. 8. 4360 |8. 3946 - 018 s O2MaN |as sa . 027 - 025 146 155 Ae 205 229 8. 2695 |8.3946 |8. 4860 |_....- &. 4360 - 019 . 025 JO2 Teele a8 2 . 027 121 131 155 205 8. 2094 |8.3215 |8.3946 |8. 4860 |_..__. - 016 - 021 . 025 AO Skaels 96 106 131 155 Rye 9. 8992 |8. 8285 |8. 5206 |8. 2605 |7. 9233 - 793 . 067 - 033 - 018 . 008 256 280 305 330 9.5078 |8.6697 |7.9151 |7.6424 |7. 9951 say 2) - 047 - 008 . 004 - 010 132 322 346 191 216 8. 9538 |8. 5871 |8. 4622 |8.3592 |8. 2280 - 090 - 039 . 029 . 023 - 017 18 208 232 257 282 8. 7345 |9.6387 |8. 6697 |7.9151 |7. 6424 -054 | .4385 | .047 | .008 | .004 107 117 3227, 346 191 7.7771 |7.6424 |7.9151 {8.6697 |9. 6387 006 | .004 -008 | .047 - 435 159 169 14 38 243 7.1819 |7.9151 |8.6697 |9.6387 |S. 7610 -002 |.008 | .047 | .485 | .058 4 14 38 243 268 8.5254 18.2746 |7.9028 |6.7753 |7. 6344 -034 | .019 | .008 | .001 - 004 299 308 333 358 202 Re Se Ts de “2 ve | 29M 9.5078 |8.9538 |8. 7345 |7.7771 |7.1819 | 8.5254 322 | .090 | .054 | .006 | .002 - 034 228 342 253 201 356 61 8.6697 |8.5871 |9.6387 |\7.6424 |7.9151 | 8.2746 -047 | .039 -435 | .004 | .008 - 019 38 152 243 191 346 52 7.9151 |8.4622 |8.6697 |7.9151 |8.6697 | 7.9028 -008 | .029 | .04 -008 | .047 - 008 14 128 38 346 322 27 7.6424 |8.3592 |7.9151 |8.6697 |9.6387 | 6.7753 -004 | .023 | .008 | .047 | .435 - 001 169 103 14 322 117 2 7.9951 |8.2280 |7.6424 |9.6887 [8.7610 | 7.6344 -010 | .017 }.004 | .485 | .058 - C04 144 78 169 117 92 158 9. 8992 |9.5079 |8.6047 |8. 2346 |8.3143 | 8. 4629 -793 | .3822 | .040 | .017 | .021 - 029 294 228 319 267 242 128 ees Hh \9. 8992 18. 4360 |7.9028 |8. 2746 | 7.9151 Be ss . 193 -027 |.008 | .019 - 008 a46 294 205 333 308 14 9. 8992 | 2225.2 8. 8500 |8. 0509 |7. 7834 | 8.5206 IS it) Ses - 071 -O11 | .006 - 033 66 nfs 271 219 194 £0 8. 4360 |8. 8500 |...... 8 2746 |8.5650 | 7.6424 OQ | AOU ai ee eeaee -019 | .037 - 004 155 89 Aa 308 284 169 7.9028 |8.0509 |8. 2746 |...... 8, 4360 | 7. 8820 -008 | .O11 OLD see - 027 - 008 27 141 52 = 155 41 8. 2746 |7. 7834 {8.5650 |8. 4360 |.....- 8.1118 -019 | .006 | .0387. | .027 | ....- - 013 52 166 76 205 ae 65 7.9151 |8.5206 {7.6424 |7. 8820 |8.1118 | .....- -008 | .033 |} .004 | .008 | .013 | ..... 346 280 191 319 295 som 254 U. S. COAST AND GEODETIC SURVEY. TaBLE 29.—Elimination factors—Continued. SERIES 163 DAYS. DIURNAL COMPONENTS. Disturbing components (B, C, etc.). Mead eee sought : hh Ki Mi 0: | 00 Pi Qu 2Q St pl Fe RA EE AP ARR 8.1495 | 8.1341 | 7.9150 | 7.4365 | 8. 4234 | 7.9579 | 7.9582 | 8.6570 | 7.0948 Ty 014 014 008 003 | .027 009 009 045 001 195 210 207 3 215 293 238 295 185 TERR th GONE ENS. Bl495u| Aoeh o 8.1495 | 7.7528 | 7.7528 | 9.0723 | 7.9150 | 7.9579 | 9.8470 | 7.5128 Oly | eee 014 006 006 | .118 008 009 703 003 165 195 192 168 199 207 993 280 350 RIM 225k Be Naha 8.1341 | 8.1495 | .2...- 7. 4365 | 7.9150 | 7.6604 | 7.7528 | 7.9150 | 8.7629 | 8.0859 014 Oey!) SUG 003 008 | .005 006 008 058 012 150 165 357 153 4 192 207 84 335 OMA Ree le. Thole 7.9150 | 7.7528 | 7.4865 | ...--- 7.7427 | 7.5513 | 8.1495 | 8.1341 | 8.4422 | 8.3724 SQOSS: || ROS) e003 Lae .006 | .004 | .014 | .014 | .028 . 024 153 168 SMe haus 156 Fi 195 210 87 338 OO! 5: LO. ee 7. 4365 | 7.7528 | 7.9150 | 7.7427 | ...--. 8.1140 | 7.8343 | 7.8631 | 8.3776 | 6.6248 MOOSTs | MOON! NOOR) (ne O06, eliekee hes .013, | .007 | .007 | .024 . 000 357 192 207 04% ae 212 220 235 292 182 PEGs 8. Phe eahde 8. 4234 | 9.0723 | 7.6604 | 7.5513 | 8.1140 | _.__-- 7. 4230 | 7.7409 | 9.8470 | 7.9852 SODTiy) || eS ROOD ST ee COL a MNOS lee ne .003 | .006 | .703 . 010 145 161 356 353 TASH ee ates 203 80 331 @qHU LS. Lee ee Bees 7.9579 | 7.9150 | 7.7528 | 8.1495 | 7.8343 | 7.4230 | ....-- 8.1495 | 8.2410 | 9.3888 SOO9L |) A008) 0062.1 eOLe |X OO7EaI Mi SOOSee came .014 | L017 245 137 153 168 165 140 ier Teese 195 72 142 IO. ee, eS 7.9582 | 7.9579 | 7.9150 | 8.1341 | 7.8631 | 7.7409 | 8.1495 | _..-.. 8.0589 | 8.5769 "009% |) X009) 1) O08) | MeO | MrOozds ey) S00 es!\ S01. samme O11 . 038 122 137 153 150 125 157 165°} 126k! 57 197 Gia aie Br BES. 8.6570 | 9.8470 | 8.7629 | 8. 4422 | 8.3776'| 9.8470 | 8.2410 | 8.0580 | ...... | 8.2562 10455) |) W708) 1) MOSS) | NODS Nh 024 HN 70820 (POL Tae) ORT epee - 018 _ 65 80 276 273 68 280 288 Edel Wiese? 250 PR Ae tos 7.0948 | 7.5128 | 8.0859 | 8.3724 | 6.6248 | 7.9852 | 9.3888 | 8.5769 | 8.2562 | ...... SO0M.|) S008, ||) O12. | 7202425) W000) +h VOL0Rel? 24500)» © OsBenl me iene an eeeee 5 175 10 25 22 178 29 218 233- 110) pees HARMONIC ANALYSIS AND PREDICTION OF TIDES. TABLE 29.—Elimination factors—Continued. SERIES 163 DAYS. SEMIDIURNAL COMPONENTS. 255 Component sought (A). Disturbing components (B, C, etc.). 2 Le |- Me No 2N Re So To de U2 v2 28M Seabees 7.4365 |7.7528 |7.9150 |7.9579 |9. 8470 |9. 0723 |9.3179 |8. 4234 |7. 0948 |7.5128 | 8.1450 ee . 003 . 006 . 008 . 009 . 703 .118 .208 | .027 . O01 . 003 . 014 357 192 207 223 280 199 299 215 185 350 26 Died 8Gorr| Vee a. 8. 1495 |8.1341 |8.1078 |8. 7464 |8.3724 |8. 7614 |9.3888 |7. 4230 |7.5513 | 7.9852 (OOS a ee eee . O14 . 014 . 013 - 056 .024 | .058 . 245 . 003 004 . 010 3 195 210 226 103 22 122 218 188 353 29 Bife(O20 18.495) [hs ee 8. 1495 |8.1341 |8. 4422 |7. 5513 |8. 4590 |8. 3724 |7.5513 |8.3724 | 7.5483 . 006 QUA A iets sate -014 | .014 - 028 -004 | .029 | .024 | .004 | .024 . 004 168 165 4 195 210 87 7 107 22 353 338 14 | ; 7.5513 |8.3724 |9.3888 | 6.3062 -004 | .024 | .245 . 000 7 338 142 179 7. 4230 |9.3888 |8. 5769 | 7. 4186 . 003 - 245 . 038 . 003 172 142 127 164 8.6570 |8. 1488 |8. 2563 | 8. 4590 - 045 014 | .018 . 029 295 265 250 107 8.1495 17.5483 |7.9852 | 7.5513 014 | .004 | .010 . 004 195 346 331 7 8.7629 |8.1668 |8.1966 | 8. 4422 058 | .015 . 016 . 028 276 246. 231 87 .|8. 4234 19.3888 |8.3724 |7.5513 |7. 4230 |8.6570 |8.1495 |8.'7629 |...... 7. 9852 |8.3386 | 7. 4230 - 027 245 . 024 . 004 - 003 . 045 AOI SP (Ulster TS he - 010 . 022 . 003 145 142 338 353 188 65 165 84 a 331 315 172 .|7. 0948 17. 4230 17.5513 |8.3724 19.3888 |8. 1488 17.5483 |8.1668 |7.9852 |_....- 8.1495 | 7.5426 - O01 - 003 . 004 024 | .245 . O14 .004 | .015 SOLO ees . O14 . 003 175 172 7 22 218 95 14 114 29 as 165 21 .|7. 5128 |7. 5513 |8.3724 |9. 3888 |8.5769 |8. 2563 17.9852 |8.1966 |8.3386 |8.1495 |...... 7. 8424 . 003 . 004 . 024 -245 | .038 . 018 - 010 . O16 . 022 OU ee ee . 007 10 7 22 218 233 110 29 129 45 195 ae 36 .|8. 1450 |7. 9852 |7. 5483 |6.3062 |7. 4186 |8. 4590 17.5513 |8. 4422 |7. 4230 |7. 5426 |7. 8424 | _..... . 014 . 010 . 004 . 000 . 003 . 029 -004 | .028 . 003 S085) | SOUP ee oS 334 331 346 181 196 253 353 273 188 339 324 ee 256 U. 8S. COAST AND GHODETIC SURVEY. TaBLe 29.—Elimination factors—Continued. SERIES 192 DAYS. DIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought \ ih 1 M O1 00 Py Qi 2Q Sy in Tite Me a a AL Ve I 7.6613 | 7.6591 | 7.0355 | 8.0828 | 7.3819 | 6.5151 | 7.0698 | 8.6281 | 7.3308 ae -005 | .005 | .001 | .012 | .002 | .000 | .o01 | .042 002 186 192 356 16 357 182 187 71 350 ERT 4 Au yl Ae TORE Wasa” | 7.6613 | 7.5891 | 7.5891 | 8.6868 | 7.0355 | 6.5151 | 9.7807 | 7.6468 005 -005| .004| .004]) .049) .001| .000| .604 004 174 186 350 10 351 356 182 265 344 TS Ae pediaeds) OU Aa TAG5OUe AGO1S) seen 8. 0828 | 7.0355 | 8.1441 | 7.5891 | 7.0355 | 8.6866 | 7.9586 PONS) | O05N) || Nee -012 |. .001, | .014 | .004 | .oon | 049 . 009 168 TaN ye as 344 4 165 350 356 80 338 Orit Ae eben rere 7.0355 | 7.5801 | 8.0828 | ...... | 7.5826 | 6.4230 | 7.6613 | 7.6591 | 8.3698 | 7.7679 SOT WOOL hy SMD gees /004 | .000 | .005 | .005 | .023 006 4 10 | AGU coe 20 1 186 192 95 354 OOKS oak we esate. 8 0828 | 7.5891 ) 7.0855 | 7.5826 | ....-. | 7.8388 | 7.3409 | 7.0344 | 8.3250 | 7.6132 OEY GEOL PSC) OO a 00% «| |. 002. | --. ODie | aaa . 004 344 350 356 SH Map 341 346 352 256 334 | PPA AP vera Uy a 7.3819 | 8.6868 | 8.1441 | 6.4230 | 7.8388 | .....- 7.1547 | 7.3491 | 9.7807 | 7.3097 .002 | .049 | L014 000 OO eas 001 002 604 002 3 9 195 359 19 185 191 95 353 Qiad 6) ose) ase kY 6.5151 | 7.0355 | 7.5891 -| 7.6613 | 7.3409 | 7.1547 | _..... 7.6613 | 8.1911 | 8.8560 /000 | .001 | .004 005 D020) eon | enya 005 016 072 178 4 10 174 i4 175 186 39 168 POMP Neuse wll ara) 7.0898 | 6.5151 | 7.0355 | 7.6591 | 7.0344 | 7.3491 | 7.6613 | ...-.- 8.0615 | 8.0931 /001 | .000 | .001 005 001 O02 O05e| aaueae 012 012 173 78 4 168 8 169 eh Po 84 162 Sy at egy er 8.6281 | 9.7807 | 8.6866 | 8.3698 | 8.3250 | 9.7807 | 8.1911 | 8.0615 | ...... 8. 2024 EOL PGOLN |) eO4ON | AO23 ile O02 Tene 60401 O1GIN| ae Ol oe meme . 016 39 95 230 285 104 265 71 D716 | ices 258 Fae eae ts CR oe 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 | 016 10 16 22 6| 26 7 192 198 102 pe . & 2 oae iain HARMONIC ANALYSIS AND PREDICTION OF TIDES. TABLE 29.— Elimination factors—Continued. SERIES 192 DAYS. SEMIDIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought (A). Ko Le Ras te ape lig) | 22 bi 8.0828 Sans. . 012 bey, 344 PDs: o's 0 yyy 4p. EOS ae wee A(T On na 16 32 Moor ee gay. oh 7.5891 7. 6613 -004 | .005 10} 174 UN ass oN zesty ey 7.0355 |7. 6591 -001 | .005 4| 168 DING. 38 id cavers 6.5151 (7. 6554 -000 | .005 178 163 1353 ee ee eee 9.7807 |8. 6778 .604 | .048 95 | 259 Siar ee eee 8.6868 |7.7679 .049 | .006 9| 354 de eee ee 9, 2922 |8. 7615 -196 | .058 104} 268 FOS at eee 7.3819. 8. 8560 -002 | .072 168 [eres Ceca 7.3308 7.1547 -002 | .001 10| 175 DS ee ee 7.6468 (6.4230 .004 | .000 16 1 735)" [eae 7.6012 |7.3097 -004 | .002 189 | 358 Me No 2N Re Se To 7.5891 |7.0355 '6.5151 |9.7807 |8.6868 |9. 2922 -004 | .001 | .000 | .604 | .049 | .196 350 | 356] 182] 265] 351] 256 7.6613 |7.6591 |7. 6554 |8. 6778 |7.7679 |S. 7615 -005 | .005 | .005 | .048 | .006 | .058 186} 192} 197} 101 92 ere 7.6613 (7.6591 |S. 3698 |6. 4230 |8. 4057 ee -005 | .005 | .023 | .000 | .025 sa 186} 192 95 86 7.6613 |....-. 7.6613 |8.1911 |7. 1547 |8. 2077 Stay Wileeeee -005 | .016 | .001 | .016 174 5 186 89] 175 80 7.6591 |7.6613 |...... 8.0615 |7.3491 |8. 0649 AOO5an| 4005) | meeees -012 | .002 | .012 GS rm lt ed) aa 8 84; 169| 74 8.3698 |8.1911 8.0615 |...... 9.7807 8. 6863 HO2SE |) OLGmsnOL2) 1 esas: -604 | .049 Ga |) (2 7le ll Me276a|e ses 265} 351 6. 4230 |7.1547 |7.3491 |9.7807 |...... 9.7807 -000 | 001 | /002 | .604 | 22... - 604 359; 185) 191]. 95) . 265 8.4057 |8.2077 8.0649 | 8. 6863,9.7807 |...-.. S025) |) O16) ee L255 049) ie GOL) ||) Sees 274| 280! 286 9 Bil! Mace 7.7679 |6. 4230 \7. 1547 |8. 6281 |7. 6613 |8. 6866 -006 | .000 | .001 | .042 | .005 | .049 354 | 359] 185 89| 174] 80 6. 4230 |7. 7679 |8.8560 |8.0768 |6.4265 |8.0959 -000 | .006 | .072 | .012 | .000 | .012 1 6| 192 96 i |e 87 7. 7679 |8.8560 |8.0931 |8. 2024 |7.3097 |8. 2350 -006 | .072 | .012 | .016 | .002 | .017 6| 192] 198] 102 7 92 6.4265 |6.8803 |7.1525 |8. 4057. |6.4230 |8. 3698 -000 | .001 | .001 | .025 | .000 | .023 359} 185/ 190] 274] 359} 265 de L2 7.3819 |7.3308 002 | .002 357 | 350 8.3560 |7. 1547 072 | .001 192 | 185 7.7679 |6. 4230 006 | .000 6| 359 6. 4230 |7. 7679 000 | .006 1| 354 7.1547 |8.8560 001 | .072 175 | 168 8.6281 |8.0768 042 | .012 271] 264 7.6613 |6. 4265 005 | .000 186 | 359 8.6866 |8.0959 049 | .012 280 | 273 pecan 7. 3097 es ma 002 lieteentl® abc PORNO Wsoree | ania ss |eaen 6 | 7 S35 17.7656 |7. 6613 006 | .005 lt 13)? 86 7.1547 16.4253 001 | .000 185 | 358 8 257 258 U. S. COAST AND GEODETIC SURVEY. TaBLE 29.—Elimination factors—Continued. SERIES 221 DAYS. DIURNAL COMPONENTS. Disturbing components (B, C, etc.). ies Tat sought . hi K | M Oy WP OOM Pi Qi | 2@ Si pl eee ee ese ee ees | eee 7.4061 | 7.4052 | 7.8848 | 8. 2672 | 8.3590 | 7.7969 | 7.7322 | 8.5324 | - 7.6535 esdide - 003 - 003 - 008 - 019 - 023 - 006 - 005 - 034 -005 a 356 353 324 28 318 321 317 247 334 BG Waele ae A Sceiaes eh ul a fe AGT S| eee ae 7.4061 | 8.0179 | 8.0179 | 9.2077 | 7.8848 | 7.7969 | 9.6969 | 7.7209 BOOS he shes - 003 - 010 - 010 - 161 - 008 . 006 - 498 - 005 4 % 356 328 32 322 324 321 251 338 Mine ences 7.4052 | 7.4061 | .....- 8.2672 | 7.8848 | 8.4189 | 8.0179 | 7.8848 | 8.6172 | 7.8313 003 CUS se | eecece 019 008 - 026 010 008 «041 007 7 | 4 . 332 36 146 328 324 5 341 Ore eeee sean sae cmt 7. 8848 | 8.0179 | 8.2672 | .....- 7.9465 | 7.3352 | 7.4061 | 7.4052 | 8.2991 | 7.8800 . 008 010 SOLO Sal ee clare, 009 . 002 003 003 - 020 - 008 6 32 28 64 174 356 353 103 190 QO Sst ete eeee 8. 2672 | 8.0179 | 7.8848 | 7.9465 | .....- 8. 2351 | 7. 8628 | 7.7946 | 8.0778 | 7.8188 019 010 | .008 (ra Wee ess - 017 007 006 - 012 - 007 332 328 | 324 296 290 292 289 219 306 1 EY SA ESM Secale 8.3590 | 9. 2077 | 8.4189 | 7.3352 | 8.2351 | .....- 6.7176 | 6.4850 | 9.6969 | 7.5854 023 161 | .026 002 NOL (mals gecies 001 000 - 498 004 42 38 | 214 186 70 Lene 182 358 109 195 Oieec} seeae Selene 7. 7969 | 7. 8848 8.0179 | 7.4061 ! 7.8628 | 6.7176 | ...... 7.4061 | 8.1112 | 8.8310 - 006 - 008 - 010 - 003 - 007 OOD eee - 003 - 013 - 068 39 36 32 4 68 178 cies 356 107 13 DA A acl Ae at 7.7322 | 7.7969 | 7.8848 | 7.4052 | 7.7946 | 6.4850 | 7.4061 | ...-.- 7.9748 | 8.0073 i - 005 - 006 - 008 - 003 - 006 - 000 S003™ || ce ees - 009 -010 43 39 36 7 71 2 4 Se 110 17 tS) (Ee ees se iain | ahi cat 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 SOOS Fs eee -014 113 109 285 257 | 141 251 253 250 sion 266 Far vereasdes Spain Sal are Ee 7.6585 | 7.7209 | 7.8313 | 7.8800 ive 8188 | 7.5854 | 8.8310 | 8.0073 | 8.1495 | ...... - 005 . 005 - 007 - 008 - 007 - 004 - 068 - 010 SUS NN occ 26 22 19 170 54 165 347 343 94 mee HARMONIC ANALYSIS AND PREDICTION OF TIDES. 259 TaBLE 29.—Elimination factors—Continued. SERIES 221 DAYS. SEMIDIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought (A). Ke Le Me No 2N Re Se TT. 2 pe v2 25M. LES Be eget ade gD lg 8. 2872 |8. 0179 |7. 8848 |7.7969 |9.6969 |9.2077 |8. 9831 |8.3590 |7.6535 |7.7209 | 8.2035 eee S - 019 - 010 . 008 . 006 - 498 - 161 . 096 . 023 . 005 . 005 . 016 seo 332 328 324 321 251 322 213 318 334 338 136 Ata es ene Sane | ae eae 7.4061 |7. 4052 |7. 4037 |8.6189 |7. 8800 |8. 6448 |8. 8310 |6. 7176 |7.3352 | 7.5854 eC ILE a ee - 003 . 003 - 003 . 042 . 008 . 044 . 068 . OOL . 002 . 004 28 micas 356 353 349 99 170 62 347 182 186 165 ioe ee soe s 8.0179 |7.4061 |...... 7.4061 |7. 4052 |8.2991 |7.3352 |8.3038 |7. 8800 |7.3352 |7. 8800 | 7.3333 - 010 HOOS = tpeceee - 003 - 003 -020 | .002 - 020 . 008 - 002 . 008 . 002 32 4 cone 356 353 103 174 65 170 186 190 168 Nears are cyan eee et 7. 8848 |7.4052 |7.4061 |...... 7.4061 |8.1112 |6.7176 |8. 1229 |7.3352 |7. 8800 |8. 8310 | 7. 0677 - 008 - 003 - 003 8 eel ol 033 - 013 - O01 - 013 - 002 - 008 - 068 . O01 36 7 4 saci 356 107 178 69 174 190 13 172 DINE aes 7.7969 |7.4037 |7.4052 |7.4061 |.....- 7.9748 |6. 43850 |7.9996 |6.7176 |8. 8310 |8. 0073 | 6.7183 006 003 003 COS Rees 009 | .000 010 O01 068 010 001 39 il 7 4 < 110 2 73 178 13 17 176 omennanre se ike! 9.6969 {8.6189 |8.2991 |8.1112 |7.9748 |..__.. 9.6970 |9. 2076 |8. 5324 |8. 0145 |8.1495 | 8.3038 498 - 042 - 020 . 013 = OOO steer e - 498 - 161 . 034 - 010 014 - 020 109 261 257 253 250 ee es 251 322 247 263 266 65 Sins clase) eee 9.2077 |7. 8800 |7.3352 |6.7176 |6. 4356 |9.6970 |.-...-. 9.6970 |7. 4061 |7.3333 |7. 5854 | 7.3352 - 161 - 008 - 002 - O01 O00 498) tees - 498 . 003 - 002 . 004 . 002 38 190 186 182 358 109 us 251 356 192 195 174 gears tees: 2 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 -O10 | .161 S498) | “SSRs - O41 -009 | .012 . 020 147 298 295 291 287 | 38 109 aks 285 301 304 103 PS eR ats em 8.3590 |8. 8310 |7. 8800 |7.3352 |6. 7176 |8. 5324 |7.4061 {8.6172 |.-.... 7. 5854 |7.8738 | 6.7176 BOZS | OOS. |) HO0S sa OO2R TOOL | 70345. FA OOSt | O40 i) ees -004 | .007 - 001 42 13 190 186 182 113 4 75 ee 195 199 178 Pinata teeta vars 7.6535 16.7176 |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 OOO | OMe ase 003 002 26 178 174 170 347 97 168 59 165 A5O0 4 162 io SoS eee eee 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 OOF |) 003 Geass 003 22 174 170 347 343 94 165 56 161 356 fs 159 PASIN Boe ee 8.20385 |7. 5854 |7.3333 |7. 0677 |6. 7183 |8. 3038 |7.3352 |8.2991 |6. 7176 |7.3294 |7.4945 | ....-. -O16 | .004 | .002 |; .001 -001 {| .020 | .002 | .020 | .001 O02 | GOS |p seciste 224 195 192 188 184 295 186 257 182 198 201 2558 260 U. S. COAST AND GEODETIC SURVEY. TaBLE 29.—Elimination factors—Continued. SERIES 250 DAYS. DIURNAL COMPONENTS. Component sought (A). Disturbing components (B, C, etc.). Ji Ki Mi O1 0O ED Sey tere ee eects IE dcp 7. 90R | 7.8898 | 8.0302 | 8.3544 Eoee - 008 - 008 - O11 - 023 : 347 334 293 41 IC a eee eH Seoe ca aes TOV ieee os 7.9011 | 8.1489 | 8.1489 O08 Soy Teens - 008 - 014 - 014 13 347 306 34 Mia espe eee TESS || 76S ies Bees 8. 3544 | 8. 0302 - 008 O08 linge eae . 023 O11 26 13 319 67 One SSS esemee 8.0302 | 8.1489 | 8.3544] _..... 7.9171 - O11 - 014 10250 Rack - 008 67 o4 41 ‘ 108 OXOS JeBoas eee ise sar 8.3544 | 8.1489 | 8.0302 | 7.9171 | ....-. - 023 . O14 - O11 OUST aI eee 319 306 293 202 Prat ok 2 Shama ee fee 8.4768 | 9.3286 | 8.5201 | 7.6029 | 8. 1443 - 030 - 213 - 033 - 004 - 014 80 66 233 192 121 Qyesas o aar ae seer 7.9350 | 8.0302 } 8.1489 | 7.9011 | 7.7748 . 009 O11 - O14 - 008 - 006 80 67 54 13 121 ZOE 5 HAR pene | 7. 8438 | 7. 9350 - 007 - 009 93 80 Sipiet soe ree 8.3527 | 9.5900 - 023 . 389 136 123 Dei oF hs SRS <2 235075 7.7796 | 7.7661 | 7.6982 | 8, 2405 | 7. 8514 - 006 006 - 005 - 017 - 007 155 83 42| 29 16 Qi 2Q $1 pl 7.9350 |-7. 8438 | 8.3527 7.7796 - 009 . 007 . 023 . 006 280 267 224 318 8.0302 | 7.9350 | 9.5900 7. 7661 O11 . 009 - 389 . 006 293 280 237 331 8.1489 | 8.0302 | 8.5519 7.6982 . 014 - O11 . 036 . 005 306 293 70 344 7.9011 | 7.8898 | 8.2274 | 8.2405 - 008 . 008 . 017 . O17 347 304 111 205 7.7748 | 7.6181 | 6.8959 7. 8514 . 006 - 004 . O01 . 007 239 226 183 277 6. 2382 | 7.3397 | 9.5900 7. 8955 . 000 - 002 - 389 . 008 359 346 123 218 bas 7.9011 ! 7.9950 | 9.2133 bs chee . 008 - 010 - 163 me 347 124 39 8. 3852 . 024 52 8. 0954 . 012 275 HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLE 29.—Elimination factors—Continued. SERIES 250 DAYS. SEMIDIURNAL COMPONENTS. Component sought (A). 261 Disturbing components (B, C, ete.). Ke Le Me N2 2N Re Se Ts he 10) v2 25M s68ya. 8. 3544 |8. 1489 /8. 0302 |7.9350 |9.5900 |9.3286 |8. 4129 |8.4768 |7.7796 |7. 7661 | 8.3023 Se ete 8.2770 | 8.1622 | 7.9899 | 7.5338 | 8.3896 | 7.7726 | 7.1486 | 8.4204 | 7.5223 Paces - O19 - O15 - 010 . 003 - 025 - 006 . OOL - 026 - 003 220 260 266 173 287 306 346 253 206 Greg cis icls mersinie site eee Sa 2ON |e eeycier 8.2770 | 8.0269 | 8.0269 | 9.2565 | 7.9899 | 7.7726 | 9.3360 | 7.3907 OLORE ce meres - O19 - O11 . O11 - 181 - 010 - 006 217 - 002 140 x 220 227 133 247 266 306 214 346 Milenio seman ncetae 8.1622 | 8.2770 | -..-.. 7.5338 | 7.9899 | 8.2044 | 8.0269 | 7.9899 | 7.5392 | 8.1019 . 015 PLO eee sie - 003 . O10 - 016 - O11 - 010 - 003 - 013 100 140 é 187 94 27 227 266 174 306 CO} reese ceace B enero 7. 9899 | 8.0269 } 7.5338 | -_...- 7. 8638 | 7.7467 | 8.2770 | 8.1622 | 7.5336 | 8. 4724 - 010 - 011 SOOS MU | eee . 007 - 006 - 019 - O15 - 003 - 030 94 133 173 87 21 220 260 167 300 OX Oia Bee aay are 7 .5338 | 8.0269 | 7.9899 | 7.8638 | .....- 8. 0954 | 7.6520 | 6.7805 | 8.14383 | 7.5129 . 003 - O11 - O10 SOO Pete - O12 - 004 - O01 . O14 - 003 187 227 266 273 : 294 313 353 260 213 1 er iescepea M y Vee t oed 8.3896 | 9.2565 | 8.2044 | 7.7467 | 8.0954 | .....- 7. 5300 | 7.8163 | 9.3360 | 8.026 - 025 . 181 . 016 . 006 ONO | odes Sec, - 003 - 007 217 - O11 73 113 333 339 66 ae 199 239 146 279 CO mas epee sae 7. 7726 | 7.9899 | 8.0269 | 8.2770 | 7.6320 | 7.5300 | ....-- 8.2770 | 7.90381 | 9.3366 - 006 - 010 - O11 - 019 . 004 SO0R I es - O19 - 008 - 217 54 94 133 140 AT 161 ee 220 127 80 DAD et Seat al Sp peel ae Pere ee 7.1486 | 7.7726 | 7.9899 | 8.1622 | 6.7805 | 7.8163 | 8.2770 | ...... 7.8741 | 8.2220 - OO1 . 006 - 010 - 015 - O01 . 007 JOLGE Ia Sees - 007 017 14 54 94 100 7 121 140 a. 5 87 40 Siete t scene eseeeee &. 4204 | 9.3360 | 7.5392 | 7.5386 | 8.1433 | 9.3360 | 7.9031 | 7.8741 | .....-. 7. 8897 . 026 «217 - 003 - 003 . 014 ~217 - 008 OGM Feces - 008 107 146 186 193 100 214 233 273 soos 312 Pls e Jose ee REE 7. 5223 | 7.3907 | 8.1019 | 8.4724 | 7.5129 | 8.0286 | 9.3366 | 8.2220 | 7.8897 | ...... - 003 - 002 - 013 - 030 - 003 - O11 217 - 017 AIS |] | otoge 154 14 54 60 147 81 280 320 48 Sosic HARMONIC ANALYSIS AND PREDICTION OF TIDES. 265 TaBLE 29.—Elimination factors—Continued. SERIES 297 DAYS. SEMIDIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought (A). i Ko Le Me Ne 2N Re Se Tes de U2 v2 2SM TE UR 2 7.5338 |8.0269 |7.9899 |7.7726 |9.3359 |9.2565 |9.1076 |8.3896 |7.5223 |7.3907 | 8. 2357 sce - 003 - O11 -010 |.006 all - 181 . 128 - 025 - 003 . 002 - 017 eae 187 227 266 306 214 247 281 287 206 346 88 3 pL a Eat Ro aestee |naomae 8. 2770 |8. 1622 |7.9326 |8.1514 |8. 4724 |8. 5711 |9.3366 |7.5300 |7. 7467 | 8.0286 OOS esos ee 019 | .015 | .009 | .014 | .030 | .037 | .217 | .003 | .006 - O11 7 a 220 260 300 27 60 94 280 199 339 81 1 Da eee ee 8. 0269 |8.2770 |...--- 8. 2770 |8. 1622 |7. 53389 |7. 7467 |8.1268 |8.4724 |7.7467 |8.4724 | 7.7179 AOI | ROL ee eee OLO LF Olan 003ics | COG) 5 | OL3. 030 | .006 030 - 005 133 P40) |) 28 220 260 167 21 54 60 339 300 41 Nene 2 VES >} 7.9899 |8. 1622 |8.2770 |....-- 8.2770 |7.9031 |7.5300 |7. 4214 |7. 7467 |8.4724 |9.3366 | 6.2014 SOLON (P0154 |) OLObM ieee .019 | .008 | .003 | .003 | .006 | .030 | .217 - 000 94 100 1400 te24 220 127 161 14 21 300 80 1 DINER G2 OEY - 4 7.7726 |7.9326 |8. 1622 |8.2770 |...-.- 7. 8741 |7. 8163 |7. 5240 |7.5300 |9.3366 |8.2220 | 7. 5050 O06) *|) \O09%|" 0152 2}: 019) -} 22. 2007 | .007 | .003' | .003 | .217 | .017 - 003 54 60 100 TAO is EES 87 121 155 161 80 40 142 Toes. 4. ees 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 G04! pssese . 217 181 026 001 | .008 013 146 333 193 233 DAR A ne 214 247 253 352 312 54 SDs Hee Meese 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 OOF 217) | eee 217 019 005 011 006 113 300 339 199 239 146 3 214 220 319 279 21 IPO ee 2 et 2 9.1076 |8.5711 |8. 1268 |7.4214 |7.5240 |9. 2566 |9.3362 |...... 7.5385 |7. 8920 |8.0039 | 7.53839 128 037 013 003 003 =| .181 le 4|t sie 003 008 010 003 79 266 306 346 205 113 TAG Ee oe 186 285 245 167 NeeAh we ccceshes: 8.3896 |9.3366 [8.4724 |7. 7467 |7.5300 |8. 4204 |8. 2770 |7.5385 |...-.. 8.0286 |8. 1647 | 7.5300 OZ 2a O300 7 006003) |) 0268 I O19) +003) eae AOL OLS . 003 73 80 300 339 199 107 140 174) | ee 279 239 161 Tape 7. 5223 17.5300 |7. 7467 |8.4724 |9.3366 |7.0150 |7.7179 |7. 8920 |8.0286 |...... 8. 2770 | 7.6680 BOOS (1003! 4) s006n" | -0s0)e|Pe2kd, . | 001) 0053 008t i Oly ee 22 - 019 - 005 154 161 21 60 280 8 41 75 Sib ese ae 140 62 CSU Se a 7.3907 |7. 7467 |8. 4724 |9.3366 |8. 2220 -002 | .006 | .030 | .217 | .017 14 21 60 280 320 PS LAS ee ne 8. 2357 |8.0286 |7. 7179 |6. 2014 |7. 5050 017. | .011 | .005 | .000 | .003 272 279 319 359 218 266 U. S. COAST AND GEODETIC SURVEY. TaBLE 29.—Elimination factors—Continued. SERIES 326 DAYS. DIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought i yy Ki Mi O1 00 Pi Q 2Q Si pi OA oS eee Opec Ieee 8.1340 | 8.0698 | 7.8631 | 7.4852 | 8.3392 | 7. 8244 | 7.6841 | 8. 2809 7.0934 ee A - 014 012 . 007 . 003 . 022 . 007 - 005 019 . OL me 210 241 235 6 249 265 296 230 190 april si Ls rs fe 81840 Hh epee 2 8.1340 | 7.7427 | 7.7427 | 9.0470 | 7. 8631 | 7.8244 | 9.0723 7. 5061 SRS AY ncaa 014 . 006 . 006 Blah - 007 007 118 - 003 150 210 204 156 219 235 265 199 340 8. 0423 O11 310 8. 3386 . 022 315 6. 6245 - 000 184 7.9254 - 008 301 Que see 8aeRee 7. 8244 | 7.8631 | 7.7427 | 8.1340 | 7.7204 | 7.4186 | ...-.. 8.1340 | 7.7258 9. 2882 - 007 - 007 - 006 .014 - 005 S003 Ahh ee sae 014 005 . 194 95 125 156 150 101 164 ae 210 144 105 DO ine Foe eet ase 7.6841 | 7.8244 | 7.8631 | 8.0698 | 7.6227 | 7.7040 | 8.1340 | ...... 7. 7948 8.3594 . 005 - 007 . 007 .012 . 004 . 005 (01 & Pie ie ass - 006 - 023 64 95 125 119 70 133 150 Sess 114 75 Speck tits sessed ee 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 6 OOGiy | ese . 006 130 161 191 185 136 199 216 246 Base 321 Places Bet tose sees 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 4006! 4) owas eee 170 20 50 45 176 59 255 285 39 abn | HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLeE 29.—EHlimination factors—Continued. SERIES 326 DAYS. SEMIDIURNAL COMPONENTS. 267 Disturbing components (B, C, etc.). Component sought (A). Ky Le Me Ne 2N R2 Se Te de 12 y2 28M. TE aye 5 pa ea 7.4352 |7. 7427 |7.8631 |7.8244 19.0720 |9.0470 |9.0037 |8. 3392 |7.0934 |7.5061 | 8.0971 LE A . 003 . 006 . 007 . 007 -118 111 -101 . 022 - 001 . 003 - 013 dae 354 204 235 265 199 219 238 249 190 340 53 BigP of AN Ca PAASO2 Weta 8. 1340 |8.0698 |7. 9526 |8.0858 |8.3386 |8.4852 |9. 2882 |7. 4186 {7.5483 | 7.9254 S{O1OB3 | eens . 014 .012 009 - 012 022 - 031 . 194 - 003 . 004 - 008 6 te 210 241 271 25 45 64 255 196 346 59 IMR oF oy ee 6 1.1427 18. 1340 |... 24. 8.1340 |8.0698 |7.0956 |7. 5483 |7.9190 |8. 3386 |7. 5483 |8.3386 | 7.5347 . 006 L OMAN We oa . 014 . 012 - 001 . 004 - 008 . 022 . 004 . 022 . 003 156 150 Me 210 241 175 | 14 34 45 346 315 28 ING a 7. 8631 |8. 0698 |8. 1340 |...... 8.1340 |7. 7259 |7.4186 |6. 7213 |7. 5483 |8. 3386 |9. 2882 | 6.3062 . 007 - 012 014 014 . 005 .003 | .001 - 004 . 022 - 194 . 000 125 119 150 eeigs 210 144 164 14 315 105 178 ONS. SR 7. 8244 |7.9526 |8.0698 |8.1340 }.._._- 7.7948 |7.7040 |7.5123 |7.4186 |9. 2882 18.3594 | 7.4010 - 007 . 009 . 012 OLN ese Ae . 006 - 005 . 003 . 003 - 194 . 023 . 003 95 89 119 | 150 3 114 133 153 | .164 105 75 148 1 eee ae ae 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 0 if ease 118 nehD) - 019 - 001 . 006 . 008 161 335 185 216 246 3 199 219 230 351 321 34 Bombo Mp 9.0470 |8. 3386 |7. 5483 |7. 4186 |7.7040 |9.0725 |...... 9.0725 8.1340 |7. 5347 17.9254 | 7.5483 hdd! . 022 . 004 . 003 . 005 7 Wal Its} aes 42 2 -118 . 014 - 003 . 008 . 004 141 315 346 196 227 161 2 199 210 332 301 14 PRS is | par 9. 0037 |8.4852 |7.9190 |6. 7213 |7.5123 |9.0472 |9.0725 |.....- 7. 7468 |7.7359 |7. 9960 | 7.0956 - 101 - 031 - 008 . 001 - 008 5 ati SOUS eee . 006 - 005 . 010 - 001 122 296 326 357 207 141 161 E 191 312 282 175 DAC Cs eee ea 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 006K S2s2e- . 008 . 016 - 003 lil 105 315 346 196 130 150 169} . 301 271 164 TCBUAR see peel tad 7. 0934 |7. 4186 |7. 5483 |8.3386 19.2882 |7.0444 |7. 4347 |7. 7359 |7.9254 |_..... 8.1340 | 7.5118 - 001 - 003 . 004 . 022 . 194 - 001 . 003 . 005 OOSiM ees . 014 - 003 170 164 14 45 255 28 48 59 SSae 150 43 iB cian SSeS 7.5061 |7.5483 |8.3386 |9. 2882 |8.3594 |7. 7843 |7.9254 |7.9960 |8. 1912 |8.1340 |...._. 7. TAIT - 003 - 004 - 022 - 194 . 023 - 006 . 008 . 010 . 016 OLA ea eee . 006 20 14 45 255 285 39 59 78 89 2108 eee 73 PASTY Cy LN 8.0971 |7.9254 |7. 5347 16.3062 |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 | <.-.: 307 301 332 182 212 326 346 185 196 317 287 Bear 72934—24}——18 268 U. S. COAST AND GEODETIC SURVEY. TaBLE 29.—Elimination factors—Continued. SERIES 355 DAYS. DIURNAL COMPONENTS. Component sought (A). Disturbing components (B, C, etc.). Mi O71 00 Pi Qi 2Q Si pl 7.9167 | 7.5111 | 7.8888 | 8. 0444 | 7.6331 | 7.6506 | 8.0032 6. 7724 - 008 . 003 . 008 . 011 - 004 . 004 . 010 - 001 222 203 19 211 224 245 206 30D 7.9464 | 6.7064 | 6.7064 | 8.4581 | 7.5111 | 7.6331 | 8.4598 | 7.5794 . 009 - 001 . O01 - 029 . 003 - 004 . 029 - 004 201 182 178 190 203 224 185 334 Teak. 7.8888 | 7.5111 | 7.7393 | 6.7064 | 7.5111 | 7. 8651 7. 9839 Seay . 008 . 003 . 005 . OO1 . 003 . 007 010 Z 341 157 169 182 203 164 313 7. 8888 | ..:.-- 6.7060 | 7.2500 | 7.9464 | 7.9167 | 6.7729 8. 1364 MOOSE Hi ase. . 001 . 002 - 009 . 008 . 001 . 014 19 ei 175 8 201 222 3) 331 7.4740 | 7.1838 7.3105 . 003 - 002 - 002 Pf 187 336 7.5554 | 8. 4598 7. 7296 . 004 . 029 . 005 214 175 324 7. 9464 | 7.4212 9. 1482 . 009 . 003 - 141 201 162 130 u eeee 7. 5984 8.3129 eS . 004 - 021 LS 141 109 7. S984 a caiccee 7. 6607 SODA a urs 005 219 Aiea 329 7.9839 | 8.1364 | 7.3105 | 7.7296 | 9.1482 | 8.3129 | 7.6607 | ...... . 010 . 014 . 002 . 005 .141 . O21 OOD RL a eee 47 29 24 36 230 251 31 eats Sop cl HARMONIC ANALYSIS AND PREDICTION OF 'TIDES. TaBLe 29.—E#limination factors—Continued. SERIES 355 DAYS. SEMIDIURNAL COMPONENTS. 269 Disturbing components (B, C, etc.). Component sought (A). | Ko Le Me Ne 2N Re So To ARs hy 2 or alee gi ps Sit Be 7. 8888 \6. 7064 |7. 5111 |7.6331 |8. 4589 |&. 4581 |8. 4553 see .008 | .001 - 003 . 004 .029 | .029 . 029 341 182 203 224 185 190 195 Jp aoe oy ed Bean 7. 8888 |....-- 7. 9464 |7.9167 |7. 8652 |8. 0217 |8. 1364 |8. 2393 SOOSRai| cee . 009 . 008 - 007 - O11 014 | .017 19 201 222 243 24 29 34 Noe se the 6. 7064 |7.9464 |... - 7. 9464 |7.9167 |6.7712 |7. 2500 |7. 4842 - O01 e OOO RF | eas -009 | .008 | .001 .002 | .003 178 159 201 222 3 8 13 IN pie oe 2) EE ae 7. 5111 17. 9167 |7.9464 |.....- 7.9464 |7.4212 |7.2957 |7. 1008 - 003 Ci OOS) OOGMa ie se . 009 - 003 .002 | .001 157 138 159 201 162 167 172 DINGS 2 oh RE tt 7.6331 |7. 8652 |7.9167 |7.9464 |...._. 7.5985 |7.5554 |7. 5017 OOF COR | OOS. O09 Wh =: 22 = -004 | .004 | .003 136 117 138 G1 se 141 146 151 Roe a < Guha eh) We 8. 4589 |8. 0217 |6.7712 |7. 4212 |7.5985 |..._.. 8. 4598 |8. 4586 . 029 . O11 - 001 - 003 O04 ees: - 029 . 029 175 336 357 198 219 185 190 SBA LR SeaHelees aie 8.4581 |8. 1364 |7.2500 |7. 2957 |7. 5554 |8. 4598 |... . 8. 4598 -029 | .014 OO 2H |e O02 en | OOF See O20) Ns ies 2 - 029 170 331 352 193 214 175 185 Ue Se ee 8. 4553 |8. 2393 |7.4842 17.1008 |7.5017 |8. 4586 18.4598 |...... . 029 . O17 003 - O01 - 003 O29 e029) | eae 165 326 347 188 209 170 175 ) ra coce Hep Eee ee &. 0444 |9. 1482 |8.1364 |7. 2500 |7. 2957 |8. 0034 17.9464 |7. 8648 . O11 . 141 014 | .002 | .002 | .010 | .009 | -007 149 130 331 352 193 154 159 164 Ge Soe ae oe 6.7724 |7. 2957 |7. 2500 |8. 13864 |9.1482 |7.0678 |7.2458 |7.3738 . OOL -002 | .002 | .014 | .141 . O01 .002 | .002 5 167 8 29 230 10 15 20 Po cersiesiosinecine se 7. 5794 |7. 2500 |8. 1364 19.1482 |8.3129 |7.6606 |7. 7296 |7. 7893 004 | .002 014 | .141 . 021 . 005 . 005 . 006 26 8 29 230 251 31 236 41 2SMEr oes skate 7.6440 |7.7296 |7.2458 |6. 7017 |7.2840 |7. 4842 |7.2500 |6. 7712 004 | .005 .002 | .001 . 002 . 003 -002 | .001 342 324 345 186 206 347 352 357 p2 v2 25M. 6. 7724 |7. 5794 | 7.6440 - OO1 . 004 . 004 355 334 18 7. 2957 7.2500 | 7.7296 . 002 . 002 . 005 193 352 36 7.2500 |8. 1364 | 7.2458 002 | .014 - 002 352 331 15 8. 1364 19.1482 | 6.7017 014 | .141 . 001 331 130 174 9.1482 |8. 3129 | 7.2840 141 | .021 . 002 130 109 154 7. 0678 |7.6606 | 7. 4842 - O01 . 005 . 003 350 329 13 7.2458 |7.7296 | 7.2500 -002 | .005 . 002 345 324 7.3738 |7. 7893 | 6.7712 -002 | .006 . OOL 340 319 3 7.7296 |8.0795 | 7.2957 005 | .012 . 002 324 303 167 Hi 7.9464 | 7.2393 eae . 009 . 002 ape 159 23 7.9464 |_..2.. 7. 5128 O09 ee rete . 004 201 Ae 44 oeBts) Webets |) S5esen 5 OPs Aas I Geade 337 316 pean 270 U. S. COAST AND GEODETIC SURVEY. TABLE 29.—Elimination factors—Continued. SERIES 369 DAYS. DIURNAL COMPONENTS. Component sought (A). a 8.3503 | 022 6.6332 | 7 000 | 358 | 8.0072. 010 | 4 1 7. 876) - 008 Disturbing components (B, C,etc.). Mi O; 00 Py Q 2Q Si 7.8740 | 7.8760 | 8.3371 | 8.2982 | 7.5509 | 7.4182 | 8.3235 007 | .008 | .022 | .020 | .004 | .003 | .021 219 | 287 1124) | (286) |) BIT, | 1 36 Rone 8.3503 | 6.6332 | 6.6332 | 8.0072 | 7.8760 | 7.5509 | 8.0074 022 | .000 | .000 | .010 | .008 | .004 | .010 290 |. 358 26) | 1356)) 287 || Weta eas Hee 20) 8.3371 | 7.8760 | 8.4104 | 6.6332 | 7.8760 | 8.3792 mf .022 | .008 | .026 | .000 | .008 | .024 g 248 73N| | 671 0 BAS) Res 69 Buasvaiee Ae 6.6329 | 6.5537 | 8.3503 | 7.8740 | 5.7100 O22 Shwe 000 | .000 | .022 | .007 | .000 112 Ge oe 4 | {178 | 290 | | 3219 0 | 7.8760 | 6.6329 | _..... 7.0438 | 7.6584 | 7.3523 | 6.8924 BOOS): ||P 000). i. .-E 4 .001 | .005 | .002 | .001 287} 356 E 354 | | 285 |) s@I5 ie Bae 8.4104 | 6.5537 | 7.0438 |... 7. 8885 | 7.6024 | 8.0074 026%!) LOODia)4e001 10 1.62) 008 | .004 | .010 293 | 182 ! 291} 221 2 6.6332 | 8.3503 | 7.6584 | 7.8885] ._.._. 8.3503 | 7. 8824 - 000) i022 ix. 0055)):. DOSi1 Lo .022 | .008 Dang 70 75 690| fsa 290 71 7. 8760 | 7.8740 | 7.3523 | 7.6024 | 8.3503 | ...... 7.5772 8008) (007 «Ii.002,° |). 004i .022 she _ Sea . 004 73| 141 145] 139 70) | aan 141 8.3792 | 5.7100 | 6.8924 | 8.0074 | 7.8824 | 7.5772 | _..... .024 | .000 | .001 | .010 | .008 | .004 | ..... 291 | 0 £))\ (358) -| (280)! vse on heme 7.9045 | 8.4169 | 7.6527 | 7.9217 | 9.0329 | 8.0586 | 7.9055 .008 | .026 | .004 | .008 | .108 | .o11 | .008 3040-) LOSHG). 112) ) ¢ 106 | «| it)! lesan ieee HARMONIC ANALYSIS AND PREDICTION OF TIDES. Ort TABLE 29.—Elimination factors—Continued. SERIES 369 DAYS. SEMIDIURNAL COMPONENTS. Disturbing components (B, C, etc.). Component sought (A). | Ke Ly M2 No 2N Re Se Ts 0) un v2 | 28M Gg ste WE eh 8.3371 |6. 6332 |7.8760 |7. 5509 |8. 0074 |8.0072 |8. 0076 |S. 2982 |5. 7099 |7. 8892 | 7.1088 Dees 022 000 008 004 010 | .010 010 020 000 008 001 248 358 287 217 358 356 354 286 0 250 175 7g SO GoSsy alae El ee 8.3503 |7. 8740 |7. 6166 |8.3758 |8. 4169 |8.4607 |9.0329 |7.8885 |6. 5537 | 7.9217 O22) | S28 2 022 007 004 024 | .026 029 108 008 000 008 112 290 219 329 110 108 106 217 291 182 106 Mocs a 6. 6332 |8. 3503 |...... 8.3503 |7. 8740 |5. 7100 |6. 5537 |6. 9049 |8. 4169 |6. 5537 |8.4169 | 6.5549 000 O22)5 fee 022 007 000 | .000 001 026 000 026 000 2 70 290 219 0 178 177 108 182 252 177 ING Se eee oe 3: 7. 8760 |7.8740 |8. 3503 |-...-- 8.3503 |7.8824 |7. 8885 |7. 8944 |6. 5537 |8. 4169 |9.0329 | 7. 6658 008 007 O220 eee 022 008 | .008 008 000 026 108 005 73 141 70 290 71 69 67 178 252 143 67 CAS (I ae aI Bea 7.5509 |7. 6166 |7. 8740 |8.3503 |...... 7.5772 \7. 6024 |7. 6268 |7.8885 |9.0329 |8.0586 | 7. 4452 004 004 007 022) P \Wasie=; 004 | .004 004 008 108 011 003 143 31 141 70 141 139 138 | 69 143 33 138 a ae SRR 8.0074 |8.3758 |5. 7100 |7.8824 |7.5772 |....-.- 8.0074 |8. 0072 |8.3235 |6.1771 |7.9055 | 6. 9049 -010 | .024 | .000 | .008 | .004 | ...-. -010 |-.010 | .021 -000 | -008 - 001 74 250 0 289 219 ate 358 356 288 181 252 177 Sere meee eh, j8-0072 |8. 4169 |6. 5537 |7. 8885 |7. 6024 |8.0074 |...... 8. 0074 (8.3503 |6.5549 |7.9217 | 6.5537 -010 | .026 | .000 |-.008 | .004 | .010 | .--.. -010 | .022 -000 | .008 - 000 = 252 182 291 221 2 eet 358 290 183 254 178 Me Se 8. 0076 |8. 4607 |6. 9049 |7. 8944 |7. 6268 |8. 0072 |8. 0074 |.....- '8. 3792 |6. 7601 |7. 9377 | 5.7100 -010 | .029 | .001 -008 | .004 |.010 | -010 | ....2 | .024 | .001 - 009 - 000 6 254 183 293 222 4 | Dili ees 291 185 256 0 ADs S25 Mei 2. 2eck 8. 2982 |9. 0329 |8. 4164 |6. 5537 |7. 8885 |8.3235 |8.3503 |8.3792 |.....- 7.9217 |7. 9044 | 7.8885 -020 | .108 | .026 | .000 | .008 | .021 O22) 4) SO2ZE GA aoe -008 | .008 - 008 74 143 252 182 291 72 | 70 69 sas 254 324 69 fabs .2 SLE sos. 5.7099 |7.8885 |6. 5537 |8. 4169 |9. 0329 |6.1771 \6. 5549 |6. 7601 |7.9217 |...... 8.3503 | 6.5542 -000 | .008 | .000 | .026 | .108 | .000 | .000 | -001 | .008 | ..... - 022 - 000 0 69 178 108 217 7S Peri; 175 LOGAN a5. 70 175 Pope i LOR i at 7.8892 |6.5537 |8. 4169 |9. 0329 18.0586 |7.9055 |7.9217 |7.9377 |7. 9044 18.3503 |....-- 7.6990 008 000 026 108 011 - 008 008 | .009 OOS 71022) 7) sees 2 005 110 178 108 217 327 108 106} 104 36 290 a 105 QS Mice Be oe ie 7.1088 |7.9217 |6. 5549 |7. 6658 |7. 4452 |6.9049 6.5537 |5.7100 |7.8885 |6. 5542 |7.6990 | .....- 001 008 000 005 003 | .001 000 000 008 000 Q05. 4) .2e2.5 185 254 183 293 222 183 182 0 291 185 255 Ses = 272 U. S. COAST AND GEODETIC SURVEY. TaBLE 30.—Products of amplitudes and angular functions for Form 245. 1 2 3 + 5 ° ° Sin Cos Sin Cos Sin Cos Sin Cos Sin. Cos 0 0. 000 1. 000 0. 000 2. 000 0. 000 3. 000 0. 000 4. 000 0. 000 5. 000 90 1 017 1. 000 035 2.000 052 3. 000 070 3.999 087 4.999 89 2 035 0. 999 070 1.999 105 2.998 140 3.998 174 4.997 88 3 052 999 105 1.997 157 2.996 209 3.995 262 4, 993 87 4 070 - 998 140 1.995 209 2.993 279 3.990 349 4. 988 86 5 087 - 996 174 1.992 261 2.989 349 3.985 436 4.981 85 6 105 995 209 1.989 314 2. 984 418 3.978 523 4.973 84 7 122 . 993 244 1.985 366 2.978 487 3.970 609 4. 963 83 8 139 -99) 278 1.981 418 2.971 557 3.961 696 4.951 82 9 156 - 988 313 1.975 469 2.963 626 3.951 782 4. 938 81 10 174 - 985. 347 1.970 521 2.954 695 3.9389 868 4, 924 80 11 191 . 982 382 1. 963 572 2. 945 763 3.927 954 4. 908 79 12 208 . 978 416 1.956 624 2.934 832 3.913 1. 040 4, 891 78 13 225 - 974 450 1.949 675 2.923 900 3.897 1125 4. 872 77 14 242 - 970 484 1.941 726 2.911 968 3. 881 1.210 4. 852 76 15 259 - 966 518 1.932 776 2. 898 1.035 3. 864 1. 294 4. 830 75 16 276 . 961 551 1.923 - 827 2. 884 1.103 3. 845 1.378 4. 806 74 17 292 - 956 585 1.913 877 2. 869 1.169 3.825 1. 462 4,782 73 18 309 - 951 618 1.902 . 927 2. 853 1. 236 3. 804 1. 545 4.755 72 19 326 - 946 651 1. 891 .977 2. 837 1.302 3. 782 1. 628 4.728 71 20 342 - 940 684 1.879 1. 026 2. 819 1.368 3. 759 1.710 4.698 70 21 358 - 934 717 1. 867 1.075 2. 801 1. 433 3. 734 1.792 4. 668 69 22 375 - 927 749 1. 854 1.124 2. 782. 1.498 3. 709 1. 873 4. 636 68 23 391 - 920 781 1. 841 Ley 2 2. 762 1.563 3.682 1.954 4. 602 67 24 407 -914 813 1.827 1. 220 2. 741 1. 627 3. 654 2. 034 4. 568 66 25 423 - 906 845 1.813 1. 268 2.719 1.690 3.625 21s 4. 532 65 26 438 . 899 877 1.798 1.315 2.696 1. 753 3.595 2.192 4. 494 64 27 454 891 908 1.782 1. 362 2.673 1.816 3. 564 2. 270 4. 455 63 28 469 - 883 - 939 1. 766 1. 408 2. 649 1. 878 3. 532 2. 347 4.415 62 29 485 . 875 . 970 1.749 1. 454 2. 624 1.939 3. 498 2. 424 4. 373 61 30 500 866 1. 000 1.732 1.500 2.598 2. 000 3.464 2. 500 4. 330 60 31 515 . 857 1. 030 1.714 1.545 2.572 2.060 3.429 2.575 4. 286 59 32 530 . 848 1. 060 1.696 1.590 2.944 2.120 3.392 2. 650 4. 240 58 33 545 . 839 1. 089 1.677 1. 634 2.516 2.179 3. 355 2. 723 4.193 57 34 559 . 829 1.118 1.658 1. 678 2. 487 2. 237 3.316 2. 796 4.145 56 35 574 . 819 1.147 1.638 1.721 2.457 2.294 3.277 2. 868 4. 096 55 36 588 . 809 1.176 1.618 1.763 2. 427 200L 3. 236 2. 939 4. 045 54 37 602 . 799 1. 204 1.597 1. 805 2.396 2.407 3.195 3. 009 3. 993 53 38 616 . 788 1. 231 1.576 1. 847 2.364 2. 463 3.152 3.078 3. 940 52 39 §29 Hel 1. 259 1.554 1. 888 Paco PA aIIZ( 3.109 3. 147 3. 886 51 AN 643 . 766 1. 286 1. 532 1. 928 2.298 PANINI 3. 064 3.214 3. 830 50 41 656 . 755 1.312 1.509 1. 968 2. 264 2.624 3.019 3.280 3.774 49 42 669 - 743 1.338 | 1. 486 2. 007 2.229 2.677 2.973 3.346 3.716 48 43 682 . 731 1.364 | 1. 463 2.046 2.194 2.728 2.925 3.410 3. 657 47 44 . 695 - 719 1.389 1. 439 2. 084 2.158 2.779 2.877 3. 473 3. 597 46 45 0. 707 0. 707 1.414 1.414 2.121 22 2. 828 2. 828 3.536 3. 536 45 Cos Sin Cos Sin Cos. Sin Cos Sin Cos Sin 1 2 3 4 5 a HARMONIC ANALYSIS AND PREDICTION OF TIDES. — 273 Tasie 30.—Products of amplitudes and angular functions for Form 245—Con. | 6 7 8 9 ° ° Sin Cos Sin Cos. Sin Cos 0 0. 000 7.000 0. 000 8.000 0. 000 9. 000 90 1 2122 6. 999 140 7. 999 157 8. 999 89 oy 244 6. 996 279 7. 995 314 8. 995 88 3 366 6. 990 419 7. 989 471 8. 988 87 4 . 488 6. 983 558 7. 980 628 8. 978 86 5 610 6. 973 697 7. 970 784 8. 966 85 6 . 732 6. 962 836 7. 956 941 8. 951 84 it 853 6. 948 | 975 7. 940 1. 097 8. 933 83 8 - 974 6. 932 1.113 7. 922 1. 253 8.912 82 9 1.095 6.914 1.251 7. 902 1. 408 8. 889 81 10 1. 216 6. 894 1.389 7. 878 1.563 8. 863 80 11 1.336 6. 871 1. 526 7. 853 erate 8. 835 79 12 1.455 6. 847 1. 663 7. 825 1. 871 8. 803 73 13 1.575 6. 821 1. 800 7. 795 2. 025 8. 769 77 14 1. 693 6. 792 1.935 7. 762 2.177 | 8. 733 76 15 1. 812 6. 762 2.071 7. 727 2. 329 8. 693 75 16 1. 929 6. 729 2. 205 7. 690 2. 481 8. 651 74 17 2. 047 6. 694 2.339 7.650 2.631 8. 607 73 18 2. 163 6. 657 2. 472 7. 608 2.781 8. 560 72 19 2. 279 6.619 2. 605 7. 564 2.930 | 8. 510 71 20 2.394 6. 578 2.736 7. 518 3.078 8. 457 70 21 2. 509 6. 535 2. 867 7. 469 3. 225 8. 402 69 22 2. 622 6. 490 2. 997 7.417 3.371 8.345 68 23 2.735 6. 444 3. 126 7. 364 3.517 | 8. 284 67 24 2. 847 6.395 3. 254 7. 308 3. 6E1 8. 222 66 25 2. 958 6.344 3.381 7. 250 3. 804 8. 157 65 26 3. 069 6. 292 3. 507 7.190 3. 945 8. 089 64 27 3.178 6. 237 3. 632 7.128 4. 086 8. 019 63 28 3. 286 6. 181 3. 756 7. 064 4, 225 7. 947 62 29 3.394 6. 122 3. 878 6. 997 4. 363 7. 872 61 30 3. 500 6. 062 4. 000 6. 928 4. 500 7. 794 60 31 3. 605 6. 000 4.120 6. 857 | 4. 635 7. 715 59 32 3. 709 5. 936 4, 239 6. 784 4, 769 7. 632 58 33 3. 812 5. 871 4.357 6. 709 4.902 7. 548 57 34 3.914 5. 803 4. 474 6. 632 5.033 | 7. 461 56 35 4.015 5. 734 4.589 6. 553 5. 162 LP 55 36 4.115 5. 663 4.702 6. 472 5. 290 7. 281 54 37 4, 213 5. 590 4.815 6.389 | 5. 416 7.188 53 38 4.310 5. 516 4.925 6. 304 5. 541 7. 092 52 39 4. 405 5. 440 5.035 6. 217 5. 664 | 6. 994 51 , | 40 4. 500 5. 362 5. 142 6. 128 5.785 | 6. 894 50 41 4. 592 5. 283 5. 248 6. 038 5. 905 6. 792 49 42 4. 684 5. 202 5. 353 5. 945 6. 022 6. 688 48 43 4.774 5.119 5. 456 | 5. 851 6.138 6. 582 47 44 4. 863 5. 035 5. 557 | §. 755 6. 252 | 6. 474 46 45 4. 950 4. 950 5. 657 | 5. 657 | 6. 364 | 6. 364 45 | | Cos. Sin. Cos. Sin. Cos. Sin. | Cos. Sin. 6 7 8 9 274 U. S. COAST AND GEODETIC SURVEY. TaBLE 31.—For construction of primary stencils. Difference. Component 2Q. Hour. d. h. d. h. d. h. d. h. d. h. d. h. d. h. dah: d. h. d. h ) 1 0 of 2 14 21 21 20% 28 20*| 3520] 4220] 4919*/ 56 19%} 63 19 +23 | —1 4 8 4 15 4 APA. BFA 548) Bel uaa) «5 43 3 +) mea nam Y all“. ese «La 2299) |) 11 11 11 10* 10* 10 10 9 g* 9 =21 || =3 18 18 18 17* 17* 17 17 16* 16* 16 +20 | —4 200 Oma 161 23) OF] | 300%) 37) 0 44 0 23* 23* 23 +19 | —5 8 8 8 7 7* 7 7 ol 6*| . 58 6* 65 6 +18 | —6 15 15 15 14* 14* 14 14 13* 13* 13 +17 | —7 22 22 21* 21* 21* 21 21 20* 20% 20 +16 | —8 3.5 OARS 17 4%) 24 4% 31 4% 38 4 45 4 52 3* -°-59 38* 66 3 +15 | —9 12 12 11* 11* 11* 11 11 10* - 10* 10 +14 |—10 19 19 18* 18* 18* 18 18 17* 17* 17 +13 |—11 4 2 12 LSE Oe Wish Sanreah Sly af 46 1 53 OF} 60 OF 67 0 +12 |—12 9 9 8* &* 8 8 8 7* T* 7 +11 |—13 16 16 15* 15* 15 15 15 14* 14* 14 +10 |—14 23 23 22% 29% 22 22 22 21* 21* 21 +9 |—15 SiG 12 6 19 5*| 26 5*| 335) 40 5] 47 54). 54 (4e) - 6104" 9 68 94 +8 |—16 13 13 12* 12* 12 12 12 11* 11* 11 7 |—17 20 20 19* 19* 19 19 19 | 18* 18* 18 +6 |—18 6 3 IBS 20° 24) Qi Os 34092 | 41) Bul”) 48) Pee 55, SEE G2 eet-c Goal +5 |—19 10 10 g* ge 9 9 Sal Se Se 8 +4 |—20 17 17 16*| 16* 16 16 15*| 15* 15* 15 +3 |—21 7 0 14 0 23% 23% 23 23 22* 22% 22% 22 +2 |—22 7 7 21° 6*| | 28 6%) 35.06 42 6 49 5%) - 56 5*| 9-63, 5*) | 70) 5. 1 j= 23 14 14 13* 13* 13 13 12* 19% 12* 12 Difference Component 2Q. Hour d. h. d. h. d. h. Cee ilt: d. h. d. h. d. h. d. h. d.h. d. h 0 70 19 7719] 8418*| 91 18*} 9818] 10518] 112 17*| 119 17*| 12617] 133 17 +23 | —1 @ 2 78 2 Sos e792 se 9or 106 1 113 O*/ 120 OF (127-0) 184 0 +22 | —2 9 9 8* &* 8 8 7 7 7 ae 7 Sool || 2 16 16 15* 15* 15 15 14* 14* 14 14 +20 | —4 23 23 22* 22% 22 22 21* 21* 21 21 +19 | —5 72 6 79 5%] "86 5*] | 93 %5*| 1005 | 107! 5) | 194 14/12) 4x) P7284) Ssh) +18 | —6 13 12* 12* 12* 12 12 11* 11* 11 11 +17 | —7 20 19* 19* 19* 19 19 18* 18* 18 18 +16 | —8 73 3 80 24 87 2% 94 2+] 101 2') 108 2} 115 14) 122 1] T20%7 | 1386 1 +15 | —9 10 g* g* Q* 9 9 S* Se 8 8 +14 |—10 17 16* 16* 16* 16 16 15* 15* 15 15. +13 |-11 74 0 23* 23% 23 23 23 22* 22% 22 22: +12 |—12 ia Sl G5) (88, Gs} 95/6), 102) (6) 109) 67) 116) Se) 123 Sap SOmr ae alee +11 |—13 14 13* 13* 13 13 13 12* 12* 12 12 +10 |—14 21 20% 20* 20 20 20 19* 19* 19 19 +9 |—15 75 4 82) 3%) > 89) 3*!) 96) 3 |) 103773) | 110) Si 7 a Paso ee tom +8 |—16 11 10* 10* 10 10 10 ge g* 9 a9 +7 |+17 18 17* 17* 17 17 16* 16* 16* 16 16 +6 |—18 76 1 83 OF}; 90 OF 97 0] 104 0 23* 23* 23* 23 +5 |—19 8 T* igs 7 7| 111 6* 118 G6 125 6* 132 6] 139 6 +4 |—20 15 14% 14* 14 14 13* 13* 13* 13 13 +3 |-21 22 21* 21* 21 21 20* 20* 20* 20 20 +2 |—22 HOES) 84 44) Ol Ae) 98°84) | 1057 747)! 1021 S*) 19) St] eo) Sa eiSo ese ealaGe a +1 |—23 12 11* 11* 11 11 10* 10* 10 10 10 HARMONIC ANALYSIS AND PREDICTION OF TIDES. TABLE 31.—For construction of primary stencils—Continued. 275 Difference. Component 2Q. Hour. d. h d. h d. h d. h GE 1D d. h. d.h d.h d. h dh 0 140 17 |. 147 16*| 154 16*) 161 16 | 16816 | 175 15*) 182.15*| 18915 | 19615 | 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 51204 4x a9 || 8 14 13* 13* 13 13 12% 12* 12 12 11* +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) 905) 1% +18 | —6 10* 10* 10* 10 10 g* 9* 9 9 ge +17 | —7 17* 17* 17* 17 17 16* 16* 16 16 15* +16 | —8} 148 O*| 150 O* 157 O* 164 0) 171 0 23* 23* 23 23 99% 415 | —9 7* 7* 7 7 7| 178 6*| 185 6* 192 6| 199 6| 206 5* +14 |—10 14* 14* 14 14 14 13* 13* 13 13 12% . +13 |—11 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 31} 200 31] 207° 2* +11 |—13 11* 11* 11 11 11 10* 10* 10 10 Q* +10 |—14 18* 18* 18 18 18 17* 17* 17 17 16* +9 |—15 | 145 1* 152 1* 159 1] 166 1] 173 O* 180 O* 187 O*| 194 0} 201 0 23* +8 |—16 8* 8* 8 8 T* UF (si u 7 | 208 6* +7 |—17 15* 15* 15 15 14* 14* 14* 14 14 13% +6 |—18 22* 22* 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 4) 209 3* +4 |—20 12* 12* 12 12 11* 11* 11* 11 M1 10* +3 |—21 19* - 19* 19 19 18* 18* 18 18 18 17* +2 |—22 147 2* 154 2* 161 2; 168 2] 175 1% 182 1* 189 1 196 1 203 1) 210 o* +1 |—23 g* 9 8* 8* 8 8 8 7* Difference. Component 2Q. Hour. d. h d. h. d. h d. h. d. h ihe Wl Gh Ip d. h. d. h d. h 0 210 14*| 217 14%; 224 14 231 14 | 238 13*| 245 13* 252 13] 259 13 | 266 13 | 273 19* eeooy eee 21* 21% 21 21 | 20* 20%) 20 20 19* 19* +22 | —2| 211 4*| 218 4* 225 4)| 232 4.) 289 3*| 246 3* 253 3] 260 3] 267 2%] 274 9% +21 | —3 11* 1 11 il 10* 10*| 10 10 g* Qx +20) —4 18* 18* 18 18 | 17* 17*) 17 17 16* 16* +19} —5 | 212 1* 219 1*| 226 1) 233 1! 240 .O*| 247 O* 254 0] 261 0 23* 23* +18 | —6 8* 8* 8 8 | 7* 7*| i 7 | 268 6*| 275 6* +17 | —7 15* 15* 15 15.| 14* 14* 14 14 13* 13* +16 | —8 22* 22 22 22 21* 21* 21 21 20* 20* +15 | —9] 213 5*) 220 5) 227 5| 234 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 11] 270 O*| 277 O* +11 |—13 g* 9 9 9 8* 8 8 8 7* 7* +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) 236 5*| 243 S*| 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) 965 2) 272 1%| 279 1* +4 |—20 10* 10 10 g* g* 9 9 9 gt 8* +3 |—21 17* 17 17 16*' 16* 16 16 16 15* 15* +2 |—22°'| 217 O0* 224 0} 231 O 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* 276 U. S. COAST AND GEODETIC SURVEY. TaBLEe 31.—For construction of primary stencils—Continued. Difference. Component 2Q. Hour. d. h. d. h. d. h. d. h. ad. Whe d. h. d. h. d. h. d. h. 0 | 280 12*| 28712] 29412] 301 11%) 308 11*}| 315 11] 322 11 | 329 10* 336 10* +23; —1 19* 19 19 18* 18* 18 18 17* +292) —2] 281 2*| 288 2] 295 2] 302 1* 309 1* 316 1) 323 11) 330 OF 337 OF +21 | —3 9* 9 9 8* 8* 8 8 7* +20} —4 16* 16 16 15* 15* 15 15 14* +19 | —5 23* 23 23 22* 22* 22 22 21* +18 | —6| 282 6*| 289 6] 296 6] 303 5*| 310 5* 317 5| 324 5) 331 4* 338 4% +17 | —7 13 13 13 12*| 12* 12 12 11* +16 | —8 20 20 20 19* 19% 19 19 18* +15] —9| 283 3. 290 3] 297 3) 304 2* 311 2%) 318 2) 325 2) 332 1% 339 1 +14 |—10 10 10 10 g* g* 9 9 8* +13 |—11 17 17 ity 16* 16* 16 16 15* +12 |—12 ! 284 0.| 291 0 23* 23* 23*| 23 23 22* +11 |—13 7 7] 298 6*| 305 6*| 312 6*| 319 6] 326 6] 333 5* 340 5* +10 |—14 14 14 13* 13* 13* 13 13 12* 12* +9 |—15 21 21 20* 20* 20* 20 ,20 19* +8 /—16| 285 4] 292 4] 299 3* 306 3* 313 3% 320 3) 327 3] 334 2% 341 2* +7 |—17 11 il 10* 10* 10* 10 10 g* 9* +6 |—18 18 18 17* 17* 17 17 17 16* 16* 16 +5 |—19| 286 1] 293 1] 300 O* 307 O* 314 0} 321 0} 328 O 23% 23* 23 +4 |—20 8 8 7 T* 7 7 7 | 3385 6*| 342 6*| 349 6 +3 |—21 15 15 14* 14* 14 14 14 13* 13* 13 +2 |—22 22 22 21* 21* 21 21 21 20* 20* 20 +1 |—23 | 287 5| 294 5] 301 4*| 308 4% 315 4] 322 4/ 329 4) 336 3* 343 3*/ 350 3 Difference. Component 2Q. Component Q. Hour. d. h. d. h. d. h. d. h. aay dake ae d. h. Gd. he | VaR 0 350 10 | 35710] 364 9% fat) 10 5 19 13%) 28 22% 38 7* 4716) 57 1 +23 | —1 17 17 16%) 5* 14 23 29 8 16*, 48 1* 10* +22 | —2] 351 0] 358 0 23% 15 23*| 20 8* is 39 2 11 19* +21 | —3 7 7| 365 6* 2 0 11 9 18 30 2* 11* 20) 58 5 +20] —4 14 14 13*| g* 18% 21 3 12 21 49 5* 14% +19 | —5 21 21 20%) 19 12 3% 12* 21* 40 6 15} 59 0 418 | —6|° 352 4] 359 4] 366 3% 3 4* 13 22 31 6* 15#/ 50 OF +17 | —7 il 11 10* 13* DOe D2 ies 16 41 1 Q* 18* +16 | —8 13 18 17* 23 13 8 16%, 32 1% 10* 19| 60 4 415 | —9|'353 11-360 1) 367 O* 4 8 17 23 2 11 19% 51 4* 13 +14 ;—10 8 8 7*| 17* 14 2% 11* 20 42 5 14 22* +13 |—11 15 | 14* 14* ay 98) 12 20*| 33 5* 14* 23 61 8 +12 |—12 22 21* 204 12* 21* 24 6 15 23%) 528s 17* +11 |—13 354 5| 361 4% 268 4% 22 15 6* 15* 34 0% 43 9 18 | 62 2* +10 |—14 12 1l* 11) 6 7 16 25 1 9* 18* 53 3* 12 +9 |—15 19 18* 18* 16* 16 1* 10 19 44 4 12* 21* +8 |—16 | 355 2) 362 1* 369 1*| diane 11 19*| 35 4% 13 225). 16a) oe +7 |—17 9 8* | 11* 20 26 5 13* 22* 54 7* 16 +6 |—18 16 15* 15% 20*| 17 5* 14* 23 45 8 16*| 64 1* +5 |—19 23 22* 22* 8 6 15 23*| 36 8* 17%) 55) 2 11 +4 |—20 356 6 363 5*| 370 5* 15* 18 0 27.9 18 46 2* 11* 20* +3 |—21 13 12* 9 1 g* 18*| 37 3 12 | 21 | 65 5* 42 |—22 20 19% | 10 19} 28 4 13 21* 56 6 15 +1 |—23 357 3 364 2* | 19* 19 4% 13 22 47 7 15*| 66 0* HARMONIC ANALYSIS AND PREDICTION OF TIDES, 277 TaBLE 31.—For construction of primary stencils—Continued. Difference. Component Q. Hour. Ge Ue d. h. dhe d. h. ahs d. h. d. h. Gh ite a> he \yaieen 0 66 9} 75 18*] 85 3% 9412] 103 21| 113 6 | 122 14*| 131 23%} 141 8 | 150 17 4-98)i[ie =) 19| 76 4 12* 21*| 104 6* 15| 123 0| 132 9 17*) 151 2* +92 !—21 67 4% 13* 22110195 | 7 15*| 114 0* ge 18| 142 3 12 +21 | —3 14 22*| 86 7* 16*| 105 1 10 18%] 133 3* 12* 21 +90 | —4 OSE ec ARS 17| 96 1* 10* 19%} 124 4 13 21%| 152 6* +19} —5| 68 8 17*| 87 2 rl 20} 115 4* 13* 22*| 143 7 16 418 | =6 18H 782.3 11* 20%] 106 5 14 23 | 134 7* 16*| 153 1* 417 | =7) 69 3% 12 21| 97 6| 14* 23*| 125 8 17| 144 2 10* E15 | = 12* 21%} 88 6* 15| 107 0| 116 8* 17] 135 2* 11 20 +15 | —9 DOM TORT 15%] 98 O# ge 18) ZC 11* 20%] 154 5* +14 |-10} 70 7 1 a 10 18*| 117 3% 12% 21| 145 6 14* +13 |—11 17| 380 1* 10* 19| 108 4 | 13 21*| 136 6* 15*| 155 0 SP Rie |) 7 ul 20; 99 4* 13* PMs Oe aii 16} 146 0* g* +11 |—13 11* 20*| 90 5 14 23! 118 7* 16*| 137 1 10 19 +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 17*| 119 2* il 20| 147 5 13* +8 |—16 15*| 82 0* 9 18i| 110588 11s 20%] 138 5* 14 23 sey leet | 7B) Gil 10 18*} 101 3* 12 21| 129 6 14% 23*| 157 * +6 |—18 10* 19| 92 4 13 21%| 120 6* 15| 139 0| 148 9 17* 45 |—19 19% 83 4% 13% 22) 11 7 16 | 130 0* ge 18 | 158 3 14 |=20'| 74. 5 14 22| 102 7% 16%} 121 1 10 19] 149 3* 12* #5, | 14* 23%| 93 8 a7 | 112 1 10* 19*| 140 4 13 22 +2 |-22| 75 0| 84 8* 17*| 103° 2* i 20] 131 4* 13% 22*| 159 7 +1 |—23 9 18| 94 3 11 20) 122 5* 4 23} 150 7* 16* Difference. Component Q. Hour d. h dihs Gaus d. h d. h Chee lite Gh lee Gh ste ds hae. 0 160 2! 169 10% 178 19%| 188 4*| 19713] 206 22] 216 6*| 225 15*| 235 O*| 244 9 e234) ST 11 20| 179 5 13*! 22%] 207 7* 16 | 226 1 9* 18* 9p || 2 20%] 170 5% 14 23 | 198 8 16%] 217 1* 10* 19 | 245 4 Hei | = 3 | 1461 "6 15 23%| 189 8* 17] 208 2 11 19*| 236 4* 13* 420 | —4 15*| 171 0] 180 9 18 | 199 2* 11% 20 | 227 5 14 | 22% +19 | —5 | 162 0* g* 18%} 190 3 12 21] 21g 5* 14* 23. | 246 8 +18 | —6 10 19} 181 3* 12* 21*| 209 6 15 | 228 0| 237 8* 17* 7 | Say 19*| 172 4 13) e228) 20062 15%] 219 0* 9 18 | 247 2* +16 | —8| 163 5 13* 22*| 191 7 16] 210 1 oe 18*| 238 3* 12 +15 | —9 14 N2Bel 18258 16*| 201 1* 10 19] 229 4 12% 21* +14 |—10 23%| 173 8*| 17| 192 2 i 19] 220 4% 13 22| 248 7 413 |-11] 164 9 17*| 183 2* 11* 20| 211 5 14 22%} 239 7* 16 2D EW, 18%} 174 3 12 20*| 202 5* 14* 23 | 230 8 17 | 249 1* +11 |—13 | 165 3% 12* 21%) 193 6 15 93%| 291 8 17*#| 240 2 11 +10 |—14 13 22| 184 6% 15*| 203 0# 212 9 18] 231 2* 11% 20% +9 |—15 22*| 175 7 | 16) 194 1 ge 18*| 222 3 12 21 | 250 5* +8 |—16] 166 7* 16%, 185 1* 10 19| 213 4 12* 21+} 241 6 15 | 17| 176 2 10* 19%] 204 4% 13 Doh 232 er 15%| 251 0* | +6 |—18] 167 2* 11%, 20) 195 5 13% 22%] 293 7* 16 | 242 1 10 45 |—19 12 20%| 186 5* 14% 23,| 214 8 16*| 233 1* 10% 19 +4 |—20 21| 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 +42 |—22 16| 178 1 oe 1s*| 206 3 12 21] 234 5% 14% 23* +1 |—23} 169 1* 10 | 19} 197 3* 12* 21%| 225 6 15 | 244 0 | 253 8* 278 U. S. COAST AND GEODETIC SURVEY. TaBLE 31.—For construction of primary stencils—Continued. Difference. Component Q. Hour d. h d. h d. h d. h GTO d. h d. h Cpe daa (he daa 0 253 18 | 263 3 | 272 11*} 281 20* 291 5] 300 14] 309 23 | 319 7*| 328 16* 338 1* P23 f 10) 254 113% 12 21 | 282 6 14* 23*| 310 8 17 | 329 2 10* +22 | —2 12% 21*| 273 6* 15 | 292 0} 301 9 17*| 320 2* 11 20 +21 | =3 92 | 264 7 15*| 283 0* g*) 18 | 311 3 12 20%] 339 5* 420 | =4 | 255 7% 16*| 274 1 10 18*| 302 3* 12% 21| 330 6 15 +19 | —5 ye) 2205) 0" 16 10* 19*| 293 4 13 21*| 321 46% 340 0 +18 | —6| 256 2 11 20 | 284 4* 13* DW Sie 7 16 | 331 O* gx eT er 11* 20*| 275 5 14 23 | 303 7 16*| 322 1 10 19 -15))) = 21 | 266 5* 14* 23*| 294 8 Ay Sey 12 10* 19*| 341 4 +15 | —9 | 257 6% 15: 276 05) 285 58* 17*| 304 2* 11 208] 3382400 13* +14 |—10 15*| 267 0* ge 18 | 295 3 11* 20*| 323 5* 14 23 eeiSq tie) (258 10 18*| 286 3* 12% 21] 314 6 14% 23*| 342 8* 211) late 10* 19 | 277 4 13 21*| 305 6* 15*| 324 0] 333 9 17* ett |=18 20 | 268 4* 13* 22 | 296 7 16 | 315 OF o* 343. 3 +10 |—14 | 259 5 14 23. | 287 7* 16*| 306 1 10 “19 | 334 3% 19* +9 |—15 14* 23%) 278 8 La 2OTaber 10* 19*) 325 4 13 22 +8 |—16 | 260 0] 269 8* 17*| 288 2* 11 20 | 316 4% 13* 22*| 344 7 +7 |-17 18 | 279 3 11* 20*| 307 5* 14 23 | © 330 | 7 16* +6 |—18 18*| 270 3* 12 21] 298 6 14* 23*| 326 8* 17 | 345 2 +5 /—19| 261 4 13 21*| 289 6% 15 | 308 0| 317 9 17*| 336 2* 11* +4 |—20 13* 22 | 280 7 16 | 299 0* 9 18 | 327 3 12 20* 13) 2 22*| 271 7* 16*| 290 1 10 19*| 318 3* 12% 21 | 346 6 +2 |—22 | 262 8 17 | 281 1* 10* 19*| 309 4 13 21*| 337 6* 15* Sel |e 17*| 272 2 11 20} 300 4* 13% 22*| 328 7 16 | 347 0 Difference. Component Q. Compotent p. Hour. d. h. d. h. davies d. h. d. h. d. h d. h dsahk d. h. | d. h 0 347 10 | 35619 | 366 3*/ 1 0] 1015*} 2011) 30 6} 40 2] 49 21% 5917 SEN ea 19*| 357 4% 13 Be 1 1 20* 16 12| 50 74) 60 3 EGY) 1) |) BV) 13* 22% 15% 11} 21 6+ 31 2 21* 17 12% +21 | —3 14 233) oOlaaS Dak 21 16*; 12 41 7* 51 3 22* +20 | —4 a*| 358 SF 17 11 DW GEl e222 21* 17 12* 61 8* +19 | —5 |! 349 9 18 | 368 2* 21 16* 12 32. 7# 42 3 22%) 18 B86 18*| 359 3 12 3 6*| 13 2* 22 17* 13} 52 8 62 4 +17 | —7 | 350) 3* 12% 21* 16* IP UNGEY TeedeaRy 1B 29* 18 13* +16 | —8 13 22) 369 6* 4. 2% 22 17* 13 43 8*| 53 4 23* 415) |) 3 29*| 360 7 16 12) | eae | 24s 2% 1s* 14 | 63 9* +14 |—10 |} 351 & 16*| 370 1* 22 ees 13 34 Sf 44 4 23* 19 +13 |—11 Ae!) SEL MDa ls otha by is} 15> 3% 23 18* 14 54 9* 64 5 +12 |—12 | 352 2* THis nea 17* 13 | 25 8] 35 4 23% 19 15 +11 |—13 12 20 Roe aes H 3* 23 18* 14 45 9*| 99/55. ol 6o9 0 +10 |—14 | SUN isis) TGS age ee 13 | 16 9| 26 4* 36 0 19* 15 10* +9 |—15 | 353. 6* 5A] LO Leet 23 18* 14 g* 46 5| 56 0 20 +8 |—16 | UGS SOS tak (be eee 7 9 V7 4) 27 0 19* 15 10*| 66 6 +7 |-17 354 1* DS Laban HE: 18* 14 OF 6387) COSFL 47 20%, 16 | | +6 |—18 | 10* AGH) Med) Hae 8 44 18 0 19* 15 10*| 57 6| 67 1* 4-5 |—19)| Diya iesee (67-4 ae a 14* 10 | 28 5*| 38.1 20* 16 11* +4 |—-20 | 355 5* TEE SIR ease ai! 9 0 19% 5 10*| 48 6 58 2 21* +3 |—21 15 ot Saab ieee 10 1975) e295 ul 20* 16 1i*| 68 7 +2 |—22t1) 300070) | SGD) Oh ale eed 1g* 15* 11 39 6} 49 2 21* 17 +1 |—23 9* Mei eos he Ls LOM S| 20rd 20* 16 11*}, 59 7} 69 2* HARMONIC ANALYSIS AND PREDICTION OF TIDES. TasBLe 31.—For construction of primary stencils—Continued. 279 Difference. Component p. Hour. d.h. aah d. h. d. h d. h d. h. d. h d. h. d. h. d. h 0 69 12* 79 8 89 3* 98 23 108 18*) 118 14 128 9*) 138 5 148 1 | 157 20* +23 | —1 22* 18 13*| 99 9 109. 4*; 119 0 19* 15 10*| 158 6 +22 | —2 70 8 80 3* 23 18* 14* 10 129 5*) 139 1 20* 16 +21 |} —3 18 13* 90 9 100 4%; 110 0 19* 15 10*; 149 6] 159 1* +20} —4 71 4 23* 19 14* 10 20 eos wees Omnele 20% 16 11* +19 | —5 13* 81 9 91 4% 101 0 19% 15 11 140 6* 150 2 21* +18 | —6 23* 19 14* 10 S| ela Tieat 20* 16 11*| 160 7 +17 | —7 72 9 82 5 92 0* 20 15* it 131 6*) 141 2 21% 17 +16} —8 19 14* 10 102 5*) 112 1 20* 16 IDES alike ese als hes} +15 |} —9 Te 5 83 0* 20 15* 11 WEP, IB) es 21* 17 12* +14 |—10 14* 10 93 5a) L037 as 21 16* 12 LAD SEN S21 aS 22% +13 |—11 74 0* 20 15* 11 INS. Gra WARY) 21* V7 12*| 162 8 +12 |—12 10* 84 6 94 1* 21 16* 12 BR ral ake Ss 22* 18 +11 |—13 20 15* il 104 6* 114 2 21* 17* 13 153 8*| 163 4 +10 |—14 (8 85 1* 21 16* 12 1A ESAT IS 22*| 18 13* +9 |—15 15* 11* 95 7 105 2* 22 iffy 13 144. 8*| 154 4 23* +8 |—16 76 1* 21 16* 12 ily ees) es) 22* 18 14 | 164 9* +7 |-17 11*) 86° 7 96. 2* 22 17* 13 | 135 8*| 145 4 23* 19 +6 |—18 21 16* 12 106 8 116 3* 23 18* 14 155 9% 165 5 +5 |—19 tt NG 87 2* 22 17* 13 126 8* 136 4 23* 19 14* “+4 |—20 17 12* 97 8 107 3* 23 18* 14 146 9* 156 5 | 166 O* +3 |—21 St 2 22 17* 13 117 8*| 127 4*) 137 0 19* 15 10* +2 |—22 12* 88 8 98 3% 23 18* 14 9* 147 5 157 0* 20 +1 |—23 22 18 13*| 108 9 118 4* 128 0 19* 15 10*| 167 6 Difference. Component p. Hour. d. h d. h. Gh) 108 Cane d. h d. h. d. h d. h. d. h. d. h 0 167 16 L77 ALE) \187 7 197 2* 206 22) 216 17*| 226 13 236 8*| 246 4 | 255 23* +23 | —1 168 1* 21 16* 12 PAV (Gy) Pale) 3h 23 18* 14 | 256 9* +22 | —2 11*| 178 7 188 2* 22 17* 13 PRL Ska) PAY @ zh 2374 19 +21; —3. 21 517 12% 198 8 208 3% 23 18* 14 247 9%) 257 5 +20 | —4 169 7 179 2* 22 17* 13 218 8* 228 4 238 0 19* 15 +19 | —5 17 12*, 189 8 199 3* 23 18* 14 9*| 248 5} 258 O* +18 | —6 170 2* 22 17* 13*| 209 9 219 4* 229 0 19% 15 10* +17 | —7 12*| 180 8 190 3* 23 18* 14 9*| 239 5 249 0% 20 +16 | —8 22% 18 13*| 200 9 210 4* 220 0 19* 15 10*| 259 6 +15; —9); 171 8 181 3* 23 18* 14 10 230 5*) 240 1 20% 16 +14 |—10 18 13%, 191 9 201 4%) 211 0 19* iy 10*/ 250 6 | 260 1* +13 |—11 ieee 23* 19 14* 10 22 Se 23a 20* 16 11* +12 |—12 13*| 182 9 192 4* 202 0 19* 15 | 10*| 241 6*/ 251 2 21* +11 |—13 23* 19 14* 10 212 SE 222 00 20* 16 TO 261, 27 +10 |—14 Lav) 183 4*| 193 0* 20 15#| il 232 6*| 242 2 21* 17 +9 |—15 19 14* 10 203) 5%) 213) 1 20* 16 252) ih | e2O 203, +8 |—16 174 5 184 0* 20 15* iAL 223 6%) 233) 22 21* V7 12* +7 |—17 14* 10 194 5* 204 1 20* 16* 12 2A es|ueZOol nO 22* +6 |—18] 175 0* 20 15* 11 214 6* 224 2 21* 17 12#| 263 8 +5 |—19 10*, 185 6 195 1* 21 16*: 12 234 7*| 244 3 22* 18 +4 |—20 20 15* 11 205 6*| 215 2 21* 17 13 254 8*) 264 4 +3 |—21 176 6 186 1* 21 16* 12 DAES TES' BRGS 3) 22* 18 ales +2 |—22 15* il 196 7 206 2% 22 17* 13 245 8*| 255 4 23* +1 |—23 iplr( Tes 21 16* 12 216 7*| 226 3 el 18 13*| 265 9* 280 U. S. COAST AND GEODETIC SURVEY. TaBLE 31.—For construction of primary stencils—Continued. Difference. Component p. Hour. d. fh: d. h. d.h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 0 | 26519] 275 14*| 28510) 295 5*| 305 1) 314 20*| 32416] 33412) 344 7* 354 3 is ae 266 oy 276 ad a 4 ou ofl p 315 oe 325 Z a1 ’ 17 12* 2) — ; 6 1 335 7 3 22* +21 | —3 | 267 0* 20 15*; 11 306 6*| 316 2 21* 17 a 12% 355 + 8* +20 | —4 HOF M27 Oe eei eas 21 16* 12} 326 7* 336 3 22% 18 +19 | —5 20 15* 11 | 297 °6*| 307 2* 22 17* 13 | 346 8*| 356 4 +18 |} —6}| 268 6| 278 1* 21 16* LON) Bes! 827 as 22% 18 13* +17 | —7 16 11*/ 288 7 | 298 2* 22 17* 13 | 337 8*| 347 4 23* +16} —8| 269 1* 21 16* 12] 308 7* 318 3 23 18* 14 | 357 9* +15 | —9 11*| 279 7 || 289 2* 22 17* 13 | 328 8*| 338 4 23* 19 +14 |—10 21 16* 12*| 299 8 | 309 3* 23 18* 14| 348 9* 358 5 +13 |—11 | 270 7 | 280 2% 22 17* 13 | 319 8*| 329 4 23* 19 15 +12 |—12 17 12* 290 8} 300 3* 23 18* 14 | 339 9% 349 5 | 359 O* ioe my | 271 ee Me mB ba Ve ie 310 iz 320 a 330 a " 19* to 10* = 2 3 5 40 5 | 350 0 2 +9 |—15 22% 18 13*| 301 “9 311 4*| 321 0 19* 15 10*| 360 8 +8 |—16 | 272 8] 282 3* 23 18* 14 9*| 331 5*| 341 1 20* 16 +7 |—17 18 13*| 292 9] 302 4* 312 0 19* 15 10*| 351 6 | 361 1* +6 |—18 | 273 3* 23* 19 14* Ny) aye Say" Bay) il 20* 16 11* +5 |—19 13*| 283 9) 293 4* 303 0 19%, 15 10*| 342 6] 352 2 21* +4 |—20 23* 19 14* 10,| 313 85*)) 323 71 20* 16 11*| 362 7 +3 |—21 | 274° 9] 284 4*| 294 0 19%, 15*) 11 | 333 6*| 343 2 21* 17 +2 —22 19 14* 10 | 304 5*) 314 1 20% 16 11*! 353) Ye |-a6an 25 +1)|—23 | 275 5) 285 0* 20 15* 11 | 324 6*| 334 2 21* 17 12* Difference. Ea Component O. | i Hour. thy ale d. h. d. h. d. h. awit. d. h. Ge ibs d. h. Ge eR 2 0 363 22*| a0) 14 22%) 293 43) 7*| 57 12 71 16*| 8521} 100 2) 114 6* ue im 364 i ob a ne a af ae 58 ee 12 86 oe 16 20* Di) = | : 2 44 1 6* 21 87 1 LOL I6)|) LS +2] | —3| 365 4 | 2 12 17 Railay iby 5ONG6e|) Saul 16 20*| 116 1 +29 =A 13*| Bera ealeh ue Zh 3l 11* 16 | 21 74 1* 88 6 102 10%) 15 +19 | —5 23% |” 16* 21 32 2 46 6* 60 11 15* 20 1OSH ty | PLES +1g |] —6] 366 9 | 4 7 18 11* 16 204! | GL yl (ono 89 10* 15 19* +17 | —7 19 21 i) ahah Sey 47 11 15* 20 90 O*} 104 5] 118 10 +1g4—8] 367 5! 5 11 16 20*, 48 1 62 5* 76 10 15 19*| 119 0 +15 | —9 14*' ikea, AD 34 10* 15 20 77) 0#| OLS LOS aes 14 +14 |—10|] 368 0* 15* 20 do 1 49 5*| 63 10 14* 19 106 0}] 120 4* +13 |—11 10* t @ 21 10* 15 19*| 64 0 78 5 92 9* 14 18* +12 |—12 20 20 WPA (Gal 3h) 50 9* 14* 19 23*| 107 4] 121 8* +11 |—13 | 369 6 8 10 14* 19%) 5 5i5 70 65 4*| 79 9 93 13* 18* +109 |—14 15* 9 OF} 23 5 37 9* 14 1s* 23*| 94 4] 108 8* 122 13 +9 /—15 | 370 1% 14* 19 23*| 52 4% 66 9 80 13* 18 22*| 123 3* eats oe RB) se) GB) Sas) amy tn 8 |aae af falls) No eae soe 3 8 1 110 3 4 7 SEG ile) ||ESPeee ese 11 9 25 13* 18* 23 68 3* 2 8 96 12* 17* 22 era bOy eee See 23*| 26 4 40 8* 54 13 17* 22%) 97 3) Lil 7 125 12 ete AG | 20 eee eee 12 13* 18 227 Desa OO MES 83 12* 17 21*| 126 2* By EPA Lae oeoeee 13 3*| 27 8% 41 13 17* 22 84 2*| 98 7* 112.12 16* SD DOM ee eieareine 18 22%). 42 3 56 7*| > 70 12* 17 21*| 113) (2 | 127 6* +1 |—23 }...------ 14 8 28 12* 17* 22 TL 2% 85 7 99 11* 16 1 Difference. —22 —23 HARMONIC ANALYSIS AND PREDICTION OF TIDES. d. h. 142 15* Component O. 21* TasiLe 31.—For construction of primary stencils—Continued. Difference. Gh TB 270 9 23 271 13* 21* 358 11* 359 2 16 360 6 282 U. S. COAST AND GEODETIC SURVEY. TABLE 31.—For construction of primary stencils—Continued. Difference. Component 2N. Hour. d. h. d. h d. h d. h d. h d. h d. h d. h d. h d.h 29 6] 4312} 5718] 72 0] 86 6] 100 12% 114 18] '129 O*} 143 6*| 157 12% BEO3 (| 20| 44.2] 58 8* 14* 20*| 101 2* 115 8* 15 21| 158 3 +22] —2] 30 10* 16* 22*| 73 4%! 87 10* 17 23| 130 5| 144 11 17 Sy = S) |p at 45 6% 59 13 LON ASS vl ODA | Mey s 19*) 145 1*| 159 7* 490 | —4 15 Zi GO) 740 Ig 15 21*| 117 3*| 131 9* 15% 21* BeiG ae —=5a | 32) eon 46nd 17 23*| 89 5% 103 11* 17*| 132 0} 146 6 | 160 12 dei) So 19% 47 14 61 7 75 13% 20} 104 2] 118 8 14 20 | 161 2* Fi | Ee SBS 15* 221 76 4 | (90 10 16 22) 133 4*| 147 10* 16* TG PS BAO ZEB Go 18] 91 OF 105 6% 119 12* 18%) 148 O*| 162 7 Bais 9 14 20] 63 2%] 77 8 14* 20*| 120 2%) 134 9 15 21 414/10} 35 44 49 10* 16% 22*/ 92 5| 106 11 17 23} 149 5 | 163 11* +13 }|—11 18*| 50 O* 64 7 78 13 LO LOT a 2 shy tse 19% 164 1% ID |-@ i ah, 15 21} 79 3] 93 9% 15* 21%| 136 3*| 150 94 16 net | —13 23 51 95 |) 65) 1 17* 23*| 108 5%) 122° 11* 18| 151 0] 165 6 +10 |—-14 | 37 13% 19*| 66 1*| 80 7% 94 14 30 | 1123 |) 1378 14 20* +9{/-15] 38 34% 52 9% 16 22 | 19594" || 109) 10 16 29% 152 4*] 166 10* +8 |—16 18) 53) Ol 67 Wer) Silt 18*| 110 O*| 124 6% 138 12* 18*| 167 1 4+7|-17| 39 8 14 20%} 82 2% 96 8 14* 20%, 139 3] 153 9 15 +6 |—18 22*| 54 4% 68 10%} | 16* 23h | Ma tlit ey Mee yal 23 | 168 5* +5|—-19| 40 12* WP ED UN RB Gig 19] 126 1*| 140 74 154 13* 19% +4|-20| 41 3] 55 9 15 21| 98 3% 112 9 15* 21*| 155 3%] 169 10 43) 91 17 23*| 70 5% 84 11* 17* 23 | 127 6 | 141 12 18 | 170 0 +2/—22|} 42 7% 456 13% 19%] 85 1*| 99 8] 113 14 20| 142 2/156 8 14* il ps 21% -57 4 | “71 10 16 22) 114 4] 128 10* 16* 22% 171 4* Difference. Component 2N. Hour. d. h d. h d. h. iby Ti d.h d. h d. h d. h d. h d. h 0 17119} 186 1) 200 7] 21413] 22819] 243 1*| 257 7%| 271 13*| 285 19*| 300 1* +23] —1] 172 9 15 21*| 215 3%] 299 9x 15% 21*| 272 4 | 286 10 16 AD? || 9 23*| 187 5% 201 11* 17* 23*| 244 6] 258 12 18 | 287 0| 301 6 +21 | —3} 173 13* 19*| 202 2] 216 8 | 230 14 20 | 259 2] 273 8 14* 20* +20] —4] 174 4] 188 10 16 22) 931 4] 245 10* 16* 22*| 288 4* 302 10* +19 | —5 18 | 189 0| 203 6% 217 12* 18%] 246 O*| 260 6% 274 13 19 | 303 1 +18 | —6] 175 8* 14% 20*| 218 2%| 232 9 15 21 | 275 3 |, 289 9 15* etl || 7 22*| 190 4%] 204 11 17 23.| 247.5] 261 11 17* 23*| 304 5* +16 | —8 | 176 13 19] 205 1] 219 7] 233 13% 19%] 262 1*| 276 7*| 290 13% 20 +15| —9}| 177 3] 191 9 15* 21*| 234 3%] 248 ge 15% 92 | 291 4 | 205 10 +14 |—10 17* 23*| 206 5*| 220 11* 18] 249 0} 263 6| 277 12 18 | 306 O* +13 |—11| 178 7%) 192 13% 20 | 221 2] 235 8 14 20 | 278 2% 292 8* 14* +12 |—12 OP || 1G 28 | Stay 1G 16 22% 250 4%| 264 10% 16* 22*| 307 5 +11 |-13! 179 12 18 | 208 O*} 222 6%] 236 12% 18*| 265 O*| 279 7] 293 13 19 +10 |—14] 180 2%} 194 g* 14* 2023703) 2510 Oy lte oa 21 | 294 i 308 9 +9 |—15 16* 22% 209 5] 223 11 17 23} 266 5] 280 11* 17* 23* +8 |—16] 181 7] 195 13 19 | 224 1] 238 7% 252 13% 19#| 281. 1*| 295 7* 309 14 +7 |—17 21} 196 3] 210 9* 15* 21*| 253 3% 267 10 16 22 | 310 4 +6 |—18 | 182 11% 17* 23% 225 5%*| 239 12 18| 268 0] 282 6| 9296 12 18* +5 |—19 | 183 1% 197 8] 211 14 20 | 240 2| 254 14* 20%, 297 2* 311 8* +4 |—20 16 22} 212 4] 296 10 16* 22*| 269 4% 283 10* 16* 23 +3 |—-21] 184 6| 198 12* 18*| 227 O* 241 6*| 255 12* 19] 284 1] 298 7 | 312 13 +2 |—99 20*| 199 2% 213 g* 14* 21} 256 3| 270 9 15 21 | 313 3* +1 |—23 | 185 10* 17 23] 2298 5) 242 11 17 23*| 285 5% 299 11* 17* HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBiE 31.—For construction of primary stencils—Continued. 283 Difference. Component 2N. Component pz. Hour. d. d. h. d. h d. h d. h. d. h. d. h d. h. d. h. d. h. 314 8 | 328 14] 342 20| 357 2 1 0 VINES) Say) (5 45 O* 5919] 74 13 +23 | —1 22 | -329 4] 343 10* 16* 8 16 2* 21 15 60 9*| 75 4 +22 | —2 315 12* 18*| 344 O* 358 6* 23 17* 31 11* 46 6 61 OF 18* +21 | —3 316 2*| 330 8* 15 21 2 13% Vians 32) 2% 21 15 76 9* +20 | —4 17 23 345 5 359 11 3 (4* 23 17 47 11* 62 6 77 «OF +19 | —5 BUTE 331 13 19* 360 1* 19 18 13* 33 8 48 2* 20* 15 +18 | —6 21*| 332 3*| 346 9* 15* 4 10 19 4* 22* 17 63 11*| 78 6 +17 | —7 318 11* 17*| 347 0 361 6 5 OF 19 34 13%, 49 8 64 2 20* +16 | —8 319 2 333 8 1 20 15* 20 10 35 4 22% 17 79 11* +15 | —9 16 22 | 348 4% 362 10* 6 6* 21 OF 19 50 13* 65 7 80 2* +14 |—10 | 320 6*| 334 12* 18*| 363 OF 21 15* 36 10 51 4 22% 17 +13 |—11 20*| 335 2*| 349 9 15 7 12 22 6 37 «OF 19 66 13*| 81 7* +12 |—12} 321 11 17 364 5 S Ga4 21 15* 52 9% 67 4 22% +11 |-13 322 1 336 7 350 13* 19* 17* 23 11* 38 6 53 (OF 19 82 13 +10 |—14 15* 21*| 351 3*| 365 9* 9 8 24° 2* 21 15 68 9* 83 4 + 9 |—15 323 5* 337 11* 18 | 366 0 23 17 39 11* 54 6 69 O0* 18* +8 |—16 20 |) 388 2) 352 8 14 10° 13%, 25 8 4) | 2% 20* 15| 84 9 +7 |—17 | 324 10 16* 22*| 367 4* ll 4* 22* 17 py INES 70 6 85 0 +6 |—18 | 325 O* 339 6*| 353 12* 18*|- 19 26 13*| 41.8 56 2 20* 15 +5 |—19 14* 21 354 3 368 9 12 10 27°=«4% 22* 17 71 11*| 86 5* +4 |—20 |] 326 5] 340 11 17 23 13. OF 19 42 13%, 57 8 (PX 22 20* +3 |—21 19 341 1*| 355 7* 369 13* 15* 28 10 43 4 22* 17 87 11* +2 |—22| 327 9* 15* 214.370" 34" 14 56 29 0* 19 58 134i 7st) on 2 +1 |—23 PRA) 3k Ode AG | Bia i ee a 21 15*| 44 9% 59 4 22* 17 Difference. Component pz. Hour. d. h. d. h. d. h. d. h d. h d. h. d. h d. h. d. h. d. h. 0 89 7* 104 2 118 20*| 133 14*) 148 9 163 3*) 177 22 192 16 207 10*| 222 5 +23 | —1 22* i6*; 119 11 134 5*| 149 0 18 178 12*| 193 7 208 1* 19* +22 | —2 90 13 105 7* 120 2 20 14* 164 9 179 3* 21* 16 ! 223 10* +21 | —3 91 4 22 16*} 135 11 150 5* 23* 18 194 12*, 209 7 | 224 1 +20 | —4 18*} 10613 | 121 7*| 136 1* 20 | 165 14*| 180 9} 195 8 21* 16 +19 | —5 92 9* 107 4 22 16*| 15111} 166 5 23* 18 | 210 12*| 225 6* +18 | —6 93 0 18*| 122 13 LB Y(e a7 fal em ay Es 20 181 14*| 196 8* 211 3 21* +17 | —7 15 108 9*| 123 3* 22 16*| 167 11 182 5 23* 18 | 226 12 +16 | —8 94 5*| 109 0 18*| 138 13 153° 7 168 1* 20 197 14*| 212 8*) 227 3 +15 | —9 20* 15 124 9 139 3* 22 16*| 183 10*| 198 5 23* 18 +14 |—10 95 11 110 5*) 125 0 18*} 154 12*| 169 7 184 1* 20 213 14 | 228 8 +13 |—11 96 2 20* 14%, 140 9 Gh) 3s 22 16 199 10*| 214 5 23* +12 |—12 17 111 11 126 5* 141 0 18 170 12*| 185 7) 200 1* 19*| 229 14 +11 |—13 97 7*| 112 2 20* 14*| 156° 9 F171 °3* 21* 16 | 215 10*| 230 5 +10 |—14 22* 16*) 127 11 142 5* 157 0 18 186 12*| 201 7 216) tL 19* +9 |—15 98 13 M13 874 128 2 20 14*| 172 9 187 3 21* 16 | 231 10% +8 |—16 99 4 22 164) 143 11 158 5* 23* 18 202 ADEE QV GedT |) 2325 1. +7 |—17 18*| 114 13 129 7*| 144 1* 20 173 14*| 188 9 203 3 21* 16 +6 |—18 100 9* 115 3% 22 16*) 159 11 174 4 23* 18 218 12*| 233 6* +5 |—19 | 101 0 18*| 13013 | 145 7} 160 1* 20 | 189 14%} 204 8* 219 3 21* +4 |—20 15 116 9 131 3" 22 16*, 175 10*| 190 5 23* 18 | 234 12 +3 |—21 102 5*) 117 0 18*| 146 12%) 161 7 176 1* 20 | 205 14 220 8*) 235 3 +2 |—22 20* 15 132 9* 147 3% 22 16 191 10*| 205 5 23* 17* +1 |—23 103 11 118 5* 133 0 18*| 162 12*| 177 7 192 1* 19*| 221 14 | 236 8* 72934—24}—__19 284 U. S. COAST AND GEODETIC SURVEY. TaBLE 31.—For construction of primary stencils—Continued. Difference. Component pz. Hour. atin d. h. d. h. dayne d. h. Canhks Gh d. h. d.) heyday h. 0 236 23 251 17*| 266.12 281 6*| 296 O*| 310 19 325 13%) 340 8 355 2 | 369 20* +23 {| —1 237 14 252 .8*| 267 2* 21 15*; 311 10 326 4 : 22 17 | 370 11* +22 / —2] 238 5 23 17*| 28212] 297 6] 312 O* 19 | 341 Va) 356 See Pees +21 | -—3 19*| 253 14 268 8*| 283 2* 21 15*| 327 9% 342 4 pS Mas il Sa +20 | —4] 239 10*| 254 4* 23 17*} 29812] 313 6] 328 O* TG 85a doe eee ae +19 | —5| 240 1 19] 26914] 284 8} 299 2* 21 15*| 343 9*) 358 4 |........ +18 | —6 16 | 25510) 270 4* 23 17*| 314 11*| 329 6| 344 O* BLS is hy ee +17 | —7 241 6*| 256 1 19*)| 285 13*| 300 8 Silay | 74a} 21 15 SOOO s| ene se +16 | —8 21* i5*| 271 10| 286 4* 23 17 | 330 11* 345 6] 360 O*......_. +15 | —9 | 24212] 257 6*| 272 1 19 | 301 13*| 316 8] 331 2* 20* ae ae +14 |—10 243 3 21* 15*| 287 10 302 4% 22% 17 346 11*| 361 6 }.......- +13 |—11 17*| 258 12 273 6*| 288 1 19 SWIMS Poa 2 nes 347 2 ZU ee RE +12 |—12 244 8* 259 3 21 15*| 303 10 318 4* 22 PEN NS62, Lee Ree +11 |—13 23 17*| 27412) 289 6*] 304 O* T9333 T3*S488 (S63 aa eee % +10 |—14 245 14 260 8*| 275 2* 21 15*| 319 10 334 4 22*| py Aly (Pea ae +9 |—15 | 246 4* 23 17*| 29012} 305 6] 320 0* 19 | 349 18%) 364 7¥*|__...22_ +8 |—16 19*| 261 14 276 8 291 2% 21 15*| 335 9*) 350 4 2S +7 |-17 247 10 262 4% 23 17*| 306 11* 321 6 336 0* 19 S6a oy eesesee = +6. |—18 248 1 19*| 277 13*) 292 8 307 2* 21 15 B51 OFS SOG ea eee +5 |—19 16 | 26310] 278 4* 23 17 | 322 11*| 337 6] 352 O* USE Rese = +4 |—20| 249 6*| 264 1 19%) 293 13*| 308 8] 323 2* 20* To SOT tase Renee +3 |—21 21* 15*| 27910] 294 4* 23 WM eeckiidesle see) aul ays (ii ee +2 |—22 250 12 265 6*| 280 1 19 309 13*| 324 8 339 2 20* hss ER ne Sea +1 |—23 | 251 3 21 15*| 29510] 310 4* 22* 17 RHSS4n is S69 ele eee : | Difference. Component N. Hour. ad. hk (is dy hk CRAIN DA d. h. doy hk d. h. d. h. ds) Reo det he 0 1 70 19 20* 39 2 58 7* 77:13 96 18*| 116 0 135 5*| 154 11 | 173 16* +23 | —1 10*| 20 16 QUE) (59) H2* i T8088 97 13* 19 |} 136 O* 155 6} 174 11* +22 | —2 PAs Sey) GA TU 40 16* 22 79 3* 98 9}. 117 14* 20} 156 1*) 175 6* +21 | —3 Suet 22 6* 4111*| 6017 22*, 99 4] 118 9*| 137 15 20*| 176 2 +20 | —4 20 DBAs 428 7, 61 12*| 8018 23*| 119 5] 1388 10*| 157 15* 21 +19 | —5 4 15* 20** 43 2 62 7* 81 13 100 18*} 120 0 139 5%*| 158 11 | 177 16* +18 | —6 5 10*| 24 16 21*| 63 3 82 8* 101 14 19*} 140 O* 159 6 | 178 11* +17 | —7 6 5* 25 11 44 16* 22 83 3*| 102 9 121 14* 20 160 1*| 179 7 +16 | —8 iu 26 6* 45 12 64 17* 23 103 4 122 9* 141 15 20*| 180 2 +15 | —9 20) 27 1* 46 7 65 12* 84 18 23%) 123 5 142 10*| 161 16 21* +14 |—10 8 15* 2 per aioe 66 8 85 13 104 18*} 124 0 143 5*) 162 11 | 181 16 +13 |—11 9 10*} 28 16 21%) 67 3 86 8* 105 14 19*| 144 1] 163 6*) 182 12 +12 |—12 10 6 29 11*|- 48 17 22 87 3* 106 9 | 125 14* 20 | 164 1*| 188 7 +11 |—13 ah pa 30 6* 49 12 68 17* 23 107 4*| 126 10 145 15* 21 | 184 2 +10 |—14 20* Ola 2 50 7 69 12* 88 18 23*| 127 5 146 10*) 165 16 21* +9 |—15 12 15* 21 51 2% 70 8 89 13%; 108 19 128 O*| 147 6 166 11 | 185 16* +8 |—16 13 11 32 16 Dict le. 90 8* 109 14 19*| 148 1] 167 6* 186 12 +7 |—17 14 6 33 11*} 52.17 22*| 91 4) 110 9*| 129 15 20} 168 1* 187 7 +6 |—18 15 1 34 6* fife 11 CMG 23 lll 4* 130 10 149 15* 21 | 188 2* +5 |—19 20* Bb) 74 54 7% 73 13 92 18* 23*| 131 5 150 10*| 169.16 21* +4 |—20 16 15* 21 Sole 74 8 93 13*} 112 19 132 O*, 151 6*| 170 11* 189 17 +3 |—21 17 11 36 16* 22 ton kos 94 8* 113 14 19#/ 152 1 171 6*| 190 12 +2 |—22 18 6 37 11* 56 17 22* 95 4 114 9% 133 15 20%} 172 2] 191 7* +1 |—23 19 1* 38 7 57 12* 76 17* 23 115 4*| 134 10 153 15% 21} 192 2* HARMONIC ANALYSIS AND PREDICTION OF TIDES. 985. TABLE 31.—For construction of primary stencils—Continued. Difference. Component N. Hour d. h. Gon 0 192 21*} 212 3 +23 ; —1 193 17 22* +22 | —2 194 12} 213 17* +21 | —3|] 195 7* 214 13 +20/} —4] 196 2*| 215 8 +19} —5 22] 216 3* +18 | —6 197 17 22% +17 | —7 | 198 12*| 217 18 +16 | —8] 199 7*| 218 13 +15} —9] 200 3] 219 8 +14 |—10 22} 220 3* +13 |—11 |} 201 17 22 +12 |—12 | 202 12*| 221 18 23*| 260 5 | 279 10*| 298 15* Dy Saupe 2:l er aoOme Sh seer. = +11 |—13 | 203 7%| 22213) 241 18*| 261 0] 280 5* 29911] 318 16* PPA Wee iy yh Bide eee ee +10 |—14 | 204 3] 223 8 242 14 19*/ 281 O* 300 6] 319 11*| 338 17 22% eee +9 |—15 22 | 224 3% 243 9} 262 14* PAU WO ME eH) 7A SBE) IRE seis s) i ee oe +8 |—16 | 205 17* 23.| 244 4% 263 9* 282 15 20*| 32% 2} 340 7* 359 13 |.......- +7 |—17 | 206 12% 225 18 23*| 264 5] 283 10*) 302 16 QUEEN Stay Sh RCO Sk eee ae +6 |—18 | 207 8] 226 13% 245 18*) 265 0) 284 5*) 303 11 | 322 16* 22) olan | see eeeee +5 |—19 | 208 3] 227 8* 246 14 19*| 285 1] 304 6* 323 12 | 342 17* 225s. +4 |—20 22*| 228 3% 247 9| 266 14* 20'| 305 1*) 324-7) 343 12%] 362 18 |..2_._-- +3 |—21 | 209 17* 23 | 248 4*| 26710] 286 15* 21 DOO 2 OLS. tl oOo lon | Meme +2 |—22 | 210 12*| 229 18 23*| 268 5 | 287 10*| 306 16 20E 345) 13 || OBS see 2 = +1 |—23 | 211 8] 230 18% 24919] 269 O* 288 6] 30711 | 326 16* 22 | MeSOO On| Rea = Difference. Component v. Hour. Gone de hei ae es d. h. (hs le Ga 1 Gea ds had. wie lean le 0 1 0 20 18*| 40 23 Gis SUS LOM AZT Sp | 41019) 16123) 82) 3 +23 | —1 il 21 15 41 19 23 82 3 102 7 | 122 11 142 15*| 162 19% 23* +22 | —2 ed 22 11 42 15 62 19 23 103 3*| 123 7*| 143 11*) 163 15%} 183 19* +21 | —3 3, 3 23°17 43 11*| 63 15*| 83 19% 23%, 124 3* 144 7* 164 11*| 184 15* +20 | —4 23%, 24 3*| 44 7% 64 11% 84 15*| 104 19* 23*| 145 4 165 8/ 185 12 +19 | —5 4 19* 23*| 45 3*| 65 7*| 8512] 10516] 12520] 146 0} 166 4/ 186 8 +18 | —6 Selsey oro 46u10 66 4 86 8} 10612); 126 16 20 | 167 0} 187 4 +17 | —7 6 12 26 16 20 67 0 87) 45 107 8h) 127120 476 20*| 188 O* +16 | —8 G8 27 12 47 16 20 88 O*| 108 4*/ 128 8* 148 12*| 168 16* 20* +15 | —9 8 4 28 8 48 12%} 68 16* 20*| 109 O* 129 4% 149 8* 169 L2* 189 16* +14 |—10 9 O* 29 4*| 49 8* 69 12%) 8&9 16* 20% 130 O*; 150 5] 170 9 190 13 +13 |—11 20*| 30 O* 50 4*| 70 8* 9u 13 110 17 21 151° 1 Tal 5) |) Lee) +12 |—12 10 16* 21 51 1 (iby) Otsu 11 13: 131 17 21 172 1 | 192 5 +11 |—13 11.13 31 17 21 72 1 G25 on V2 9132513 152 17* 1* 193 1* +10 |—14 12 9 32 13 52 17 21 93° IF), 113 5*|) 133" 9F) Vss ia) 173 v7 21* +9 |—15 135 38) Cai GS iBesy Sai ales 21*| 114 1* 134 5*, 154 9*| 174 13*| 194 17* +8 |—16 14 14 34 S* 654 9% «6974 13%) «94 17% PA*| 135 1*| 155 6 | 17510) 195 14 +7 |-17 GAEAN ety Wea in Gite aisha) a aye (Sole iy AEE) 22} 156 2] 176 § | 196 10 +6 |—18 15 17* 22 56 2 76 6 9610 | 116 14} 136 18 22) ltd) 197 6% +5 |—19 16 14 36 18 22 U1 2 97 6 117 10 | 1387 14 157 18* 22*| 198 2* +4 |—20 17 10 14 57 18 22 98 2*| 118 6*| 138 10*) 158 14*| 178 18* 22* +4 |—21 18 6 38 10* 58 14*| 78 18* 22*| 119 2*| 139 6*| 159 10*| 179 14* 199 19 +2 |—22 19 2*| 39 6% 5910* 79 14% 99 18* 22*| 140 3 160 7 180 11 } 200 15 +1 |—23 22% 40 2*| 60 6* 80 10%} 10015] 120 19 23 161 3 181 7 | 201 11 286 U. S. COAST AND. GEODETIC SURVEY. TABLE 31.—For construction of primary stencils—Continued. Difference. Component v. ] Hour. daht apie d. h d. h d. h d. h d. h d. h d. h 0 202 7* 222 11* 242 15*| 262 19%} 282 234% 303 3] 323 7* 343 11* 363 16 +23 —1 203 3* 223) ti5 243 11%) 263 15: 283 19% 23 | 324 4] 344 8] 36412 +22 —2 23% 224 3* 244 7% 264 12 284 16] 304 20) 325 0| 345-4] 365 8 +21 —3 204 20 225 0 245 4 265 8 |° 28512) 305 16 20} 346 0| 366 4* +20 —4 205 16 20 246 0 266 4] 286 8} 306 12*| 326 16* 20*| 367 0* +19 —5 206 12 226 16 20 267 O*| 287 4% 307 8*| 327 12*| 347 16* 20* +18 —6 207 8* 227 12*| 247 16* 20*| 288 O* 308 4% 328 8*| 348 12% 368 17 +17 —7 208 4*| 228 &*| 248 12%) 268 16* 20*, 309 1} 329 5} 349 9! 369 13 +16 —8 209 O*| 229 4* 249 8* 269 13 |‘ 289 17 21 We 380er1 ii B50) Ao! asO 9 +15 —9 21 230 1 250 5 270 9 290 13 310 17 21 OL la eee ae he +14 —10 210 17 21 ml it Qe ee 9) eS AS 4 ermal DUE Seis ey es +13 —11 211 13 231 17 QI) QT 2 E292 oF ps1 2609F)) 1332) 13452 ates +12 —12 212 9*| 232 13*| 252 17% QU 29Zki Ae), 1313) S5*essaeeg soso mboml eens +11 —13 NIB Gf 233 OF 253 13* 273 17* QU) 3l4 2) | sd 4eeGn 54: LON See ele +10 —14 214 1* 234 5% 254 10 274 14 294 18 22) \iid30) £2) (MOOOV EO ee seceee +9 —15 22 235 2 255 6 275 10 295 14 315 18 22,1) 5800 pone eee +8 —16 215 18 22 256 2 276 6 296 10 | 316 14*| 336 18* OR Peep we +7 —17 216 14 236 18 225) 277 82* 297 6F) Sl7 OF Soils lon totems +6 —16 217 10*) 237 14*| 257 18% 22%; 298. 2* 318 6%). 338 10%) 358 14" -.-. 2... +5 —19 218 6* 238 10*| 258 14* 278 18* 22* 10 319i Br biwdo9) elt Oe al eee eee +4 —20 219 2* 239 6* 259 11 279 15 | 299 19 23 340ue3' ||, (OGD Naieee enon = +3 —21 23 240 3 260 7 280 11} 30015] 32019 23| 361 3 jocceeseee +2 —22 220 19 23 261 3 281 7} 301 11 | 321 15%) 341 19% 23%) aegis +1 —23 221 15 241 19 23%) oe282053) fdO2 wih nO22u ble S42) dots O2 Lo aeeeeee Difference. Component 2MK. Hour d. h. d. h. d. h d. h Gls 10. d. h d. h d. h d. h d. h. 0 ta0 22657. 44 0 65 17 87 10 109 3 130 20} 152 13 174 6] 195 23 +23 | —1 11* 23 «4% 21* 66 14* 88 7*| 110 O* 131 17*| 153 10*} 175 4 | 196 21 +22 | —2 2 9* 24 2% 45 19*| 67 12* 89 5% 22%) 132 15*| 154 8* 176 1* 197 18* +21 | —3 37 25 0 46 17 68 10 DON oe) TU 206) Tsetse) dooms 23 | 198 16 +20 | —4 4 4* 22 47 15 69 8 91 1 112 18 | 134 11 156 4] 177 21 | 199 14 +19 | —5 5 2a) 26094) 480240) 700 5s 22%) 113 15*| 1385 8*) 157 1% 178 18*| 200 11* +18 | —6 6 0 27:17 49 10 is 92 20} 11413] 136 6* 23*| 179 16*| 201 9* +17 | —7 22 28 15 50 8 (2a 93 18 115 11 137 4 158 21 180 14 | 202 7 +16 | —8 7 19* 29 12% iby Gy 22%| 94 15*| 116 8*| 138 1*) 159 18%] 181 11*| 203 4* +15 | —9 Salis, SOMO 52) Shile WepeOtl) “Obese ATE 6s 23*| 160 16%} 182 9*) 204 2* +14 |—10 9 15 31 8 53 1 74 18 96.11} 118 4] 139 21) 16114] 183 71} 205 0 +13 |—11 10.12%) 32 5* 22* 75 16 97 9 119 2) 140 19 16212} 184 5 22 +12 |—12 1104) 38s Sy 542204) 7ERISs) 98) 6% 23%] 141 16*| 163 9* 185 2*| 206 19* +11 |—13 12 8 34 1 55. 18 eau 99 4 120 21 142 14 164 7 186 0*| 207 17* +10 |—14 13 6 23 56 16 73 9 100 2) 121 19 143 12 165.5 22 | 208 15 +9 |—15 14 3%) 35 20% bya alesse) cas 23*| 122 16%) 144 9% 166 2* 187 19*| 209 12* +8 |—16 iG) al 36 18*) 58 11*) 80 4% 101 214 123 14%] 145 7*| 167 O* 188 17*| 210 10* +7 |—17 23 37 16 Fa, a 81 2 102 19 124 12 146 5 22; 18915} 211 8 +6 |—18 16 20*} 38 13*| 60 6* 23*| 103 16*| 12510] 147 3] 168 20} 19013 | 212 6 +A |—19 17 18* 39.11*| 61 4*| 82 21%) 104 14%, 126 7* 148 O*| 169 17*| 191 10*) 213 3* +4 |—20 18 16 40 9 62 2 83 19 105 12 L275 22 170 15 192 8 | 214 1 +3 |—21 19 14 41. 7 63 0 84 17 106 10 128 3 149 20 | 171 18 193 6 23 +2 |—22 20 11%} 42 4% 21* 85 14+} 107 7*- 129 O* 150 17%] 172 10*| 194 3%*) 215 20* +1 |—23 21 9 43. 2 64 19 86 12% 108 5* 22*) 151 15*) 173 8*| 195 1*) 216 18* if HARMONIC ANALYSIS AND PREDICTION OF TIDES. 987 TaBLE 31.—For construction of primary stencils—Continued. Difference. Component 2MK. Component MN. Hour. d. h. d. h d. h. d. h d. h d. h dhs dev ah. d. h d. h 0 217 16 239 9 261 2 282 19 304 12 326 5 347 22 369 15 1 70)) 23720 +23 | —1 218 14 | 240 7 262 0| 283 17 305 10 320 3 348 20 | 370 13 12*| 24 19* +22 | —2 219 11*) 241 4* 21*| 284 144 306 7*| 328 O* 349 17%.......-- 2 11*| 25 18* +21 | —3 220 9 242 2 263 19%| 285 12*| 307 5* 225) S50 1oe | Lae 2 eae 3 il 26 18 +20, |) —4)] 9221 7 | 243 0 | ' 264 17 | } 286 10 | 308 3.) -329 20) 351 13 |-cae. 2 410*| 27 17 +19 | —5 222 4* 21*| © 265 14*| 287 -74~ 309 OF 3380 174%) 352 10*|.2....-.: 5 9* 28 16* +18 | —6 223 2*| 244 19*| 266 12*| 288 5* 22%! 2331 154) e853! WSF LeRe. aes 6 9 29 16 +17 | —7 224 0 245 17 267 10 | 289 3 310 20 | 332 13 B04 6) yee. eas 7 8 30 15 +16 | —8 22 | 246 15 2680 S| 9290. > Bo) Aol Sal Sess Wile) PShb) 4a eee ae 8 7*| 31 14¥ +15 | —9 225 19*| 247 12*| 269 5* 22%|>) 312 15*) 13384 8*) 356 1]... 22.22. 9) (64) 3213" +14 |—10 | 226 17 248 10 270 3 291 20 | 313 13*| 335 6* 237 | See cee 10 6 33 13 +13 |—11 227 15, 249 8 271 “2 292 18 314 11 336 4 Bod Ziyi | eee il 5% 34 12% +12 |—12 | 228 12*| 250 5%* 22%| 1293 f5*) 9315 84 1337 ~ FF) e858) 18%). - 12 4% 35 11* +11 |—13 229 10*| 251 3*| 272 20*| 294 13*| 316 6* 23*! 9859) 16%|-2 ees 13 4] 3611 +10 |—14} 230 8 252) 1 273 18 295 11 317 4] 388 21 360 4k) see. ee 14 3 37 10 +9 |—15 23 his 22*| 274 16 295 9 318 2 339 19 Bods) 2 pe gs 15 2 38 9*F +8 |—I6 232 3*| 253 20*] 275 13*| 297 6* 23%) » 340) IGF) 6862 WOR pre - teen 16) 2 39 8* +7 i/—17 |) 233 1) 25418! 27611); 298 4] 319 21) 34114] 363 7#.-....... 17 1} 40 8 +6 |—18 23) F255 16) 277 3901) £299 p27) 320 9)! w S42 12) 1), B64 | Sul aee es 18 O* 41 7* +5 |—19 | 234 20* 256 13%) 278 6* 235| 321 16*| e343 OF) e365 | QE eee. ers 23*| 42 6* +4 |—20 235 18*| 257 11*| 279 4% 300 21 322 14*| 344 7* 366 O*/.2..:..2.. 19 23 43 6 +3 |—21 236 16 256 9 280 2 301 19 323 12 345 5 7 | Re eae 20 22 | 44 5 +2 |—22 | 237 13*| 259 6% 23*| 302 16*| 324 10 346 3 357) 20) eee A 21 21*| 45 4* +1 |—23 238 11*| 260 4*| 281 21*| 303 14% 325 7*l 347 OF 368 17%|......-.. 22 21 45 4 | Difference Component MN. Hour. d. h. d. h. d. h. d. h d. h d. h d. h d. h d. h d. h 47 3 70 10 93 17 PL7 120 140 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 20 211 234 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 | —38 50 1 TBA tS 95 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 O*| 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 i7il 8 194 15 217 22 241 § | 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 1 105 8 129 14*| 152 21*/- 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 i1 223 18 247 14270 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% 23*| 272 6* +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 ASI jy: 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 &9 20 ibBy Ss) 136 10 159 16* 23*| 206 6*| 229 13*| 252 20*| 276 3* +3 |—21 67 12 90 19 114 2 437 29. 160 16 |. 183 23 207. 6 230 13 253 20 | 277 3 +2 )—22 68 11* 91 18*; 115 -1*/.138 8*) 161 15*| .ig4 22*| 208 5*| 231 12 254 19 | 278 2 +1 |—23. 69 10*| 9217 116 O*| 139 7*| 162 14*| 185 21*) 209 4*| 232 11*| 255 18*| 279 1* 288 U. S. COAST AND GEODETIC SURVEY. TaBLE 31.—For construction of primary stencils—Continued. Difference. Component MN. Component M. Hour dane d. h d. h d. h d. h. d. h. d. h. d. h. d. h. d. h. 0 280 O* 303 7* 326 14% 349 21* IPE) 29 22%, 59 11% 89 0] 11813] 148 1* +23 | —1 PAHO! | Geigy 327 14 350 21 15* 31 4 60 17 90 5*| 119 18*) 149 7 +22 | —2 23%] 305 6%) 328 13*/ 351 20 2 21 32 10 61 22* 91 11 121 0 | 150 12* +21 | —3 282 22% 306 5*| 329 12% 352 19* 4 2% 33 15* 63 4 92 17 122 5* 151 18 +20} —4)] 283 22] 307 5] 33012! 353 19 5 8 34 21 64 9% 93 22% 123 11] 153 0 +19} —5 | 284 21 308 4] 3381 11 354 18 6713*, 36 2%) 65 15 95 4 124 16%) 154 5* +18 | —6! 285 20%). 309 3%) 332 10* 355 17* 719 37. 8 66 20*| 96 9*) 125 22] 155 11 +17) —7| 286 20; 310 2* 333 9 356 16% 9 O* 38 13% 68 2 9715} 127 3*| 156 16* +16 | —8] 28719) 311 2) 334 9) 357 16 10 6 39 19 69 7* 98 20*| 128 9 | 157 22 +15 | —9 288 18% 312 1*| 335 8*) 358 15% 11 12 41 0* 70 13 100 2 129 14%) 159 3% +14 |—10 | 289 17%) 313 O* 3386 7*) 359 14% 12 17%*| 42 6 71 19 101 7*| 130 20] 160 9 +13 |—11 290 17 314 0 337 7 360 14 13 23 43 11* 73 OF} 102 13 1382 2 | 161 14* +12 |—12 | 291 16 23 | 3388 6) 371 13 15 4% 4417 74 6 103 18*| 133 7*| 162 20 +11 |—138 292 15*| 315 22%] 339 5*| 362 19% 16 10 45 22% 75 11* 105 0 134 13 | 164 1* +10 |—14 | 293 15] 316 22] 340 5] 363 11% 17 15%) 47 4 76 17 106 5*| 135 18*) 165 7 +9 |—15 294 14 317 21 341 4 364 11 18 21 48 OF 77 22*| 107 11 137 0 | 166 12* +8 |—16 | 295 13%) 318 20*) 342 3*| 365 10*/ 20 2* 49 15 79 4 108 16*) 138 5*) 167 18 +7 |—17 | 296 12%) 319 19% 343 2*) 366 9*| 21 8 50 20*, 80 9%) 109 22 139 11 | 168 23* +6 |—18 | 29712] 32019] 344 2) 367 9 22 13%) 51 2% 8115) 111 3%) 140 16* 170 5 +5 |—19] 298 11*, 32118] 345 1 368 8 23 19 53 8 82 20*/ 112 9%) 141 22 | 171 10* +4 |—20 | 299 10*| 322 17*) 346 O*| 369 7%} 25 O* 54 13% 84 2 113 15 143 3*| 172 16* +3 |—21 300 10 | 32317) 347 0| 370 7 26 6 55 19 85 7 114 20%) 144 9 | 173 22 +2 |—22 301 9 OZAUT OSES. Da | eee 27 11* 57 0* 86 13 116 2 145 14%) 175 3* +1 |—23 |] 3802 8* 325 15*] 348 22%)__...._.. 28 17 58 6 87 18*| 117 7*| 146 20/176 9 | Difference. Component M. Component MK. - Hours. ds ihe d. h. d. h. d. h d. h d. h d. h. d. h. do he | ah 0 177 14*| 207 3] 286 16] 266 4*| 295 17*| 325 6 | 354 19 1 0 46 5* 92 9* +23 | —1 178 20 | 208 8* 237 21*| 267 10 | 296 23 326 11*| 356 O* 2 0 48 3* 94 7* +22 | —2 180 1*} 20914] 239 3] 268 15*| 298 4% 32717] 357 6 3 22 50 2] 96 6 +21 | —3 181 7 210 19*| 240 8% 269 21 299 10 | 328 22%) 358 11* 5 20 52 0] 98 4 +20) —4 182 12*, 212 1] 241 14) 271 2*| 300 15*| 330 4 359 17 7 18*| 53 22 | 100 2 +19} —5 183 18 | 213 7]! 242 19*| 272 8] 301 21 331 9*) 360 22*| 9 16*) 55 20*| 102 O* +18 | —6 184 23*) 214 12*| 244 1] 273 14] 303 2*| 33215) 362 4 11 14*| 57 18*) 103 22* +17) —7| 186 5} 21518) 245 6* 274 19*| 304 8 | 333 21] 363 9* 13 18 59 16*) 105 20* +16 | —8| 187 10*| 216 23*| 2946 12| 276 1 305 13* 3385 2% 364 15 15 H 61 15 | 107 18* +15! —9| 188 16 | 218 5| 247 17*| 277 6*| 30619] 336 8] 365 20*, 17 9 63 13 | 109 17 +14 |—10] 189 21*| 219 10*) 248 23 | 27812} 308 O*| 337 13*| 367 2 19 7* 65 11} 111 15 +13 |—11 191 3] 22016] 250 4% 279 17%) 309 6] 33819] 368 7*| 21 5*| 67 9* 113 13 +12 |—12 192 9] 221 21*| 251 10] 280 23} 310 11%) 340 O*| 369 13 | 23 3%) 69 7¥* 115 11* +11 |—138 193 14*, 223 31] 25216 | 282 4*| 311 17/] 341 6] 87018*/ 25 2 71 5*) 117 9* +10 |—14 194 20 | 224 8* 253 21%) 283 10] 312 23 | 342 11*)._.....-. 27 0 73 4/)119 7* SORTS | 296 2 E225 0 P5955 0S) | eRe Tae ee S14) ek ods! 17 | eee 28 22 75 2) 12t° 6 +8 |—16 197 7 | 226 19%] 256 8*] 285 21 3L5 10s) 344) 224) Ss 30 20*, 77 0} 123 4 +7 |—17 198 12%) 2287 1 | 257 14 | 287) 24) 816 W5*) 346 4 |i) lll u.8. 32 18%) 78 22%) 125 2 +6 |—18 | 19918 | 229 6* 258 19%} 288 8 | 317 21 SSI cal Oe a se 34 16*| 80 20*) 127 0* +5 |—19 | 200 23%} 23012] 260 1 |~ 289 13*| 319 2*| 348 15 |_.._.2..- 36 14%) 82 18*| 128 22* +4 |—20 | 202 5] 231 17*| 261 6*) 29019] 320 8] 349 20*/..-....-. 38 13 84 17 | 130 20* +3 |—21 | 203 10* 232 23*| 262 12] 292 O*| 321 13%] 351 2 /|......... 40 11 86 15 | 132 19 +2 |—22 | 204 16] 234 5] 263 17*| 293 6*| 32219] 352 7¥#......... 42 9 88 13 | 134 17 +1 )—23'| 205 21*| 235 10*|’ 264 23 | 294 12 | 324 O*| 353 134/:....... 44 7* 90 11*) 136 15 HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLE 31.—For construction of primary stencils—Continued. Difference. Component MK. Hour. d. h d. h. d. h d. h d. h 0 138 13%) 184 17 230 21 217° 1 a2e. 9 +23 | —1 140 11*| 186 15*| 232 19%; 278 23 320) 93 +22 | —2 142 9*) 188 13*| 234 17*| 280 21*| 327 1 +21 | —3 144 8 190 11*| 236 15*| 282 19*| 328 23* +20) —4 146 6 192 10 238 14) 284 17%) 330 21* +19 | —5 148 4 194 8 | 240 12 286 16 332 19* +18 | —6 150 2*| 196 6 242 10 288 14 334 18 +17| —7! 152 O*| 198 4*| 244 8 290 12 336 16 +16} —8 153 22*| 200 2*| 246 6% 292 10*| 338 14 +15 | —9 155 21 202 OF 248 4%) 294 8*| 340 12% +14 |—10 157 19 203 23 250 2* 296 6*| 342 10* +13 |—11 V59: V7 9205 21e| 3252 11 298 5 | 344 8* +12 |—12 161 15*| 207 19 253 23 +11 |-—13 163 13*) 209 17*| 255 21 +10 |—14 165 11*| 211 15*| 257 19* +9 |—15 167 10 213 13%) 259 17* +8 |—16 169 8 215 12 261 15* +7 |—17 171 6 217 10 263 14 +6 |—18 173 4*| 219 8 265 12 +5 |-19 175 2*| 221 6*| 267 10 +4 |—20; 177 OF 223 4*| 269 8* 315 12 361 16 +3 |—21 178 22*| 225 2*| 271 6*| 317 10*| 363 14* +2 |—22 | 180 21} 227 1] 273 4% 319 8* 365 12* +1 |—23 182 19 228 23 21d) 321 6*| 367 10* Difference. Component A. Hour d. h. d. h. d. h. devant Ge Ips 0 220 7*| 275 10*| 330 13 10 58 20* > +23 ; —1 | 22215] 277 17*| 332 20 26H, 619 7s +22 | —2] 224 22) 280 O*| 335 3 4 17* 63 19 +21 | —3 227 5 | 282 7*| 337 10 7 4* 66 6 +20) —4| 22912) 284 14* 339 17* 9 15* 68 17 +19 | —5 | 231 19 286 22 | 342 O* 1) es 71 4 +18} —6| 234 2* 289 5] 344 7* 14 13* 73 15 +17 | —7 236 9% 291 12 | 346 14* Ot 16 2 +16} —8} 2388 16*| 293 19 | 348 22 19 11* 78 13 +15} —9 |} 240 23%; 296 2# 351 5 21 23 81 0 +14 |—10| 243 7] 298 9* 353 12 24 10 83 11* +13 |-11 245 14 | 300 16*| 35519 |. 26 21 85 22* +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*| 314 11 369 14 41 15*| 100 16* +6 |—18 | 261 15*| 316 18*| 371 21 44 2* 1038 4 +5 |—19 | 263 22*, 319 1¥*......... 46 13*| 105 15 +4 |—20| 266 6}, 321 8¥........: 49 0* 108 2 +3 |—21 268 13%) (323 15*| S05. 522 51 11*| 110 13 +2 |—22 | 270 20 | 325 22*)......... 53 22%, 113 0 +1.|—23 | 273 3] 328 6 |-........ 56 OF 115 11 Component X. d. h GaP Tay Oe w at Gales 369 9 1 0 55.0] 110 2* od 7 me Ary on. 7 18 12, OF 2 Be o/- Bae 411*| 5914] 114 16* ee Cy 6 18*) 61 21 117 (+O a ae 9 1* 64 4*| 119 7 2 Be Oe 11 8*| 66 11*| 121 14 cea Bee 13 16 68 18*| 123 21 Laie. ames 15 23 71 1*; 126 4 it. Sp ge th 18 6 73 8*| 128 11* ORCS Ae 20 13 75.16} 130 18* iY. pa 22 20*| 77 23 133 1* fe Specie 25 34-80 6 |e) 135.8% 27 10*| 8213] 137 15* 28 17*| 84 20 | 139 23 32 O0*| 87 3*| 142 6 34 8 89 10*) 144 13 36 15 91 17*| 146 20 38 22 94 O* 149 3* 41 5 96 8| 151 10* 43 12*| 9815] 153 17* Saga gecise 45 19*| 100 22] 156 0* a eee 48 2% 103 5] 158 7* ace Cee 50 9* 10512) 160 15 e2SRi 2 oe 52 16*} 107 19*| 162 22 Component MS. (5 Wea en Beli We Ghia C5 ie 117 22 | 176 23*| 236 1 295 2* 120 9 179 10*| 238 12) 297 13* 122 20*) 181 21*| 240 23 | 300 0* 125 7*| 184 9 | 243 10*| 302 11* 127 18%) 186 20 | 245 21*| 304 23 130 5*| 189 7] 248 8*| 307 10 132 16*| 191 18} 250 19*| 309 21 135 3*| 194 5] 253 6*) 312 8 137 14*| 196 16 | 255 17*| 314 19 140 1*| 199 3] 258 4% 317 6 14213} 201 14 | 260 15*| 319 17 145 0} 204 1*| 263 3] 322 4 147 11 | 206 12*| 26514] 324 15* 149 22 | 208 23*| 268 1) 327 2* 152 9] 211 10*) 27012) 329 13* 154 20 | 213 21*| 272 23} 332 O*. 157 7}. 216 8* 27510] 334 11* 159 18 | 218 19*| 277 21] 336 22* 162 5*| 221 6% 280 8 | 339 9* 164 16%) 223 18 | 282 19*| 341 20* 167 3*| 226 5| 285 6*| 344 8 169 14*| 228 16 | 287 17*| 346 19 172 1*| 231° 3| 290 4*| 349 6 174 12*| 233 14 | 292 15*| 351 17 289 d. h. 165 5 167 12 169 19* 172 2* 174 9* 176 16* 178 23* 181 7 183 14 185 21 188 4 190 11* 192 18* IGS ips 197 .8* 199 15* 201 23 204 6 206 13 208 20 211 3 213 10* 215 17* 218 0* h. h. 354 4 356 15 359 2 361 13 364 0* 366 11* 368 22% 371 9* 290 U. S. COAST AND GEODETIC SURVEY. TABLE 31.—For construction of primary stencils—Continued. : Compo- Difference. Component L. Component P. nent T Hour. d. h. d. h d. h. d. h. @. oR. d. h. d. h. d. h d. h. 0 1 0 63 8 126 23 190 14 | 254 5]! 317 20 1 0] 358 16 1 0 +23 ! —-1 2 8*° 65 23%) 129 14%) 193 5¥* 256 20* 320 11% 8 15*| 373 21 16 6 +22 - —2 i 10) 68 15 132 6 195 21 | 25912] 323 3 23) 20 | eras haat 46 16* +21 —3 7 16 be a 134 22 198 12*| 262 3* 325 18*| 39 2 |. 02... 2. Viens +20 —4 10. 7 73 22*| 137 13%) 201 4% 264 19%) 328 10#| 54 7 ]..-.__... 107 13* 4-19 —5 12 23 76 14 140 5 203 20 | 26711} 331 2 69) B23 eee Sea 138 0 +18 —6 15 14* 79 5*| 142 20% 206 11*) 270 2% 333 17*| 84 17*)_.-...... 168 10* +17 —7 18 6% 81 21%) 145 12*| 209 3) 27218] 336 9 OO Qaxie Sede se 198 21 +16 —8 20 22 84 13 148 4 Zbl 195) 4275 VOR! S39) fs) MS) Tak ee 229 7* +15 —9 23 13* 87 4*| 150 19%) 214 10*) 278 1%) 341 16* 130 9*._..._... 259 18 +14 —10 26 5 89 20 153 11 217 2} 28017] 344 8 | 145 14*). 22. LL... 290 4* +13 —il 28 21 92 12 156 2*| 219 17*| 283 8* 346 23*| 160 20 |......... 320 15 +12 —12 31 12* 95 3*| 158 18%) 222 9 286 O* 349 15*| 176 1 |.......-- soleale +11 —13 34 4 97 19 161 10 225 1) 28816) 352 7! 191 6*._2..__.2 381 12 +10 —14 36 19*| © 100 10%) 164 1%} 227 16% 291 7* 354 22%) 206 11*).2......-]..-...--- +9 —15 39 11*| 1038 2%) 166 17 230 88 293 2ah) 357 Tas) el Qi. Tis Sea ee Re ae +8 —16 42 3 105 18 169 9 233 O°} ‘296 15°) °360 |G") 236 220) 532). sf tsk ae —17 44 18* 108 9% 172 OF 235 15%) 299 G6* 362 21%) 252 BF). 2 ses) 285.8] +6 —18 47 10 iil 1 174 16 238 74} (301 227) 9365 139! 267 18s) See tele +5 —19 50 2 113 17 177-74! 240 22%] 304 13*) ‘368 (4%) (282 19k) aoe 2 See eee +4 —20 SQN] IG -S¥} 79 23B*P) (243 144) 307 Se 370 20%) “297 19a), Been 2 eros ele +3 —21 55° 9 119 0 182 15 246 6} 309 21°)..-...-..- SES (OR Uet. s san a ee ee 2 —22 3S OF) OT 2T 15%). 185 Gs) 248 20) SED 124) eae oe 328 5* = Sete) hapa vate Be aie +1 —23 60 16%, 124 7* 187 22 ZOU AS Wl STO ee ee AB HOPE Bice ricco pa bape tok. | Difference Soni _| Component K. Component 2SM. Hour d. h. d. h. d. h d. h. d. h. d. h. d. h. d. h. Gd.) 1. | jae 0 1 0 1 0} 358 16 1 0 29 22%) 59 11*| 89 O]} 118138} 148 1*| 177 14* +1 |—23 16 6 8 15*| 373 21 TSR aah ct 60 17 90 5*| 119 18% 149 7 | 178 20 +2 |—22 AG 16% 23 (208). | 28! 2 21 32 10 61 22*, 91 11 121 0 | 150 12*| 180 1* +3 |—21 UG BON ZA NOL. trees 4 2%) 33 15%) 63 4 9217 | 122 5* 15118] 181 7 +4 )/—20)| O07 13%) 54 ti eo 8 5 8 34 21 64 9* 93 22% 123 11] 153 0 | 182 12* +5 |—19 | 138 0 69 12) Jet. eas 6 13%; 36 2% 65 15 95 4] 124 164 154 5*| 183 18 +6 |—18 | 168 10*| 84 17%): 22.222: 719 37 8 66 20%, 96 9* 125 22 | 155 11 | 184 23* +7 |—17 | 198 21 OO (23))| ee FAS 9 OF 38 138%, 68 2 9715] 127 3* 156 16*/ 186 5 +8 |—16 |' 229 7# 115) 4.) 220.2222: 10 6 39 19 69 7*| 98 20*| 128 9] 157 22 | 187 10* +9.|—15 |-°259 18 | 130 “OF S222) Lo. 11 12 41 0* 7013] 100 2] 129 14% 159 3*) 188 16 +10 |—14 | 290 4* 145 14*|__..._... 12 17*| 42 6 7119] 101 7* 13020] 160 9 | 189 21* +11 |—13 | 32015): 160 20 |..-..2... 13 23 43 114) 73 O*| 102 138 | 1382 2] 161 14*/ 191 3 15 4% 44 17 74 6 | 103 18*| 133 7% 162 20 | 192 9 16 10 45 22*| 75 11*| 105 0] 1384138] 164 1*| 193 14* 17 15*| 47 «4 76.17 | 106 5* 135 18* 165 7 | 194 20 18 21 48 9+ 77.22*| 10711] 187 0] 166 12%) 196 1* 20 2%) 49 15 79 4 108 16*| 138 5* 16718 | 197 7 21 8 50 20%} 80 9*/° 109 22 | 189 11 | 168 23*) 198 12* ate Lilt =O) | Bese): oh DBT | BH) Max VEG: 22 18%, 52 2% 8115! 111 3* 140 16% 170 5 | 199 18 A219.) 5!) oe. 24 282 13%) Sa. VoES 23 19 53 8 82 20*| 112 9% 144 22] 171,10*! 200 23* +220.) —4 | Bee) Te. 297 (19) bees see 25 OF 54 18%) 84 2) 113 15 ):°143 38% 172 16*/ 202 5 +21.) —3 |..-....0- 53 TI tes Sn 26 6 55 19 85 7*| 114 20*| 144 9 | 173 22-| 203 10* ECA in| Se, cee 32 SiO ee ies 27.11% 57 OF 8613 116 2] 145 14%) 175 3% 204 16 Se} a Ui a ci SASL Os Cee eee 28 17 58 6 87 18*| 117 7* 146 20) 176 9 | 205 21* | | HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLE 31.—For construction of primary stencils—Continued. 291 Difference. Component 2SM. Component J. Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h d. h d. h 0 207 3) 23616] 266 4* 295 17*| 325 61] 354 19 eG, 26are SL 18.) 77 8* +1 |—23 208 8*| 237 21*| 267 10 296 23 326 11*| 356 O0* 13* 71 ie 52 19*| 78 10* +2 |—22 209 14 239 3 268 15*| 298 4% 327 17 357 6 215 28 6 53 21 79 12 +3 |—21 210 19*| 240 8*| 269 21 299 10 328 22%) 358 11* By LZ 29 7* 54 22%) 80 13* +4 |—20 D2 1 241 14 271 2% 300 15*) 330 4 359 17 4 18* 30 9% 56 O*) 81 15 +5 |—19 213% 242 19%] 272 8 301 21 331 9% 360 22* 5 20 31 11 Dey 2; 82 17 +6 |—18 214 12%) 244 1 273 14 303 2*| 332 15 362 4 6 21* 32 12* 5S 3*| 83 18* +7 |—17 215 18 245 6*| 274 19* 304 8 333 21 363 9* 7 23* 33 14 59 «5 84 20 +8 |—16 216 23*| 246 12 216 1 305 13%; 335 2*| 364 15 CO) pal 34 16 60 6*| 85 21* +9 |—15 218 5 247 17%| 277 6* 306 19 336 8 365 20* Lom 25 35 17%* 61 8*) 86 23* +10 |—14 219 10*| 248 23 278 12 308 O*| 337 13*/- 367 2 ll 4 36 19 62 10 88 1 +11 |—13 220 16 250 4%*| 279 17#| 309 6 338 19 368 7* 12 6 37 20* 63 11*; 89 2* +12 |—12! 221 21% 251 10] 280 23 | 310 11*| 340 0O*| 369 13 13 7*| 38 22% 6413] 90 4 +13 |—11 | 223 3 252 16 282 4% 311 17 341 6 370 18* 14 9 40 0 65 15 91 6 +14 |—10 224 §8# 253 21*| 283 10 312 23 DAD is | ae ee eee 15 10* 41 1* 66 16*| 92 7* +15 | —9 225 14 255 3 284 15*| 314 4% 343 17 |_......-. 16 12* 42,3 67 18 93 9 +16 | —8 226 19%) 256 8*/ 285 21 315 10 CAA IES 17 14 43 5 68 19%*| 94 10* +17 | —7 228 1 257 14 281) 28) SIGMESs| 346) 4a eee ae 18 15* 44 6* 69 21*| 95 12* +18 | —6 | 229 6% 258 19*| 288 8| 317 21 Ae 2.) ee 19 17 45 8 70 23 | 96 14 +19 | —5 230 12 260 1 289 Sal 319) P24) S48 othe le ees 20 18* 46 9% 2 O* 97 15* +20 | —4 231 17*| 261 6*| 290 19 320 8 O49 20%) 2 PES 21 20* 47 11* oF 21> 98517 +21 | —3 232 23%, 262 12 POZE Os ozletac oo leon ane eee 22 22 48 13 74 4 99 18* +22 |} —2 234 5 263 17*| 293 6*| 322 19 OZ AEE sae 23 23* 49 14* 75 5*| 100 20* +23 | —1 235 10*| 264 23 294 12 B24 Os saa loses eee es PA 1 50 16 76 7 | 101 22 Difference. Component J. Hour d. h. d. h. Gh. lie d. h. asthe Gane d. h d. h d. h. d. h. 0 102 23*| 128 14%) 154 5* 179 20*| 205 11*| 231 2 256 17 282 8 307 23 | 383 14 +1 |—23 |} 104 1 129 16 T5570 180 22 206 13 232 4 257 18*| 283 9*| 309 O*| 334 15* +2 |—22 105.3 130 18 156 8*| 181 23*) 207 14%) 233 5% 258 20*| 284 11*) 310 2 | 335 17 +3 |—21 106 4*| 131 19% 157 10%) 183 1 208 16 234 7 259 22 285 13 311 4 | 336 18* +4 |—20 107 6 132 21 158 12 184 3 209 18 235 8% 260 23% 286 14%} 312 5*) 337 20* +5 |—19 | 108 7*| 133 22*) 159 138*| 185 4% 210 19% 236 10*| 262 1 287 16 | 313 7 | 338 22 +6 |—18 109 9* 135 O* 160 15 186 6 241 21 237 12 263 3 288 18 314 8*) 339 23* +7 |—17 110 11 136); 2 161 17 187 7*| 212 22% 238 13%) 264 4% 289 19%! 315 10*| 341 1 +8 |—16 111 12*| 137 3% 162 18*| 188 9*| 214 O*| 239 15 265 6 290 21 316 12 | 342 3 +9 |—15 112 14 138 5 |. 163 20 189 11 215 240 17 266 7*| 291 22%) 317 13*| 343 4* +10 |—14 113 16 139 6*| 164 21% 190 12%; 216 38*| 241 18*| 267 9* 293 O*| 318 15] 344 6 +11 |—13 114 17*| 140 &*| 165 23*) 191 14 217, 95 242 20 268 11 294 2 319 17 | 345 7* +12 |—12 115 19 141 19 167 1 192 16 218 6*| 243 21*| 269 12*| 295 3*| 320 18%) 346 9* +13 |—11 116 20*, 142 11% 168 2* 193 17*| 219 8*| 244 23% 70 14 296 5 321 20 | 347 11 +14 |—10 417 22%) 143 13 169 4 194 19 220 10 246 1 271 16 297 6*| 322 21%) 348 12* +15 | —9 119 0 144 15 170 6 195 20*| 221 11*| 247 2% 272 17+} 298 8*) 823 23%) 349 14 +16 | —8 120 1*|. 145 16*| 171 7% 196 22*| 222 13 248 4 273 19 299 10 325 1 | 350 16 +17 | —7 121 3°} 146 18 172 9 198 0 223 15 | 249 6 274 20%) 300 11%} 326 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 3 225 18 251 9 277 0 302 15 328 6 | 353 20* +20) —4.| 124 8 149 23 }.175 14] 201 5 226 19%) 252 10%; 278 1% 303 16*| 329 7*| 354 22* +21 | —3 123 9* 151 O* 176 15*| 202 6% 227 21*| 253 124] 279 3 304 18 330 9 | 356 0 +22 | —2 126 14*) 452 2 177 17 203 8 228 23 254 14 280 5 805 19*| 331 10*| 357 1* +23} —1 127 13 | 1153 4 178 18*| 204 9* 230 O*| 255 15*| -281 6*| 306 21* 332 12*| 358 3 292 U. S. COAST AND GEODETIC SURVEY. TaBLE 31.—For construction of primary stencils—Continued. Difference. |Comp. J. Component OO. Hour d. h. d. h. d. h. h. h. d. h. d. h. d. h. d. h. | De ee 0 359 5 10) 13 22 27 2 40 6* 53 10*| 66 14*| 79 18*| 92 22*| 106 2* +1 |—23| 360 6* 7*| 14 11* 15* 19* 23*| 67 3* 80 7*| 93 11* 15* +2 |—22] 361 8 20%} 15° OF} 28 4% 41 8* 54 12* 16* 20*| 94 1) 107 5 +3 |—21 | 362 9* 2 OF 13* 17* 22 55 2 68 6 81 10 14 18 +4 |—20 | 363 11* 23 16 3 29 7 42 11 15 19 23 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* +8 |—16 | 367 18 5 3% 18 7* 31 11* 15* 19% 23*| 84 3% 97 8 | 110 12 +9 |—15 | 368 19* 16* 20*| 32 1 45 5 58 9 71 13 17 2b +10 |—14 |} 369 21 6 6 19 10 14 18 22 72 2 8 6 98 10 14 +11 |—13 | 370 22* 19 23 33 3 46 7 59 11 15 19 23*| 112° 32* Ady 12 aie lee Clihas) 20 12 16* 20*, 60 OF 73 4* 86 8* 99 12* 16* Seah ih ie eae 21*| (21 1% 34 5*| | 47 OF 13* 17* 21*| 100 1*| 113 5* = 14 ))——10) |e. et 8 10* 14* 18* 22%/ 61 2*| 74 6* 87 11 15 19 154) 9 ose esse 234) 1722 3*| 358 48 12 16 20 88 0] 101 4) 114 8 AG) S89) fe = hte Se 9 13 17 21 49 1 62 5 75 9 13 17 21 SLT) RG ee Sk 10 2 23 6 36 10 14 18 22 89 2* 102 6*) 115 10* sal 8) — Gi) foes i ae 15 19 23*| 50 3*| 63 7* 76 11* 15* 19* 23* tal Ol Oy) [ete eae Tae 24 8s 37) 12% 16* 20*| 77 OF 90 4*| 103 8#| 116 12* +20 | —4 Jeol... 17* 21*| 38 1% 51 5* 64 9* 13* 18 22) 117 2 +21 | —3 |.....-..- 12 6* 25 10* 15 19 23 78 3 91 7] 104 11 15 SPs || Mesilla ay 20 26 0 39 4 52 8 65 12 16 20; 105 0; 118 4 2301) eek as 13 9 13 17 21 66 1 79 5 92 9* 13* 17* Difference Component OO. Hour. d. h d. h d. n. | d. h d. h d.h d. h.| d.h. 0 145 14*| 158 18*) 171 22*| 185 2*| 198 6% 211 11 | 224 15 | 237 19 +1 |—23 146 4/ 159 8] 172 12 16 20} 212 0] 225 4] 238 8 +2 |—22 17 21) 173 1) 186 5| 199 9 13 17 21 +3 |—21 147 6] 160 10 14 18 22 | 213 2% 226 6*| 239 10* +4 |—20 19* 23%) 174 3% 187 7*| 200 11* 15* 19* 23* +5 \—19 148 8* 161 12* 16* 20*| 201 OF 214 4*| 227 8*| 240 12* +6 j—18 21*| 162 1% 175 5*| 188 9* 14 18 22) 241 2 +7 I-17 149 11 15 19 234] °':202)°3 | "2159 71) 8228 a8 15 +8 |—16 150 0] 163 4]; 176 8] 189 12 16 20 | 229 0} 242 4 +9 |—15 13 17 21} 190 11] 203 5*| 216 9* 13* 17* +10 |—14 151 2*| 164 6*| 177 10* 14* 18* 22%; 230 2*| 243 6* +11 |—13 15* 19* 23*| 191 3*) 204 7* 217 11* 15* 19* +12 |—12 152 4* 165 8*| 178 12* 16* 21]) 218 1] 231 5] 244 9 +13 |—11 18 22 | 179 2) 192 6] 205 10 14 18 22 +14 |—10 153 7 | 166 11 15 19 23 | 219 3| 232 7} 245 11 +15 | —9 20) 167 0; 180 4] 193 8} 206 12* 16* 20*} 246 O* +16 | —8 154 9% 13* 17* 21*| 207 1*| 220 5*| 233 9* 13* +17 | —7 22*| 168 2* 181 6* 194 10* 14* 18* 22*| 247 2% +18 | —6 155 11* 15* 19* 23*| 208 4] 221 8 | 234 12 16 +19 | —5 156 1{ 169 5} 182 9] 195 13 17 21 | 235 2 | 248 5 +20 | —4 14 18 22 |} 196 2) 209 6] 222 10 14 18 +21 | —3 157 3} 170 7} 183 11 15 19* 23*| 236 3*| 249 7* +22 | —2 16* 2 184 OF 197 4* 210 8¥| 223 12* 16* 2 +23 | —1 158 5*| 171 9* 13* 17* 21*| 224 1* 237 5* 250 9* HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLeE 31.—For construction of primary stencils—Continued. Difference. Hour. d. h. 250 23 +1 |—23 | 251 12 Component OO. 293 d. h d. h d. h. 290 11 | 303 15 | 316 19 291 0} 304 4] 317 8* 302 12* 17 21 303 2 329 10 Ghd Te 356 7 Gh Tie 369 11 20*| 370 0* 294 U. S. COAST AND GEODETIC SURVEY. COMPONENT J. TaBLE 32.—Divisors for primary stencil sums. Series. 29 58 87 105 | 1384 | 163 | 192 | 221 279 | 297 | 326 | 355 | 369 Hour Oe eee We 30 59 87 | 106 | 134] 164) 192] 221) 250] 279| 298] 826) 355 370 Dig eh eae 31 59 89 | 106 | 1385] 164 | 193 | 222} 250] 280| 298] 327} 356 369 ese Na ae 28 58 86 | 104] 1384] 162) 192} 220] 250 | 278 | 296 | 326) 354 369 Bsa est ee AN) 30 59 88 | 106} 135 | 165) 192} 222) 251 | 280! 299] 326| 356 370 Fa eA SH SER 29 59 88: 104] 1385] 163 | 193 | 222) 250) 280] 297) 327] 355 369 ty) a crane Sel) 28 59 87 | 105] 1384] 163 | 193 221 | 251 | 278 | 297 | 326) 355 370 Gere es sey ee 30 57 88 | 106! 184] 165} 192] 222| 250] 280] 298] 326 | 356 369 CU eateneh aan: 28 58 87 104 | 134 163 193 221 250 279 297 | 327 | 354 369 Coles Fseioned a) Velleate its 29 58 88 | 106} 184] 164) 193 | 222] 251 | 279) 298 | 326] 356 371 CO) alae i A 29 57 87 | 105) 184] 163 | 192} 222] 250) 280} 297 | 326) 355 369 TOE ae oes ee, 28 58 86 | 104; 134] 162} 193 | 220} 250] 278 | 297 | 326) 354 368 nH ie rears ened 30 59 88 | 107 | 134] 164) 193 | 223 251 | 280 | 299} 327) 357 370 I -pacaeneseiea ohare 29: 57 87 1 104} 184! 162) 191] 221 | 250! 279| 296! 326) 354 368 TSS RRS 28 58 85 | 104 | 133) 162 | 191 | 220| 250) 278) 297 | 325] 354 368 Paes Sa 30 58 88 | 106 | 134] 164) 192; 223) 250] 280) 297] 327) 356 369 LAS) Be aE a) Ps Dh 29 58 87 | 105} 135 | 162) 192) 220] 251! 279 | 296 | 327) 355 369 LG ee elie 28 58 86} 105) 133} 163 | 191 | 220} 250] 279 | 297; 325) 355 369 IY (ea Nae at Oey 30 57 87 | 105; 1384] 163 | 192) 221 | 250} 280! 296 | 326} 355 368 Maes ae ears Sera 28 58 86 | 104) 134] 162) 192 | 220) 250} 278} 296 |} 325] 355 369 TS false eee a 29 58 87 | 106.| 133 | 163} 191 | 221 | 249] 280) 297 | 325) 356 369 PA Jay to ebay fe ark 29 57 87 | 104) 134] 162) 191 | 220} 249} 279) 296) 326] 354 368 DSRS UE i 28 58 85 | 104! 133 |° 162) 191 | 219 | 249 277 | 296 | 325} 354 369: Pp AONE SARIN ES 30 58 88 | 106 | 134} 164) 192 | 222| 249) 279! 208 | 326] 356 369 Pate as a ge late 28 57 86} 104] 184] 161 | 191 | 219 | 249] 277) 295) 325) 353 368 COMPONENT K. Series 14 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour ORS. 15 30 59 88 | 106 | 185} 164] 193} 221 | 250) 279} 297) 326) 355 369 Lippe Were 14 30 59 88 | 106 | 135] 164] 193 | 222) 251} 279] 297 | 326) 355 369 PAP eases 14 29 59 88 | 106} 1385] 164) 193 | 222] 251 | 280} 298) 327 | 355 369 Oar eeee 14 29 59 88} 106 | 185} 164) 193 | 222) 251) 280] 298) 327) 356 370 Ape EO 14 29 57 87 | 105; 184] 163 | 192] 221 | 250 | 279; 297 | 326)! 355 369 Devinn. 14 29 58 88 | 105} 134] 163) 192) 221 | 250) 279) 297 | 326) 355 369 Gree 14 29 58 87 | 106 |} 1385] 168 | 192] 221 | 250 | 279] 297! 326) 355 369 hoes 14 29 58 87 | 105 | 1385] 164) 193] 221 | 250] 279| 296] 325) 354 368 Sune nee 14 29 58 873) 105) |) F354 1645) 193)) 22257 251 | 280298) | Sere eecon) 369 bo Nee ee 14 29 58 87 | 105 | 184] 164] 193 | 222) 251) 280] 298) 327) 356 370 LOWE Aa 14 29 57 86 | 104] 133} 163] 192) 221 | 250 | 279} 297} 326) 355 369 ai oie 14 29 58 8%) 105) |) MSS 62 ))) 192) 2200 2501 e279) 2978s oeenlimioas 369 pee See 14 29 58 87 | 105! 184] 163 ' 192] 221! 250) 279! 297} 326! 355 369 Tey Arete ee 14 29 58 87 | 105} 1384] 163] 192] 222; 250] 279; 297 | 326) 355 369 PAre ae & 14 29 58 ST 1050) 1349" 163) | 192) 2220 251) 280s 2978 Seon meaoo 369 dls} at Sl Fe 13 28 57 86 | 104} 183 | 162) 191 | 220| 250 | 279) 297) 326) 355 368 LEE ar 14 29 58 86 | 104] 133 |} 162] 191 | 220) 249] 279} 297 | 326] 355 369 (a aaeie 14 29 58 87 | 105 | 1383 | 162) 191 | 220] 249 | 279} 297 | 326) 355 369 d hoje em 14 29 58 87 | 105 | 134] 163 | 191 | 220] 249) 278 | 297!) 326| 355 369 14 29 58 87 | 105} 1384] 163] 192) 221 | 250} 278 | 297 | 326) 355 369 14 29 58 87 | 105} 134] 163] 192] 221} 250 | 279} 297} 326| 355 369 14 28 57 86 104 133 162 191 220 249 278 296 325 355 369 14 29 58; 86] 104] 133 | 162) 191) 220} 249 | 278 | 296 | 325) 355 369 25s Ln) 14 29 58 | 87 | 105 | 1384] 162] 191 | 220] 249] 278 | 296 | 325] 354 369 TaBLE 32.—Divisors for primary stencil swms—Continued. HARMONIC ANALYSIS AND PREDICTION OF TIDES. COMPONENT L. 295 Series. 29 58 87 105... 134 | 163.) 192 | 221 | 250 | 279). 297.| 326 | 355 | 369 Hour GOs = eae ~ de 2S 59 87} 105 | 133 | 163.) 191 | 221 | 250) 279) 297] 326) 355 369 143 Be ee ae 29 59 87 | 106+) 134] 164), 192)) 222) 251°) 279"! 297) 3261 355 | --369 Zip Vie ae i ae 29 58 87 | 106) 184) 163] 192] 221) 250; 250] 298) 326) 356 370 Ss Se oe So! | 30 58 87 | 105} 1384] 163] 192] 221 | 250] 279] 298} 326] 356 370 eae ae a oe 30 58 88} 106) 135.) 164) 192) 222] 250] 279: 297) 326) 355 370 agg nee ER Ma. 29 58 88 | 106) 1384] 164) 192| 222] 250] 280) 298] 327] 356 369 (C) lie ene gee tee 29 57 SGrle TOS5s), 133.) WE3y), PLE 220 2498) 279s) 297) 325.) 355 368 Pe cage yee 30 59 &8 | 196 | 135) 164] 193] 222) 250) 279) 298) 326] 356 369 PBR eS eras ce: 30 58 88 |. 105) 1385] 164] 193} 222) 251 280 | 298) 327) 357 37 | eo Oe Ne 29 57 87 | 104) 133], 163] 191} 221 | 250] 279) 296) 326 | 355 369 5 Se Se a a 39 58 87} 105} 134) 164] 192] 221 | 249] 279] 296) 326) 354 368 ee Heese = Soke 29 58 87 | 105} 134) 162] 192) 222) 259] 280} 297) 326) 355 369 1 CARERS Be aN ee ee 29 58 87 | 104] 1384) 162). 192] 221] 250] 279) 297} 326] 355 369 11S LAY Singha Sa eee 29 58 88} 105] 135} 163} 192) 220] 250} 279) 296) 326] 354 368 1: A) Be 0 sa a 29 58 88 | 105! 134) 163! 193} 221! 250] 280! 297) 327: 355 370 (Ge ae 28 58 86 | 105) 1384] 163] 192} 221 250} 279.) 297) 327] 3551 --369 5 |g) ses See 28 58 86 | 104] 134] 162] 191 220 | 249! 278) 296] 325) 333 367 die ot eee ee 28 57 86 | 104) 134] 162) 192) 220] 250] 278) 297| 326] 355 369 QR Eek es 29 58 87 | 105| 134] 162} 192) 220] 259) 278) 296) 326] 355 369 Bs Ue yee 2 = bs 29 58 87 | 105} 1385) 163) 192} 221) 250] 279) 297) 326) 354 369 AD Sai epee: 3. oe 28 58 86 |} 105} 134) 163) 192] 221] 250} 279) 297) 327) 355 369 A Ny see oi! es 28 58 86 | 104). 132] 162; 191) 219} 249] 277) 296) 324] 354 368 2A Sa Re 29 58 87 | 105) 184) 163] 193} 221 | 251) 279) 297) 325) 355 369 ERE pete /s 29 58 87 | 105} 1384] 163) 193) 221) 251} 279) 298) 326) 355 370 COMPONENT M. Series. | 15 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour Late 15 29 59 87} 105) 1385} 164} 192] 222] 250] 279) 297) 325] 355 369 ih See ee 15 29 57 87 | 105] 134} 163 | 192] 221] 250] 279} 296) 326! 354 369 Den Fae 15 28 58 86; 105} 134] 162} 192} 221) 250} 279] 296) 325] 354 369 Bee .3-55 16 29 59 88, 107) 1385 | 165.) 1938.{ 222°) 251 ') 281) 299°! 328 | 357 371 (nee 16 30 58 87} 106} 1385] 164] 193) 222) 251 | 280] 297) 326) 355 370 (ae ee 15 28 57 86} 104) 1384] 163] 192] 221 | 250] 278] 296 | 325| 354 368 Ges Jas 15 29 58 87} 106} 134] 1638} 192] 222] 250! 280} 297) 326] 355 369 Chere Bee 16 29 58 87} 105| 1384] 168} 192) 221) 250] 279] 296) 326) 354 369 82 aches 16 29 59 87.| 106) 135 | 164] 193] 221} 251) 280] 298] 326 | 355 370 © oe tee 15 29 58 Sil, LOG Wady) 1655), 193s) 2230) Zale) 20s) 298s 327 357 371 10% 2 1.82 15 29 57 87 | 105) 134), 163) 192) 221] 250] 279] 296] 326] 354 368 ah See ee 15 28 57 86}. 104.) 183] 162} 192; 221 | 250] 278. 296} 325 | ° 354 369 15 29 58 87 | 105) 183] 162] 191} 220) 250) 280] 297] 326) 355 368 15 30 59 88} 105) 134] 163] 192] 221), 250) 279{| 298) 327) 355 369 15 29 58 87.1. 105 |; 1384.1 163 |) 1921 220) 250) 278 | 2971 326 | 356 369 14 29 58 87 | 104) 13 163 | 192) 222) 250] 279) 298) 3261] 356 369 15 29 57 87 | 104] 1383] 162] i191} 220) 249) 278] 296] 326) 354 368 DE ote 15 29 59 87 | 105| 134] 162] 192] 220} 250] 279] 298] 326) 355 369 Ce Bee 14 29 58 SGale 10556 A835) 1635\y 191i, 2207), 2492) 278.) 297, 326.) 355 368 WO ef. 32 15 30 58 88 | 105) 135.| 163] 192} 221) 250) 279] 2971 326) 356 369 8 ee ee 14 28 57 86 |. 103). 1383] 162] 191.) 220.) 249) 277! 296] 325} 354 368 he -). 3 14 29 58 87} 105) 133] 162] 192] 221) 250) 280] 298] 3271! 356 369 De = i 3 15 30 59 88 |. 105}. 134}, 163} 192} 221 | 249] 279] 298] 327 | 355 369 23 ae 15 29 58 87) 105.) 134] 163] 1921] 220) 250) 278! 296] 325) 355 369 296 TABLE 32.—Divisors for primary stencil sums—Continued. U. S. COAST AND GEODETIC SURVEY. COMPONENT N. Series. | 15 29 58 87 105 |. 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour feats 16 29 58 87 | 105} 134] 163] 191 | 220, 250} 279| 297) 327] 356 370 7 aes 16 29 58 88 | 106] 135] 165] 194] 223) 252] 281 | 299| 327) 357 370 Deel 15 29 57 87 | 105] 1383] 162} 191 | 220) 248] 278] 296] 324] 354 367 Sa ahs 16 30 58 88 | 106| 134| 163} 192} 221] 249| 279] 297} 326) 355 370 As aia 16 30 58 87 | 105} 1385] 164] 193] 2238) 252] 282) 299) 328) 357 371 ert 15 30 59 88} 106| 134) 164] 192} 222|) 250| 279| 297) 326) 355 369 Genwi 15 29 58 87 | 105| 133] 163] 191 | 221 | 249) 278] 296) 324] 354 367 ce ae 15 29 58 87 | 105); 133] 163] 191] 220) 250} 279] 298] 326) 357 370 Beicrixe 14 29 58 88 | 107) 185] 164] 194] 223] 251) 281 | 299) 3271! 356 370 QO Daa 15 30 58 88] 105} 134] 163} 192} 221 | 249) 279} 297) 325) 354 368 Oo se 15 30 58 88 | 105} 1384] 163 191 | 221} 249| 279] 297) 326) 356 370 Pe 15 30 58 86 | 106} 1385} 165] 193] 224] 252) 281} 299) 328) 357 371 Pe 15 28 59 87 | 106} 134] 164] 192) 220) 250) 278) 297) 3257) 355 368 ils pee 15 28 58 86 | 104} 1383} 161] 191! 219] 249) 277| 295 | 324) 354 368 nea ae 14 28 57 s6 | 104] 133) 161] 191! 220} 250! 279] 297) 326| 354 369 Ga ee ) 14 29 58 88 105 135 164 194 222 251 280 | 298 327 355 370 IGe ees eto 29 58 86 104 134 162 191 220 249 277 295 325 353 368 1 (eee aes 15 28 58 86 | 104) 134] 162] 191) 220) 249) 278| 296| 327) 355 370 Se ose 15 28 58 87 | 105 | 1384] 164) 193} 222) 252) 280) 298] 327) 356 371 1 ee Se 15 29 59 87} 105} 1384] 163] 192] 220] 250) 278] 296) 325] 354 367 20 in bss 14 28 57 86 | 104] 183] 161] 191] 219] 249) 277) 295) 325] 353 367 21. tas 14 28 57 86 | 103 | 1383) 161] 192] 220] 249) 279) 297|] 326) 354 368 DPA Lite 16 30 59 88 | 106) 137) 165 | 194] 223] 252 | 281 | 298) 328] 356 370 Do scones 15 29 58 Be 1047" 1837)" 162) 191 2207 249) 27a 295s ie s2on ioe 367 COMPONENT 2N. Series. 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour (Pip ta Ca date 28 58 86.) 105 | 135) 163} 193 | 222) 251) 280} 299] 3827) 357 371° re oh yea tersettats 30 58 88 106 135 165 194 223 252 281 299 329 357 371 AMIE ie nage CAN at 28 58 87 | 105] 134] 164] 193 | 222] 250! 279| 297) 325) 353 368 SHAE Cat sts 30 59 88 | 106} 135 | 164] 193 221) 251] 280) 298] 326] 356 370 CMRI Uitte Bt 29 57 86 | 104| 132] 161] 190| 220] 249| 278] 295] 325] 353 368 inet Ga RE 28 58 86 | 105| 134] 163] 192] 222] 251 | 280] 298| 326| 356 369 Gisced eres 30 58 88 | 106] 185] 164] 194] 222] 252} 281) 298] 328) 356 370 REE RS nd SS BS 29 59 88 106 135 165 193 223 251 280 | 297 325 354 368 be argh oa aah 29 59 88 106 135 163 192 | 220 249 279 | 296 325 355 368 tS Semper nee IS hc 29 57 86 | 104] 133 | 162] 191 | 220] 250) 278| 296 | 326] 354 369 1 KU) TS RN nk 29 58 87 | 106} 185] 164] 193) 223] 251] 280) 298] 327] 357 370 dL Nae HB aha slg 29 58 87 | 106} 135] 164] 194) 222] 251) -280) 298} 326] 355 369 PDs als bates tolene 29 58 88 | 106] 185] 165) 193 | 221 | 249] 277) 295) 325] 354 368 3 29 59 87} 105} 134] 162) 190} 219] 248) 278 | 296) 325] 354 368 - 29 57 86 | 1041 183] 161! 191 | 220! 250) 278! 297] 3261! 355 370 TUS hye 2 ha 29 58 87 | 105] 133] 163] 192] 222] 250| 280) 298) 327) 356 370 See oe ee 29 58 87.) 104} 1384] 163] 192} 221] 251 | 279) 297) 325) 354 368 d iy RE Seapets 29 58 88] 105} 184} 163 | 192| 220} 249] 278] 296} 326) 355 369 1) a ee SEN 29 59 87 | 104} 1382] 161) 189} 219} 248) 278] 296] 325) 354 368 Qe Peete sae 29 57 86 | 103 | 1383} 161] 191 | 220] 249] 278| 297! 326] 355 369 QO ae Seeders 30 59 88 | 106; 1384] 164] 193] 222] 251 | 280; 299] 328) 357 371 Bate eet ee ae 28 58 87 | 104) 1384] 163} 192] 221} 250] 279] 297] 325 | 354 367 PPAR LYS acta? Blaine 30 58 87 | 106 | 185] 163} 192| 220] 249] 278) 296| 326) 355 369 2a cwiste ew oceans 28 56 85 | 103 | 131} 161 | 189) 219] 248) 277) 295) 325) 354 368 a HARMONIC ANALYSIS AND PREDICTION OF TIDES. 297 TaBLE 32.—Divisors for primary stencil sums—Continued. COMPONENT O. Series. | 14 29 58 87 105 | 184 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 me Hours Oa 13 29 58 87 | 106] 1385) 164] 192) 222} 251 | 279} 298 | 327) 355 369 1 Cains a 14 29 59 88 105 134 164 192 221 251 280 298 327 355 369 Do as 14 28 57 86 105 133 162 192 221 250 279 296 325 354 368 Oe Lae 14 30 57 87 105 134 164 193 221 251 280 297 326 356 370 Be UR 14 29 58 87 106 135 163 193 222 |. 250 280 297 325 354 369 (aye a 14 29 59 87 105 135 163 192 222 251 280 297 326 355 369 (yas ee 14 29 58 87| 105 | 1384] 164] 193 | 222) 251 | 279] 297 | 326| 355 369 Wee nae 14 28 58 87 105 135 164 192 222 251 280 297 327 355 369 {agit 14 29 58 88 | 106 | 134) 164] 193} 221 | 250] 278] 296) 325) 355 368 eee tt 15 36 58 SM LOG W495) L634 T1938 221 2500s 2790 297 38265 isbo 369 OL Soe 14 29 58 87 | 105} 185) 164 193 | 221) 249] 279) 297] 326) 356 370 1h) | Speen 14 30 59 88 | 107| 186] 164] 192] 222) 251] 280) 2991 327} 356 370 tie se 14 29 59 87 105 135 163 192} 221 250 | 279 297 327 | 355 369 11 eee 13 28 57 87 104! 1382 161 189 | 219! 248 276 295 324} 253 367 i aaa 14 29 58 S| LOs i tase LGan) 1O2n 2200 250 2i9) ome raat tooo 369 il ae se 14 29 58 87 105 133 162 192 221 250 | 280) 297} 326) 355 370 Li ae 14 30 58 87 104 134 163 192} 221 250 279 | 298] 3251] 355 369 Fe, LE NE 14 29 58 86 | 104) 1383] 161) 191 | 220 |° 248) 277) 296) 325) 354 369 ihe 13 28 58 Si 04 18455 16S) LOI oot 25On 20 e297. eset) ooo 369 1G peers 14 29 58 88 | 104) 1383] 163} 192) 221) 250! 278) 297) 326) 355 368 7) | ea 15 29 58 87) 105 | 134] 163) 192] 220) 250) 279) 296) 326) 355 369 Qe cee 14 29 58 87 105 133 163 192 220 249 279) 297 326 | 356 370 7 ASA 15 30 57 86 105 134 162 191 | 221 250 279 298 | 326 355 369 oseisae 14 28 58 86 | 104) 1384] 162) 192] 221) 249) 279| 297) 326) 355 369 COMPONENT OO. Series. 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour (EE Sa 29) 58 86 | 104} 184] 163) 192} 221 | 250] 280) 298] 326) 355 369 TBA Be ee 30 60 88 | 107 | 186] 164) 193 | 221) 250! 279} 297} 326) 355 369 Pe le Rhy 1 aE 29 57 86} 103) 133] 162) 192} 220) 250| 280) 297] 327) 355 369 Se oe ae 31 60 89} 107 | 1387| 166} 194] 223] 251] 281 | 298) 327] 355 370 EUR Op oe eae 30 58 87 104 | 182} 162 191 220 | 249] 278) 297] 326] 355 369 AU Oe 29 59 88 | 106) 1385] 166) 194) 223; 251.) 280) 298| 326] 355 369 Ge ERP 28 58 86 104 132 161 190 219 | 249 | 277| 297 325 355 368 OR abies 29 58 88 | 105} 1385| 165) 194] 223.) 251) 280] 298) 327] 355 369 Set SUM. 18) 29 59 88} 105} 1384] 163] 191] 220} 249) 278) 297) 326] 355 369 CRS a ECR 29 58 87 105 134 163 193 223 251 280 298 326 | 355 369 QS ee a Y 28 58 87 105 133 162 191 219 | 248 277 | 296 325 | 355 368 1G Rae erat Ue 28 57 87 | 104] 134) 162) 193 | 222] 251) 280) 299) 327] 355 369 TOAIEOI ee aes BS 29 58 88} 106} 185} 163) 193] 221) 250| 278) 296} 326) 355 369 29 57 87! 104] 133! 162) 192! 221 | 250! 280} 297! 328] 356 370 30 59 88 107 135 164 192 222 | 250 278 296 | 325 | 354 369 28 57 85 | 104] 132} 162} 191) 220] 250) 279} 297) 327] 356 369 29 58 87 106 185 164 193 222 251 279 | 297 326 354 369 29 57 86 | 104] 133 161) 190 | 220) 249] 278) 296; 326] 355 369 30 58 88} 106 | 135] 165] 193] 224) 252| 281} 298| 327 | 356 370 28 57 85| 104] 1382| 161 | 189] 218} 248] 277) 295} 323] 354 368 29 58 87 106 135 164 193 222 252 | 280] 298 326 | 356 369 28 58 86} 104] 133] 161} 190} 218 | 248] 277} 295) 324) 384 368 29 57 87 | 105} 135) 163) 1938] 222) 251) 281} 298| 327) 356 370 29 58 87} 105) 134) 163] 191} 220) 249] 278) 295) 325] 354 369 298 U. S. COAST AND GEODETIC SURVEY. TABLE 32.—Divisors for primary stencil swums—Continued. COMPONENT P. Series 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 Hour Ee Me a ET 29 58 87 | 105) 135) 164) 193] 222) 251) 280) 298] 3827) 356 369 Te eee 29 58 87 | 105] 134] 163 | 192] 221] 250] 279| 297} 297) 354 368 Se ee ee a 29 58 87 | 105 | 134} 163] 192] 222] 251) 280] 298) 327) 355 369 ae le, weg Lie 29 59 88 | 106} 135} 164] 193 | 222) 251} 280] 298) 327) 356 370 ares: eerie 29 58 87 | 105} 134] 164] 193 | 222 | 251} 280] 297) 326) 355 369 RAP Pes te eee eS 29 58 87. | 105} 134} 163} 192] 221] 250} 279] 296) 325) 354 368 Gee ce leh: di 29 58 87 | 105 | 134} 163 / 192] 221 | 251 | 279] 297 | 326] 355 369 pes a Mes Le 29 58 88 | 106] 135) 164] 193 | 222] 251) 279} 297) 326] 356 370 RAE ee eee 29 58 87 | 105] 134) 63] 192] 221) 249] 278) 296) 325] 354 368 QP us! Mienpecel 33nd 29 58 87} 105] 134} 164] 193) 221} 250) 279! 297) 326] 355 369 SC ere ae 29 58 87 | 105 | 134! 163] 192] 220} 249) 279] 297 | 326] 355 369 1) eae eee 29 58 88 | 106) 135 | 164] 192} 221] 250) 279) 297 | 326] 356 370 PARR foe Dene 29 58 87} 105] 134] 163] 191] -220| 249) 278} 296 | 325) 354 368 Re EIN 29 58 87) 105] 134! 162] 1921! 221 | 250) 279! 297 | 326! 355 369 Ys aR a ae 30 59 88} 106] 135} 163] 192) 221} 250) 280); 298} 327) 356 370 GPS heb ear 29 58 87 | 105} 133] 162} 191) 220} 249} 278) 296 | 325) 354 368 MG pe erie 29 58 87 | 106 | 134} 163] 192] 221). 250) 279} 297) 326] 355 370 i by (oe eet omer 29 58 87 | 104] 133} 162] 192] 221] 250) 279} 297) 326] 355 369 30 59 87 | 105) 134} 163] 192) 221 | 250) 279} 298) 327] 356 370 29 58 86 | 104] 133 | 162] 191 | 220} 249] 278) 296 | 325) 354 368 29 57 86 | 104] 134] 163] 192 | 221) 250) 279) 297 | 326) 355 369 29 57 86.| 104] 133 | 162] 191 | 221) 250) 279) 297 | 326) 355 369 28 57 86 | 104 | e133 | 162] 191 | 220) 249} 278) 296 | 325) 354 368 28 58 87 | 105] 1384] 163] 192} 221] 250] 279} 298) 327)| 356 370 COMPONENT Q. Series. 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 29 59 88 | 106 | 136} 164] 194] 222] 250] 280] 297] 326] 355 368 29 58 86 | 104) 1383 | 162) 191 | 221 |, 250 | 280] 298} 327] 357 370 29 59 88 | 106 | 1385] 165] 193} 222] 251} 280] 298} 326] 354 368 28 56 86 | 103 | 132) 161} 190 | 220). 249 |) 278 | 297|] 326) 354 369 30 59 89 | 107 | 136} 166] 195} 225] 253] 282) 299) 328) 356 370 29 58 87 | 105} 133} 162) 191 | 219} 249) 277) 296 | 325] 254 369 Gee eae dos} ah) 59 88 | 107} 136], 165) 195 | 224; 254 / 281 | 300) 328] 356 371 Case ps 30. 58 87] 104] 133 | 162] 191} 219] 248; 277} 295) 324) 354 369 Boa swesatias 28 58 87 | 105} 135} 164) 194] 223] 251] 280} 298) 326] 356 369 eae Gee Some er 29 59 88 | 106} 135} 163) 192) 221} 248] 278 | 296 |) 325] 355 369 NO OS Sel es 28 58 86 | 104} 134] 163} 192] 221 | 250] 280} 298) 327) 355 370 AU Aya Serene pete 29 58 88 | 106 | 134] 164) 192] 220! 249} 277! 295] 324] 353 368 1 Oe 29 57 86} 104] 133] 163] 191] 220} 250) 279} 297) 327] 356 370 a sy ae chee yes 30 59 89 | 107! 136} 165! 192] 2211! 250} 2781! 296) 3251! 354 369 LW Sach is aah Pa 29 58 87 | 104] 133 | 161) 191 | 220; 250) 279} 297) 327) 356 371 1s Sela ee 30 59 88 | 107] 136] 164) 194] 222] 251] 280} 297 | 326) 355 368 SEE Reereaae ee 29 58 86 | 104] 133] 161) 190] 219} 248] 278} 296) 325) 355 368 iO Seca creses 29 59 87] 106} 1385] 164] 193] 223) 251 | 280} 298] 326] 355 368 DR See acne see 28 57 85 | 103] 131 | 160) 188] 218} 247] 277} 295) 324) 354 367 Sted AAR ae 29 58 87 | 105] 134] 164] 193) 223] 252] 281} 299) 328) 356 370 PA IGA MD, bre ten 29 57 86 | 104] 132] 161] 190| 218 | 247] 276} 294) 324) 353 367 7 See Besse aes 30 57 87 | 105} 134] 164} 193 | 222] 252] 281} 299} 328) 356 370 PAGER De arash 29 58 87 | 105} 134] 162) 191.) 220] 249) 277) 295) 325) 354 368 Dee |: prema ae 27 56 85 | 103] 1383] 162] 192] 221 | 251 | 280] 298] 327) 357 370 HARMONIC ANALYSIS AND PREDICTION OF TIDES. 299 TABLE 32.—Divisors for primary stencil sums—Continued. COMPONENT 2Q. Series. 29 58 87 105 | 134 | 163 | 192°) 221°] 250 | 279 | 297 | 326 | 355 | 369 25 50 83} 113 | 142) 167) 192} 217 | 242) 279) 309) 334] 359 371 25 59 | 101] 116} 141] 166) 191 | 216} 255} 293) 308} 333} 358 370 36 77} 102} 117}. 142) 167} 192) 233) 269) 293) 309) 334) 359 371 39 64 89') 104} 129) 154] 196} 2301) 255} 279 | 295} 320) 345 366 25 50 75> 90) 115 }' 159 | 192} 217') 242) 266 282°} 307) 355 370 25 501° 75 90} 13 167 | 192} 217) 242) 266) 282) 332). 358 370 Gh. 3 eS 2 SE 25 50 83} 113] 142-)' 167 | 192} 217} 241] 277] 309 | 334) .358 370 (RO 2 Sem 25 60'| 102} 117) 142) 167) 192 | 217 | 254) 293} 309} 334.) 358 370 tS a 36 76: 101} 116] 141] 166) 191; 232) 267) 292} 308°} 333.) 357 369 Oe. 1 Seb 39 64 89} 104) 129) 154] 197) 230} 255 | 280] 296] 320} 345 365 NUE. SOG. 1 1 20 50 75 90} 115} 159) 191 | 215) 240} 265} 281} 305] 353 369 1 CS 8 Se 25 50 75 90} 1386} 167) 192) 216) 241 | 266} 283) 331) 358 370 ae aa 25 50 83] 113! 142) 167! 192] 216! 241 | 2771 307} 332! 357 369 LS See ae 25 60} 101} 117} 142] 167) 191} 216) 254) 293) 308) 333) 358). 370 1 ee ee 37 771 101} 117) 142) 167) 191 | 231} 268) 293] 308} 333 | 358.) .. 370 1 ee Sea 38 63 87 | 103) 128} 153} 194] 229) 254] 279 | 294] 319) 344 364 Te. 51 POh. ee 25 50 74 90} 115] 159) 191) 216} 241 | 266) 281) 306] 354 370 2 Es 25 49 74 90} 186] 1657) 190) 215} 240) 265] 280) 330 | .357 369 Gu FLGGG 2 25 49 81) 113 | 142] 166] 191 | 216) 241 |) 278) 308 |} 333 | .358 370 19s GOR a 25 58 |° 101 | 117). 142] 166) 191 |} 216] 254.) 292, 3077} 332 |. 357 369 ZAR, a a a 36 76 | 101) 117} 141) 166] 191} 231) 268} 293] 308) 333 | 358.| 370 7 37 62 87 | 103) 127 | *152| 194 || 229} 254 | 279] 294) 319] 344 364 Daas INE «ole fs 24 | 49 74 90} 114] 158} 191 | 216 | 241 | 266} 281'| 306 |. 354 370 Dee be 8 24) 49 74 90} 185 |.166) 191) 216 | 241) 266) 281) 331 | 358 370 COMPONENT R. Series. 29 58 87 | 105 | 134 | 163 | 192 | 221 | 250 | .279 | 297 | 326 | 355%) 369 Hour. Oe fs ate OF 30 597} 88) 106}. 135} 164) 193 222} 251] 280} 298) 327) 356 370 1 [Sapa ope ce) Se ad 29 59 88 | 106) 135'}) 164°} 193 |' 222)): 251.) 280 298] 326). 355 369 7 ies a 29 58 88} 106) 135] 164} 193 | 222] 251) 279) 297] 326} 355 369 he Se ale Bee 29 58 87 | 105 |. 135'|' 164} 193} 2211) 250) 279} 297} 326) 355 369 aR TS OSE = 29 58} 87} 105} 134°} 163 | 192] 221} 250) 279] 297) 326). 355 369 Be. Sesh =ict- oe 29 58 86 | 104]. 133°) 162')) 192 |! 221:} 250°) 279} 297] 326) 355 369 OG...) ARE 28 57 86 | 104) 133°) 162} 191 |’ 221} 250 279) 297] 326) 355 369 A a ee ae ae 8 29 58 87 | 105.) 1384] 163} 192] 221} 251 { 280} 298] 327) 356 370 2 Sera 1S eae 29 58 87 | 105] 184) 163 | 192} 221} 250) 280.) 298) 327] 356 370 ea art SS hoi 29 58 87} 105) 134] 163} 192] 221] 250] 279} 298) 327 | 356 370 ai) eae Ree: | aS 29 58 87 | 105) 1384) 163) 192) 221) 250) 279} 297 | 327 | 356 370 11 Page Re. tat 29 58 87} 105) 134] 163}. 192) 221 | 250] 279] 297] 326}. 356 370 1 re Reo. OS 29) 58 87 | 1051 184) 163'}' 192) 221°] 250) 2791 297 | 326! 354 368 11 Spe ges =e ee 29 58 87} 105 | 184) 163) 192} 221} 250) 279 | 296) 325] 354 368 0 Te Ae 29 58 87} 105 | 134) 163} 192 | 221 | 249) 278) 296] 325] .354 368 LS Re 2 +} aE HG 29 58 87} 105 |; 1384) 163 |) 191 |) 220) 249°} 278° 296°) 325] 354 368 1 Se See ae nie 29 58 87) 105) 1383'} 162 )' 191 | 220} 249} 278] 296] 325] 354 368 a aa EQ 5 42 88 29 57 86} 104) 1383} 162] 191 | 220} 249) 278} 296) 325) 354 368 TS. RG: 508 29°) 58 87 | 105} 134) 163) 192) 221} 250] 279% 297 | 326 | 355 369 12 Saee a A 29 58 87} 105 | 134) 163} 192) 221} 250) 279 | 297] 326| 355 369 20..... Rak: SSS 29 58 87} 105} 134] 163} 192]! 221}: 250} 279 | 297} 326 | 355 369 7 ee A tk Se 29:| 584) 87} 105) 134) 163} 192) 221} 250°} 279) 297] 326 355 369 5 AR 8 ae 297° 58 87} 105) 134) 163] 192] 221) 250] 279.| 297:) 326) 355 369 bec ct faut 5 eS 29 58 87} 105) 134) 163} 192) 221} 250 279 | 297) 326) 355 369 72934—24}——_20 3800 U. S. COAST AND GEODETIC SURVEY. TABLE 32.—Divisors for primary stencil sums—Continued. COMPONENT T. . : ; | HARMONIC ANALYSIS AND PREDICTION OF TIDES. TABLE 32.—Divisors for primary stencil sums—Continued. COMPONENT uz. 301 Series. 192 | 221 | 250 192 | 223] 252 194 | 223] 252 192 | 221] 250 190] 219 | 249 192 | 222] 249. 193! 220} 250 190 | 220} 250 193 | 223} 251 194 | 222] 250 191 | 219} 249 191 | 221 | 251 192} 222{ 249 193 | 220} 250 191:} 221} 251 193 | 222} 250 193 | 222] 250 191 | 219} 249 190 | 220}; 249 193 | 222] 250 193.|) 221:}* 251 191 | 221} 250 193 | 222] 250 192 | 220) 249 190 | 219} 24! 355 369 302 U. S. COAST AND GHODETIC SURVEY. | TABLE 32.—Divisors for primary stencil swms—Continued. COMPONENT >». Series. 29 58 87 105 | 134 | 163°] 192'|'221 | 250-| 279 | 297 |: 326 | 355 1° 369 30 59 89 | 107} 185°) 164) 193 |! 222) 251 | 279 | 297] 326 | 355 369 29 59 88 | 105} 1384) 164) 193) 222) 251 | 280) 299] 328) 356 372 28 57 87} 105 | 134] 162°} 191 | 220) 249) 278) 295 | 324] 353 367 29 58 87 | 106} 135] 165) 194 | 224) 252) 281) 299) 328} 357] . 371 30 58 87 | 106 |}. 185'} 163} 192] 221) 250) 279) 297) 326] 355 370 29! 58 87 | 106 | 1841 163) 192) 220) 250! 278) 296) 325) 354 368 (aR oe Me 28 58 87}. 106} 136) 165°) 193°) 223} 252 | 282} 2997) 328 357 371 We cos OR. Le 29 58 87 | 104} 134} 163) 192] 221) 249) 278) 296) 325} 354 369 fe aI LS See el 29 58 87°} 105°} 135°) 165°) 194 | 222) 251 | 280) 298) 326 | .355 370 (ae Wa a 29 58 87 |} 105 ‘| 133)) 162) 192.], 222 || 251} 280) 298.) 327) 357 371 SO! 2. SBOE EN 29 57 86} 104] 183} 162.) 191 | 220) 249) 278) 296) 325] 353 366 1, Ua SM 28 58 87 |} 105} 134')' 163)) 193)| 223] 252 280°) 298°) 327 | 356 369 1, i ae a8 29 58 86 |» 104} 133°) 162) 190] 219] 249) 279} 297) 326) 355 369 2S ae on See a 30 59 88 | 106 }° 135) 163) 192:} 221 | 250} 280) 297) 326) 355 368 TA oe =| 29" 58 87 | 104} 1384) 163 | 193 | 221 | 250; 280] 299.) 328 | 357 370 Oa Gh 28 57 86 | 104] 1382] 162) 190) 219] 248) 276 | 295) 324] 354 367 $62. BG. 30 59 88 | 106} 135} 164} 193.) 222 |) 250) 279 298) 3271) 355 369 3 17 Sa Coa ge 28 57 86} 104} 133°) 161 | 191 | 220] 250) 279) 298) 327) 356 370 29 58 87 | 104] 133 | 162) 191 | 220] 248 | 277 | 295) 325) 354 367 30 58 87 | 106} 135) 164] 193 | 222) 251) 280] 297) 326) 356 369 28 57 86; 104] 182:| 161] 190] 219) 249) 277) 295) 325) 354 368 30 59 87 | 105-} 134] 163} 191 | 220} 249) 278| 296) 324) 353 368 29 58 88 | 105'|. 185 | 164] 193] 222.) 251) 281) 298) 328] 357 371 29 58 86 | 104] 133) 162} 191 | 219} 248] 277) 295) 323 |. 352 367 COMPONENT MK. Series. 29 58 87 105 | 134 | 163 | 192 | 221°) 250 } 279 | 297 | 326 | 355 | 368 30 59 88} 105] 135} 164} 192) 222) 251) 279] 297) 325) 355 368 29 58 88} 106] 1385] 163} 192) 222) 251 280] 298) 327] 356 369 29 58 88 | 105] 134] 164] 192} 221} 249) 278) 297) 326) 355 369 30 58 86 | 104) 133] 163} 191 | 221} 250] 278) 296) 325) 355 368 29 58 87} 106} 184] 163] 192] 222 |) 251) 280) 298] 326] 356 369 29 58 87! 105} 1384!) 164} 192) 220} 250!) 279) 298! 327) 356 369 Geils Reo. aie 30 58 86 | 105 | 134) 164) 193] 221 | 250] 279) 297) 326] 356 369 hes NS eS 30 58 87 | 105} 183] 163 192) 221 | 251) 279] 297) 326] 355 369 Beane 2 ae 30 59 88} 106} 135] 164) 193] 221) 251) 280) 299%; 327) 355 369 OE. A-BGR. D8 28 57 86) 105} 1384) 164) 192] 220} 249] 278) 296) 325] 354 369 pA IRE ee a ae es 58 86 | 104) 133) 163) 192] 221) 251 |) 279) 297) 326) 355 369 11) SAP a) Co 29 59 88} 106} 134) 162) 192] 221} 250} 279) 298) 326) 354 369 (7S a 28 58 86 | 105} 134) 163] 192] 220} 250} 279) 297) 326) 355 369 1S eS Se ae 29 58 86 | 105} 133) 162) 192] 221) 250) 278) 297) 326) 355 369 1 PSS SAee ae mie 29 59 88} 106] 134) 163 | 193) 222] 250} 280] 297) 326) 355 370 LG: INS. URE 29 59 87 | 106| 1385) 163) 192) 221 | 250} 280| 297) 327] 355 369 EG. 23 ROE eB 28 57 86 | 105} 133) 162) 192) 220] 249] 278! 296) 326) 355 369 AG. 5 ARES 1 29 59 87 | 104] 134) 163} 193) 222) 250) 279} 296) 325) 354 369 aS. GE Le 29 58 88} 105] 134] 162) 191 |) 220) 249} 279) 296] 326) 354 368 EQ ot eR BS 28 57 87} 105} 135} 163} 193 | 2217) 250) 279] 297] 327) 356 370 PAUSREE ES ee ae 29 57 86} 103} 133) 162} 191 | 221) 249) 278} 296} 325} 354 369 Ze os IS 29 58 88] 105] 134) 162] 191) 221 | 250) 280] 297) 326] 355 369 ESE Se AS 28 57 87} 105} 1385) 163} 191) 221} 250) 280} 297] 327 | 355 369 2 Se age 29 57 87 | 104] 134] 163) 192) 221) 249] 278] 297) 325) 355 370 j HARMONIC ANALYSIS AND PREDICTION OF TIDES. TaBLE 32.—Divisors for primary stencil COMPONENT 2MK. sums—Continued. 303 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 29 | 58 86 104 134 163 191 221 250 278 297 325 354 368 30 59 87 105 135 164 192 221 251 280 299 327 356 370 30 58 87 105 134 163 ; 192 221 250} 279 296 326 355 368 29 57 87 104 134 164 192 222 251 279 297 326 356 369 30 58 88 105 135 164 192 221 251 279 297 326 355 369 30 59 88 106 135 164 193 222 251 280 |- 297 327 356 370 29 57 87 105 134 162 192 221 |. 249 278 | 296 326 356 369 29 57 87 104 134 162 191 221 249 278 296 325 354 368 29 57 86 104 133 162 191 220 249 279 297 326 355 369 29 58 87 105 134 162 192 221 249 279 296 326 355 369 28 57 87 105 135 163 192 222 250 279 297 326 354 369 1 = seg ao 28 57 86 104 133 162 | 191 220 | 249 278 296 | 325! 353 368 1c, s Set ea 29 59 88 106 135 163 193 222 250 280 298 328 356 371 LS (ee sa 28 58 87 105 134 162 192 221 249 278 296 325 354 368 AOE Le BAS) 8 28 57 87 104 133 162 191 220 250 279 297 326 355 369 AOE 2 So REE. 3 29 58 87 105 133 163 192 220 250 279 297 | 326 354 369 OR ee BER 2 28 58 87 106 134 163 193 221 250 279 297 326 355 370 7.3: Sage yo a 28 57 87 104 133 162 191 | .219 249 278 296 325 354 368 COMPONENT MN. Series 29 58 87 105 134 163 192 221 250 279 297 326 355 269 Hour US See yee 28 56 85 |} 104 134 163 193 223 253 283 301 328 356 370 i See eee 30 60 90 109 139 166 194 222 250 277 296 325 355 370 GPS ae ae 28 56 84 101 129 158 188 218 248 278 297 326 355 369 2 IR BIO i 30 60 91 109 139 168 198 226 254 281 299 326 355 370 VV eas ee ae 30 59 87 104 132 159 187 217 248 277 296 325 355 369 See Bae 28 57 88 106 136 165 195 225 256 283 301 328 356 369 i oS a ee 30 61 90 109 136 164 192 220 247 277 295 325 355 369 Wipe eS wet 28 56 83 101 130 160 190 221 250 280 298 327 355 368 oe SOR OE eee 30 61 90 109 138 168 196 224 251 279 296 325 355 369 (Se ee 29 57 84 102 129 157 187 218 247 277 295 325 355 369 AV eee ae ee 30 60 89 108 137 167 198 228 |. 255 283 300 328 356 369 See ee 31 61 89 107 134 162 190 218 247 277 295 325 356 370 We ef! ae ol 28 55 84 102 132 162 193 222 252 282 299 327 355 368 Tie ht area 30 59 89 107 137 165 193 220 248 276 294 324 355 369 ae ee eae 28 55 83 100 128 159 189 218 248 27 296 327 355 368 pee 2 aaa aae A 30 59 89 107 13 168 198 225 253 281 298 326 356 370 122 ee eee 30 5 86 103 131 159 187 216 246 276 294 325 355 363 a U7 (2 eae ai 28 56 86 104 135 165 195 224 254 282 299 327 355 368 Te Sad ik 29 59 89 107 135 163 190 218 246 276 295 325 354 369 eee Oe ee 27 55 83 100 131 161 190 220 250 280 299 328 355 369: 20ee hee he 29 59 89 108 138 168 195 223 251 279 296 325 354 369 7S eae ae a 28 56 84 101 129 157 186 216 246 276 295 325 354 369 C.F Se ne a ee 28 58 88 107 137 167 196 226 254 282 299 327 354 369 Zoe up Sa 2 29 59 88 105 133 161 188 216 246 276 | 295 325 354 369 304 TaBLE 32.—Divisors for primary stencil sums—Continued. U. S. COAST AND GEODETIC SURVEY. COMPONENT MS. Series. 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 30 59 88 | 106] 135] 164] 192] 222) 250] 279] 297) 326) 354 369 30 58 88 | 106] 134] 164} 192] 221) 250) 279) 296] 326] 355 369 29 58 87} 105] 134] 163] 192] 222) 250} 280} 298] 327} 355 369 30 58 88) 107] 1385] 165] 194) 223 | 252} 281} 298] 328) 356 370 29 58 87 | 105] 1384] 163] 191 | 221 | 249) 279| 296} 325] 354 368 30 58 88 | 105) 1384} 164) 192} 221 | 250) 280) 297] 327) 356 370 Basagessasonse 29 58 88 | 105] 185 | 164) 194] 223] 252 | 281 | 299 | 328) 356 371 URS GRE ee 30 58 87 | 105] 134} 162) 192] 220] 250} 278| 296 | 326) 354 368 SBR OS eeE Nae ae 29 58 87 | 104) 134] 162] 191} 220] 249; 278] 296] 326) 355 369 Oe tecicee nes 29 57 87} 104) 133] 162} 192] 220) 250) 279] 297] 326] 355 369 is tabeacesee 30 59 88} 106; 186] 164] 193 | 222) 251) 279! 298} 327] 355 370 a eae Se aa Oe 30 58 88 | 105} 134} 163] 192} 220] 250] 278) 296) 326) 354 368 1 SI eal SII 28 58 87 | 104] 134] 162] 192] 221] 250} 279) 298| 326) 356 370 I aag ae GAS ae 28 57 86] 105! 134] 163! 193) 221! 251 | 279! 297) 3251! 355 369 id bakgaaceode 29 59 87 | 105) 135] 163} 192] 221) 250) 278) 297] 325} 354 369 a Ra aa se 29 58 87 | 105] 1384} 163] 192] 220] 250) 278) 296 | 325] 355 369 Uassassdaasoss 28 58 86 | 104} 134} 163] 193 | 222} 251} 280) 299) 327{ 356 371 AY (GATS SO SE, 29 58 87 | 106) 135] 163] 193} 221) 251) 279) 297} 326} 355 369 USE geee Gomaegae 28 58 86 | 104] 133} 162} 190] 220] 248) 278 | 296] 324] 354 367 Noses eoabenhe 28 57 86} 104) 132} 162} 191 | 220) 249) 279| 297} 326) 356 369 PiDedaoecnvincase 29 59 87} 106] 134] 164] 192|) 222) 250} 279] 298) 326) 355 369 DU Ao tome alas 29 58 86} 105) 133 | 162 | 191} 220] 249) 278) 296} 325} 354 367 CPAs aee Gad CUES 28 58 86 | 104] 133! 162) 190] 220} 248; 278} 296) 325) 355 368 PEG sod oangaSue 28 57 86 | 105] 133 163 | 192} 221) 250) 280| 297 | 326) 356 369 COMPONENT 28M. Series. 29 58 87 105 | 134 | 163 | 192 | 221 | 250 | 279 | 297 | 326 | 355 | 369 28 57 87} 106} 136] 164] 192} 220; 250) 280; 299] 329] 356 369 30 60 88 | 106] 183; 163) 193 | 223 | 252) 280) 297 | 325} 355 370 28 56 86} 105; 1385] 165) 193 | 220} 249) 279) 297 | 327] 356 370 30 60 89; 107] 135] 163} 193 223) 253) 281 | 298] 326] 355 370 28 55 84} 103] 183] 163] 191 | 219) 247) 276| 294} 324] 353 367 30 60 90} 107 | 135} 163} 193] 223} 253] 282} 299] 326} 355 370 Cie ata 28 56 84] 103 | 133[{ 163] 192] 220) 248| 277 | 295 | 325] 355 370 fee A 30 60 90} 109) 137] 164} 193 | 223} 253} 283; 300} 328] 356 370 tse en SSeauees 29 57 85} 103} 1383] 163] 193 | 221 | 249) 277] 295] 325] 355 370 Bheedherscssqne 29 59 89} 108; 1387} 165) 193 | 222) 252) 282; 300] 328) 356 370 It igaege Sel aa 30 59 86} 104} 133] 163] 193) 223) 251 | 279] 295] 325] 355 370 UB esse accsadne 28 57 87} 106] 1386] 164] 192) 220} 249; 279 298] 327] 355 369 aires Sb onnone 30 59 87} 104] 132] 161] 191) 221 | 249] 277} 295] 323] 353 367 Ve}seeee asedosee 28 57 87 | 105! 1385 | 164! 191 | 219! 249) 2791 298] 328! 356 369 Ae eet eae 30 60 88} 105] 133} 162} 192) 222) 251 | 279{| 297] 325) 355 369 OEE eeleciee 27 55 85} 103] 133} 163} 191 | 219) 247! 277} 296] 326] 355 368 SB ges 2 ohh ee 30 60 89) 106) 134] 162} 192 | 222) 252} 281 | 298] 326) 355 369 LY aaa 8 Senki pee 28 56 84} 102; 132] 162} 191) 219) 247} 276] 295] 325) 355 368 1 SME Sede eae 30 60 90) 108} 135} 163) 192} 222] 252} 282} 300} 328} 355 369 TOES OS Mena oe 29 57 85} 102) 132} 162] 192) 220) 248] 276] 295] 325) 355 369 PAL WORDS sin ence 29 59 89} 107) 135] 163] 190} 220) 250) 280] 298} 326| 354 367 72 Ns gS Ae per pete! 29 57 85 | 102) 132) 162) 192) 221 | 249 | 276] 295] 325) 355 369 Jefe GOCE Bee 28 58 88} 106] 1385] 163] 191 | 220) 250) 280} 299} 327] 355 368 2oeee chee wae 30 58 86} 103) 132} 162} 192) 222) 250} 278} 295} 325} 355 369 ’ | HARMONIC ANALYSIS AND PREDICTION OF TIDES, 805 TABLE 33.—For construction of secondary stencils. Compo- | nent A. J Ss L CoD 00 25M K and P Rand T MS N Compo-| Differ-| Compo-| Differ-| Compo-| Differ-| Compo-| Differ-| Compo-| Differ-| Compo-| Differ- Page. |nent ‘A,| ence, |nent A,| ence, |nent A,| ence, |nent A,| ence, |nent A,| ence, |nent A,| ence, hours. | hours.| hours. | hours. | hours. | hours.| hours. | hours.| hours. | hours.| hours. | hours. a5 = + + — — Eee 0-23 3 0-23 0 0-23 0 0-23 0 0-23 0 0-23 0 Beals oe 10- 3 9 0-23 1 0-23 1 0-23 0 0-23 0 0-23 1 Se ee Fs 16- 4 15 0-23 2 0-23 1 0-23 1} 17-21 0 0-23 1 OT ee es 23- 5 21 0-23 3 0-23 2 0-23 1 0-23 1 0-23 i Dare ashen 5- 6 3 0-23 4 0-23 2 0-23 1 0-23 1 0-23 2 GiecNe 0-23 10 0-23 5 0-1 2 0-23 1 0-23 il 0-23 2 Teta oe 19-12 16 0-23 6 0-23 3 0-15 1 0-23 1 0-23 3 BN soe 1-12 22 j-11 6 0-23 3 0-23 2 0-23 2 0-23 3 Cerra 8-13 4 0-23 7 0-23 4 0-23 2 0-23 2 0-23 3 LORE a 14 10 0-23 8; 0-23 4 0-23 2 0-23 2 0-23 4 1 ee 0-23 17 0-23 9 0-23 5 0-23 2 0-23 2 0-23 4 As ee 3-20 23 0-23 10 0-23 5 0-23 3 0- 23 2 0-23 5 LS eee ee 10-21 5 0-23 il 0-23 6 0-23 3 0-23 3 0-23 5 tan se as 16-22 11 0-23 12 0-23 6 0-23 3 0-23 3 | 12-20 5 Paes 23 17 0-23 13 0-23 7 0-23 3 0-23 3 0-23 6 1 eee 6- 4 0 6 3 13 0-23 a 0-23 4 0-23 3 0-23 6 Wes 8 12-5 6 0-23 14 0-23 8 0-23 4 0-23 3 0-23 7 Dyes ek 545% 19- 6 12 0-23 15 0-23 8 0-23 4 0-23 4 0-23 7 BO ase 1-7 18 0-23 16 0-8 8 0-23 4 0-23 4 0-23 8 207 sl oas 2 8 0 0-23 17 0-23 9 0-23 4 0-23 4 0-23 & DA ISR eeae ees 0-23 7 0-23 18 0-23 i) 0-23 5 0-23 4 0-23 8 Da Nes teen 21-14 13 0-23 19 0-23 10 0-23 5 0-23 4 0-23 9 BOs a ae 4-14 19 49 19 0-23 10 0-23 5 0-23 5 0-23 9 PA eee 10-15 1 0-23 20 0-23 11 0-23 5 0-23 5 0-23 10 Denes ae 0-23 8 0-23 21 0-23 il 0-23 6 0-23 5 0-23 10 74 ESE 23-21 14 0-23 22 0-23 12 0-23 6 0-23 5 0-23 10 O(a cee 6-22 20 0-23 23 0-23 12 0-23 6 0-23, 5 0-23 i Pts eae ere 12-23 2 0-23 0 0-23 13 0-23 6 0-23 6 0-23 11 Oss Acie 19- 0 8 0-23 il 0-23 13 0-23 7! 023 6 0-23 12 51) Rema 1 14 0-23 2 0-23 14 0-23 7 0-23 6 0-23 12 St ea rae & 6 21 8&1 2 0-23 14 0-23 7 0-23 6 0-23 12 Sea aaa 15- 7 3 0-23 3 0-15 14 0-23 7 0-23 6 0-23 13 oS 4 Sebes 21- 8 9 0-23 4 0-23 15 0-23 7 0-23 7 0-23 13 By eRe 4-9 15 0-23 5 0-23 15 0-23 8 0-23 7 0-23 14 Someeceaee 0-23 22 0-23 6 0-23 16 0-23 8 0-23 7 0-23 14 BOATS. 17-15 4 0-23 7 0-23 16 0-23 8 0-23 7 2-21 14 Sic pen ene 23-15 10 0-23 8 0-23 17 0-23 8 0-23 7 0-23 15 Ste Seat Bie 6-16 16 6- 8 8 0-23 17 0-23 9 0-23 8 0-23 15 SO Sarena a 12-17 22 0-23 te 0-23 18 0-23 9 0-23 8 0-23 16 Ate einane 0-23 5 0-23 10 0-23 18 0-23 9 0-23 8 0-23 16 ¢: 8 cra 2-23 11 0-23 11 0-23 19 0-23 9 0-23 8 13-15 16 pA ire ae 8-0 17 0-23 12 0-23 19 0-23 10 0-23 8 0-23 17 ASN Ce ees 15- 1 23 23 13 0-23 20 0-23 10 0-23 9 0-23 17 0. NSS a 21- 2 5 0-23 14 0-23 20 0-23 10 0-23 9 0-23 18 Lic ae Bs 0-23 12 0-23 15 0-23 20 0-23 10 0-23 9 0-23 18 Cs ae ea ae 10- 8 18 10-23 15 0-23 21 0-23 10 Q-23 9 0-23 19 C19 (Ie AEE ae 17-9 0 0-23 16 0-23 21 0-23 11 0-23 9 0-23 19 i ke a ea aed 23-10 6 0-23 17 0-23 22 0-23 il 0-23 10 0-23 19 AME Aalto 6-11 12 0-23 18 0-23 22 0-23 il 0-23 10 0-23 20 is eee Paps 0-23 19 0-23 19 0-23 23 0-23 11 0-23 10 0-23 20 Ea a aot Boek 19-16 1 0-23 20 0-23 23 0-23 12 0-23 10 0-23 21 S252 cece 2-17 7 0-23 21 0-23 0 0-23 12 8-16 10 0-23 21 G3) ese 7-14 12 0-23 21 0-23 0 0-23 12 0-23 11 0-23 21 306 U..S. COAST AND GEODETIC SURVEY. ee TaBLe 33.—For construction of secondary stencils—Continued. Compo- nent A L M N O Compo- eon MK pag | MN 2MK v PRS oN Compo-| Differ-| Compo-| Differ-| Compo-| Differ-) Compo-| Differ- Gamanae Differ-! Compo-| Differ- Page. |nentA,| ence, |nent A,| ence, |nent A,| ence, |nent A,| ence, |nent A,| ence, | nent A,| ence, hours. | hours.) hours. | hours.} hours. }hours.| hours. jhours. | hours. | hours.| hours. | hours. = = + ar ae + TAGE 2 23-10 0 0-23 1 27-0 0 0-23 0 0-23 0} - 0-23 0 eae ir 20-8 1 0-23, 2] 11-23 1 0-23 1 0-23 1 0-23 0 Cee es | 17-5 2 0-23 4 2-14 2 0-23 1 0-23 1 0-23 0 TORSO HRS 15- 3 3 0-23 5 | 17-6 3 0-23 1 0-23 2 0-23 0 Sanne Sr 12-1 4 0-23 7 9-21 4 0-23 2 0-23 2 0-23 0 i ne 9-22 5 0-23 8 0-13 5 0-23 2 7-8 2 0-23 0 (eee 7-20 6 0-23 10 15- 4 6 0-23 3 0-23 3 0-23 0 Boe ne 4-17 7 5- 0 11 6-19 7 0-23 3 0-23 3 0-23 0 OU ae 2-15 8 0-23 13} 22-11 8 0-23 3 0-23 4 0-23 0 OH ec eas 23-12 9 18- 8 14 13- 2 9 0-23 4 0-23 4 0-23 0 Wisse 20-10 10 | . 0-23 16 418 10 0-23 4 0-23 5 0-23 1 ae 18- 7 11 7-15 17| 20-9 11 0-23 5 0-23 5 0-23 1 Seas rae 15- 5 12 0-23 19} 11-1 12 0-23 5 0-23 6 0-23 1 1 eee > 12- 2 13 19-22 20 2-16 13 2-10 5 0-23 6 0-23 I 1! yaar 10- 0 14 0-23 22] 17-2 14 0-23 6 0-23 7 0-23 } 17 See 7-21 15 0-23 0 9-23 15 0-23 6 0-23 7 0-23 1 Lear 419 16 0-23 1 0-15 16 0-23 7 0-23 ts) 0-23 1 PRN TSE Aa 2-17 17 0-23 3.| 15- 6 17 0-23 7 0-23 8 | 0-23 1 1956 a 23-14 18 0-23 4 6-21 18 0-23 8} 21-5 8} 0-23 aes! Oe Ler a keg 21-12 19 0-23 6 | 22-13 19 0-23 8 0-23 9 0-23 1 7 A BI 18- 9 20 0-23 7 13- 4 20 0-23 8 0-23 9 0-23 1 Dac aden 15- 7 21 0-23 9 4-20 21 0-23 9 0-23 10 0-23 I 7 Aare ee 13-5 22 0-23 10 19-11 22 0-23 9 0-23 10 0-23 1 7) ole ae: 10- 2 23 0-23 12 11-3 23 0-23 10 0-23 11 0-23 1 ps. en 7-0 0 0-23 13 2-18 0 0-23 10 0-23 ll 0-23 1 0-23 15 17-10 1 0-23 10 0-23 12 0-23 i 0-23 16 eo 2 0-23 11 0-23 12 0-23 1 0-23 18 0-16 3 0-23 11 0-23 13 0-23 1 6-0 19 |. 15-8 4 0-23 12 0-23 13 0-23 1 0-23 21 6-23 5 0-23 12; 023 14 0-23 2 19- 7 22 22-15 6 0-23 12 0-23 14 0-23 94 0-23 0} 13-6 7 0-23 13 | 11-4 14 0-23 2 7-15 1 4-22 8 0-23 13 0-23 15 0-23 Zi 0-23 3} 19-13 9 0-23 14 0-23 15 0-23 2 20-22 4 1i- 5 10 0-23 14 0-23 16 0-23 z 0-23 6 2-20 11 2-20 14 0-23 16 0-23 We 0-23 8 17-12 12 0-23 15 0-23 17 0-23 2 0-23 9 & 3 13 0-23 15 0-23 17 0-23 2 0-23 li 0-18 14 0--23 16 0-23 18 0-23 2 0-23 12 15-10 15 0-23 16 0-23 18 0-23 z 0-23 14 6- 1 16 6- 9 16 0-23 19 0-23 z 0-23 15 |. 21-17 17 0-23 i7 0-23 19 0-23 2 0-23 17 13-8 18 0-23 17 0-23 20 0-23 2 0-23 18 40 19 0-23 18 0-23 20 0-23 2 0-23 20 | 19-15 20 0-23 18 1-23 20 0-23 2 0-23 21 10- 6 21 0-23 19 0-23 21 0-23 2 0-23 23 2-22 22 0-23 19 0-23 21 0-23 2 18-16 0} 17-13 23 0-23 19 0-23 22 0-23 2 0-23 2 8-5 0 0-23 20 0-23 22 0-23 3. 6-23 3 0-20 1 0-23 20 0-23 23 0-23 3. 0-23 5 | 15-12 2 0-23 21 0-23 23 0-23 3 19- 7 6 6- 3 3 0-23 21 0-23 0 0-23 3 0-23 8} 0-23 4 0-23 21 0-23 0 0-23 3 5 ORES iS ea) WU ee A TE Pe Rs Se el IN Le HARMONIC ANALYSIS AND PREDICTION OF TIDES, 307 TaBLEe 33.—For construction of secondary stencils—Continued. Component A....... O Component B....... p Q 2Q Compo-| Differ- |Compo-| Differ- | Compo-| Differ- | Compo-| Differ- | Compo-| Differ- Page. nent A,| ence, jnent A,| ence, |nent A,| ence, jnent A,| ence, jnent A,| ence, hours. | hours. | hours. | hours. | hours.'| hours. | hours. | hours. | hours. | hours. 2 0-23 3 | 18-22 5 | 23-11 6 | 12-17 di 8 6- 3 9 6- 8 17 9-20 18} 21-5 19 13 | 18-12 15} 18-6 6 7-17 CN a ae (Tees a" 18 7-22 21 7-16 18} 17-5 19 6 20 23), 19-8 3; 19-1 6 2-14 7| 15-18 8 5 7-18 9 7-11 18} 12-0 19 1-6 20 10 | 19-3 15 | 19-21 6 | 22-10 7| 11-18 8 15 7-13 21 7-19 19 | 20-6 7A) bee Serene VEN aay LY 20} 19-23 3| 19-5 ul 6-18 Silbsesseoelceceade 2 8 9| . 8-15 19 16- 3 20 4-7 21 7 0-23 16 | 20-0 7 1-13 8| 14-19 9 12 8- 5 22 8-10 19 |} 11-23 20 0- 7 21 18 | 20-15 4 20 7 | 21-8 8 9-19 9 23 8 1 10 8-18 20} 19-7 2 levee ise aya Mca se vale 4} 20-10 16 | 20-4 8 5-17 9| 18-19 10 9 9-20 22 9-14 20°} 15-2 21 3- 8 22 15 | 21-6 4} 21-23 8 0-12 9| 13-20 10 20 9-15 10 9 20 | 10-22 21 23- 8 22 1] 21-1 16 | 21-7 9 8-20 LO: |eeeeeces|bcrteees 7 9-11 22 9-17 21 18- 6 22 7-8 23 12 0-23 5] 21-3 9 4-16 10 | 17-20 11 17 | 10-8 11) 10-12 21} 13-1 22 2-9 23° 22 | 22-17 17 22 9 | 23-17 10 | 12-21 11 4) 10-3 23 10-21 22 | 22-9 25) eee sey eee 9 | 22-13 5| 22-6 10 7-19 il 20-21 12 14) 10-22 11 10-16 22.| 17-5 23 6- 9 0 19} 22-8 17 | 22-2 10 3-14 11 15-21 12 1 10-18 23} 10-11 22 | ,12- 0 23 1-9 0 6.) 23-3 5 | 23-10 11} 11-22 DD b betorarerrerte| SEE s Se iL 11-13 11 11-19 23.) 20-8 0 9-10 1 17 23 17 | 23-5 11 6-18 12 | 19-22 13 22 0-23 QO} 11-15 23 | 16-4 0 5-10 10 3 | 23-20 6 | 23-0 ii 1-13 12 | 14-22 13 8; 11-5 12, 11-23 0 0-10 Lilie tes soe 14 0-15 18 0-9 12} 10-21 13} 22-23 14 19 |) 12-1 0| 12-18 0). 19-7 1 8-11 2 0 0-10 6 0-4 12 5-17 13 | 18-23 14 6 | 12-20 12 | 12-14 0} 15-2 1 3-11 2 ibl 0- 6 18 0-12 13 |) 13-23 V4. |e ece corde 16), 12-15 0 | 12-22 1.) . 23-1 De berateraremrareilsic des cate 21 1 6 1-7 13 8-20 14} 21-0 15 3 0-23 13 | (13-17 1 18- 6 2 7-12 3 8 1-22 19 1-3 13 4-16 14} 17-0 15 13 13- 8 1 13 1 14-1 2 2-12 3 19 1-17 7 1-11 14.| 12-0 15m) erotcravcravsrel dads noe AG). a see ED 13-7 0] 13-3 13 13-21 2.|. 22— 9 3 10-12 4 VSR 2 A Cs a ee 2-11 5 2-13 19 2-16 14 7-19 15 | 20-1 16 AS oo. MAS | Sake | Ue 14-15 10 | 14-22 1 14-16 2.) 17-5 3 6-13 4 A9e > as bon FADO 2-0 16 2- 8 7 2 14 3-15 15 | 16-1 16 HO Sissies eee pamaegaa 14-4 21 14-18 3} 14-0 3 1-13 CN SSG SRONe KEES See EIR AGS a ne 2-8 2 2-3 19 2-10 15 | 11-23 16 0-1 17 ES et Sg ea 0-23 8 0-23 2) 14-20 3] 21-8 4 9-13 5 (CG) OE Erinn 4-16 12 4-23 a 4 13 5-16 14) 17-3 15 308 U. S. COAST AND GEODETIC SURVEY. TaBLE 34.—For summation of long-period components. ASSIGNMENT OF DAILY SUMS FOR COMPONENT Mf. Component division. Days of series. LD eee NEARER ANE pa Aes amen teem air T2855 82* 110 137 164* 192 219 246 274 301 328 356 111 138 166 193 220 248* 275 302 330* 357 112 139 167 194 221 249 276 303 331 358 113 141* 168 195 223* 250 277 304* 332 359 114 142 169 196 224 251 278 306 333 360 115* 143 170 197* 225 252 279 307 334 361 117 144 171 199 226 253 281* 308 335 363* 118 145 172 200 227 254 282 309 336 364 119 146 174* 201 228 256* 283 310 337* 365 120 147 175 202 229 257 284 311 339 366 121 149* 176 203 230* 258 285 312* 340 367 122 150 177 204 232 259 286 314 341 368 123* 151 178 205* 233 260 287 315 342 369 125 152 179 207 234 261 289% 316 343 ..... 126 153 180 208 235 262 290 317 344 ..... 127 154 182* 209 236 263* 291 318 345* .... 128 155 183 210 237 265 292 319 347 .... 129 156* 184 211 238* 266 293 320 348 .... 130 158 185 212 240 267 294 322* 349 .... 131* 159 186 213 241 268 295 323 350 es 133 160 187 215* 242 269 297* 324 351 .... 134 161 188 216 243 270 298 325 352 .... ae se ee Sa Mea Salata ee wie ee 26* 53 80 108* 1385 162 189* 217 244 271* 299 326 353 .... Be aol ciclo aes quia slayste/sicl saree exsie ara ange 27 54 81 109 186 163 191 218 245 273 300 327 355* .... ASSIGNMENT OF DAILY SUMS FOR COMPONENT MStf. Component division. Days of series. Weep sacaesh-.adec o aseredseneasooeocsseees 1 30 60* 89 119 148 178 207 237 266 296 325 355 1 [re el Cu SB PP ae © Ok ontlts Sy eee 2 31 641 90 120 149* 179 208* 2388 268* 297 327* 356 PGS OE SB oc Ne See oy erete Hein SHS e A Cepe ee 3 33* 62 92* 121 151 180 210 239 269 298 328 357 BAe sScebicesc he Me ou tk ane Shee 4 34 63 93 122 152 181* 211 240* 270 300* 329. 359* AES BASES CE abe 4- Seaee Ae ace cee setae 5* 35 65* 94 124* 153 183 212 242 271 301 330. 360 EAE icicle (5 uC eae, 4 SEL Ne os 7 36 66 95 125 154 184 213* 243 272* 302 332* 361 Geom ceet rents veges sl stcinth Se eee re tec ase 8 37* 67 96* 126 156* 185 215 244 274 303 333 362 1, SERS ON 3) I oR ok cd a Ree ly eat 9 39 68 98 127 157 186 216 245 275 304* 334 364* CE NS eR See apo Berka, ON earner We |S ue) CRA Bs 2 A 10 40 69* 99 128* 158 188* 217 247* 276 306 335 365 CS RN BE TT IO Se aN Ung A 12* 41 71 100 130 159 189 218 248 277 307 336* 366 1) JR os © a! Salas ane a lesen | eae «Se Ub 13 42 72 101* 131 160* 190 220* 249 279* 308 338 367 1 ET Sea a ie S29S9 SAR No gp ge op pee 600090 30.00 30 ee why hy | 30 00 Number , of Pitch. teeth, ea 40 | 24" 120 24 120 24 72 24 72 24 75 30 75 30 a ies | 7: 30) 75 30 rye 75 | 30° 75 30 ah 60 |" 48° 120 48 84 48 84 48 “esol 180 | 48" 60 48 “pag 240 | 48. 60 48 a 60 | 48° 120 48 Seon i'l ones SOOM aera 46 40 60 40 60 40 46 40 72 24 72 24 ipsant TION| 180) 110 30 110 30 ce ay it ek 30. 75 30 75 30 75 30 Period of rotation. 24 X 366 24 X 366 Remarks. Hand crank shaft for operating machine. Spur gear on shaft 1. Spur-stud gear. ene gear on shaft 2. Short horizontal shaft. Bevel gear on shaft 2. Revel gear on shaft. 3, Diagonal shaft connecting with front com- ponent section. Bevel gear on shaft 3. Bevel gear on shaft 4, Short vertical shaft through desk top. Bevel gear on shaft 4. Bevel gear on shaft 5. Short horizontal shaft. Bevel gear on shaft 5. Bevel gear on shaft 6. Main vertical shaft of dial case. Releasable bevel gear on shaft 6. Bevel gear on shaft 7. Intermediate shaft to hour hand. Bevel gear on shaft 7. Bevel gear on shaft &. Hour-hand shaft. Releasable bevel gear on shaft 6. Bevel gear on shaft 9. Intermediate shaft to minute hand. Bevel gear on shaft 9. Bevel gear on shaft 10. Minute-hand shaft. Releasable bevel gear on shaft 6. Bevel gear on shaft 11. Intermediate shaft to day dial. Worm screw, 0.55 inch diameter, 18 threads to inch on shaft 11. Worm wheel, 6.47 inch diameter, on shaft 12. Day dial shaft. Spur gear at top of shaft 6. Spur-stud gear. Spur-stud gear connected with gear 25 by ratchet wheel and pawl. Spur gear atlower end of feeding roller. Bevel gear on shaft 3. Bevel gear on shaft 13. Main vertical shaft of front component section. Spur gear on shaft 13. Spur stud gear. Spur stud-gear on shaft 14. 9 Front vertical shaft of rear component section. Bevel gear on shaft 14. Bevel gear on shaft 15. Main connecting horizontal shaft of rear com- ponent section. Bevel gear on shaft 15. Bevel gear on shaft 16. Rear vertical shaft of rear component section. 314 U. S. COAST AND GEODETIC’ SURVEY. TasBie 38.—U. 8S. Coast and Geodetic Survey tide-predicting machine No. 2. COMPONENT GEARS AND MAXIMUM AMPLITUDE SETTINGS. “Component. Teeth in gear wheels. Theoretical F Compo- | Gear speed Maximum speed enieet Intermediate shatts.| nent per dial /£'Tor per | amplitude per hour. | * ; shafts. hour. Gia) SE EUTLES of cranks. I il MOL IV ° 2 2 Units. 15. 5854433 107 90 52 119 | 15. 5854342 —0. 08 1.4 15. 0410686 61 73 51 85 | 15. 0410959 + .24 11.0 30. 0821372 122 80 96 146 | 30. 0821918 + .48 3.9 29. 5284788 104 61 56 97 | 29.5284773 =r 01 2.4 14, 4920521 103 85 59 148 | 14. 4920509 — 0 1.0 28. 9841042 103 74 59 85 | 28. 9841017 — .02 20.0 43. 4761563 86 62 70 67 | 43. 4761675 + .10 1.4 57. 9682084 118 74 103 85 | 57. 9682035 — .04 4.0 86. 9523126 140 62 86 67 | 36. 9523351 + .20 1.0 115. 9364168 118 37 103 85 | 115. 9364070 ap ot) 0.4 28. 4397296 65 46 53 79 | 28. 4397358 + .05 6.0 27. 8953548 68 58 46 58 | 27. 8953627 + .07 1.0 13. 9430356 92 89 58 129 | 13. 9430363 + .01 9.0 16. 1391016 134 131 71 135 | 16.1391009 — .ol 0.8 14. 9589314 91 73 50 125 | 14. 9589041 — .24 4.8 13. 3986609 84 88 51 109 | 13.3986656 + .04 3.0 12. 8542862 127 114 50 130 | 12. 8542510 — .3l 0.6 30. 0410686 85 50 43 73 | 30. 0410959 + .24 0.4 15. 0000000 63 75 50 84 | 15. 0000000 - 00 2.0 30. 0000000 70 70 70 70 | 30. 0000000 . 00 9.8 60. 0009000 75 45 60 50 | 60. 0000000 | - - 00 1.0 90. 0000000 90 48 80 50 | 90. 0000000 - 00 0.4 29. 9589314 81 50 45 73 | 29. 9589041 — .24 1.0 29, 4556254 131 65 57 117 | 29. 4556213 — .04 0.4 27. 9682084 125 §2 74 121 | 27. 9681516 — .50} 1.2 28. 5125830 89 69 70 95 | 28. 5125258 + .02 2.0 13. 4715144 69 70 41 90 |} 13. 4714256 — .75 0.8 44, 0251728 120 81 105 106 | 44. 0251572 — .14 1.9 42. 9271398 81 52 79 86 | 42. 2271020 — .33 1.4 57. 4238338 135 42 53 89 | 57. 4237560 — .68 0.7 58. 9841042 118 61 62 61 58. 9841440 — .35 2.0 31. 0158958 69 47 50 71 | 31. 0158825 — .12 1.4 1. 0980330 84 45 1 51 1. 0980392 + .05 4.0 1. 0158958 149 80 1 55. 1. 0159091 store les ~2.0 ‘0. 5443747 93 4) 1 125 0. 5443902 + .14 3.0 149 a8 as See ; 0. 0410686 51 { 5s as iba \ 0.0410738| + .05 8.0 0. 0821372 51 149 1 125 0. 0821477 + .09 3.0 *Designed for one-half of speed of Mo. HARMONIC ANALYSIS AND PREDICTION OF TIDES. TABLE 39.—Synodic periods of components. DIURNAL COMPONENTS. 315 See Ji Ky Mi QO; (exe) Py Qi 2Q Si Days. Days. Days Days Days Days. 1G ee Oa A PGs Ae Le all ee he no Ge alg Fag lag abt Ag des Nh PON oP (hoe Le Meee 2 vee 13.777 ATU NTSISYBY NPR Ae aS Se A aN kd Ae a ok 2S ts |W Ue Or eee Se 9.133 13. 661 Doi use Ai wen oS NERA DUGG ier eM seaie F Rba ae REINS STR Ms ess Ne Se OO2LL. 64 27. 093 13. 661 9.133 ONSZO4 | EU EO EP ae Ee ee ee eee Bo Lee ma a 23. 942 182. 621 32, 451 14. 765 PACA Ah ape te ICR SN Be ee Aer Sey (AV eee aero 6, 859 9.133 13. 661 27, 555 5, 474 OU AR iced Wied SRD NE Nop dh bey re be Be ZAG ace alligate 5, 492 6. 859 9.133 13. 777 4. 566 RCO ad aN | peels ecole reiapall eae carafe Srestee 2 25.622 | 365. 243 29. 803 14, 192 13. 168 S245 ity SHE AG{991) [Sessile sae Plesea cee are 7. 096 9. 557 14, 632 31,812 5. 623 10. 085 205 892 24. 302 9, 814 SEMIDIURNAL COMPONENTS. Ke Le Me Ne 2N Re Se T: de 10) v2 Days. | Days. | Days. | Days. | Days Days. | Days. | Days. | Days. | Days. | Days Bagels har QT OOS) Uisegsme sel > sears = Seeded 3 S| Sete is Sasa scrste cvs eatral ei Cvicreete Kile tee = Arle ens oe clits deuce fe ses Messe. 13. 661 Ae LAO | Re ee ae erste teal er eke er mpacase atte ere sage vape cela a eer fis core tacaoe | Pkeiagais: eves | ees tenes | Masri Nee eee 9, 133 TE CCC 2-H eae Sie | 8 8 A ee ee Ae en ee pee ey LE A ARR SB ONES 2. 6. 859 9.185 | 13.777 DT OOD | eae) Ze eal A ea Sree epee UPN ce ee Ee) ac Rates. 365. 225 | 29.263 | 14.192 9. 367 GEQOLA, RE Nid ae Se Pa, SD ERE SA et eh AS Sen o-hi= 182.621 | 31.812 | 14.765 9. 614 Feld |p 860: 2509 ese 3. 28s Sal esas Salles were -yss gotnaleapes ee Bots eee: 121.748 | 34.847 | 15.387 9. 874 269) | 1821630036: 259P eee a Ale SE eee eee ivieoecud 23.942 | 205. 892 | 31.812 14. 765 9.614 | 25.622 PUB | PRESS eae Sec albe abouatledasee eS TN Ea 7. 096 9.614 | 14.765 | 31.812 | 205. 892 7. 236 Tockess || WISER WTO O35) loc sesallessoce er ee 9, 557 14.765 | 31.812 | 205. 892 24. 302 9, 814 HOLOSSE LORS | las S0bn P27 Doon eee 28M --.-| 16.064 10. 085 7.383 5. 823 4. 807 15. 387 14,765 | 14.192 9.614 | 4.922 | 5,992 TaBLe 40.—Day of the common year corresponding to day of month. [For leap year increase all numbers after February 29 by one day.] Day of month. Jan. Feb. Mar.} Apr. May. June. | July. Aug. Sept. | Oct. Nov. Dec. rae poe AES oA SIT: 1311321 1260 91 121 152 182 213 244 274 305 ©6335 BE Se) SSE ee ey By eRe 92) 22.153 188 214 245 275 306 336 dase eseeneeas soe aes 3 deb BY 93 123 154 184 215 246 PLM) «By BRVE Be Oe ee nee ge Naty a8} 94 124 155 185 216 247 277 =308 = 338 es oe Cae ee ae ee ee 5 36 64 95) (125 156 186 217 248 278 309 339 ee as 96 126 157 187. 218 249 279 310 340 (ee OoS te OO) 97 127° 158 188 219 250 280 311 341 8 39 67 98 128 159 189 220 251 2816, 9312 342 9 40 68 OOP 129) 160 190) 22153 252) 282 33 343 10 41 69 100 130 = 161 LOY 91222" 4)253 283 314 344 LL] basi ho 9 sane ey Oe UR ete nena Sead iL ee e170) (ih ey UST Ko 192 223 254 284 315 345 R702 ey ee ae eee ee 1 pati ad LOD) 4132) 1163 193. 224 255 285 316 346 1 Oe ES blest Saree) 138 44 72 | 103 133 164 194. 225 256 286 317 347 ARS BPE RCI ES MER AD EL FNL 145 445° 17380 ©) ¢ S104 7 134 165 195 226 257 287 «©6318 = 3348 Gare ea Sets Sere pene 15 46 74 | 105 1385 166 196 227 258 288 319 349 TGS 2k SE De NE a Ot 1G AO eS alas BTS) | CGY) | 197 228 (259 289 320 350 (So SASS a eee a eee: WA AS ABE MOA BYE EY IDS) RS Aa) 290° 321 «35 TE oo su Ses Oe ame eee eae 130 49 77) 108) 138) eel69) 2) 199) 230) 261 291 322 352 1G) Se Sk es Re ee rey LOM SON esis) elOS eres Omes LAO 200 231 262 292 323 353 DAD acts pe RS IE Re PA al 7) LO ) 140% 7 20232 ebe 293 324 354 Pail. Gio eee te Bee By PAL TD iil Ve le 202 233 264 Ae ybo wey SI} DE 1 A IM SRT 7h 5 22,53 81 INP IEEE ile} 203 234 265 295 326 356 PRs ROE Ee a oe 23h) 540 82) M3) 143 3 074 204 235 266 296 327 357 Bie Oe Se emeagey & RM Eile ots 24 55 8&3 114 144) «175 205 336 267 297 328 358 SAR 0 eal ale ee ai ba a 25 56 84 Toys tlebey igo) 206° 237 268 298 329 359 PAR) oy a) Se OA Sea a 26) Oo So 116 146 177 207 238 269 299 330 360 DAN EEN ON ee gS 27) COSh BOL eT o. 14s F178 208 239 270 1010 ets ts 9 so GAS} EE AE Toe og ei QR SOP Sil LES ase V9 209 240 271 CO BBR Bl OAD i SU ate Bee PS a pe a a 29 88 119 149 180 210 241 272 302 333 363 SSS es BAe ae Si ene 30 89 120 150 181 21 242 oto 303 334 364 GF ne adie ee ean 31 90 151 212 243 304 365 72934—24}——21 Part III.—TIDAL HARMONIC CONSTANTS. TIDAL HARMONIC CONSTANTS. The tidal harmonic constants given on the following pages have been compiled from various sources, which are indicated by the references at the bottom of each column. All amplitudes have been reduced to feet, and the epochs have been referred to the local meridians. Values inclosed in parentheses have been inferred. _ The combined length of the series, together with the first and last year of the observations from which the constants were derived, is indicated for each station. In combining the results from several series of observations at any place the usual practice has been to take separately the direct means of the amplitudes and epochs. A more precise method was used for combining the results for the components Mf, MSf, and Mm as derived from the different series of observations at the stations in India, where observations covering periods of many years have been analyzed. This method consists of the use of the following formulas: 72 rw LER hn oT Mean amplitude of component A = al C. >H, cos ms) + G >H, sin ) and Lip as SUL ese Mean epoch of component A=tan Sie n being the number of individual results combined and H, and x, the amplitude and epoch, respectively, of any component A as derived from each series of observations. The sources from which the constants have been compiled, together with the corresponding reference number used in the compilation, are given below. The cooperation of all those who have so cour- teously furnished the harmonic constants for various ports is very much appreciated by the U. S. Coast and Geodetic Survey. It is. hoped that this cooperation will be continued, and all persons who: may secure additional constants for any station are invited to send a copy of the same to the Director, U. S. Coast and Geodetic Survey, Washington, D. C. References. Analysis by U. 8. Coast and Geodetic Survey. Proceedings of the Royal Society of London, vol. 45, 1888-89. Proceedings of the Royal Society of London, vol. 39, 1885.. Annual tide tables of Russian Hydrographic Office for year 1910. Resultater af Vanstands-Observationer paa den Norske Kyst, Hefte VI,, 1904. Bihang till Kongl. Svenska Vetenskaps-Akademiens Hadlinger, vol. 15, part I, No. 11, 1889-90. (There is probably an error of 180° in the: kappas of the diurnal components which has not been corrected.) 316 rent a at i a a Ou ODF S|—#" ” SS SS (or) wa HARMONIC ANALYSIS AND PREDICTION OF TIDES. ont (7) Dr. W. Bell Dawson, superintendent tidal survey of Canada. _ (8) Tables des Marees des Colonies Francaises de L’ Atlantique, pour 1923. (9) a Lemos, Engenheiro Residente do Service de Mares e Correntes, Rio de aneiro. Capt. Abel Renard, chief of hydrography, Buenos Aires, Republic of Argentina. Anuario Hidrografico de la Marina de Chile, 1912. Notices to Mariners, U. 8. Hydrographic Office, 1896. Russian Year Book of Tides for the Pacific Ocean for 1917. Report of International Geodetic Association, Paris, 1900. (In this report all amplitudes are represented as being in meters, but this is assumed to be an error as to the Japanese stations, for which the amplitudes appear to be in feet. The stations in Chosen were indicated by latitude and ae only, so names giving the general localities have been sup- plied. Manuscript table from Tokyo, Japan, dated May 24, 1894. Journal of the College of Science, Imperial University of Tokyo, April 3, 1911, by S. Hirayama. Miscellaneous Reports on Hydrography No. 9, by 8S. Ogura, hydrographic engineer of the Japanese Navy, 1921. Proceedings of the Royal Society of London, Series A, vol. 83, 1909. Proceedings of the Royal Society of London, vol. 71, 1902. The Governor, Province of Macao. Tables des Marees des Colonies Francaises des Mers de Chine, 1923. Royal Survey, Bankok, Siam. ee E. Young, deputy surveyor general, Taiping, Straits Settlement, 10. Dr. J. P. Van der Stok in Mededeelingen en Verhandelingen, Koninklijk Nederlandsch Meteorologisch Instituut, No. 102, year 1910. Zeemansgids voor den Oost Indischen Archipel, Deel IV, 1922. Zeemansgids voor den Oost Indischen Archipel, Deel III, 1921. Zeemansgids voor den Oost Indischen Archipel, Deel V, 1922. Zeemansgids vocr den Oost Indischen Archipel, Deel I, 1921. Getijtafel voor Palembang, 1912, door het Koninklijk Magnetisch en Meteorologisch Observatorium te Batavia. The Netherlands Government, through the secretary of the American legation. Getijtafel voor het Westgat Soerabaia, 1923, door het Koninklijk Mag- netisch en Meteorologisch Observatorium te Batavia. Wind and Weather, Currents, Tides and Tidal Streams in the East Indian Archipelago, by Dr. J. P. Van der Stok, 1897. Zeemansgids voor den Oost Indischen Archipel, Deel VI, 1920. Annalen der Hydrographie for 1891. The German Government, through the American Embassy. The German Tide Tables for 1903. Nature for July 30 and August 6, 1903. Proceedings of the Royal Society of London, Series A, vol. 86, 1911. R. W. Chapman, M. A., and Captain Inglis, in paper read before the Australian Association for the Advancement of Science, December 13, 1895. H. B. Curlewis, Government astronomer, Australia. Report of the Surveyor General of New Zealand for 1922. Records of the Survey of India, annual reports for various years. Annales Hydrographiques, Volume de 1908, 1909, and 1910. E. Nevill, Government astronomer, Natal Observatory, Africa. W. H. Finlay, M. A., in a pamphlet of Approximate Tidal Constants, 1887. Annalen der: Hydrographie for 1903. Annales Hydrographiques for 1911. Harmonische Analyse und Theorie Sterneck, 1922. Annalen der Hydrographie for 1914. Die Gezeitenerscheinungen in der Adria, by Dr. Robert Daublebaky v. Sterneck, 1919. der Mittelmeergezeiten von Dr. Robert ) G. Magrini, director, Hydrographic Office, Venice. Italian Government, through the American Ambassador. Report of International Geodetic Association, Stuttgart, 1898. : Recherches Hydrographiques sur le Regime des Cotes, No. 993 Service Hydrographiques de la Marine, Paris, 1916. U. S. COAST AND GEODETIC SURVEY. Annales Hydrographiques for 1901. Getijconstanten voor Plaatsen Langs de Kusten en Benedenrivieren in Nederland Berekend uit de Waterstanden van het Jaar, 1906, door M. H. Van Beresteyn. The Deutsche Seewarte. Finnlandische Hydrographisch-Biolgische Untersuchungen No. 2, by Von Rolf Witting, Helsengfors, 1908. The British Admiralty. Dr. Arthur T. Doodson, Tidal Institute, University of Liverpool. Proceedings of Royal Irish Academy, 3d series, vol. 1, 1889-1891. Inferred from the British Tide Tables. National Antartic Expedition, 1901-1904, Physical Observations, pub- _ lished by the Royal Society of London, 1908. Deuxieme Expedition Antarctique Francaise, 1908-09. Etude sur les Marees par R. E. Godfrey. Miscellaneous Reports on Hydrography No. 10, by S. Ogura, hydro- graphic engineer of the Japanese Navy, 1923. 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COAST AND GEODETIC SURVEY. “107 °d oes syuouodm0D poried-3u0] pus punoduiod 10,4 1 (1 ) (11) (21) (21) (LT) (#1) (¢9) (#1) (cT) (Cae ee COUCHES § “qquoUl [ “q4uom T ea a ea Seca > Na “sieok % “qquoul [ TARR NEA aH [et aa la | ee el cc cane “syquoul Z “qquou T cn aa a aoe pat yysueT, Be ehh mpmared rer saa Ceara Searles cee a alRaaee fk Oe rei ir a ee | ones See Bee Seer re Prt rok oe oa Se 2681 e68T weeeaeeeeee ee #gQTIOg eee eee ee ind tie ir GIRO see Neca ae |, 889°0, | F6E_ ] 88:0 | 102. | 08 908, OLS OS esse Sale stro |v BS potce Bcllaecee 980°0 [tree Fie OGOR Oi eam eo eaaee oa G9g"E | 912 86 'G geo’g rccrcccfrcet et ecfrcstee feces BAneea | SAaeereal tee aser nonaen tecesncelseteeaes gz Pr ae shite tad eacgecslinc ne ke | ce ermal [Sea Maaco cea Lea nae ga ne tes c0E COSNO = | ee €E #0 Ste LEE°O | SFE €£°0 9 O8E "0 | LIZ QUEHORS Ace eters ees 612 = | 20'T 91Z Itt G1Z #0 81Z £160 | E12 68 °0 £02 96°0 pee \| OFO'T | Fz 16°0 £26 PLIT | 008 GLO ple a Seon oe oe Seiatbil es i ee es ee 882 ceo 0 ee ee ee ee a ee 912 00 ee ee Tes ween tesla ence ee el www etree eee ie ‘baq | aq | baq | ‘yaaq | ‘baq | ‘00g | ‘baq | ‘200g | ‘baq | “gaaqg | “boq | ‘qaaqg | ‘bod | “700q | “bod | “799q baq | "pag bad | *7aq » H ” H ” = H ” H ” H a H » H ” H > : = ” H ; ca oe cer | “GH 68 BET | “CE TE CST | a ae cer | “(co ee cet | “(om ee ocet | m9 oer | “Cae oer | “CH ee ose | “Ca 8h cet N 102 o¥8) 80g pueyjuy 9 pueyuy "NPT oF 8) ‘N (70 oF8) *N JOT oF8) "N 186 oF 8) “NI (81 oF8) *N 8&3 of 8) “N 19% o¥8) 809 pueloy Her ae 299 809 pueyoy 809 pueluy 80g pus]uy 809 pusluy BIg 99 pueluy “qnem0 du00, SUlByoyo x if H pus ony |‘emITysg 0,018 57 ‘Tels LTSnsqi ‘8po0jeg puso] ‘omoy, | ‘nsj}-oml1yg TION Tysesry | OULyNS}TA *ponulju0g—pusjsy nysuoy ———————————— *ponuyjmuop—NVd VE 347 HARMONIC ANALYSIS AND PREDICTION OF TIDES. 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S}U9UI8]}}05 AoUyIO 1 Ueqg JOALY UIVAVS Joquin f{ “NI /69 of 9) “NI JZP o88) ON 10 ge) qvay-[e-39249 SPIBIYS 1Ssomulo14g rugnoutmoay | ‘aeysururuy BIS OF MM sreoevD ;“u0q TATA i "US }surued ‘uoy ‘eissny | -stq ‘qesnjI0g HeIGTD *pueyjoog *‘purlsuy "RISV (99) “IBdA T ST6L BIT 12d ‘0 GIG | 4600 __ LPT 910 0 OST $6 0 Pr 680 0 8ST 850 0 10z 680 ‘0 €81 10€ 0 (64 960 0 OIL TLS 0 86 $00 0 69 90 0 62 660.0 L8G cst 0 831 P88 7% 16 €20 0. GEL 880 0 CPL G8Z 0 261 PLE 0 PSI 910 0 “bag WEEN fi zt H \LSIT TVINAWATddAS—SLNVLISNOO OINOWHVH TYCIL [JopsO Ie[NZO1 UL WOT}AOSUI 10; 04R] 00} PoAtoooyy] “CAA ,0T 089 “NI 6P 09) BURINY) “Ig .UMO} -93.1004) ‘VOL -leuly q4nog See et ee qysae7 “yuou0du10 HARMONIC ANALYSIS AND PREDICTION OF TIDES, Compound and long-period components. 403 [Upper line for each station gives the amplitudes (H) in feet: lower line hrs a epochs (x) in degrees. For other components and references see pp. 319 to 402.] Compound components. Long-period components. Stations. MK | 2MK |} MN MS | 2SM Mf MSf | Mm Sa Ssa Arctic regions. Monta Corzer, Grant Wand.) <2 5) -sacne| rites sic sao wae eo roaece|aee cal satdas al Usidec| ss ose. 0.199 | 0.143 MNT Ghani thst dey hin 'leretnraterete miele) e\wlehtil hate atelier | w\nlaidlapeiall niosataa(cpai| tate al Sole ai] Salim eM et mle 203 335 CAL ONIISKAY A VEUUSSIA 9144 Sale lilo s oe |mctara sielaeiaia clef eaten shearer ogee dulien Sai cu aa 0. an S.aRee 3 ale Ea ha © RD eaeenion |aaae cee eee aan Bec eee kale lal Me I ROI BE 2 i Sie Dao Le HCC rat MEHSSLA AAS A cieceratsicrtes ao rtatarel= ol ifoeleieia aiererstete Wie U0 Sse hos EOe: Son seociseo| daposse 0.180 | 0.220 286 Sc6d|honeseallbocesoe 1 OOS Boece (ane debe saben een 148 120 WHEGOs INOEWAY 3552 vecreroridie Ss a| tects aerate oa dl eecew's 0.094 | 0.074 | 0.092 | 0.048 | 0.051 | 0.482 | 0.200 a cmisteeral| aisha eras aici 173 0 227 162 193 249 188 HN CIGeN OLWAY <2 +2 fen 2 Se Geer |esees solesese bee cee-s 0.051 | 0.034 | 0.010 | 0.089 | 0.079 | 0.348 | 0.226. SEE Ra Peepers ees eee 62 64 206 260 213 202: Hal clad Paw NORWAY = i nnevete co sa|tenies wa|aieds oaftscreare ot 0.154 | 0.077 | 0.125 | 0.049 | 0.094 | 0.374 | 0.187 SSSR eral Seeboss me coos 28 144 78 277 171 264 218 OC OFPNIORWAN es sae Sabre a eocleniecice « feelers HELE oe done 0.139 | 0.062 | 0.121 | 0.181 |....... 0.359 | 0.147 SSCS Re sacs Ceol sus oacs 346 168 189 289) [esi Sees 245 189 British America, East Coast. Forteau Bay, Belle Isle Strait...| 0.034 | 0.044 | 0.033 | 0.022 | 0.013 | 0.088 | 0.053 |0.070 | 0.217 | 0.092 230 185 11 290 | 48 301 34 231 231 203 SeMUOlTse Ne WwlOUnG And 22 a\c)cela nic | seem alewrsaioe ol ecieisise | Cente ate lee wits Seu [ ce oaks |e ee 0. 200 | 0.071 ete | Sica rete crea ass faere ee perl Cee ee is eeu al Veron | CIO aL 268 217 South West Point, Anticosti | 0.007 | 0.010 | 0.015 | 0.069 | 0.037 | 0.045 | 0.021 | 0.038 } 0.108 | 0.067 Island. 130 66 223 22 118 208 120 84 128 194 Father Point, St. Lawrence | 0.036 | 0.031 | 0.020 | 0.065 | 0.061 | 0.037 | 0.056 | 0.064 | 0.159 | 0.189 River. 154 105 9 77 124 195 348 196 151 156 Quebec, St. Lawrence River....| 0.171 | 0.168 | 0.316 | 0.400 | 0.078 | 0.164 | 0.529 | 0.270 | 0.502 | 0.42 354 317 249 323 | €9 73 54 33 74 116 Charlottetown, Prince Edward | 0.030 | 0.054 | 0.621 | 0.050 | 0.032 | 0.041 |-0.042 | 0.047 | 0.165 | 0.102 Island. 202 136 335 211 155 234 172 297 232 168 St. Paul Island, Cabot Strait....| 0.015'| 0.011 | 0.011 | 0.016 | 0.010 | 0.041 | 0.037 | 0.056 | 0.142 | 0.083 82 353 124 334 | 290 177 206 299 228 154 Halifax, Nova Scotia.........-. 0.031 | 0.010 | 0.056 | 0.058 | 0.010 | 0.045 | 0.036 | 0.046 | 0.129 | 0.079 &6 30 318 152 279 140 211 206 249 184 St. John, New Brunswick... ..- 0.069 | 0.034 | 0.054 | 0.187 | 0.096 | 0.049 | 0.083 | 0.045 | 0.130 | 0.150 135 116 112 242 194 161 114 129 125 120 United States, East Coast. | Becsiraveusts 10 ks Means Mm aes a tae fete a fee alee let ti es La a Rae ter ain eas I 0.105 | 0.044 Be et a 0 8 DUH a a | peg a ie eo Se polisaik V5l 9 ae ae 309 EU OVHE DE aye OL ay ged We le ee Le | bd Sah PR OROUAMROLOZ Sig Bae Shee tel OL eae 0.163 | 0. 086 HES ahs). oes Ace 17 WO BO VEL TNS ae telah 5 8G 178 87 s Eye tava CC gee ee ea Lh | Seyi ga ea YP okay Ge Lents! Vee | 0.142 | 0.060 Boston (Commonwealth Pier), |.::...-|-...---)--:---- OROD 2) SS eee NBs, ae Wl aleve 0.159 | 0.049 PUL ASS SMR iin aT oui ky eed ual etme d ai. SRA NICS dyes LOS, SoH P| SoG UES Te ee CE 143 87 Ne MOD GH OND AaTn s\ si rusie pai Aes skeralin fa Noe Sabre me Ae ORE y rt ekd Sih ey Pale eo 0. 144 | 0. 067 TO ON GIAPY BE) Sey GaP ee Ue SL SCDOT a Pe ts MO CG Me IE eco US | 153 | 145 TB AAUSUI) TRIS TAD Soha ea AN Saf ees eae raat: BO ih hy ddl Ur Dede baal OL 0.194 | 0.059 aiafe ode ined a Yioisie MEAP Ieee Ucn atuiotnluea Se Wnt cdl Gioi Sieh apg ERC eee pe No) a: 87 Brovidence, Reals sie. eo: Pee CAR em SUR AT ae uae cf ai tel Ket Pea A De 0. 244 | 0,04 AO EDD ee Eg A a PCN SSN er eel rear F707 18 New London (Fort Trumbull), |.-..-.-|----2-| --eofeeee feted By Hse Sake 0,242 | 0.124 (Chovawa te S50) VV 2 2 US Ia CBR UE | Acar amy Us Pesca eee ey mea ary Len ee ey OO Wad \aresoaa|ecemencdiem Loo 19 News Hondon(mavall) station) lus. sc |ese een eoseetnie! selene rele lf enlh dye ILS cL ane Pay | 0.238 | 0.100 pra piers teeta tl een dr clea es a Lal led tetraceomael lara ated | steele. os | 152 43 404 U. S. COAST AND GEODETIC SURVEY. Compound and long-period components—Continued. [For other components and references see pp. 319 to 402.] Stations. Compound components. MK | 2MK | MN | MS | 28M Long-period components. Mf | MSf United States, East Coasi—Con. Willets Pointi(U.S. E. Wharf), ING a Fort Hamilton, N. Y......---. New York (Governors Island), ING Ya New York (The Battery), N. Y- Albany, Hudson River, N. Y.. Sandy Hook, N. J.........--.-- Atlantie City (Million Dollar Pier), N. J. Philadelphia (Chestnut St.), Pa- Breakwater Harbor, Del....... Old Point Comfort, Va.......-. Richmond, James River, Va... Washington (navy yard), D.C. Wasanetod (7th St. Wharf), Baltimore (Fells Point), Md...].......|-.----- Baltimore (Fort McHenry), Md. Wilnepeten, Cape Fear River, Charleston (custom house wharf), S. C. Tybee Light, Savannah River, a. Savannah (Fort Jackson), Ga Savannah (water works), Ga... Fernandina (Fort Clinch), Fla. Fernandina (Desoto Street), Fla. Key West (Fort Taylor), Fla.. Key West (Curry’s Wharf), Fla. Pensacola (Warrington Navy Yard), Fla. wee ee nl e see eee fee ee nee) Ue UZU |--~-... = welwle ala} = inte elm we jtsteeee ee ees (ars RFS rag Bo) Fo at THis DSTO Hatt ae ea Se tay a Tt ae UP Ree Sees eid 0. 069 | 0. 095 134 336 Mm:| Sa Ssa 0. 094 | 0.274 304 145 35 peak HARMONIC ANALYSIS AND PREDICTION OF TIDES. Compound and long-period. components—Continued. [For other components and references see pp. 319 to 402.] 405 Compound components. Stations. United States, East Coast—Con. Port) Mads ha. cess rocorcccesc Galveston (Fort Point), Tex... Galveston (Doswell’s Wharf), Tex.” Galveston (20th St.), Tex..... West Indies. ASSAM SAR AMAS. 6-2 SUE: St. Thomas Harbor, Virgin eres eat Ol HSIANGSMmUMeNe eS Tiel df. Poked scc| smc. 0. South America. Georgetown, British Guiana...| 0.033 | 0.013 290 348 Itaqui, Maranhao, Brazil....... Pernambuco (Recife Arsenal), Brazil. Rio del Janeiro (arsenal), Brazil. Rio Grande do Sul, Brazil..... Buenos Aires, Argentina =e65eHS) (Shoo 60d) Gaac os Mar del Plata, Argentina.......|....-.- Puerto Quequen, Argentina....|.....-.|......- Puerto Belgrano, Bahia Blanca, PATPCINAMAL MN fh) Vids Carls sec ncale a lone San Antonio, Argentina. .......}..-.---|.....-. Comodoro Rivadavia, Argen- tina. ; Puerto Deseado, Argentina....].......|..-...- Panama Canal Zone, West Coast. Naos Island, Canal Zone Walnanaiso Chiles: oo. acs er 0.155 120 0. 041 9 Balbony@anall Zone so xicexcleai2|etent nim niais)a =| we nimararn Long-period components. 25M Mf | MSf Mm Ssa 406 U. S. COAST AND GEODETIC SURVEY. Compound and long-period components—Continued. [For other components and references see pp. 319 to 402.] Stations. Mexico, West Coast. Mazatlan, Mexico United States, West Coast. Han Dievo,|Califse..sscseesoseae San Francisco (Fort Point), Calif. San Francisco (Presidio), Calif. . Sausalito; Calif: 2.2.2 -cc-ceenee Humboldt Bay (South Jetty Landing), Calif. Astoria, Oreg Port Townsend, Wash...-...... Seattle (Madison St.), Wash. .-. British America, West Coast. Victoria, Vancouver Island....- Sand Heads, Fraser River, British Columbia. Vancouver, Burrand Inlet, British Columbia. Clayoquot, Vancouver Island, British Columbia. Wadhams, Rivers Inlet, British Columbia. Prince Rupert, Chatham Sound, British Columbia. Port Simpson, Chatham Sound, British Columbia. Alaska. Juneau, Gastineau Channel, Stephens Passage. ; Skagway, Lynn Canal Anchorage Compound components. 2MK | MN 2SM Mf MSf | Mm 0.029 | 0.046 | 0.052 | 0.079 66 142 255 138 0.035 | 0.077 | 0.059 | 0.077 14 149 286 103 0.048 | 0.101 | 0.067 | 0.078 356 130 335 316 0.051 | 0.088 | 0.86 | 0.091 189 165 257 114 0.079 | 0.088 | 0.070 | 0.076 131 187 184 141 0.092 | 0.085 | 0.060 | 0.093 145 174 234 168 0.060 | 0.092 ; 0.069 | 0.087 44 180 78 183 Long-period components. Ssa HARMONIC ANALYSIS AND PREDICTION OF TIDES. 407 Compound and long-period components—Continued. [For other components and references see pp. 319 to 402.] Compound components. Long-period components. Stations. MK | 2MK | MN MS 28M Mf MSf | Mm Sa Ssa Alaska TCG. ao oR BE ne Ree BRE Se ORE Se | Genel see ein Aiea IEE, seve IA eo el lect erica 0. 335 | 0.176 SNE Sy eiete, ae arene tote a tena Ye all peeve seal erasares cra al a ete 1a | (ay eerataias= 196 112 Japan, Hokushu Island. MET sol abe e ier ore Bneland -- 2.2.20. sce] sce clone see HO S250 ain O FeLi) meets reer aia ate arepea eel eee See Cee eta Mey Bean } 254 PUSH | sect at elise a eco a | eens eee cleemene Thiveuponteebingl andes. 2 e|. eee valle We Aes ; 0.406 | 0.137 | 0.041 | 0.056 | 0.126 | 0.362] 0.142 Soe aoe Seeoree Gaeta 1 208 216 282 21 279 239 189 Seelam (uinub Ma Of miOhyGe) sit ae toa oe eee ee le Sc cclleCa ste s e OE I e a ee 0.485} 0.058 SEDER LS 71 <3 Rik EAA as NR bey ee eta eyo (UTE SD] ea Le eran | Ieee et 240 183 OAM OCOLAM GN se vac. ction is secs c|asee eS e 0.061 | 0.062 | 0.063 | 0.038 | 9.061 | 0.128 | 0.256 | 0.124 erdeiees [er abv ane 88 170 325 8 104 145 263 151 BintreipunSiMek Scotlands cltcciie doer otal ee Cael edn io tela Wn labo eee 0.177 | 0.082 Pes ea ais | ele Rivas ear eis eS iced eae [ees remeron fthcy Grey rtd B= A 220 113 Stromness, Orkney Islands-..-...}-.--.-.|.--.-.. 0.030 | 0.099 | 0.068 | 0.066 | 0.932 | 0.025 | 0.444 | 0.099 abo irape line Sk Shard 220 287 255 323 263 135 238 202 Ramccromertroland) ee EB Rhee ee BV Rare 329 Halim alher ae are ue tire eee eh eee ee are 371 Jeleheitenol iene ee 383 slope eyes s es ae 340 TEVO 1) 31 GRR eae re ane Se ae 396 LOH SUTTELS Ta 17 Cee eee PURE Hee ee tee eran 342 TdSOnS travt eee eee eee 322 I. Pra RN Gy Sane 2a re al ea ny a 385 Imilandy Seah: soe a8 ee ae a ee ae 345, 351 Hire earn Oe. 2 US a eee el pe pe 400 1 8 721 ee ra UNE TSE TATE | OY SS 393 d. 319 340 376 364 Juan de Fuca Strait (Wash.) _______--------- 335 KK. Kei Slam senwwe ue. as eee wy Oe ee 371 Keer rielenvils] am die sai ae Ale alee ae ee 391 KS Up IS lan Cae Lae ea pas ea eae eee 352 KOM AKS NS] an Ge Sea Cee Ue eed el ae 338 Komodovisland 2325). tees ee ae 380 EGO {212 a aR kere oy Gm ea NS AS? SEE ee eee 358 KGS Kaye aye ee ee 339 L. Take Superior: a-3 2 2 Se eos ee ee 401 Heytellsland=s) ie Bee 364 Lombok Island 379 TE OUTS TEVA ee a ees eae a LO a eee eter 328 AUZOMVES sn Ges sees Mee LE tok La ee ae 363 ynn@ anal = sass Sad Be eee es 337 M. Madagascar 2.22225 ee ee ee ae ee 390 (Wisin ett) ioe hate Pe ee 323 Malay Archipelago nesses ee eee ae 365 NMialayebenins il aes mail seo eee a ee 363 DMilarey ari Ge aie Weer ae tel ies eee See oe 326 Miassachusettiss =: 5 eet lene at eee 323 IMG UTI bUSoS] aride sees eee es ie ee 390 Mediterranean seasesees see ee ee 392 416 INDEX. Page. Ss. Page IVGelvaillesislar Gee saan ioee ey oO eum he wesc. SPO Sirs eh wadsaes) Igy ee 322: IIS AKO IDE COLo Yh ee Uae ee 3297 | (Stacia: 222 5.2 ee eee 330 Wiest Cast sg ee a ag ar See 334), )) Samar TST aT Gl peices isn! | lege ebay all aes 364 Michigan________ RA A SE ee Be eae 401-9) Samoan Islands: 22) yeaa eee eee 383 VEIT MESO bas ee ee Ee BES eo ED 401") Sannak Islands = 502) eens 338 IVIRSSISSIO Oe ae oe NM aU A A a aS 328..| Santa, Rita Tsland 402 ee a ee 364 DWE OR OCCON TS MAINT Tay Noy aot ne ee eA egal UNE paso 392 21, (SCO GLa Ee 2 Ne Age Ua sap a 400, 402 Omega ee ON PEP DN Ue Neca oe a 391 N. ShikokuIsland (2 2 eee ee 350 INamSeT Slam dGiee ee ne Saar oe oe SOG ST ama 8 a ee i Et ae ee 363 SIN St Urn 2a eS irra GS a has 368: | “‘Sitberia soo eld ee 340 New britaimilslan dae ie ee see ae 383° | ‘Siberia CArctic) ee ee 320 RING ye SIS GUT yi te es hp ae 323. || Somaliland: Mrench... 3) {eee 389 ING veer aa A a aS OE ENS RE $22).|./ South Africas) ee 391 INVe wy Gra rn ae ee Neg oeld 381 -|> South-America: 2) Se eee 330: INewilerse ye a ee ee 320 | South Carolina__________- 327 ING Ww Yiornkico we 2s. ae ee 324 | South Georgia Islands 333: New Zealand 384 | South Natuna Islands 368. INOnt HG arora Sei a ae CGPS OE B27 |SSpaiimis E ESEe 395 INOLUHMDS VOT oes CL aly i ie EN he 820) 1) Spitz berger es ei ee eae 321 North Natuna Islands_-_-_-___----- eI i 368: | Stephens Passaces. ee eee 337 INorthsSomerset=22246- ars ce Sees 320 | Strait of Bab-el-Mandeb______-_____________- 389: ISG ALON aT SOV Hao Les Le Ai rN ie 389) | Straits Settlements 22sec eee 402. INO wa yrs ehrce RR oy ES RE Re 399. | Sumatra ss. ee 372: SING aye CAM CUT) eae an tee aE I ee ee 32l.| Sumba Islands» ss 2202 ee nae 380 TNO NA WI SKE ONE Es ee ea ne RIT I a 2 ee 323) 1) Sumbawa Islam 22 05s eee ee 379 INtSH aga UR ayaa tee eee ia ee ay 839) ‘Sweden: 20s ss. 20 aoe se eae 398. 0. Omnbulslarad ast coemner Mb cee a SNe kee 371 f T. Oregon______-_-___--------------------------- 885 | Taiwan Teland... se 5 357 P. 1 TMOXaS soe Ta ee ee ea 328. : Miger Islands. 2/0) Su eee 380 Panama, Canal Zone, East Coast_-_--------- 329'5) Tiimor Island) 7 Se ee ee 381 Panama, Canal Zone, West Coast__--_------- B80) ais 2s ee ae ae 392° iPaternosteruslandsas eis = eee eee 309: Murkey Soo 0 eee eee 389: TRETITAS VV aaa es see UE ees NI IE ee 325 AEA ETA TAL UGE Gots seeere we a en ae ee 320 erie Strait eres Ce we NAO A TU Ve 336 U. YEN SP eR Rela Eo a es ME Us lp thao ee nt 389 : Pescadores Islands -_--__-----------------__-- 357 | United States, East Coast_________________-- 323. Bhilippinehislan GSEs see eee ee 363 | United States, West Coast___________________ 334 Bontar WS ar Gee eae ee ere ian eee eke 381 ||) Uruguay. 208s a2 ee ee 331 AER OT tC TATA CRY ee oe eee ees st gi ag 320 d ERO ye| Bil {EXO 00) (0 Upsala cea lou ec eee 320 Vv TExaye Roy d oe (Olay Lope eek ah eal ME est Nh a eA 329 : Portier eek eae eel as ee ee ha eee B95; 4 025 aniiir-priralihs learn Cl Soca s a ea 330: Jetosiallliforal ej byaxolse ye ee eee 37905): Wireman 20s are ae eee eee 325: IPribiloflslandseesse es ee oe ae 339 Prince Hdwardslsland 2 eee eee 323 W Prince) Wilham) Sound? este eee 337 Y Puget Sound :(Wash:)-22- 22) oho ee GBI) IT \yiclareateq nore 10), (Ol Ses 325: Q Washington. (State)L = sae 335 Quebec i 399 | West Indies___--..---------------_----------- 329 Ce Ta ETRE EGR GL cae aaa Winter Harbor... 0S eee R. Ried Seanitrnta liga vse OMe Mant MeaeL CEU 389 Y. TRG bao ISI GING Les Es eee 390 , TR ode sl slain sok Ts 324. | Yugoslavia... 2 2 eee 392. UTS icyou nate ata ae ae MON ed BSR aT RAE ie 321, 402) ||) WukonlRiver Deltas: sass ea 339 O s 3 ee a ee ae oth i é y vice ee a en neh ae jmvarer senirastmpoaleeenslte sThareleo ei Loewen hea ne beara eachraas paw reas bbe Nene h-akhY Roth MPAA LAA RAPA 7p Lm rahe Mtatntn tees Coseek nat tas aati yer nerves ath orp am art bet ae ord keg gry bch bone aa hsarhat dey M Bath ot rf