C at the intersection of the ecliptic and the projection of the moon's orbit, and the lunar inter- section A at the intersection of the celestial equator and the projection of the moon's orbit. For brevity these three points are sometimes called respectively "the equinox," "the node," and "the intersection." The vernal equinox, although subject to a slow westward motion of about 50 " per year, is generally taken as a fixed point of reference for the motion of other parts of the solar system. The moon's node has a westward motion of about 19° a year, which is sufficient to carry it entirely around a great circle in a little less than 19 years. 20. The angle co between the ecliptic and the celestial equator is known as the obliquity of the ecliptic and has a nearly constant value of 23 K . The angle i between the ecliptic and the plane of the moon's orbit is also constant with a value of about 5°. Figure 1. The angle / which measures the inclination of the moon's orbit to the celestial equator might appropriately be called the obliquity of the moon's orbit. Its magnitude changes with the position of the moon's node. When the moon's ascending node coincides with the vernal equinox, the angle / equals the sum of co and i, or about 28}£°, and when the descending node coincides with the vernal equinox, the angle /equals the difference between co and i, or about lS}i°. This variation in the obliquity of the moon's orbit with its period of approximately 18.6 years introduces an important inequality in the tidal movement which must be taken into account. 21. In the celestial sphere the terms "latitude" and "longitude" apply especially to measurements referred to the ecliptic and vernal equinox, but the terms may with propriety also be applied to meas- urements referred to other great circles and origins, provided they are sufficiently well defined to prevent any ambiguity. For example, we may say "longitude in the moon's orbit measured from the moon's HARMONIC ANALYSIS' AND PREDICTION OF TIDES 7 node." Celestial longitude is always understood to be measured toward the east entirely around the circle. Longitude in the celestial equator reckoned from the vernal equinox is called right ascension, and the angular distance north or south of the celestial equator is called declination. 22. The true longitude of any point referred to any great circle in the celestial sphere may be defined as the arc of that circle intercepted between the accepted origin and the projection of the point on the circle, the measurement being always eastward from the origin to the projection of the point. The true longitude of any point will generally be different when referred to different circles, although reckoned from a common origin ; and the longitude of a body moving at a uniform rate of speed in one great circle will not have a uniform rate of change when referred to another great circle. 23. The mean longitude of a body moving in a closed orbit and referred to any great circle may be defined as the longitude that would be attained by a point moving uniformly in the circle of reference at the same average angular velocity as that of the body and with the initial position of the point so taken that its mean longitude would be the same as the true longitude of the body at a certain selected position of that body in its orbit. With a common initial point, the mean longitude of a moving body will be the same in whatever circle it may be reckoned. Longitude in the ecliptic and in the celestial equator are usually reckoned from the vernal equinox T , which is common to both circles. In order to have an equivalent origin in the moon's orbit, we may lay off an arc &, T' (^g. 1) in the moon's orbit equal to fi, T in the ecliptic and for convenience call the point T' the referred equinox. The mean longitude of any body, if reckoned from either the equinox or the referred equinox, will be the same in any of the three orbits represented. This will, of course, not be the case for the true longitude. 24. Let us now examine more closely the spherical triangle ftTA in figure 1 . The angles co and i are very nearly constant for long periods of time and have already been explained. The side & T, 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 18.6 years, causes a constant change in the form of the triangle, the elements of which are of considerable im- portance in the present discussion. The value of the angle /, the supplement of the angle Q> A T , has an important effect upon both the range and time of the tide, which will be noted later. The side A T, designated by v, is the right ascension or longitude in the celestial equator of the intersection A. The arc designated by £ is equal to the side Q> T— side Q, A and is the longitude in the moon's orbit of the intersection A. Since the angles i and co are assumed to be constant, the values of /, v, and £ will depend directly upon N, the longitude of the moon's node, and may be readily obtained by the ordinary solution of the spherical triangle Q> T A. Table 6 give the values of /, v, and £ for each degree of N. In the computation of this table the value of co for the beginning of the twentieth century was used. However, the secular change in the obliquity of the ecliptic is so slow that a difference of a century in 8 U. S. COAST AND GEODETIC SURVEY the epoch taken as the basis of the computation would have resulted in differences of less than 0.02 of a degree in the tabular values. The table may therefore be used without material error for reductions pertaining to any modern time. 25. Looking again at figure 1, it will be noted that when the longitude of the moon's node is zero the value of the inclination I will equal the sum of co and i 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 I will be equal to angle co— angle i. The angle / will be always positive and will vary from co— i to co+i. When the longitude of the moon's node equals zero or 180°, the values of v and £ will each be zero. For all positions of the moon's node north of the Equator as its longitude changes from 180 to 0°, v and £ will have positive values, as indi- cated in the figure, these arcs being considered as positive when reckoned eastward from T and T', respectively. For all positions of the node south of the Equator, as the longitude changes from 360 to 180°, v and £ will each be negative, since the intersection A will then lay to the westward of T and T'. DEGREE OF APPROXIMATION 26. The problem, of finding expressions for tidal forces and the equilibrium height of the tide in terms of time and place does not admit of a strict solution, but approximate expressions can be ob- tained which may be carried to as high an order of precision as desired. In ordinary numerical computations exact results are seldom ob- tained, the degree of precision depending upon the number of decimal places used in the computations, which, in turn, will be determined largely by the magnitude of the quantity sought. In general, the degree of approximation to the value of any quantity expressed numerically will be determined by the number of significant figures used. With a quantity represented by a single significant figure, the error may be as great as 33 % percent of the quantity itself, while the use of two significant figures will reduce the maximum error to less than 5 percent of the true value of the quantity. The large possible error in the first case renders it of little value, but in the latter case the approximation is sufficiently close to be useful when only rough results are necessary. The distance of the sun from the earth is popularly expressed by two significant figures as 93,000,000 miles. 27. With three or four significant figures fairly satisfactory approxi- mations may be represented, and with a greater number very precise results may be expressed. For theoretical purposes the highest at- tainable precision is desirable, but for practical purposes, because of the increase in the labor without a corresponding increase in util- ity, it will be usually found advantageous to limit the degree of precision in accordance with the prevailing conditions. 28. Frequently a quantity that is to be used as a factor in an expres- sion may be expanded into a series of terms. If the approximate value of such a series is near unity, terms which would affect the third decimal place, if expressed numerically, should usually be re- tained. The retention of the smaller terms will depend to some ex- HARMONIC ANALYSIS' AND PREDICTION OF TIDES 9 tent upon the labor involved since their rejection would not seriously affect the final results. 29. The formulas for the moon's true longitude and parallax on pages 19-20 are said to be given to the second order of approximation, a fraction of the first order being considered as one having an approxi- mate value of 1/20 or 0.05, a fraction of the second order having an approximate value of (0.05) 2 or 0.0025, a fraction of third order having an approximate value of (0.05) 3 or 0.000125, etc. As these formulas provide important factors in the development of the equations repre- senting the tide-producing forces, they determine to a large extent the degrees of precision to be expected in the results. DEVELOPMENT OF TIDE-PRODUCING FORCE FUNDAMENTAL FORMULAS 30. The tide-producing forces exerted by the moon and sun are similar in their action and mathematical expressions obtained for one may therefore by proper substitutions be adapted to the other. Be- cause of the greater importance of the moon in its tide-producing effects, the following development will apply primarily to that body, the necessary changes to represent the solar tides being afterwards indicated. 31. The tide-producing force of the moon is that portion of its gravitational attraction which is effective in changing the water level on the earth's surface. This effective force is the difference between the attraction for the earth as a whole and the attraction for the differ- ent particles which constitute the yielding part of the earth's sur- face; or, if the entire earth were considered to be a plastic mass, the tide-producing force at any point within the mass would be the force that tended to change the position of a particle at that point relative to a particle at the center of the earth. That part of the earth's surface which is directly under the moon is nearer to that body than is the center of the earth and is therefore more strongly attracted since the force of gravity varies inversely as the square of the dis- tance. For the same reason the center of the earth is more strongly attracted by the moon than is that part of the earth's surface which is turned away from the moon. 32. The tide-producing force, being the difference between the attraction for particles situated relatively near together, is small com- pared with the attraction itself. It may be interesting to note that, although the sun's attraction on the, earth is nearly 200 times as great as that of the moon, its tide-producing force is less than one-half that of the moon. If the forces acting upon each particle of the earth were equal and parallel, no matter how great those forces might be, there would be no tendency to change the relative posi- tions of those particles, and consequently there would be no tide- producing force. 33. The tide-producing force may be graphically represented as in figure 2. Let 0=the center of the earth, C=the center of the moon, P=any point within or on the surface of the earth. Then OC will represent the direction of the attractive force of the moon upon a particle at the center of the earth and PC the direction of the attractive force of the moon upon a particle at P. Now, let the magnitude of the moon's attraction at P be represented by the length of the line PC. Then, since the attraction of gravitation varies inversely as the square of the distance, it is necessary, in order to represent the attraction at on the same scale, to take a line CQ of such length that CQ : CP=CP 2 : CO 2 . 10 HARMONIC ANALYSIS! AND PREDICTION OF TIDES 11 34. The line PQ, joining P and Q, will then represent the direction and magnitude of the resultant force that tends to disturb the posi- tion of P relative to 0, 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 0, will have no tendency to change the position of P relative to 0. The remaining force PQ will tend to alter the position of P relative to and is the tide-producing force of the moon at P. The force PQ may be resolved into a vertical component PR, which tends to raise the water at P, and the hori- zontal component PT, which tends to move the water horizontally. Figure 2. 35. If the point P' is taken so that the distance CP' is greater than the distance CO, the tide-producing force P'Q' will be directed away from the moon. While at first sight this may appear paradoxical, it will be noted that the moon tends to separate from P f , but as is taken as the point of reference, this resulting force that tends to separate the points is considered as being applied at the point P' only. 36. To express the tide-producing force by mathematical equations, refer to figure 2 and let r= OP = distance of particle P from center of earth, b= PC = distance of particle P from center of moon, d= OC = distance from center of earth to center of moon, z= COP = angle at center of earth between OP and OC. Also let M=mass of moon, E^mass of earth, a = mean radius of earth, /jl= attraction of gravitation between unit masses at unit distance. #=mean acceleration of gravity on earth's surface. Since the force of gravitation varies directly as the mass and inversely as the square of the distance, Attraction of moon for unit mass at point in direction 0C= ixM (2) 12 U. S. COAST AND GEODETIC STJBVEY Attraction of moon for unit mass at point P in direction PC=-yr (3) 37. Let each of these forces be resolved into a vertical component along the radius OP and a horizontal component perpendicular to the same in the plane OPC, and consider the direction from toward P as positive for the vertical component and the direction corresponding to the azimuth of the moon as positive for the horizontal component. We then have from (2) and (3) Attraction at in direction to P=—w cos z (4) Attraction at perpendicular to OP=—p sin z (5) Attraction at P in direction to P=-p- cos CPR (6) Attraction at P perpendicular to OP=^'-jj- sin CPR (7) 38. The tide-producing force of the moon at any point P is measured by the difference between the attraction at P and at the center of the earth. Letting F v = vertical component of tide-producing force, and F a = horizontal component in azimuth of moon, and taking the differences between (6) and (4) and between (7) and (5), we obtain the following expressions for these component forces in terms of the unit ix\ F 5 /M=M(^™- c -^i) (8) F. h=M(*^-%*) (9) 39. From the plane triangle COP the following relations may be obtained : b 2 =r 2 +d 2 -2rd cos z=d 2 [l-2(r/d) cos z+ (r/d) 2 ] (10) sin CPR=sin CPO=(d/b) sin 2=n — n , ,,>. , -, , , N211 (11) v ' ; [1— 2{r/d) cos z+(r/d) 2 ]i v ' cos <7Pfl=(l-sin 2 CPR)*= n 57 T z ~ r ^ d (12) v ' [1 — 2 {r/d) cos z+ (r/d) 2 ]* v ' 40. In figure 2 it will be noted that the value of z, being reckoned in any plane from the line OC, inay vary from zero to 180°, and also that the angle CPR increases as z increases within the same limits. Sin z and sin CPR will therefore always be positive. As the angle OCP is always very small, the angle CPR will differ by only a very small amount from the angle z and will usually be in the same quad- rant. In obtaining the square root for the numerator of (12) it was therefore necessary to use only that sign which would preserve this HARMONIC ANALYSIS! AND PREDICTION OF TIDES 13 relationship. The denominators of (11) and (12) are to be consid- ered as positive. 41. Substituting in equations (8) and (9) the equivalents for b, sin CPR, and cos CPR from equations (10) to (12), the following basic formulas are obtained for the vertical and horizontal components of the tide-producing force at any point P at r distance from the center of the earth: „ , Mf cos z—r /d 1 /10V F - /*-;* |_{i-2(r/d) cos s +(rid) 2 y*~ cos 2 J (13) n i M\~ sin z 1 ,1 AS F °l»=l[i { l-2(rld) cos Z +(rld)>}l ~ Sm Z \ (14 > 42. To express these forces in their relation to the mean accelera- tion of gravity on the earth's surface, represented by the symbol g r we have glv=E/a*, or »/g=a 2 /E (15) in which E is the mass and a is the mean radius of the earth. Sub- stituting the above in formulas (13) and (14), we may write F. /9 =(MIE) (oft* [(^(^^W^ 2 ] (16) F a lg= (M/E) {aid)* [ {1 _ 2m Zs\+m^ ~^ z ] < 17 > 43. Formulas (16) and (17) represent completely the vertical and horizontal components of the lunar tide-producing force at any point in the earth. If r is taken equal to the mean radius a, the formulas will involve the constant ratio M/E and two variable quantities — the angle z which is the moon's zenith distance, and the ratio a/d which is the sine of the moon's horizontal parallax in respect to the mean radius of the earth. Because of the smallness of the ratio a/d it may also be taken as the parallax itself expressed as a fraction of a radian. The parallax is largest when the moon is in perigee and at this time the tide-producing force will reach its greatest magnitude. A more rapid change in the tidal force at any point on the earth's surface is caused by the continuous change in the zenith distance of the moon resulting from the earth's rotation. The vertical com- ponent attains its maximum value when z equals zero, and the hori- zontal component has its maximum value when z is a little less than 45°. Substituting numerical values in formulas (16) and (17) and in similar formulas for the tide-producing force of the sun, the fol- lowing are obtained as the approximate extreme component forces when the moon and sun are nearest the earth: Greatest F v /#=.144X10- 6 for moon, or .054 X10" 6 for sun (18) Greatest F a /#=.107X10" 6 for moon, or .041 X10~ 6 for sun (19) The horizontal component of the tide-producing force may be meas- ured by its deflection of the plumb line, the relation of this component to gravity as expressed by the above formula being the tangent of the angle of deflection. Under the most favorable conditions the 246037—41 2 14 U. S. COAST AND GEODETIC SURVEY greatest deflection due to the moon is about 0.022" and the greatest deflection due to the sun is less than 0.009" of arc. 44. To simplify the preceding formulas, the quantity involving the fractional exponent may be developed by Maclaurin's theorem into a series arranged according to the ascending powers of r/d, this being a small fraction with an approximate maximum value of 0.018. Thus : = 1 + 3 cos z (r/d) {1-2 (r/d) cos z-\-(r/d) 2 y + 3/2 (5 cos 2 2-1) (r/d) 2 + 5/2 (7 cos 3 s-3 cos 2)(r/c0 3 +etc. (20) 45. Substituting (20) in formulas (16) and (17) and neglecting the higher powers of r/d, we obtain the following formulas: F v /g=Z (M/E)(a/d) 2 (cos 2 2-1/3) (r/d) + 3/2 (M/E)(a/d) 2 (5 cos 3 2-3 cos 2) (r/d) 2 (21) F a /-90 o +2/:)] (49) F v t 2 /g=3/2 £7 cos 2 FX [(c/d) 3 cos 4 \I cos (2T-2s+2/i+2£-2*>-2£) + 1/2 (c/d) 3 sin 2 / cos (?T+2h-2v) + (c/d) 3 sin 4 \I cos (2T+2s+2/^-2£-2*>+2£) (50) c Figure 4. Disregarding at this time the slow change in the function of I, the variable part of each term of the above formulas may be expressed in one of the following forms — (c/dy, (c/d) 3 cos A, (c/dy cos (A-\-2k), or (c/dy cos (A— 2k), in which A includes all the elements of the variable angular function excepting the multiple of k. 62. The following equations for the motion of the moon were adapted from Godfrey's Elementary Treatise on the Lunar Theory: s' = true longitude of moon (in radians) =s (mean longitude) +2e sin (s— £>) + 5/4 e 2 sin 2(s— p) (elliptic inequality) + 1 5/4 me sin(s— 2h-\- / p) (evectional inequality) + 1 1 /8 m 2 sin 2 (s — h) (variational inequality) (51) 20 U. S. COAST AND GEODETIC SURVEY c/d= (true parallax of moon)/(mean parallax of moon) = unity -\-e cos (s— p)-\-e 2 cos 2(s— p) (elliptic inequality) + 15/8 me cos (s—2h-\-p) (evectional inequality) + m 2 cos 2(s—h) (variational inequality) (52 ) in which s' = true longitude of moon in orbit (referred to equinox) s=mean longitude of moon h=mesm longitude of sun p=mean longitude of lunar perigee e = eccentricity of moon's orbit =0.0549 m= ratio of mean motion of sun to that of moon =0.0748 The elements e and m are small fractions of the first order and the square of either or the product of both may be considered as being of the second order. In the following development the higher powers of these elements will be omitted. 63. Since k has been taken as the difference between the true and the mean longitude of the moon, we may obtain from (51) k=2e sin (s— p) + 5/4 e 2 sin 2(s— p) + 15/4 me sin (s— 2h+p) + ll/8 m 2 sin 2(s-h) (53) The value of k is always small, its maximum value being about 0.137 radian. It may therefore be assumed without material error that the sine of k or the sine of 2k is equal to the angle itself. Then sin 2k=2k=4:e sin (s— p)+5/2 e 2 sin 2(s— p) + 15/2 me sin (s-2A+p) + ll/4 m 2 sin 2(s-h) (54) cos 2k=l-2 sin 2 k=\-2k 2 = 1 -4e 2 +4e 2 cos 2 (s-p) (55) terms smaller than those of the second order being omitted. 64. Cubing (52) and neglecting the smaller terms, we obtain (c/d) 3 =l + 3/2 e 2 +3e cos (s-p) + 9/2 e 2 cos 2(s-p) +45/8 me cos (s— 2/^+p) + 3 m 2 cos 2(s—h) (56) Multiplying (54) and (55) by (56) (c/dy sin 2fc=4e sin (s— p) + 17/2 e 2 sin 2(s— p) + 15/2 me sin (s-2/*,+p) + ll/4 m 2 sin 2(s-h) (57) (c/dy cos 2fc=l-5/2 e 2 + 3 e cos (s-p) + 17/2 e 2 cos 2(s-p) +45/8 me cos (s— 2/^+p) + 3 m 2 cos 2(s— h) (58) 65. From (56), (57), and (58), we may obtain the following general expressions applicable to the further development of formulas (48) to (50). Negative coefficients have been avoided by the introduction of 180° in the angle when necessary. (c/dy cos (A-2k) = (c/dy cos 2k cos A+ (c/d) z sin 2k sin A = (1-5/2 e 2 ) cos A + 7/2 e cos (A-s+p) + l/2e cos (^l+s-p+180 ) + 17/2 6 2 cos (A-2s+2p) + 105/16 me cos (A-s+2h-p) + 15/16 me cos (A+s-2h+p+180°) +23/8 m 2 cos (A-2s+2h) + l/8 m 2 cos (A+2s-2h) (59) HARMONIC ANALYSIS' AND PREDICTION OF TIDES 21 (c/df cos A= (1 + 3/2 e 2 ) cos A + 3/2 e cos (A— s+p) + 3/2 e cos (A+s— p) + 9/4 e 2 cos (A-2s+2p) + 9/4: e 2 cos (A+2s-2p) +45/16 me cos (A— s+2/i— p)+45/16 me cos (A+s— 2h+p) + 3/2 m 2 cos (A-2s+2h) + 3/2 m 2 cos (A+2s— 2h) ' (60) (c/d) 3 cos (A+2k) = (c/df cos 2£ cos A— (c/d) 3 sin 2£ sin A = (1-5/2 e 2 ) cos A + 7/2 e cos (A+s-p) + 1/2 e cos (A-s+p+l$0°) + 17/2 e 2 cos (A+2s-2p) + 105/16 me cos (A+s — 2^+jp) + 15/16 me cos (A-s+2/^-2?+180°) +23/8 m 2 cos (A+2s-2h) + l/8 m 2 cos (A-2s+2/i) (61) 66. After suitable substitutions for A have been made in the three preceding equations they are immediately applicable to the final expansion of the several terms in formulas (48) to (50), excepting the first term of (48) for which formula (56) may be used directly. Each term in the expanded formulas given below represents a constituent of the lunar tide-producing force and for convenience of reference is designated by the letter A with a subscript. There are also given the generally recognized symbols for the principal constituents, and when such a symbol is enclosed in brackets it signifies that the term given only partially represents the constituent so named. 67. Formula for long-period constituents of vertical component of principal lunar tide-producing force: ^30 /0=3/2 £7(1/2-3/2 sin 2 7) X (A) [(2/3-sin 2 /){ (1 + 3/2 e 2 ) permanent term (A) +3 e cos (s— p) Mm (A) +9/2 e 2 cos (2s -2p) (A) +45/8 me cos (s— 2h+p) (A) +3 m 2 cos (2s-2h)} MSf (A) + sin 2 7{ (1-5/2 e 2 ) cos (2s-2£) Mf (A-j) +7/2 e cos (3s— p — 2£) (A 8 ) +1/2 e cos (s+p+18Q°-2Z) (A 9 ) +17/2 e 2 cos (4s—2p—2£) (Ao) +105/16 me cos (3s-2h+p-2£) (A n ) +15/16 me cos (s+2h-p+180°-2$ (A 12 ) +23/8 m 2 cos (4s-2/i-2£) (A 3 ) + 1/8 m 2 cos (2h-2£) }] (62) 68. Formula for diurnal constituents of vertical component of principal lunar tide-producing force: F m /g=3/2 C7sin2FX (A u ) [sin I cos 2 1/2/ {(1-5/2 e 2 ) cos (T-2s+h+90 o + 2£-v) O l (A 15 ) +7/2 e cos (T-3s+h+p + 90°+2^-p) Q { (A lt ) +1/2 e cos (T-s+h-p-90°+2^-v) [M,] (A l7 ) +17/2 6 2 cos (T-4s+h+2p+90°+2^-v)_. 2Qx (As) + 105/16mecos(7 7 -3s+3/i- : p + 90 o + 2£-*>)_ Pl (A 9 ) +15/16 me cos (T-s-h+p-90°+2£-v) (A 20 ) +23/8 m 2 cos (T-4s + 3h+90°+2£-v) )-___ [KJ {^23) +3/4ecos(T-s+A,+^-90°-^) [MJ W24) +3/4 e cos (T+s+/i,-^-90 -^) J t (-A25) +9/8 e 2 cos (T-2s+ft+2p-90°-*>) (A 2e ) +9/8 e 2 cos (r+2s+/*,-2p-90 o -j/) (^27) +45/32 me cos (T-s+3h-p-90°-v) X i (A 28 ) +45/32 me cos (T+s-Zi+p-gO -*') O x (A») +3/4 m 2 cos(T-2s+3/>,-90 o -*>) MP X {^30) +3/4m 2 cos(T+2s-A-90 o -*>)} SOx +sin I sin 2 }£/ (^31) {(1-5/2 e 2 ) cos (r+2s+^-90°-2^-^)___ OOi (Aw) +7/2 6 cos (T+3s+/i-^-90 o -2^-i/) KQ X (^33) +1/2 e cos (T+s+h+p + 90°-2£-p) (^34) +17/2 e 2 cos (T+4s+h-2p-90°-2£-p) (A 35 ) +105/16 me cos (T+3s-h+p-90°-2Z-v) (A Z6 ) +15/16 me cos (T+s+3^-^ + 90°-2^-^) (Am) +23/8 m 2 cos (T+4s-h-90°-2£-v) (A 38 ) +1/8 m 2 cos (T+3&-90°-2£-*/)}] (63) 69. Formula for semidiurnal constituents of vertical component of principal lunar tide-producing force: F m /g=S/2 Ucos 2 YX [cos 4 il{ (1-5/2 e 2 ) cos (2T-28+2h+2£-2v) M 2 + 7/2 e cos (2T-Z8+2h+p+2£—2v) N 2 + 1/2 e cos (2T—8+2h—p+180°+2Z—2v) _ [L,] + 17/2 e 2 cos (2T-4s+2h+2p+2£-2v) __. 2N 2 + 105/16 me cos (2T-3s+4^-p+2£-2z/> . v 2 (A \Am, <4n (A ti (A a (A44 (A« (A a {A„ (A ts (A49 (A m (An (A 52 (A S3 (A 5 4 (A m (4m (A sa (Am \Aqo (A a (A 62 (An 15/16 me cos (2T—8+p+180° + 2Z—2v) . X +23/8 m 2 cos (2T—4s+4th+2Z—2v) \x 2 + 1/8 m 2 cos (2T+2£-2*>)} + sin 2 /{ (1/2 + 3/4 e 2 ) cos (2T+2h-2v) [K 2 ] +3/4 e cos (2T-s+2h+p-2v) [L 2 ] + Z/4 e cos (2T+s+2h-p-2v) KJ 2 + 9/8 e 2 cos (2T- 2s+2h+2p-2v) + 9/8 e 2 cos (2T+2s+2h-2p-2v) +45/32 me cos (2T-s+4h-p-2v) +45/32 me cos (2T+s-\-p-2v) + 3/4 m 2 cos (2T—28+4h—2v) + 3/4 m 2 cos (2T+2s-2*>)} + sin 4 y r { (1-5/2 e 2 ) cos (2jT+2s+2^-2^-2^) + 7/2 e cos {2T+Zs+2K-p-2$-2v) + 1/2 e cos (2T+s+2/>,+2?+180 o -2£-2j>) + 17/2 e 2 cos (2r+4s+2/i-22?-2£-2?) + 105/16 me cos (2T+3s+p-2£-2v) + 15/16 me cos (2T+s+4/i-p + 180 o -2£-2*>) +23/8 m 2 cos (2T+4s-2£-2v) + 1/8 m 2 cos (2T+4^-2^-2z.) }] (64) 70. Arguments. — Except for the slow changes in the values of /, £, and v which result from the revolution of the moon's node, each term other than the permanent one in the three preceding formulas is an harmonic function of an angle that changes uniformly with time. This angle is known as the argument of the constituent, also as the equilibrium argument when obtained in connection with the develop- HARMONIC ANALYSIS AND PREDICTION OF TIDES 23 merit of the equilibrium tide. By analogy, the argument of the per- manent term may be considered as zero, the cosine of zero being unity. 71. The argument serves to identify the constituent by determining its speed and period and fixing the times of the maxima and minima of the corresponding tidal force. It usually consists of two parts represented by the symbols V and u. When referring to a particular instant of time such as the beginning of a series of observations, the V is written with a subscript as F - The first part of the argument in- cludes any constant and multiples of one or more of the following astronomical elements — T, the hour angle of the mean sun at the place of observation; s, the mean longitude of the moon; h the mean longitude of the sun; and p, the longitude of the lunar perigee. The second part u includes multiples of one or both of the elements £ and v, which are functions of the longitude of the moon's node and vary slowly between small positive and negative limits throughout a 19-year cycle. In a series of observations covering a year or less they are treated as constants with values pertaining to the middle of the series. They do not affect the average speed or period of the constituent. Their values corresponding to each degree of N, the longitude of the moon's node, are included in table 6, formulas for their computation being given on p. 156. 72. The hourly speed of a constituent may be obtained by adding the hourly speeds of the elements included in the V of the argument. These elementary speeds will be found in table 1. The period of a constituent is obtained by dividing 360° by its speed. The approxi- mate period is determined by the element of greatest speed contained in the argument. Thus, the hour angle T has a speed of 15° per mean solar hour and all constituents with a single T in their argu- ments have periods approximating one day, while constituents with arguments containing the multiple 2T have periods approximating the half day. Next to T, the element of greatest speed is s the mean longitude of the moon, and long-period constituents with a single s in their arguments will have periods approximating the month and with any multiple of s the corresponding fraction of a month. The arguments and speeds of the constituents are listed in table 2. Numerical values of the arguments for the beginning of each calendar year from 1850 to 2000 are given in table 15 for con- stituents used in the Coast and Geodetic Survey tide-predicting machine. Tables 16 to 18 provide differences for referring these arguments to any day and hour of the year. 73. In order to visualize the arguments of the constituents depend- ing primarily upon the rotation of the earth, some have found it convenient to conceive of a system of fictitious stars, or "astres fictifs" as they are sometimes called, which move at a uniform rate in the celestial equator, each constituent being represented by a separate star. Thus, for the principal lunar constituent we have the mean moon and for the principal solar constituent the mean sun, while the various inequalities in the motions of these bodies are served by imaginary stars which reach the meridian of the place of observation at times corresponding to the zero value of the constituent argument. For the diurnal constituents the argument equals the hour angle of the star but for the semidiurnal constituents the argument is double the hour angle of the star. 24 U. S. 00 AST AND GEODETIC SURVEY 74. Coefficients. — The complete coefficient of each term of formulas (62) to (64) includes several important factors. First, the basic factor U, which equals the ratio of the mass of the moon to that of the earth multiplied by the cube of the mean parallax of the moon, is common to all of the terms. This together with the common numerical coeffi- cient may be designated as the general coefficient. Next, the function involving the latitude Y is known as the latitude factor, each formula having a different latitude factor. Following the latitude factor is a function of /, the inclination of the moon's orbit to the plane of the . earth's equator, which may appropriately be called the obliquity factor, each factor applying to a group of terms. Lastly, we have an indi- vidual term coefficient which includes a numerical factor and involves the quantity e or m. Since these factors are derived from the equa- tions of elliptic motion, they will here be referred to as elliptic factors . The product of the elliptic factor by the mean value of the obliquity factor is known as the mean constituent coefficient ((7). Numerical values for these coefficients are given in table 2. Since all terms in any one of the formulas have the same general coefficient and latitude factor, their relative magnitudes will be proportional to their constitu- ent coefficients. Terms of different formulas, however, have different latitude factors and their constituent coefficients are not directly comparable without taking into account the latitude of the place of observation. 75. The obliquity factors are subject to variations throughout an 18. 6-year cycle because of the revolution of the moon's node. Dur- ing this period the value of / varies between the limits of co — i and co-H, or from. 18.3° to 28.6° approximately, and the functions of I change accordingly. In order that tidal data pertaining to different years may be made comparable, it is necessary to adopt certain stand- ard mean values for the obliquity factors to which results for different years may be reduced. While there are several systems of means which would serve equally well as standard values, the system, adopted by Darwin in the early development of the harmonic analysis of tides has the sanction of long usage and is therefore followed. By the Darwin method, the mean for the obliquity factor is obtained from the product of the obliquity factor and the cosine of the elements £ and v appearing in the argument. This may be expressed as the mean value of the product J cos u, in which J is the function of / in the coefficient and u the function of £ and v in the argument. Since u is relatively small and its cosine differs little from unity, the result- ing mean will not differ greatly from the mean of J alone or from the function of i" when given its mean value. 76. Using Darwin's system as described in section 6 of his paper on the Harmonic Analysis of Tidal Observations published in volume I of his collection of Scientific Papers (also in Report of the British Association for the Advancement of Science in 1883), the following mean values are obtained for the obliquity factors in formulas (62) to (64). These values were used in the computation of the corresponding constituent coefficients in table 2. The subscript is here used to indicate the mean value of the function. For terms A x to A 5 in formula (62) [2/3-sin 2 /] =(2/3-sin 2 «) (1—3/2 sin 2 t) = 0.5021 (65) For terms A 6 to A 13 in formula (62) [sin 2 / cos 2£] =sin 2 co cos 4 #=0.1578 (66) HARMONIC ANALYSIS AND PREDICTION OF TIDES 25 For terms A u to A 2X in formula (63) [sin 7 cos 2 \I cos (2£— p)]o=sin co cos 2 |co cos 4 £i=0.3800 (67) For terms A 22 to A zo in formula (63) [sin 27 cos 4,= sin 2co (1-3/2 sin 2 i)=0.7214 (68) For terms A zi to ^4 38 in formula (63) [sin 7 sin 2 \I cos (2£ + ^]o=sin w sin 2 \v> cos 4 ^=0.0164 (69) For terms A 39 to ^4 4 e in formula (64) [cos 4 \I cos (2£- 2 */)]()= cos 4 |co cos 4 ^=0.9154 (70) For terms A i7 to A 55 in formula (64) [sin 2 7 cos 2Ho=sin 2 o> (1-3/2 sin 2 i) =0.1565 (71) For terms A 5& to A &3 in formula (64) [sin 4 fcf cos (2£+2*>)] =sin 4 £« cos 4 -^=0.0017 (72) 77. The ratio obtained by dividing the true obliquity factor for any value of 7 by its mean value may be called a node factor since it is a function of the longitude of the moon's node. The symbol generally used for the node factor is the small/. The node factor may be used with a mean constituent coefficient to obtain the true coefficient corresponding to a given longitude of the moon's node. Node factors for the several terms of formulas (62) to (64) may be expressed by the following ratios: J(A 1 ) to f(A 5 ) =/(Mm) = (2/3-sin 2 7)/0.5021 (73) J(A 6 ) toJ(A l3 ) =/(Mf) = sin 2 1 /0.1578 (74) j{A u ) to j(A 2l ) =/(O0 =sin I cos 2 ±1 /0.3800 (75) j(A 22 ) toJ(A 30 ) =/(J0 =sin 27 /0.7214 (76) /(^3i) to f (A 2B ) =/(OO0 = sin 7 sin 2 \I /0.0164 (77) /CAso) to/(A 46 ) =/{M 2 ) =cos 4 17 /0.9154 . (78) /(A 47 ) to/(^ 55 ) =sin 2 7 /0.1565 (79) /(^1 56 ) to/(A 3 )=sin 4 1 7 /0.0017 (80) Node factors for the middle of each calendar year from 1850 to 1999 are -given in table 14 for the constituents used in the Coast and Geodetic Survey tide-predicting machine. These include all the factors above excepting formulas (79) and (80). However, since formula (79) represents an increase of only about one per cent over formula (74), the tabular values for the latter are readily adapted to formula (79). Node factors change slowly and interpolations can be made in table 14 for any desired part of the year. For practical purposes, however, the values for the middle of the year are generally taken as constant for the entire year. 78. The reciprocal of the node factor is called the 1 reduction fact or and is usually represented by the capital F. Applied to tidal coeffi- cients pertaining to any particular year, the reduction factors serve to reduce them to a uniform standard in order that they may be comparable. Logarithms of the reduction factors for every tenth of a degree of 7 are given in table 12 for the constituents used on the tide-predicting machine of this office. 79. Formulas (62), (63), and (64), for the long-period, diurnal, and semidiurnal constituents of the vertical component of the tide-pro- ducing force may now be summarized as follows: Let 7£= constituent argument from table 2 (7= mean constituent coefficient from table 2 / =node factor from table 14 26 U. S. OOAST AND GEODETIC StTRVEY Then Fm /flf=3/2 £7(1/2-3/2 sin 2 F) 2/(7 cos # (81) Fm lg=3/2 U sin 2F 2 fC cos # (82) F vZ2 lg=3/2 U cos 2 F S/(7 cos E (83) Latitude factors for each degree of F are given in table 3. The column symbol in this table is Y with annexed letter and digits corre- sponding to those in the designation of the tidal forces. Thus, F p30 represents the latitude factor to be used with force F v30 , its value being equal to the function (1/2 — 3/2 sin 2 F). Taking the numerical value for the basic factor £7 from table 1, the general coefficient 3/2 U is found to be 0.8373 X10~ 7 . HORIZONTAL COMPONENTS OF FORCE 80. The horizontal component of the principal part of the tide- producing force as expressed by formula (25), page 14, is in the direc- tion of the azimuth of the tide-producing body. This component may be further resolved into a north-and-south and an east-and-west direction. In the following discussion the south and west will be considered as the positive directions for these components. Now ]et F s2l /.g= south component of principal tide-producing force F w z )g=west component of principal tide-producing force A = azimuth of moon reckoned from the south through the west. From formula (25), we then have F sZ /g=S/2 (M/E) (a/d) z sin 2z cos A (84) F w3 /£=3/2 (M/E) (a/df sin 2z sin A (85) 81. Referring to figure 3, page 16, the angle P'PM equals A, the azimuth of the moon. Now T , keeping in mind that the angle MPC is the supplement of A, the angle PCM equals t , and the arcs MC and PC are the respective complements of D and Y, we may obtain from the spherical triangle MPC the following relations : sin z cos A= — cos Y sin D+sin Y cos D cos t (86) sin z sin ^L=cos D sin t (87) Multiplying each of the above equations by the value of cos z from formula (31), the following equations may be derived: sin 2 z cos ^1=2 sin z cos z cos A = 3/4 sin 2Y (2/3-2 sin 2 J9) — cos 2 Y sin 2D cos t + 1 /2 sin 2 Y cos 2 D cos 2 1 (88) sin 2z sin A=2 sin z cos z sin A =sin Y sin 2D sin t + cos Fcos 2 Dsin2* (89) 82. Substituting in (84) and (85) the quantities from equations (88) and (89), we have F s * lg = 9/8 (M/E) (aldf sin 2 Y (2/3 - 2 sin 2 £>) F s30 /g -3/2 (M/£')(a/e0 3 cos2Fsin2Z>cos* F sZ1 /g + 3/4 (M/E) (a/df sin 2F cos 2 £> cos 2t F s32 lg (90) F w3 /g=S/2 (M/E) (a/df sin F sin 2D sin * F w31 /g + 3/2 (M/#)(a/d) 3 cosFcos 2 Psin2* F^ lg (91) HARMONIC ANALYSIS AND PREDICTION OF TIDES 27 The south component is expressed by three terms representing respec- tively the long-period, diurnal, and semidiurnal constituents. For the west component there are only two terms — the diurnal and semidiur- nal, there being no long-period constituents in the west component. Each term has been marked separately by a symbol with annexed digits analogous to those used for the vertical component to indicate the class to which the term belongs. 83. Comparing formula (90) for the south component with formula (32) for the vertical component, it will be noted that the same functions of D and t are involved in the corresponding terms of both formulas, and that the terms differ only in their numerical coefficient and the latitude factor. Allowing for these differences, summarized formulas analogous to those given for the vertical component (page 26) may be readily formed. In order to eliminate the negative sign of the coefficient of the middle term, 180° will be applied to the arguments of that term. With all symbols as before, we then have F s zo /flf=9/8 U sin 2Y2fC cos E (92) F s u /0=3/2 Ucos 2F2/(7cos (#+180°) (93) ^32 lg= 3/4 U sin 2 Y 2 jC cos E (94) 84. Comparing the two terms in formula (91) for the west com- ponent with the corresponding terms in formula (32) for the vertical component, it will be noted that the D functions are the same but that in (91) the sine replaces the cosine for the functions of t. It may be shown that the corresponding development of these terms will be the same as for the vertical component except that in the developed series each argument will be represented by its sine instead of cosine. In order that the summarized formulas may be expressed in cosine functions, 90° will be subtracted from each argument. With the same symbols as before and allowing for differences in the latitude factors,, we obtain F w n /flf=3/2 U sin Y S jC cos (#-90°) (95) F wZ 2 /sr=3/2 U cos Y 2 jC cos (#-90°) (96) 85. Formulas for the horizontal component of tide-producing force in any given direction may be derived as follows: Let A equal the azimuth (measured from south through west) of given direction, and let F a30 /g, F aZ1 /g, and F aZ2 /g, respectively, represent the long-period, diurnal, and semidiurnal terms of the component in this direction. Then F m /g=F m /gXcosA (97) Fan lg=F sZl /gXcos A+F w3l /gXsin A (98) F a ,2 lg=F s v IgXcos A+F wZ2 /gXsin A (99) As the long-period term has no west component, the summarized formula for the azimuth A may be derived by simply introducing the factor cos A into the coefficient of formula (92) . For the diurnal and semidiurnal terms it is necessary to combine the resolved elements from the south and west components. 86. Referring to formulas (93) to (96) and considering a single constituent in each species we obtain the following: 28 U. S. COAST AND GEODETIC SURVEY Diurnal constituent, 3/2 UjC [cos 2Fcos A cos (£'+180°) + sin F sin A cos (E-90°)] = 3/2 UjC (—cos 2F cos A cos #+ sin F sin A sin E) = 3/2 £7jf(7 P 1 cos (E-XO (100) in which P,= (cos 2 2F cos 2 A+sin 2 Y sin 2 A)* (101) Xi = tan" sin Y sin A (102) cos 2 F cos A Semidiurnal constituent, 3/2 UjC [sin F cos F cos A cos E+cos F sin A cos (E-90 )] = 3/2 UjC cos F (sin F cos A cos E+sin .A sin E) = 3/2 UjfC P 2 cos (E-X 2 ) (103) in which P 2 =cos F (sin 2 F cos 2 A+sin 2 A)* (104) X>=tan ; sin A sin F cos A (105) 87. Summarized formulas for the horizontal component of the tide-producing force in any direction A may now be written as follows: Paso /flf=9/8 U sin 2F cos i2/0 cos E P a3 i /fl=3/2 C/Pi 2/0 cos (E-XO P a32 /flf=3/2 ?7P 2 S/Ccos (#-X 2 ) (106) (107) (108) the values for P x , P 2 , Xi and X 2 being obtained by formulas in the preceding paragraph. P x and P 2 are to be taken as positive and the following table will be found convenient in determining the proper quadrant for X x and X 2 . A quadrant North latitude South latitude quadrant x 2 quadrant X, quadrant x 2 quadrant 1 2 3 4 2 or 1 1 or 2 4 or 3 3 or 4 1 2 3 4 3 or 4 4 or 3 1 or 2 2 or 1 2 1 4 3 For the Xi quadrant the first value of each pair is applicable when the latitude does not exceed 45° north or south. Otherwise the second value is applicable. EQUILIBRIUM TIDE 88. The equilibrium theory of the tides is a hypothesis under which it is assumed that the waters covering the face of the earth instantly respond to the tide-producing forces of the moon and the sun and form a surface of equilibrium under the action of these forces. The theory disregards frictjdn and inertia and the irregular distribution of the land masses of the earth. Although the actual tidal movement HARMONIC ANALYSIS AND PREDICTION OF TIDES 29 POLE Figure 6. 246037—41 3 30 TJ. S. COAST AND GEODETIC SURVEY of nature does not even approximate to that which might be expected under the assumed conditions, the theory is of value as an aid in visualizing the distribution of the tidal forces over the surface of the earth. The theoretical tide formed under these conditions is known as the equilibrium tide, and sometimes as the astronomical or gravita- tional tide. 89. Under the equilibrium theory, the moon would tend to draw the earth into the shape of a prolate spheroid with the longest axis in line with the moon, thus producing one high water directly under the moon and another one on the opposite side of the earth with a low water belt extending entirely around the earth in a great circle midway between the high water points. It may be shown mathe- matically, however, that the total effect of the moon at its mean dis- tance would be to raise the high water points about 14 inches above the mean surface of the earth and depress the low water belt about 7 inches below this surface, giving a maximum range of tide of about 21 inches. The corresponding range due to the sun is about 10 inches. Figures 5 and 6 illustrate on an exaggerated scale the theoretical disturbing effect of the moon on the earth. In the first figure the moon is assumed to be directly over the equator and in the last figure the moon is approximately at its greatest north declination. 90. With the moon over the equator (fig. 5), the range of the equi- librium tide will be at a maximum at the equator and diminish to zero at the poles and at any point there will be two high and low waters of equal range with each rotation of the earth. With the moon north or south of the equator (fig. 6), a declinational inequality is introduced and the two high and low waters of the day for any given latitude would no longer be equal except at the equator. This inequality would increase with the latitude and near the poles only one high and low water would occur with each rotation of the earth. Although latitude is an important factor in determining the range of the equilibrium tide, it is to be kept in mind that in the actual tide of nature the latitude of a place has no direct effect upon the rise and fall of the water. 91. A surface of equilibrium is a surface at every point of which the sum of the potentials of all the forces is a constant. On such a surface the resultant of all the forces at each point must be in the direction of the normal to the surface at that point. If the earth were a homogeneous mass with gravity as the only force acting, the surface of equilibrium would be that of a sphere. Each additional force will tend to disturb this spherical surface, and the total deforma- tion will be represented by the sum of the disturbances of each of the forces acting separately. In the following investigation we need not be especially concerned with the more or less permanent deformation due to the centrifugal force of the earth's rotation, since we may assume that the disturbances of this spheriodal surface due to the tidal forces will not differ materially from the disturbances in a true spherical surface due to the same cause. 92. The potential at any point due to a force is the amount of work that would be required to move a unit of matter from that point, against the action of the force, to a position where the force is zero. This amount of work will be independent of the path along which the unit of matter is moved. If the force being considered is the gravity of the earth the potential at any point will be the amount HARMONIC ANALYSIS AND PREDICTION OF TIDES 31 of work required to move a unit mass against the force of gravity from the point to an infinite distance from the earth's center. For the tide-producing force, the potential at any point will be measured by the amount of work necessary to move the unit of mass to the earth's center where this force is zero. 93. Referring to formula (21) for the vertical component of the tide- producing force, if the unit g is replaced by the unit /x from equation (15), the formula may be written as follows: Fv= ^T (cos2 s-l/3>+^^(5 cos 3 2-3 cos z)r 2 (109) 94. Considering separately the tide-producing potential due to the two terms in the above formula, let the potential for the first term involving the cube of the moon's distance be represented by V 3 and the potential for the second term [involving the 4th power of the moon's distance by V 4 . In each case the work required to move a unit mass against the force through an infinitesimal distance — dr toward the center of the earth is the product of the force by — dr, and the potential or total work required to move the particle to the center of the earth may be obtained by integrating between the limits rand zero. Thus V 3 =-^ (cos 2 z- 1/3) f\-dr 3/xM 2d' (cos 2 2-l/3)r 2 (110) Vi= — ^-y^ (5 cos 3 2 — 3 cos z) r 2 dr = f^l(5 cos 3 2-3 cos zY (111) 95. At any instant of time the tide-producing potential at different points on the earth's surface will depend upon the zenith distance (2) of the moon and may be either positive or negative. It will now be shown that the average tide-producing potential for all points on the earth's surface, assuming it to be a sphere, is zero. Assume a series of right conical surfaces with common apex at center of earth and axis coinciding with the line joining centers of earth and moon, the angle between the generating line and the axis being z. These conical surfaces separated by infinitesimal angle dz will cut the surface of the sphere into a series of equipotential rings, the surface area of any ring being equal to a 2 ir r 2 sin z dz. The average potential for the entire spherical surface may then be obtained by summing the products of the ring areas and corresponding potentials and dividing the sum by the total surface area of the sphere. Thus Average V. 3 = . , 3 - I (cos 2 2—1/3) sin z dz = ^yrT- 1/3 cos 3 2+1/3 cos zT=0 (112) 32 U. S. 00 AST AND GEODETIC SUBVEY Average ^=iir (5 cos 3 2—3 cos 2) sin 2 dz = ^^[-5/4 cos 4 2+3/2 cos 2 zT=0 (113) 96. Let V, represent the potential due to gravity at any point on the earth's surface. Since the force of gravity at any point on or above the earth's surface equals nE/r 2 , the corresponding potential becomes Tr C °>dr uE ,.. ^ JX Vf=^J ,72=^- (114) If the earth is assumed to be a sphere with radius a, the gravitational potential at each point will equal pE/a, which may be taken as the average gravitational potential over the surface of the earth. 97. For a surface of equilibrium under the combined action of gravity and that part of the tide-producing force involving the cube of the moon's distance the sum of the corresponding potentials must be a constant, and since the average tide-producing potential for the entire surface of the earth is zero (par. 95), the constant will be the average gravitational potential or pE/a. Then from (110) and (114) we have Vt+V^-0 (cos* z-l!sy+^=^ (115) Transposing and omitting common factor n, we may obtain -^^=3/2(M/^(aAQ 8 (cos» 2-1/3) (116) Let r=a+h (117) so that h represents the height of the equilibrium surface as referred to the undisturbed spherical surface of an equivalent sphere. Then ^^=jj^==hla-3(h/ay+ + F '=S (5 cos3 3_3 cos zy+ir^ir (12 °) HARMONIC ANALYSIS AND PREDICTION OF TIDES 33 (r-a)a" =1/2 ^ M/E ^ a / d y^ cos 3 2 _ 3 cos g) (121 ) Letting r=a-{-h 4 and expanding the first member of the above formula, it becomes equal to A 4 /a after the rejection of the higher powers of this small fraction. The formula may then be written h 4 /a=l/2 (M/E)(a/dy(5 cos 3 2-3 cos z) (122) 99. Formulas (119) and (122) involving the cube and 4th power of the moon's parallax, respectively, represent the equilibrium heights of the tide due to the corresponding forces, the heights being expressed in respect to the mean radius (a) of the earth as the unit. In deriving these formulas the centrifugal force of the earth's rotation was dis- regarded and the resulting heights represent the disturbances in a true spherical surface due to the action of the tide-producing force. It may be inferred that in a condition of equilibrium the tidal forces would produce like disturbances in the spheroidal surface of the earth and the h of the formulas may therefore be taken as being referred to the earth's surface as defined by the mean level of the sea. 100. The extreme limits of the equilibrium tide, applicable to the time when the tide-producing body is nearest the earth, may be obtained by substituting the proper numerical values in formulas (119) and (122). They are given below for both moon and sun. From formula (119) involving the cube of parallax — Greatest rise =1.46 feet for moon, or 0.57 foot for sun (123) Lowest fall =0.73 foot for moon, or 0.28 foot for sun (124) Extreme range=2.19 feet for moon, or 0.85 foot for sun. (125) From formula (122) involving the 4th power of parallax — Greatest rise =0.026 foot for moon, or 0.000025 foot for sun (126) Lowest fall =0.026 foot for moon, or 0.000025 foot for sun (127) Extreme range =0.052 foot for moon, or 0.00005 foot for sun. (128) 101. A comparison of formulas (23) and (119), the first expressing the relation of the vertical component of the principal tide-producing force to the acceleration of gravity (g) and the other the relation of the height of the corresponding equilibrium tide to the mean radius (a) of the earth, will show that they are identical with the single excep- tion that the coefficient of the height formula is one-half that of the force formula. Therefore the development of the force formula into a series of harmonic constituents is immediately applicable in obtain- ing similar expressions for the equilibrium height of the tide. Using a notation for the height terms corresponding to that used for the force terms, let h zo /a, h u /a, and h 32 /a represent, respectively, the long- period, diurnal, and semidiurnal terms of the equilibrium tide involv- ing the cube of the moon's parallax. Then referring to formulas (81) to (83) we may write h 30 /a=3/4 17(1/2-3/2 sin 2 Y) S/C cos E (129) hn /a=3/4 U sin 2F 2 jG cos E (130) h 32 /a=3/4 U cos 2 YXfC cos E (131) the symbols having the same significance as in the preceding discussion of the tidal forces. 34 U. S. COAST AND GEODETIC SURVEY TERMS INVOLVING 4TH POWER OF MOON'S PARALLAX 102. Formulas (24) and (26) represent the vertical and horizontal components of the part of the tide-producing force involving the 4th power of the moon's parallax. This part of the force constitutes only about 2 percent of the total tide-producing force of the moon and for brevity will be called the lesser force to distinguish it from the principal or primary part involving the cube of the parallax. The vertical component F vi /g has its maximum value when z equals zero and, if numerical values pertaining to the moon and sun when nearest the earth are substituted in formula (24), the extreme values for this component are found to be 0.37X10" 8 for the moon and 0.35X10 -11 for the sun. The horizontal component F a4 /g has its greatest value when z equals about 31.09° and the substitution of numerical values in formula (26) gives the extreme value of this component as 0.26 X10" 8 for the moon or 0.24 XKT 11 for the sun. 103. Substituting in (24) the value of cos z from (31), the vertical component of the lesser force is expanded into four terms as follows: Fvi lg= 15/4 (M/E) (a/dy sin F(cos 2 F— 2/5) sin D(5 cos 2 D—2) _ F v40 /g +45/8 (M/£;)(a/^) 4 cosF(cos 2 r-4/5)cosL>(5cos 2 Z>-4)cos/ F Hl /g +45/4 (M/E)(a/dy sin F cos 2 F sin D cos 2 D cos 2t F H2 /g + 15/8 (M/E) (a/dy cos 3 F cos 3 D cos 3/ F vi3 /g (132) These four terms represent, respectively, long-period, diurnal, semi- diurnal, and terdiurnal constituents, according to the multiple of the hour angle t involved in the term. Each term is followed by a symbol which is analogous to those used in the development of the principal force. 104. Each term in formula (132) may be further expanded by means of the relations given in formulas (39) and (42). Expressing these terms separately we have — F m /0=15/4 (M/E) (a/dy sin F(cos 2 F-2/5)X [3 (sin 7-5/4 sin 3 I) cos 0'-90°) + 5/4 sin 3 I cos (3j-90°)] (133) F vil / + 3£-3iO (A 4 ) +6 cos (3T-2s + 3/i-;H-180 o + 3£-3i>) (A 5 ) +127/8 6 2 cos (3T-5s+3/i+2p + 3£-3*>) (Ae) +75/8 me cos (3T-4s+5A-;p + 3£-3*>)} + cos 4 -i-7sin 2 |-7 (At) {3(l + 26 2 ) cos (ZT-s+3h+£-Zv) (As) +96 cos (3T-2s + 3h+p + £-3v)}] (140) 106. All of the constituent terms in formulas (137) to (140) are relatively unimportant but they are listed in table 1 because of their theoretical interest. The only one of these terms now used in the prediction of tides is (A2) representing the constituent M 3 which has a speed exactly three-halves that of the principal lunar constituent M 2 . Term (Ai) is of interest in having a speed exactly one-half that of M 2 and is sometimes called the true Mi to distinguish it from the composite M x which is used in the prediction of tides and which will be described later. 36 U. S. COAST AND GEODETIC SURVEY 107. For simplicity and the purposes of this publication, the mean values of the obliquity factors in the terms of the lesser tide-producing force will be taken as the values pertaining to the time when / equals eo or 23.452°, excepting that for constituent M 3 and associated terms the mean has been obtained in accord with the system described in paragraph 75. The corresponding node factors (paragraph 77) may then be expressed by the following formulas in which the denominators are the accepted means of the obliquity factors: J(A 64 ) to J(A m ) = (sin 7-5/4 sin 3 /)/0.3192 (141) j(A, 7 ) to/(^ 68 )=sin 3 I/0.0630 (142) j(A m ) to f(A 70 ) =sin 2 / cos 2 iJ/(U518 (143) j(A n ) to f(A n )*= (1-10 sin 2 i/+15 sin 4 i/) cos 2 § J/0. 5873 (144) f(A u ) to J(A 75 ) = (1 - 10 cos 2 i/+ 15 cos 4 i J) sin 2 i//0.2147 (145) j(A n ) toJ(A 7S ) =sin I cos 4 K/0.3658 (146) J(A 79 ) to J (Ago) = (cos 2 i/-2/3) sin / cos 2 i//0.1114 (147) J(A 8l ) = (cos 2 |7- 1/3) sin I sin 2 i//0.0103 (148) J(A 82 ) to/(Ae) =/(M 3 ) =cos 6 i//0.8758 (149) f(A 87 ) to /(As) = cos 4 J7 sin 2 J//0.0380 (150) Comparing formulas (149) and (78), it will be noted that the node factor for M 3 is equal to the node factor for M 2 raised to the 3/2 power. Computed values applicable to terms A s2 to A 86 are included in table 14 for years 1850 to 1999, inclusive. 108. For the tabulated constituent coefficients of the terms in formulas (137) to (140) there are included not only the elliptic and mean obliquity factors but also such other factors as may be necessary to permit the use of the general coefficient (3/2 U) of formulas (81) to (83) for the vertical component of the principal tide-producing force. The common coefficient (M/E) (a/c)* of formulas (137) to (140) is equal to U multiplied by the parallax a/c, and the latter together with the necessary numerical factors is included in the constituent coefficients in table 2. Formulas (137) to (140) may then be summarized as follows: F m /0=3/2 U sin F(cos 2 F-2/5) S/O cos E (151) F m /flf=3/2 U cos F(cos 2 F-4/5) 2/(7 cos E (152) Fv42 /if =3/2 U sin Y cos 2 F 2fC cos E (153) ^43 /0=3/2 U cos 3 F 2/(7 cos E (154) 109. It is to be noted that in formulas (151), (152), and (153), the maximum value of the latitude factor in each is less than unity, being HARMONIC ANALYSIS AND PREDICTION OF TIDES 37 0.4, 0.2754, and 0.3849, respectively, if the sign of the function is disregarded. In formula (154), as in the corresponding formulas for the principal tide-producing force, the maximum value of this factor is unity. In comparing the relative importance of the various con- stituents of the tide-producing force the latitude factor should be in- cluded with the mean coefficient. Attention is also called to the fact that the relative importance of the constituents involving the 4 th power of the moon's parallax is greater in respect to the vertical com- ponent of the tide-producing force than in respect to the height of the equilibrium tide. In table 2 the mean coefficients are taken com- parable in respect to the vertical component of the tide-producing force and the constituent coefficients pertaining to the lesser force are therefore 50 percent greater than they would be if taken comparable in respect to the equilibrium tide. 110. The south and west horizontal components of the lesser tide- producing force may be obtained by multiplying formula (26) by cos A and sin A, respectively. Using the same system of notation as before, we then have F si /g=S/2 (M/E) (a/dy sin z (5 cos 2 z-l) cos A (155) F wi /g=S/2 (M/E) (a/dy sin z (5 cos 2 z-l) sin A (156) 111. By means of the relations expressed in formulas (31), (86), and (87), the above component forces may be separated into long- period, diurnal, semidiurnal, and terdiurnal terms as follows: South component, ^ S 4o lg=- 15/4 (M/E) (a/dy cos F(cos 2 F-4/5) sin D(5 cos 2 Z>-2) (157) F M /0=45/8 (M/E) (a/dy sin F(cos 2 F-4/15) cos D(5 cos 2 £>-4) cos t (158) F si2 lg= -45/4 (M/E) (a/dy cos F(cos 2 F-2/3) sin D cos 2 D cos 2t (159) F M /g= 15/8 (M/E) (a/dy sin F cos 2 F cos 3 Z> cos 3* (160) West component, F wi i lg= 15/8 (M/E) (a/d) 4 (cos 2 F-4/5) cos D(5 cos 2 Z>-4) sin t (161) Fun lg= 15/4 (M/E) (a/dy sin 2F sin D cos 2 L> sin 2t (162) F wi3 /g= 15/8 (M/E) (a/dy cos 2 F cos 3 L> sin 3* (163) 112. Comparing formulas (157) to (160) for the south component force with the corresponding terms of (132) for the vertical com- ponent, it will be noted that they differ only in the latitude factors and in sign for two of the terms. With adjustments for these dif- ferences the summarized formulas (151) to (154) are directly applicable for expressing the corresponding terms in the south component. Thus F s40 /0=3/2 U cos F(cos 2 F-4/5) SjfC cos(#+180°) (164) F sil /£=3/2 U sin F(cos 2 F-4/15) 2/0 cos E (165) F SA2 /flf=3/2 Ucos F(cos 2 F-2/3) XJC cos (£4-180°) (166) F si5 /g=Z/2 U sin F cos 2 F XJC cos E (167) 113. For the west component there is no long-period term. Com- paring (161) to (163) with the corresponding terms of (132), it will be noted that the /-functions are expressed as sines instead of cosines but they may be changed to the latter by subtracting 90° from each 38 U. S. COAST AND GEODETIC SURVEY argument. With this change and allowing for differences in the latitude factors and numerical coefficients, the summarized formulas for the west component will be similar to those for the vertical com- ponent and may be written as follows: Fuu /g=l/2 U (cos 2 F-4/5) XfC cos (#-90°) (168) F w42 lg=l/2 U sin 2F XfC cos (E-90°) (169) Fvm /flf=3/2 U cos 2 F XfC cos (#-90°) (170) 114. To obtain the horizontal component of the lesser force in any direction, the same procedure may be followed as was used for the principal tide-producing force (paragraphs 85 to 87) . With the same system of notation we then have F ai0 /g=3l2 Ucos F(cos 2 F-4/5) cos AXfC cos (#+180°) (171) F ail /<7=3/2 U P x XJC cos (E-Xi) (172) F a u /flf=3/2 UP 2 2fC cos (E-X 2 ) (173) F a43 /g=3/2 UP 3 2fC cos (E-X,) (174) in which P 1 = [sin 2 F(cos 2 F-4/15) 2 cos 2 A+l/9(cos 2 F-4/5) 2 wa*A]* (175) P 2 =cos F[(cos 2 F-2/3) 2 cosM+4/9 sin 2 F sm 2 A]^ (176) P 3 = cos 2 F(sin 2 F co&A+wtfA)* (177) v _ _ t (cos 2 F— 4/5) sin A /i7q\ Al_tan 3 sin F(cos 2 F-4/15) cos A (178) v _, 2 sin F sin A ,., ,__. X2 = tan -3(cos 2 F-2/3)cos^ < 179 ) ^^tan- 1 ■ *v A a (180) sm F cos A v ' The proper quadrants for X h X 2 , and X 3 will be determined by the signs of the numerators and denominators in the above expressions, these signs being respectively the same as for the sine and cosine of the corresponding angles. 115. Comparing formula (122) for the equilibrium height of the tide due to the lesser tide-producing force with formula (24) for the vertical component of the force, it will be noted that they are the same with the exception that the numerical coefficient of the former is one-third that of the latter. With this change, the summarized formulas (151) to (154) for the vertical force may be used to express the corresponding equilibrium heights. Following the same system of notation as before, we have h 40 /a=l/2 £7 sin F(cos 2 F-2/5) 2/(7 cos E (181) ha /a=l/2 Ucos F(cos 2 F-4/5) 2/(7 cos E (182) h 42 /a=l/2 U sin F cos 2 F 2/(7 cos E (183) h AZ /a=l/2 U cos 3 F 2/(7 cos E (184) It is to be noted that the equilibrium height of the tide due to the principal tide-producing force when measured by the mean radius of the earth as a unit is one-half as great as the corresponding vertical component force referred to the mean acceleration of gravity as a unit, while the equilibrium height due to the lesser tide producing force similarly expressed is only one-third as great as the corresponding force. In table 2, the coefficients (C) of the constituents derived HARMONIC ANALYSIS AND PREDICTION OF TIDES 39 from the lesser force are made comparable with the others in respect to the vertical component force rather than in respect to the equi- librium height. SOLAR TIDES 116. Since the tide-producing force of the sun is similar in action to that of the moon, the formulas derived for the latter are applicable, with suitable substitutions, to the solar forces. Referring to formulas (62), (63), and (64), let Uhe replaced by U\ representing the product (S/E) (a/ci) 3 in which $ is the mass of the sun and (a/ci) its mean parallax. Also replace e by €,, the eccentricity of the earth's orbit; / by co, the obliquity of the ecliptic; s by h, the mean longitude of the sun; and p by p h the longitude of the solar perigee. For the solar forces the arcs £ and v become zero and all terms representing the evectional and variational inequalities are omitted. 117. Making the changes indicated the solar constituents are now expressed in the following formulas. Each term is marked for iden- tification by the letter B with the same subscript used for the corre- sponding term in the lunar tide. The usual constituent symbol is also given for the more important terms. Using the same system of notation as before, Solar F m /g = 3/2 U x (1/2-3/2 sin 2 F)X (B{) [(2/3-sin 2 co){ (1 + 3/2 e\) permanent term (B 2 ) +3 ei cos (h—pi) (B 3 ) +9/2 e\ cos (2h-2p 1 )} (B 6 ) + sin 2 co{ (1-5/2 e\) cos 2/* Ssa (fit) +7/2 ei cos (Sh—pi) (B 8 ) +1/2 e x cos (&+?, + 180°) (B 9 ) +17/2 e\ cos (4h-2p l ) }] (185) Solar F m /g=S/2 V x sin 2FX (B u ) [sin co cos 2 ico{ (1-5/2 e\) cos (T-h+90°) Pi (Bib) +7/2e lC os (T-2h+p 1 -\-90°) ir, (Bu) +1/2 e, cos (T-^i-90 ) (B 17 ) +17/2 e\ cos (T- 3^+2^ + 90°) } (B 22 ) + sin2co{ (1/2 + 3/4 e\) cos (T+h-90°) [KJ (B 23 ) +3/4 e x cos (T+^-90 ) (RO +3/4 e t cos (T+2/i-^ 1 -90 o ) ft (B 25 ) +9/8 e\ cos (T-h+2p l -90°) (J5 26 ) +9/8 e\ cos (T+3/1-2^-90 )} (fti) +sin co sin 2 ico{ (1-5/2 e\) cos (T+3/*,-90°)___ ft (5 82 ) +7/2 6 X cos (T+4A- j p 1 -90°) (£33) +1/2 6! cos (T+2h+p 1 +90°) (S 34 ) +17/2 6 2 ! cos (T-}-5h-2p 1 -90 )}} (186) Solar F m /g=d/2 17, cos 2 FX (5 39 ) [cos 4 ico{ (1-5/2 e 2 cos (2T) S 2 (#40) +7/2 «j cos (2T-h+p 1 ) T 2 (B 41 ) +1/2 6! cos (2T^ r h-p 1 +lS0°) R 2 (B i2 ) +17/2 e\ cos (2T-2/*,+2pO} (B#) +sin 2 co{ (1/2 + 3/4 e\) cos (2T+2A) [K 2 ] (5«) +3/4 6, cos (2T+h+p x ) (5 49 ) +3/4 e x cos (2T+3^-2?i) (Formula continued on next page) 40 U. S. COAST AND GEODETIC SURVEY (B 50 ) + 9/8 e\ cos (2T+2^0 (fti) +9/8 e\ cos (2T+4h-2 Pl ) } (B 56 ) + sin 4 ia>{ (1-5/2 e 2 i) cos (2T+4&) (B 57 ) +7/2 6i cos (2T+5h-p 1 ) (B 58 ) +1/2 e x cos (2T+ 3/^+^ + 180°) (B m ) +17/2 6 2 ! cos (2T+6^-2b)}] (187) 118. The general coefficient for the solar tide-producing force differs from that of the lunar force in the basic factor. From the fundamental data in table 1, the ratio of UJU is found to be 0.4602. This ratio, which will be designated as the solar factor with symbol S', represents the theoretical relation between the principal solar and lunar tide-producing forces. In computing the constituent coefficients of the solar terms for use in table 2, the solar factor was included in order that the same general coefficient may be applicable to both lunar and solar terms. All of the summarized formulas involving the coefficients and arguments of table 2 are therefore applicable to both lunar and solar constituents. For the solar constituents, however, the node factor (f) is always unity since co, the obliquity of the ecliptic, may be considered as a constant. 119. By substituting solar elements in formulas (137) to (140) the corresponding solar constituents pertaining to the 4th power of the sun's parallax are readily obtained. Since the theoretical magnitude of the lesser solar tide-producing force is less than 0.00002 part of the total tide-producing force of moon and sun, it is usually disregarded altogether. However, certain interest is attached to three of the constituents which are considered in connection with shallow water and meteorological tides (p. 46). These are constituents Sa, Si, and S 3 , corresponding respectively to terms A M , A 7U and A 82 of the lunar series. They are listed in table 2 with reference letter B and cor- responding subscripts. Sa has a speed one-half that of constituent Ssa represented by term B 6 of formula (185). Its theoretical argu- ment as derived from term A 6i contains the constant 90°, but being considered as a meteorological rather than an astronomical consti- tuent, this constant is omitted from the argument. Constituents Si and S 3 have speeds respectively one-half and three-halves that of the principal solar constituent S 2 . 120. The arguments of a number of the solar constituents include the element p 1 which represents the longitude of the solar perigee. As this changes less than 2° in a century, it may be considered as practically constant for the entire century. Referring to table 4 it will be noted that p t changes from 281.22° in 1900 to 282.94° in 2000. The value of 282° may therefore be adopted without material error for all work relating to the present century. With p x taken as a constant, it will be found that a number of terms in table 2 have the same speeds and may therefore be expected to merge into single constituents. Thus, constituents receiving contributions from more than one term are as follows: Sa from terms B 2 , B 8 , and B u ; Ssa from terms B 3 and B G ; P x from terms B u and B 25 ; Si from terms B l6 , B 23 , and B n ; \f/ 1 from terms B 2i and # 33 ; <£i from terms B 26 and B 3l ; S 2 from terms B Z9 and B 50 ; and R 2 from terms B iX and B i8 . A few other solar terms also merge. HARMONIC ANALYSIS AND PREDICTION OF TIDES 41 THE Mi TIDE 121. The separation of constituents from each other by the process of the analysis depends upon the differences in their speeds. Constit- uents with nearly equal speeds are not readily separated unless the analysis covers a very long series of observations but they tend to merge and form a single composite constituent. In formula (63), terms Aw and A 23 have nearly equal speeds, one being a little less and the other a little greater than one-half the speed of the principal lunar constituent M 2 . These two terms are usually considered as a single constituent and represented by the symbol Mi. Neglecting for the present the general coefficient and common latitude factor, the two terms may be written as follows: term A u =l/2 e sin / cos 2 \I cos (T-s + h—p — 90° + 2S—v) (188) term A 23 =3/2 e sin / cos / cos (T— s+h+p — 90° — v) (189) The latter term, having a coefficient nearly three times as great as that of the first term, will predominate and determine the speed and period of the composite tide while the first term introduces certain inequalities in the coefficient and argument. 122. For brevity, let A and B represent the respective coefficients of terms A u and A 23 and let 6=T-s+h+p-90°-v (190) itude of the lunar perigee reckoned from P = P+Z (191) Also let P equal the mean longitude of the lunar perigee reckoned from the lunar intersection. Then We then have term A U =A cos (6—2P) =A cos 2P cos d+A sin 2P sin d (192) term A 2Z =B cos 6 (193) M 1 =A l6 J r A 23 =(A cos 2P+B) cos 6+ A sin 2P sin B = (A*+2AB cos 2P+B*)i cos [f-^ 1 A ^^p+ J = 6 sm Iws 2 H cQs {T _ s + h+p _ 90 o_ v _ Qu) (194) in which ^b/^^i™ 2p + 9 / 4 StJ (195) Q u =tsai ^ r , — 2~rin ^td (196) ^ 3 cos I /cos 2 fi + cos 2P v ' If / is given its mean value corresponding to w, formula (195) may be reduced to the form 1/Q a = (2.310+1.435 cos2P)* (197) Values of log Q a for each degree of P based upon formula (197) are given in table 9. 42 U. S. OOAST AND GEODETIC SURVEY 123. The period of the composite constituent M 1 is very nearly an exact multiple of the period of the principal lunar constituent M 2 , and for this reason the summations which are necessary for the analysis of the latter may be conveniently adapted to the analysis of the former. With other symbols as before, let e=T-s + h-90°-\-£- v (198) Terms A w and A 2 3 may then be combined as follows: term A 16 =A cos (0— P) =A cos P cos 6+ A sin P sin 6 (199) term A 2Z =B cos (0+P) =B cos P cos 6—B sin P sin (200) M 1 =A l6 +A 23 =(A J r B) cos P cos 6+ (A— B) sin P sin = (A 2 +2AB cos 2P+B 2 )- cos p-tan-^^^tan pYl - e Sm — C -^-cos(T-s + h-90°+Z-v+Q) (201) in which ^.^JSt^) (202) If 7 is given its mean value corresponding to co, formula (202) may be reduced to the following form which was used for computing the values of Q in table 10. tan Q= 0.483 tan P (203) 124. Formulas (194) and (201) are the same except in the method of representing the argument. The elements +p — Q u in the first formula are replaced by +I-+Q in the latter, but it may be shown from (196) and (202) that Qu+Q=P=p-k (204) P-Q U =Z + Q (205) The complete arguments are therefore equal but in formula (201) the uniformly varying element p has been transferred from the V of the argument and included in the value of Q where it is treated as a constant for a series of observations being analyzed. The speed of the argument as determined by the remaining part of the V is then exactly one-half that of the principal constituent M 2 and with this assumption the summations for the latter may be adapted to the analysis of the former. It is to be noted, however, that the u in this case has a progressive forward change of nearly 41° each year. The true average speed of this constituent is determined by the V of formula (194) which includes the element p. 125. The obliquity factor for the composite M x constituent may be expressed by the formula sin I cos 2 ^/X l/# a - According to the work of Darwin (Scientific Papers by Sir George H. Darwin, vol. 1, p. 39) the HARMONIC ANALYSIS AND PREDICTION OF TIDES 43 mean value of this factor is represented by the product sin co cos 2 \ & cos 4 ^XV2.307, which equals 0.3800X1.52, or 0.5776. When deriving the node-factor formula for M l7 Darwin inadvertently omitted the factor V2.307 and obtained the approximate equivalent of the following: VA , N sin I cos 2 |7 ir . sin I cos 2 |7 /n /on _ N r(Mi) = i 2?—^ 4T^ X 1/Va = n oqaa X 1/Oa (206) JK J sin w cos 2 -^ (o cos 4 ^ '^ 0.3800 '^ v J Comparing the above with formula (75), it will be noted that /(M 1 )=/(0 1 )Xl/«. (207) Factors pertaining to constituent M 1 in tables 13 and 14 are based upon the above formulas. 126. Because of the omission of the factor y 2.307 from formula (206), the node factors for Mi which have been in general use since this system of tidal reductions was adopted are about 50 percent greater than was originally intended, while the reciprocal reduction factors are correspondingly too small. This constituent is relatively unimportant and no practical difficulties have resulted from the omis- sion. The M : amplitudes as reduced from the observational data are comparable among themselves but should be increased by 50 percent to be on the same basis as the amplitudes of other constituents. The predicted tides have not been affected in the least since the node factors and reduction factors are reciprocal and compensating. The theoretical mean coefficient for this constituent with the factor V2.307 included is 0.0317; but in order that this coefficient may be adapted for use with the tabular node factors when computing tidal forces or the equilibrium height of the tide, the coefficient 0.0209 with the factor V2.307 excluded should be used. 127. Although Mi is one of the relatively unimportant constituents and the error in the node factor has caused no serious difficulties, it may be questionable whether it should be perpetuated. It is obvious, however, that any change in the present procedure would lead to much confusion unless undertaken by general agreement among all the principal organizations engaged in tidal work. By making any change applicable to the analysis of all series of observations beginning after a certain specified date it would be possible to interpret the results on the basis of the period covered by the observations without the neces- sity of revising all previously published amplitudes for this constituent. THE L 2 TIDE 128. The composite L 2 constituent is formed by combining terms An and A& of formula (64). Neglecting the general coefficient and common latitude factor these terms may be written term ^l 41 = l/2 e cos 4 \I cos (2T-s+2h-p J r 180° + 2£-2v) (208) term A iS =3/4: e sin 2 I cos (2T-s + 2h+p-2v) (209) A reference to table 2 will show that the mean coefficient of the first term is about four times as great as that of the latter term. The first 44 U. S. COAST AND GEODETIC STJBVEY term will therefore predominate and determine the speed of the composite constituent. 129. With other symbols as before, let A and B represent the respective coefficients of the two terms and the argument of the first term. We then have A 41 =A cos 6 (210) A 48 =£ cos (0+2P-18O°) = -P cos (0+2P) (211) L 2 =A i i-\-A&= (A— B cos 2P) cos 6+B sin 2P sin 6 = (A 2 -2AB cos 2P+P 2 )* cos [fl-tan- 1 ^ b^os^p ] 4 -i = 1/2 e C< ^ ? /cos (2T-s+2^-^+180 o +2^-2^- J R) (212) in which 1/P a =(l-12 tan 2 \I cos 2P+36 tan 4 \T) 1 (213) 7?_f -i sin 2P (214) it -tan 1/6cot a i /_ CO s2P Values of log P a and P computed from the above formulas are given in tables 7 and 8, respectively. 130. The obliquity factor for the composite L 2 constituent may be expressed by the formula cos 4 -Ji"Xl/P a - The mean value of 1/P a is approximately unity, and in accord with the Darwinian system the mean for the entire obliquity factor is taken as the product cos 4 \ co cos 4 -^, which equals 0.9154 and is the same as the mean value of the obliquity factor for the principal constituent M 2 . Multiplying this by the elliptic factor \e gives 0.0251 as the mean constituent coefficient. 131. The node factor formula for constituent L 2 based upon the above mean for the obliquity factor is as follows: /(L 2 )=^|J X 1//W(M 2 ) X 1/2?, (215) Node factors for constituent L 2 based upon the above formula are included in table 14 for the middle of each year from 1850 to 1999, inclusive. The logarithms of the reciprocal reduction factors covering the period 1900 to 2000 are contained in table 13. LUNISOLAR Ki AND K 2 TIDES 132. Lunar diurnal term ^L 22 of formula (63) and solar diurnal term P 22 of formula (186) have the same speed. Together they form the lunisolar K x constituent. Also, lunar semidiurnal term A i7 of formula (64) and solar semidiurnal term P 47 of formula (187) have speeds exactly twice that of constituent Ki and together form the lunisolar K 2 constituent. In order that the solar terms may have the same general coefficient as the lunar terms, the solar factor Ui/Uj which will be designated by the symbol S', will be transferred from the general coefficient of the solar terms and included in the constituent coefficients. Then, neglecting the general coefficient and HARMONIC ANALYSIS AND PREDICTION OF TTDES 45 latitude factors common to the terms combined, we have the following formulas in which numerical values from table 1 have been sub- stituted for constant quantities. term A 22 = (1/2 + 3 /46 2 ) sin 2/ cos (T+h-90°-v) = 0.5023 sin 2/ cos (T+h-90°-v) (216) term 5 22 =(l/2+3/4ef)S" sin 2 co cos (T+h-90°) = 0.1681 cos (T+A-90 ) (217) term^4 7 =(l/2 + 3/4e 2 ) sin 2 / cos (2T+2h-2v) = 0.5023 sin 2 / cos (2T+2h—2v) (218) term £ 47 = (l/2 + 3/4e 2 )&' sin 2 w cos (2T+2h) = 0.0365 cos (T+2h) (219) 133. Taking first the diurnal terms, let A represent the lunar co- efficient 0.5023 sin 2/ and let B represent the solar coefficient 0.1681. We then have A 22 =A cos (T+h-90°-v) =A cos v cos (T+h-90°)+A sin v sin (T+A-90 ) (220) B 22 =B cos (T+h-90°) (221) Ki=(A cos y+£) cos (T+^-90°)+^l sin r sin (T+^-90°) _ = (A 2 +2AB cos *>+J9 2 )^ cos [^+^-9 °- t a n " 1 ^^rg] =Oi cos (T+A-90 -/) (222) in which C x =(A 2 -\-2AB cos v+Brf = (0.2523 sin 2 2/+0.1689 sin 2/ cos I/+0.0283)* (223) , , _, Asiny , _, sin 2/ sin p , N / = tan 1 - 1 r^=tan l ~-. — pry , n 00/) - (224) -A cos y-h-D sin 2/ cos v+0.3347 Values of v f for each degree of N, which is the longitude of the moon's node, are included in table 6. 134. The obliquity factor for K x will be taken to include the entire coefficient (A 2 -\-2AB cosv-\-B 2 )? and its mean value will be taken as the mean of the product (A 2 -{-2AB cos v +i? 2 )' cos v' . From (224) we may obtain cos / = (A cos v+B)l(A 2 +2AB cos v+B 2 ^ (225) Then for mean value of coefficient of K x [(A 2 +2AB cos v+B 2 )i cos v'] =[A cos v+B] = [0.5023 sin 2/ cos p-f-0.1681]o= 0.5305 (226) the numerical mean for sin 2/ cos v being obtained from formula (68) . For the node factor of K x divide the coefficient of (222) by its mean value and obtain f(K 1 ) = (0.2523 sin 2 2/+0.1689 sin 2/ cos y+0.0283)*/0.5305 = (0.8965 sin 2 2/+0.6001 sin 2/ cos ^+0.1006)* (227) 246037—41 4 46 U. S. COAST AND GEODETIC SURVEY The node factors for the middle of each year 1850 to 1999 are included in table 14. Logarithms of the reciprocal reduction factors for each tenth of a degree* of 7 are given in table 12. 135. The semidiurnal terms A i7 and 7? 47 may be combined in a similar manner. Letting A represent the lunar coefficient 0.5023 sin 2 / and B the solar coefficient 0.0365, we have Aa = A cos (2T+2h-2v) =A cos 2v cos (2T+2h)+A sin 2v sin (2T+2h) (228) B A7 =B cos (2T+2h) (229) K 2 =(A cos 2v+B) cos (2T+2A)- = {A 2 -\-2AB cos 2v+B 2 )* cos A sin 2v sin (2T+2A) ^. cos 2i>+7? J = C 2 cos (2r+2^-2/ / ) (230) In which C 2 = (A 2 +2AB cos 2*/+5 2 )* = (0.2523 sin 4 1+0.0367 sin 2 7 cos 2^+0.0013)* (231) rt „ . , ^4 sin 2i> . , sin 2 7 sin 2v 2v =tan _1 -j 7z — r^==tan _1 A cos 2v+B sin 2 7 cos 2^+0.0727 (232) Values for 2v" for each degree of N are included in table 6. 136. The obliquity factor for K 2 will be taken to include the entire coefficient (A 2 -\-2AB cos 2v-\-B 2 )* and its mean value will be taken as the mean of the product (A 2 -\-2AB cos 2v-\-B 2 )* cos 2v" . Prom (232) cos 2v"= (A cos 2v J r B)/(A 2 +2AB cos 2v+B 2 )i (233) Then for the mean value of coefficient of K 2 [(A 2 +2AB cos 2v+B 2 )* cos 2v"] =[A cos 2v+B] = [0.5023 sin 2 7 cos 2*>+0.0365] =0.1151 (234) the numerical mean for sin 2 7 cos 2v being obtained from formula (71). For the node factor of K 2 divide the coefficient of (230) by its mean value and obtain /(K 2 ) = (0.2523 sin 4 7+0.0367 sin 2 7 cos 2^+0.0013)* /0.1151 = (19-0444 sin 4 7+2.7702 sin 2 7 cos 2i>+0.0981)* (235) See table 14 for node factors and table 12 for reciprocal reduction factors. METEOROLOGICAL AND SHALLOW- WATER TIDES 137. In addition to the elementary constituents obtained from the development of the tide-producing forces of the moon and the sun, there are a number of harmonic terms that have their origin in meteorological changes or in shallow-water conditions. Variations in temperature, barometric pressure, and in the direction and force of the wind may be expected to cause fluctuations in the water level. Although in general such fluctuations are very irregular, there are some seasonal and daily variations which occur with a rough periodic- ity that admit of being expressed by harmonic terms. The meteoro- logical constituents usually take into account in the tidal analysis are HARMONIC ANALYSIS AND PREDICTION OF TIDES 47 Sa, Ssa, and Si with periods corresponding respectively to the tropical year, the half tropical year, and the solar day. These constituents are represented also by terms in the development of the tide-producing force of the sun but they are considered of greater importance as meteorological tides. Ssa occurs in the development of the principal solar force while Sa and Si would appear in a development involving the 4th power of the solar parallax (par. 119). In the analysis of tide observations both Sa and Ssa are usually found to have an appreciable affect on the water level. Constituent Si is relatively of little im- portance in its effect on the height of the tide but has been more noticeable in the velocity of off-shore tidal currents, probably as a result of periodic land and sea breezes. 138* The shallow- water constituents result from the fact that when a wave runs into shallow water its trough is retarded more than its crest and the wave loses its simple harmonic form. The shallow-water constituents are classified as overtides and compound tides, the over- tide having a speed that is an exact multiple of one of the elementary constituents and the compound tide a speed that equals the sum or difference of the speeds of two or more elementary constituents. 139. The overtides were so named because of their analogy to the overtones in musical sounds and they may be considered as the higher harmonics of the fundamental tides. The only overtides usually taken into account in tidal work are the harmonics of the principal lunar and solar semidiurnal constituents M 2 and S 2 , the lunar series being designated by the symbols M 4 , M 6 , and M 8 , and the solar series by S 4 , S 6 , and S 8 . The subscript indicates the number of periods in the constituent day. These overtides with their argu- ments and speeds are included in table 2a, the arguments and speeds being taken as exact multiples of those of the fundamental con- stituent. There are no theoretical expressions for the coefficients of the overtides but it is assumed that the amplitudes of the lunar series undergo variations due to changes in the longitude of the moon's node which are analogous to those in the fundamental tide. The node factors for M 4 , M 6 , and M 8 , respectively, are taken as the square, the cube, and the fourth power of the corresponding factor for M 2 . For the solar terms this factor is always zero. 140. Compound tides were suggested by Helmholtz's theory of sound waves. Innumerable combinations are possible but the prin- cipal elementary constituents involved are M 2 , S 2 , N 2 , K 2 , and Oi. Table 2a includes the compound tides listed in International Hydro- graphic Bureau Special Publication No. 26, which is a compilation of the tidal harmonic constants for the world. The argument of a compound tide equals the sum or difference of the arguments of the elementary constituents of which it is compounded. The node factor is taken as the product of the node factors of the same con- stituents. Table 2a contains the arguments, speeds, and node factors of these tides. 141. Omitted from table 2a are a number of compound tides which have the same speeds as elementary constituents included in table 2. Thus, 2MS 2 , compounded by formula 2M 2 — S 2 , has the same speed as constituent ju 2 represented by term A A5 of formula (64). Considered as a compound tide there would be a small difference in the u of the argument and also in the node factor. Since there is no practical way of separating the elementary constituent from the compound 48 U. S. 00 AST AND GEODETIC SURVEY tide of the same speed, this has been treated solely as an elementary constituent. Constituent MSf represented by term A 5 of formula (62) has the same speed as a compound tide of formula S 2 — M 2 . This con- stituent is relatively unimportant and it makes little difference whether treated as an elementary or a compound tide. Following the pre- vious practice in this office it is treated in the harmonic analysis as a compound tide with corresponding argument and node factor. When included in the computation of tidal forces, however, the argument and node factor indicated in table 2 should be used. ANALYSIS OF OBSERVATIONS HARMONIC CONSTANTS 142. In the preceding discussion it has been shown that under the equilibrium theory the height of a theoretical tide at any place can be expressed mathematically by the sum of a number of harmonic terms involving certain astronomical data and the location of the place. It has also been pointed out that for obvious reasons the actual tide of nature does not conform to the theoretical equilibrium tide. How- ever, the tide of nature can be conceived as being composed of the sum of a number of harmonic constituents having the same periods as those found in the tide-producing force. Although the complexity of the tidal movement is too great to permit a theoretical computation based upon astronomical conditions only, it is possible through the analysis of observational data at any place to obtain certain constants which can be introduced into the theoretical formulas and thus adapt them for the computation of the tide for any desired time. 143. In the formulas obtained for the height of the equilibrium tide each constituent term consists of the product of a coefficient by the cosine of an argument. For corresponding formulas expressing the actual height of the tide at any place, the entire theoretical coeffi- cient including the latitude factor and the common general coefficient is replaced by a coefficient determined from an analysis of observa- tional data for the station. This tidal coefficient, which is known as the amplitude of the constituent, is assumed to be subject to the same variations arising from changes in the longitude of the moon's node as the coefficient of the corresponding term in the equilibrium tide. The amplitude pertaining to any particular year is usually designated by the symbol R while its mean value for an entire node period is represented by the symbol H. Amplitudes derived directly from an analysis of a limited series of observations must be multiplied by the reduction factor F (par. 78) to obtain the mean amplitudes of the harmonic constants. For the prediction of tides, the mean ampli- tudes must be multiplied by the node factor/ (par. 77) to obtain the amplitudes pertaining to the year for which the predictions are to be made. 144. The phases of the constituents of the actual tide do not in general coincide with the phases of the corresponding constituents of the equilibrium tide but there may be lags varying from to 360°. The interval between the high water phase of an equilibrium con- stituent and the following high water of the corresponding constituent in the actual tide is known as the phase lag or epoch of the constituent and is represented by the symbol k (kappa) which is expressed in angular measure. The amplitudes and epochs together are called harmonic constants and are the quantities sought in the harmonic analysis of tides. Each locality has a separate set of harmonic con- stants which can be derived only from observational data but which remain the same over a long period of time provided there are no 49 50 U. S. COAST AND GEODETIC SURVEY physical changes in the region that might affect the tidal conditions. 145. If we let y x equal the height of one of the tidal constituents as referred to mean sea level, it may be represented by the following formula: Vi=jU cos (E—k)=JH cos (V+u—k) (236) The combination symbol V-\-u is the equivalent of E and represents the argument or phase of the equilibrium constituent. 146. Formula (236) is illustrated graphically in figure 7 by a cosine curve with amplitude jH. The horizontal line represents mean sea level and the vertical line through T may be taken to indi- cate any instant of time under consideration. If the point M repre- sents the time when the constituent argument equals zero, the interval from M to the following high water of the constituent will be the epoch k. The interval from the preceding high water to M is measured by the explement of k which may be expressed as — k. The phase of the constituent argument at time T is reckoned from M and is expressed by the symbol (V-\-u). The phase of the constit- FlGURE 7. uent itself at this time is reckoned from the preceding high water and therefore equals (V-\-u—k). OBSERVATIONAL DATA 147. The most satisfactory observational data for the harmonic analysis are from the record of an automatic tide gage that traces a continuous curve from which the height of the tide may be scaled at any desired interval of time. This record is usually tabulated to give the height of the tide at each solar hour of the series in the kind of time normally used at the place. It is important, however, that the time should be accurate and that the same system be used for the entire series of observations regardless of the fact that daylight saving time may have been adopted temporarily for other purposes during a portion of the year. When the continuous record from an automatic gage is not available, hourly heights of the tide as observed by other methods may be used. The record should be complete with each hour of the series represented. If a part of the record has been lost, the hiatus may be filled by interpolated values; or, if the gap is very extensive, the record may be broken up into snorter series which do not include the defective portion. HARMONIC ANALYSIS 1 AND PREDICTION OF TIDES 51 148. If hourly heights have not been observed but a record of high and low waters is available, an approximate evaluation of the more important constituents may be obtained by a special treatment. The results, however, are not nearly as satisfactory as those obtained from the hourly heights. 149. Although the hourly interval for the tabulated heights of the tide has usually been adopted as most convenient and practicable for the purposes of the harmonic analysis, a greater or less interval might be used. A shorter interval would cause a considerable increase in the amount of work without materially increasing the accuracy of the results for the constituents usually sought. However, if an attempt were made to analyze for the short period seiches a closer interval would be necessary. An interval greater than one hour would lessen the work of the analysis but would not be sufficient for the satisfactory development of the over tides. 150. In selecting the length of series of observations for the purpose of the analysis, consideration has been given to the fact that the pro- cedure is most effective in separating two constituents from each other when the length of series is an exact multiple of the synodic period of these constituents. By synodic period is mean the interval between two consecutive conjunctions of like phases. Thus, if the speeds of the two constituents in degrees per solar hour are represented by a and b, the synodic period will equal 360°/ (a^b) hours. If there were only two constituents in the tide the best length of series could be easily fixed, but in the actual tide there are many constituents and the length of series most effective in one case may not be best adapted to another case. It is therefore necessary to adopt a length that is a compromise of the synodic periods involved, consideration being given to the relative importance of the different constituents. 151. Fortunately, the exact length of series is not of essential im- portance and for convenience all series may be taken to include an integral number of days. Theoretically, different lengths of series should be used in seeking different constituents, but practically it is more convenient to use the same length for all constituents, an excep- tion being made in the case of a very short series. The longer the series of observations the less important is its exact length. Also the greater the number of synodic periods of any two constituents the more nearly complete will be their separation from each other. Con- stituents like S 2 and K 2 which have nearly equal speeds and a synodic period of about 6 months will require a series of not less than 6 months for a satisfactory separation. On the other hand, two constituents differing greatly in speed such as a diurnal and a semidiurnal con- stituent may have a synodic period that will not greatly exceed a day, and a moderately short series of observations will include a relatively large number of synodic periods. For this reason, when selecting the length of series no special consideration need be given to the effect of a diurnal and a semidiurnal constituent upon each other. 152. The following lengths of series have been selected as conform- ing approximately to multiples of synodic periods involving the more important constituents— 14, 15, 29, 58, 87, 105, 134, 163, 192, 221, 250, 279, 297, 326, 355, and 369 days. The 369-day series is considered as a standard length to be used for the analysis whenever observations covering this period are available. This length conforms very closely with multiples of the synodic periods of practically all of the short- 52 U. S. COAST AND GEODETIC SURVEY period constituents and is well adapted for the elimination of seasonal meteorological effects. When observations at any station are avail- able for a number of years, it is desirable to have separate analyses made for different years in order that the results may be compared and serve as a check on each other. Although not essential, there are certain conveniences in having each such series commence on January 1 of the year, regardless of the fact that series of consecutive years may overlap by several days because the length of series is a little longer than the calendar year. 153. If the available observations cover a period less than 369 days, the next longest series listed above which is fully covered by the observations will usually be taken, any extra days of observations being rejected. However, if the observations lack only a few hours of being equal to the next greater length, it may be advantageous to extrapolate additional hourly heights to complete the larger series. The 29-day series is usually considered as a minimum standard for short series of observations. This is a little shorter than the synodical month and a little longer than the nodical, tropical, and anomalistic months. It is the minimum length for a satisfactory development of the more important constituents. 154. For observations of less than 29 days, but more than 14 days, provisions are made for an analysis of a 14-day series for the diurnal •constituents and a 15-day series for the semidiurnal constituents, the first conforming to the synodic period of constituents K : and O u and the latter to the synodic period of M 2 and S 2 . Through special treatment involving a comparison with another station, it is possible to utilize even shorter series of observations. This treatment is rarely required in case of tide observations but is useful in connection with tidal currents where observations may be limited to only a few ■days. SUMMATIONS FOR ANALYSIS 155. The first approximate separation of the constituents of the observed tide is accomplished by a system of summations, separate summations being made for all constituents with incommensurable periods. Designating the constituent sought by A, assume that the entire series of observations is divided into periods equal to the period of A and each period is subdivided into a convenient number of equal parts, the subdivisions of each period being numbered consecutively beginning with zero at the initial instant of each period. All subdivi- sions of like numbers will then include the same phase of constituent A but different phases for all other constituents with incommensurable speeds. The subdivisions will also include irregular variations arising from meteorological causes. By summing and averaging separately all heights corresponding to each of the numbered subdivisions over a sufficient length of time, the effects of constituents with incommensur- able periods as well as the meteorological variations will be averaged out leaving intact constituent A with its overtides. 156. The principle just described for separating constituent A from the rest of the tide is applicable if the original periods into which the series of observations is divided are taken as some multiple of constituent A period. In general practice, that multiple of the constituent period which is most nearly equal to the solar day is taken as the unit. This is the constituent day and includes one or more HARMONIC ANALYSIS' AND PREDICTION OF TIDES 53 periods according to whether the constituent is diurnal, semidiurnal, etc. The constituent day is divided into 24 equal parts, the beginning of each part being numbered consecutively from to 23 and these are known as constituent hours. 157. To carry out strictly the plan described above would require separate tabulations of the heights of the tide at different intervals for all constituents of incommensurable periods, a procedure involving an enormous amount of work. In actual practice the tabulated solar hourly heights are used for all of the summations, these heights being assigned to the nearest constituent hour. Corrections are afterwards applied to take account of any systematic error in this approximation. 158. There are two systems for the distribution and assignment of the solar hourly heights which differ slightly in detail. In the system ordinarily used and which is sometimes called the standard system, each solar hourly height is used once, and once only, by being assigned to its nearest constituent hour. By this system some constituent hours will be assigned two consecutive solar hourly heights or receive no assignment according to whether the constituent day is longer or shorter than the solar day. In the other system of distribution, each constituent hour receives one and only one solar hourly height neces- sitating the occasional rejection or double assignment of a solar hourly height. The difference in the results obtained from the two systems is practically negligible but the first system is generally used as it affords a quick method of checking the summations. STENCILS 159. The distribution of the tabulated solar hourly heights of the tide for the purpose of the harmonic analysis is conveniently accom- plished by a system of stencils (fig. 10) which were devised by L. P. Shidy of the Coast and Geodetic Survey early in 1885 (Report of U. S. Coast and Geodetic Survey, 1893, vol. I, p. 108). Although the original construction of the stencils involves considerable work, they are serviceable for many years and have resulted in a very great saving of labor. These stencils are cut from the same forms which are used for the tabulation of the hourly heights of the tide and 106 sheets are required for the summation of a 369-day series of observa- tions for a single constituent. Separate sets are provided for different constituents. Constituents with commensurable periods are included in a single summation and no stencils are required for constituents, Si, S 2 , S 4 , etc. 160. The use of the stencils makes a standardized form for the tabulation of the hourly heights essential. This form (fig. 9) is a sheet 8 by 10}i 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 consecu T tively from h at midnight to 23 h at 11 p. m. When the tabulated heights are entered, each day is indicated by its calendar date and also by a serial number commencing with 1 as the first day of series. The days on the stencil sheets are numbered serially to correspond with the tabulation sheets and may be used for any series regardless of the calendar dates. 161. The openings in the stencils are numbered to indicate the constituent hours that correspond most closely with the times of the 54 U. S. 00 AST AND GEODETIC STXRVEY height values showing through the openings when the stencil is applied to the tabulations. Openings applying to the same constituent hour are connected by a ruled line which clearly indicates to the eye the tabular heights which are to be summed together. For convenience in construction two stencil sheets are prepared for each page of tabulations, one sheet providing for the even constituent hours and the other sheet for the odd constituent hours. 162. The stencils are adapted for use with tabulations made in any kind of time provided the time used is uniform for the entire series of observations. For convenience the tabulations are usually made in the standard time of the place. The series to be analyzed, however, must commence with the zero hour of the day and this is also taken as the zero constituent hour for each constituent. Suc- cessive solar hours will fall either earlier or later than the correspond- ing constituent hour according to whether the constituent day is longer or shorter than the solar day. 163. For the construction of the stencils it is necessary to calculate the constituent hour that most nearly coincides with each solar hour of the series. Let a=speed or rate of change in argument of constituent sought in degrees per solar hour. p= number of constituent periods in constituent day; 1 for diurnal tides, 2 for semidiurnal tides, etc. sh= number of solar hour reckoned from at beginning of each solar day. shs= number of solar hour reckoned from at beginning of series. dos= day of series counting from 1 as the first day. ch= number of constituent hour reckoned from at beginning of each constituent day. chs= number of constituent hour reckoned from at beginning of series. Then Q (? f\ 1 constituent period= solar hours. (237) 1 constituent day = solar hours. (238) a 157? 1 constituent hour == — - solar hours. (239) d 1 solar hour =tt^ constituent hours. (240) Ibp Therefore, (chs) =^(shs) = ~[24{ (dos) - 1 } + (sh)} (241) 164. The above formula gives the constituent hour of the series (chs) corresponding to any solar hour of the series (shs). The observed heights of the tide being tabulated for the exact solar hours of the day, the (shs) with which we are concerned will represent successive integers counting from at the beginning of the series. The (chs) as derived from the formula will generally be a mixed number. As HARMONIC ANALYSIS' AND PREDICTION OF TIDES 55 it is desired to obtain the integral constituent hour corresponding most nearly with each solar hour, the (chs) should be taken to the nearest integer by rejecting a fraction less than 0.5, or counting as an extra hour a fraction greater than 0.5, or adopting the usual rule for computations if the fraction is exactly 0.5. The constituent hour of the constituent day (ch) required for the construction of the stencils may be obtained by rejecting multiples of 24 from the (chs). 165. In the application of the above formula it will be found that the integral constituent hour will differ from the corresponding solar hour by a constant for a succession of solar hours, and then, with the difference changed by one, it will continue as a constant for an- other group of solar hours, etc. This fact is an aid in the prepara- tion of a table of constituent hours corresponding to the solar hours of the series, as it renders it unnecessary to make an independent calculation for each hour. Instead of using the above formula for each value the time when the difference between the solar and con- stituent hours changes may be determined. The application of the differences to the solar hours will then give the desired constituent hours. 166. Formula (241) is true for any value of (shs), whether integral or fractional. It represents the constituent time of any instant in the series of observations in terms of the solar time of that same instant, both kinds of time being reckoned from the beginning of the series as the zero hour. The difference between the constituent and the solar time of any instant may therefore be expressed by the following formula: Difference^^W - (ska) =^^(shs) (242) 167. If the constituent day is shorter than the solar day, the speed a will be greater than 15p, and the constituent hour as reckoned from the beginning of the series will be greater than the solar hour of the same instant. If the constituent day is longer than the solar day the constituent hour at any instant will be less than the solar hour of the same instant. At the beginning of the series the difference between the constituent and solar time will be zero, but the difference will increase uniformly with the time of the series. As long as the difference does not exceed 0.5 of an hour the integral constituent hours will be designated by the same ordinals as the integral solar hours with which they most nearly coincide. Differences between 0.5 and 1.5 will be represented by the integer 1, differences between 1.5 and 2.5 by the integer 2, etc. If we let d represent the integral difference, the time when the difference changes from (d— 1) to d, will be the time when the difference derived from formula (242) equals (d—0.5). Substituting this in the formula, we may obtain (sAs)= ^S^-°- 5) (243) in which (shs) represents the solar time when the integral difference between the constituent and solar time will change by one hour from (d—1) to d. By substituting successively the integers 1, 2, 3, etc., for d in the formula (243) the time of each change throughout the series may be obtained. The value of (shs) thus obtained will 56 U. S. GO AST AND GEODETIC SUBVEY generally be a mixed number; that is to say, the times of the changes will usually come between integral solar hours. The first integral solar hour after the change will be the one to which the new difference will apply if the usual system of distribution is to be adopted. In this case we are not concerned with the exact value of the fractional part of (shs) but need note only the integral hours between which this value falls. 168. If, however, the second system of distribution should be desired, it should be noted whether the fractional part of (shs) is greater or less than 0.5 hour. With a constituent day shorter than the solar day and the differences of formula (242) increasing positively, the application of the differences to the consecutive solar hours will result in the jumping or omission of a constituent hour at each change of difference. Under the second system of distribution each of the hours must be represented, and it will therefore be necessary in this case to apply two consecutive differences to the same solar hour to represent two consecutive constituent hours. The solar hour selected for this double use will be the one occurring nearest to the time of change of differences. If the fractional part of the (shs) in (243) is less than 0.5 hour, the old and new differences will both be applied to the preceding integral solar hour; but if the fraction is greater than 0.5 hour the old and new differences will be applied to the integral solar hour following the change. 169. With a constituent day longer than the solar day and the differ- ences of formula (242) increasing negatively, the application of the differences to the consecutive solar hours will result in two solar hours being assigned to the same constituent hour at each change of differences. Under the second system of distribution this must be avoided by the rejection of one of the solar hours. I n this case the integral solar hour nearest the time of change will be rejected, since at the time of change the difference between the integral and the true difference is a maximum. Thus, if the fractional part of the (shs), is less than 0.5 hour, the preceding solar hour will be rejected; but if the fraction is greater than 0.5 hour the next following solar hour will be rejected. 170. Table 31, computed from formula (243), gives the first solar hour of the group to which each difference applies when the usual system of distribution is adopted. Multiples of 24 have been rejected from the differences, since we are concerned only with the constituent hour of the constituent day rather than with the constituent hour of the series, and these differences may be applied directly to the solar hours of the day. For convenience equivalent positive and negative differences are given. By using the negative difference when it does not exceed the solar hour to which it is to be applied, and at other times using the positive difference, the necessity for adding or rejecting multiples of 24 hours from the results is avoided. 171. The tabulated solar hour is the integer hour that immediately follows the value for the (shs) is formula (243). An asterisk (*) indicates that the fractional part of the (shs) exceeds 0.5, and that the tabular hour is therefore the one nearest the exact value of (shs). If the second system for the distribution of the hourly heights is adopted, the solar hours marked with the asterisk will be used with both old and new difference to represent two constituent hours, or will be rejected altogether according to whether the constituent day HARMONIC ANALYSIS' AND PREDICTION OF TIDES 57 is shorter or longer than the solar day. If the tabular hour is un- marked, the same rule of double use or rejection will apply to the untabulated solar hour immediately preceding the tabular unmarked hour. For the ordinary stencils no attention need be given to the asterisks. By the formula constituents with commensurable periods will have the same tabular values, and no distinction is made in the construction of the stencils. Thus, stencils for constituent M serve not only for M 2 but also for M 3 , M 4 , M 6 , etc. 172. For the construction of a set of stencils for any constituent a preliminary set of the hourly height forms is prepared with days of series numbered consecutively beginning with 1 and each hourly height space numbered with its constituent hour as derived by the differences in table 31. The even and odd constituent hours are then transferred to separate sets of forms and the marked spaces cut out. In the Coast and Geodetic Survey this is done by a small machine with a punch operated by a hand lever. Spaces corresponding to the same constituent hour are connected by ruled lines which are num- bered the same as the hours represented. Black ruling with red numbering is recommended, the red emphasizing the distinction between these numbers and the tabulated hourly heights which are to be summed. 173. When in use the stencils are placed one at a time on the sheets of tabulated heights, with days of series on stencils matching those on the tabulations, and all heights on the page corresponding to each constituent hour are then summed separately. For constituent S no stencils are necessary as the constituent hours in this case are identical with the solar hours. For constituents K, P, R, and T with speeds differing little from that of S, the lines joining the hourly spaces frequently become horizontal and the marginal sum previously ob- tained for constituent S becomes immediately available for the sum- mation at hand. In these cases a hole in the margin of the stencil for the sum replaces the holes for the individual heights covered by the sum. SECONDARY STENCILS 174. After the sums for certain principal constituents have been ob- tained by the stencils described in the preceding section, which for con- venience will be called the primary stencils, the summations for other constituents may be abbreviated by the use of secondary sten- cils which are designed to regroup the hourly page sums already ob- tained for one constituent into new combinations conforming to the periods of other constituents. Certain irregularities are introduced by the process, but in a long series, such as 369 days, these are for the most part eliminated, and the resulting values for the harmonic con- stants compare favorably with those obtained by use of the primary stencils directly, the differences in the results obtained by the two methods being negligible. For short series the irregularities are less likely to be eliminated, and since the labor of summing for such a series is relatively small, the abbreviated form of summing is not recommended. As the length of series increases the saving in labor by the use of the secondary stencils increases, while the irregularities due to the short process tend to disappear. It is believed that the use of the secondary stencils will be found advantageous for all series more than 6 months in length. 58 U. S. 00 AST AND GEODETIC SURVEY 175. In the primary summations there are obtained 24 sums for each page of tabulations, representing the 24 constituent hours of a con- stituent day. In general each sum will include 7 hourly heights, and the average interval between the first and last heights will be 6 con- stituent days. A few of the sums may, however, include a greater or less number of hourly heights within limits which may be a day greater or less than 6 constituent days. 176. Let the constituent for which summations have been made by use of the primary stencils be designated as A and the con- stituent which is to be obtained by use of the secondary stencils as B. For convenience let it be first assumed that the heights included in the sums for constituent A refer to the exact A-hours. This assumption is true for constituent S but only approximately true for the other constituents. It is now pro- posed to assign each hourly page sum obtained for constituent A to the integral J5-hour with which it most nearly coincides. Constituent A and constituent 5-hours separate at a uniform rate, and the proposed assignment will depend upon the relation of the hours on the middle day of each page of tabulations. The tabulated hourly heights on each full page of record run from zero (0) solar hour on the first day to the 23d solar hour on the seventh or last day of the page. The middle of the record on each such page is therefore at 11.5 solar hours on the fourth day, or 83.5 solar hours from the beginning of the page of record. 177. Let a and b represent the hourly speeds of the constituents A and B, respectively, and p and pi 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(ft-l)-|-83.5] or (168ft— 84.5) solar hours (244) from the beginning of the series. Since one solar hour equals a/15p constituent ^L-hours (formula 240), the middle of page n will also correspond to (168ft— 84. 5)r^-constituent ^4-hours (245) from the beginning of the series. As there are 24 constituent hours in each constituent day, the middle constituent ^i-day of each page will commence 12 constituent A-hours earlier than the time represented by the middle of the page, or at [(168ft-84.5)y^ - 12] constituent ^1-hours (246) from the beginning of the series. 178. The 24 integral constituent ^1-hours of the middle constituent day of the page will therefore be the integral constituent ^i-hours which immediately follow the time indicated by the last formula. The numerical value of this formula will usually be a mixed number. Let / equal the fractional part, and let m be an integer representing the number of any integral constituent hour according to its order in the middle constituent day of each page. For each page m will have HARMONIC ANALYSIS AND PREDICTION OF TIDES 59 successive values from 1 to 24. The integral constituent 4-hours falling within the middle constituent day of each page of tabulations will then be represented by the general formula. [(168ft— 84.5)~— 12— -f+m] constituent ^4-hours (247) from the beginning of the series. 179. The relation of the lengths of the constituent A- and constit- uent 5-hours is given by the formula 1 constituent 4-hour =- — constituents-hours (248) The constituent 5-hour corresponding to the integral constituent 4,-hour of formula (247) is therefore [(168ft— 84.5)y| 12-/+m]^constituent 5-hours (249) from the beginning of the series. The last formula will, in general, represent a mixed number. The integral constituent 5-hour to which the sum for the constituent A- hour is to be assigned will be the nearest integral number represented by this formula. Let g be a fraction not greater than 0.5, which, applied either positively or negatively to the formula, will render it an integer. 180. The assignment of the hourly page sums for constituent 4- hours to the constituent 5-hours may now be represented as follows,, multiples of 24 hours being rejected: [(168ft— 84.5)^ — 12— jf+ra— multiple of 24] constituent .4-hour lbV (250) sum to be assigned to [{(168ft— 84.5)^-— 12— j+m}2—±g— multiple of 24] constituent B- hour. 15p Pia (251) The difference between the constituent 4.-hour and the constituent 5-hour to which the 4-hour sum is to be assigned is [{(168n-84.5) I |^-12-/+m}{^-l}±flf-multiple of 24] (252) By means of the above formula table 33 has been prepared, giving the differences to be applied to the constituent 4-hours of each page to obtain the constituent 5-hours with which they most nearly coincide. 181. For the construction of secondary stencils the forms designated for the compilation of the stencil sums from the primary summations may be used. Because of the practical difficulties of constructing stencils with openings in adjacent line spaces it is desirable that the original compilation of the primary sums should be made so that each alternate line in the form for stencil sums is left vacant. As with the: 60 U. S. COAST AND GEODETIC SURVEY primary stencils, it will generally be found convenient to use two stencils for each page of the compiled primary sums, although in some cases it may be found desirable to use more than two stencils in order to separate more clearly the groups to be summed. The actual construction of the secondary stencils is similar to that of the primary stencils. A preliminary set of forms is filled out with constituent B- hours as derived by differences from table 33 applied to the constit- uent A-hours. The odd and even constituent 5-hours are then transferred to separate forms and the spaces indicated cut out. The openings corresponding to the same constituent 5-hour are connected with ruled lines and numbered to accord with the constituent hour represented. The page numbering corresponding to the page num- bering on the compiled primary sums and referring to the pages of the original tabulated hourly heights is to be entered in the column provided near the left margin of the stencil. 182. In using the stencils each sheet is to be applied to the page of compiled primary sums having the same page numbering in the left- hand column as is given on the stencil. The primary sums applying to the same constituent 5-hour are added and the results brought together in a stencil sum form, where the totals and means are ob- tained. A table of divisors for obtaining the means may be readily derived as follows: In a set of stencil sum forms corresponding to those used for the compilation of constituent A primary sums the number of hourly heights included in each primary sum is entered in the space corresponding to that used for such primary sum. The secondary stencils for constituent B are then applied and the sums of the numbers obtained and compiled in the same manner as that in which the constituent B height sums are obtained. The divisors having been once obtained are applicable for all series of the same length. 183. In the analysis the means obtained by use of the secondary stencils may be treated as though obtained directly by the primary summations except that a special augmenting factor, to be discussed later, must be applied. The closeness of the agreement between the hourly means obtained by use of the secondary stencils and those obtained directly by use of primary stencils will depend to a large extent upon the relation of the speeds of constituents A and B. The smaller the difference in the speeds the closer will be the agreement. 184. To determine the extreme difference in the time of an indi- vidual hourly height and of the 5-honr 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 ^L-hours, and that on the middle day of the page of tabulated hourly heights one of the integral 5-hours coincides exactly with an ^1-hour. At the corresponding ^.-hour, one ^4-day later, the 5-hour will have increased by 24 - i — constituent 5-hours. Rejecting a multiple of 24 hours, this becomes 24( 1 ), so that at the end of one A-day after the coincidence of integral hours of constituents A and B the constituent A hourly height will differ in time from the integral constituent 5-hour to which it is to be assigned by 24( — 1 ) constituent 5-hours. HARMONIC ANALYSIS AND PREDICTION OF TIDES 61 At the end of the third ^4-day this difference becomes 72( — 1 ) constituent 5-hours. The same difference with opposite sign will apply to the third constituent day before the middle day of the page. Now, taking account of the fact that the 5-hour on the middle day of the page may differ by an amount as great 0.5 of a 5-hour from the integral .A-hour, and that the integral A-hour may differ as much as 0.5 of a constituent A, or 0.5 pb/Pia of a constituent B hour from the time of the actual observation of the solar hourly height, the extreme difference between the time of observation of an hourly height and the time represented by the 5-hour with which this height is grouped by the secondary stencils may be represented by the formula ± |" 72 (— /x/1 )+°- f ("^ +1 )l constituent ^-hours. (253) The differences may be either positive or negative, and in a long series it may reasonably be expected that the number of positive and negative values will be approximately equal. 185. The above formula for the extreme difference furnishes a criterion by which to judge, to some extent, the reliability of the method. Testing the following schedule of constituents for which it is proposed to use the secondary stencils, the extreme differences as indicated are obtained. The differences are expressed in con- stituent 5-hours and also in constituent 5-degrees. It will be noted that one constituent hour is equivalent to a change of 15° in the phase of a diurnal constituent, 30° in the phase of a semidiurnal constituent, etc. Constituent A .__. ._ . J S 00 2SM K, Ki R 2 T 2 P, Difference in hours . . 3.58 54 1.36 41 1.20 18 1.20 36 1.10 33 1.10 33 1. 20 D inference in degrees 18 L 2MK MS X2 MK MN vz N 2 Difference in hours .. ..... .. 1.09 65 1.18 35 1.43 64 1.24 74 1.26 38 1.45 Difference in degrees 44 Constituent A.^ Constituent B.. . . M2 2N pi Q 2Q 1.21 36 1.02 31 3.42 51 3.79 57 6.58 99 186. In the ordinary primary summation the extreme difference between the time of the observation of a solar hourly height and the intregal constituent hour to which it is assigned is one-half of a con- stituent hour and, represented by constituent degrees, it is 7.5° for diurnal, 15° for semidiurnal, 22.5° for terdiurnal, 30° for quarter 246037—41 5 62 U. S. COAST AND GEODETIC SURVEY diurnal, 45° for sixth-diurnal, and 60° for eighth-diurnal constituents. By the above schedule it will be noted that the extreme difference exceeds 60° in only a few cases. The largest difference is 99° for constituent 2Q when based upon the primary summations for O. This is a small and unimportant constituent, and heretofore no analysis has been made for it, the value of its harmonic constants being in- ferred from those of constituent O. Although theoretically too small to justify a primary summation in general practice, the lesser work involved in the secondary summations may produce constants for this constituent which will be more satisfactory than the inferred constants. FOURIER SERIES 187. A series involving only sines and cosines of whole multiples of a varying angle is generally known as the Fourier series. Such a series is of the form h=H Q +C 1 cos 0+<7 2 cos 20+<7 3 cos 30+ + /Si sin d+S 2 sin 20+£ 3 sin 30+ (254) It can be shown that by taking a sufficient number of terms the Fourier series may be made to represent any periodic function of 0. This series may be written also in the following form: h=H +A x cos (d+a 1 )+A 2 cos (20+a 2 )+^ 3 cos (30+a 3 ) + (255) in which A m =[CJ+S m 2 V and ^--tan" 1 j£ m being the subscript of any term. 188. From the summations for any constituent 24 hourly means are obtained, these means being the approximate heights of the constituent tide at given intervals of time. These mean constituent hourly heights, together with the intermediate heights, may be represented by the Fourier series, in which Ho = 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 constituent day. The values of corresponding to the 24 hourly means will be 0°, 15°, 30°, 330°, and 345°. Formula (254), or its equivalent (255), is the equation of a curve with the values of as the abscissae and the corresponding values of h as the ordinates. If the 24 constituent hourly means are plotted as ordinates corresponding to the values of0°, 15°, 30°, _ _ _ _ for0, it is possible to find values for H , C m , and S m , which when substituted in (255) will give the equation of a curve that will pass exactly through each of the 24 points representing these means. 189. In order to make the following discussion more general, let it be assumed that the period of has been divided into n equal parts, and that the ordinate or value of h pertaining to the beginning of each of those parts is known. Let u equal the interval between these ordi- nates, then n u=2ir, or 360° (256) Let the given ordinates be ho, h lf h 2 h ( n _i) corresponding to the abscissae o, u, 2u (n—1) u, respectively. HARMONIC ANALYSIS AND PREDICTION OF TIDES 63 It is now proposed to show that the curve represented by the following Fourier series will pass through the n points of which the ordinates are given : h=H +C 1 cos 6+C 2 cos 20+ C k cos k 6 + Si sin 0+S 2 sin 20+ 8 t sin Id m=A m=Z =#o+ S C m cos mHS Sm sin md (257) m=l m=l 71/ 71 — 1 in which the limit k — -x if n is an even number, or k= — ~ — if n is an 71 7\ — 1 odd number; and the limit 1=^—1 if n is even, or — ^— if n is odd. 190. By substituting successively the coordinates of the n given points in (257) we may obtain n equations of the form m=/c m=l h & =H -\- XI @m cos mau^r 2 $m sin mau (258) m=l m=l in which a represents successively the integers to (n—1). By the solution of these n equations the values of n unknown quantities may be obtained, including H and the (n—1) values for C m and S m . It will be noted that the sum of the limits k and I of (257) or (258) equals (n—1) for both even and odd values of n. 191. The reason for these limits is as follows: A continued series 2 C m cos mau may be written C x cos a u-\-C 2 cos 2 a u-\- +<% cos n a u + C (w -m cos (fH~l) a u-\-C (n+ <2) cos (n + 2) a u-\- J rC 2n vos2 nau + (7 ( 2n+D cos (2n+l) au+(7 (£ „ +2 ) cos (2n + 2) an+____ + (7 3w cos 3 ?i a it + (259) Since n u=2tt and a is an integer, the above may be written [<7i + CV hl) + <7(2rc+i) + ] cos a u + [<7 2 +<7 ( „ + 2) + <7 (2n+2) + ] cos 2 au + + [ > or — ~— > according to whether n is even or odd. Similarly, the continued series 2 S m sin m a u may be written 64 U. S. COAST AND GEODETIC SURVEY [&+&,+&+ IsinO + [Si + S&.+ 1) + $(2n+l) + — ^(w-i) — S ( 2n-i) — ^^w-d — ] sin a n + [$2+ $01+2) ~\- S (2n +2) + — S(n-2) — S(2n-2) — $(3re-2) _ ] sill 2 tt U + [Si + $( n + 1) *f" $(2rc+ ?) + — Stn-v — Svn-D — Sun-i)— ] sin Z a u (262) The first term in the above equals zero. The remaining terms will take complete account of the series S S m sin m a u, if 1=-^— 1 when ?i is even, or — ~— when n is odd. n From the foregoing it is evident that the limit of m will not exceed ■_• 192. If we let u and a represent any angles with fixed values, m and p any integers with fixed values, and a an integer having successive values from to (n— 1), it may be shown that XI sm (a m ^+a)= . A sm [f (w— 1) m tt+a] (263) a=o sm 2 n?< 16 = ^r^ , , N sin | n m ^ rl , „ N , , /rtnj v > , cos (a m u J r a) = — :— == — cos if (n—1) mu-\-a] (264) £zi sin | m u J a=(n-i) _ _ sin i n (np— m \ u cos i (n—1) (p — m) u 2_, sm a p u sin a m n=| f — . , , =-4 ^-^ — a^ sm f (p—m) u ! sin | n (p+ro) n cos | (n—1) (p+m) m , . 2 sin J (p+m) ^ a=<5~l) s i n 1 ^ (np — m ) u cos l( n _l) (p — m ) u 2_, cos a p n cos a m n= | f — . , , ^ ^— ^- — t^i sm | (p— m) n ii sin j n (p-\-m) u cos § (n—1) (p+m) u , . " t " 2 " sin± (^+m)u ^ bb; a=(n-i) . s i n i Ti, U)— m ) u sin | (n—1) (p—m) u 2_\ sm a p u cos a m u=\ - — . , , r — — t^i sm \ (p—m) u t sin j n (p + m) u sin § (n—1) (p+m) u ( i ~ 2 sin | (p+m) u {Z or n n=27r, then formulas (263) to n ' (267) may be written as follows: . / m \ a=(n _ 1) sinm 7rsin(m7r-- ttJ S sin am w= ^ — L (268) a=o * the above (268) to (272) become equal to zero. Thus, a=(n-l) S sm a m ^^o a=o a=(n-l) S cos a m u=0 a=o a=(n-l) y^, sin a 2? w sin a m u=0 f (273) a=o a=(n-l) 2_j cos a p u cos a m u— a=o a=(n-l) y^, sin a 2? w cos a m u=0 a=o 195. If ^? and m are equal integers and do not exceed -~, formulas (270), (271), and (272) will contain the indeterminate quantity ^— = ~> and also when p and m each equal -& the indetermin- sin *- -ir n . ... sin (p-\-m)r ate quantity ¥—. — (— =7^ M J (p+m) sin ^— j — -7r 66 U. S. 00 AST AND GEODETIC SURVEY Evaluating these quantities we have sin (p — m)ir"l 7r cos {p — m) if and 3 p — m 7r p—m sin 7r L N - cos 7r n (p— m)=o n n _\{p—m)=o sin (p-\-m)T sin p-\-m hi n- cos (p+-m)- COS 7T = ti (274) n (275) 7i J(p + m)=n n n _\(p-\-m)=n In (275) it will be noted that when the integers p and m each equal Ti ■x, n must be an even number, and therefore cos rnr is positive, while cos t is negative. 196. Assuming the condition that p and m are equal integers, each less than ^, we have by substituting (274) in (270), (271), and (272), a=(n-l) _ a=(n-l) 22 sin a p u sin a m u= 23 sm2 a m u= i n (276) a=o a=o a=(n— 1) a=(n-l) 22 cos a P u cos a m u= S cos2 a m u =h n (277) a=o a=o a-*(n— 1) a=(n— 1) y^, sin a £> u cos a m tt= 22 sin a m u cos a m u=0 (278) a=o a=o 71 197. Assuming the condition that p and m are each equal to ~ we have by substituting (274) and (275) in (270), (271), and (272), = (n-l) y^, sin 2 a m u=\ n-\-\ n cos ir=0 a = (n-l) 22 cos2 a m u= 2 n—\ n cos ir=n a=o a=(n-l) >! sin a m u cos a m w=0 a=o (279) (280) (281) 198. Returning now to the solution of (258), by substituting the successive values of a from to (n — 1), we have K =H o +C x cos + <7 2 cos 0+ + C k cos + Si sin + £ 2 sin 0+ +£, sin hi=H +Ci cos w+(7 2 cos 2w+ + C* cos &u -f-'S'i sin u-\-S 2 sin 27/+ -j-Sj sin lu h 2 =H -\-C l cos 2w+(7 2 cos 4it-f + (7* cos 2ku + # 1 sin 2?i+# 2 sin 4u+ +5, sin 2Ztt r ( 282 ) A (n _i)=i7o+(7i cos (n— l)u+(7 2 cos 2(n— l)u-\- + (7 b cos (ti— l)ku -\-Si sin (71—1)^+^2 sin 2(n— l)u-{- . -j-Si sin (n— \)lu HARMONIC ANALYSIS' AND PREDICTION OF TIDES 67 199. To obtain value of H , add above equations a=(n-l) 2] K=n H a=o a=(n-l) a=(n-i) a=(n-l) + C\ 2j cosau-fft S cos 2 a u-{- -{-C k 2 cosakv a=o a=o a=o a=(n-l) a=(n-l) a=(n-l) + #i 22 sin a 7i+# 2 2D sin 2 a 7/ -f + $* S sin a Z u a=o a=o a=o m = /c a=(n-l) m = Z a=(n-l) =n# +X)6m S C0S ftw^+Si 2D sin a m ^ (283) m = l a=o m=l a = o a=(n— 1) a=(n— 1) From (273), S cos a m ^ an d 2D sm a m u each equals zero, a=o a=o since neither k nor Z, the maximum values of m exceeds ■=- Therefore a=(n-l) S A a =w H (284) a=o and 1 a=(n-l) V =i S Ai . (285) "' a=o 200. To obtain the value of any coefficient C, such as C Pi multiply each equation of (282) by cos a p u. Then h cos 0=H cos + it-f + ft cos A: u cos p u -\-Si sin u cos 2? it +$2 sin 2u cos #> w+ -{-Si sin Z u cos p u h 2 cos 2^> u=H cos 2p 7/, + ft cos 2u cos 2^ 7^+ft cos 4t/, cos 2^? m+ -\-C k cos 2& 7/, cos 2p u -\-Si sin 2u cos 2^> it+$ 2 sin 4^ cos 2p u-\- -{-Si sin 21 u cos 2^> u h {n _i) cos (ti— 1) 2? u=H cos (n— 1) p u + ft cos (ti— 1) ucos (ti— 1) £>^-f ft cos 2 (ti— 1) t^cos (n— 1) pu-{- + ft cos {n— 1) £ 16 cos (7i— 1) p t/, + /Si sin (n— 1) t^cos (n— 1) pu-\-S 2 sin 2 (ti— 1) t^cos (ti— 1) £>7/+__ -f&z sin (ti— 1) I u cos (7i—l) p u (286) Summing the above equations a=(n-l) a=fn-l) 2D ^« cos a p u—H 2D cos a p u a=o a=0 a=(n-l) a=(n-l) + ft 2D cos a u cos a 2? ^+*S'i 2D sin a 7/, cos a 2? u a=o a=o (Formula continued next page) 68 U. S. COAST AND GEODETIC SURVEY a=(n-l) a=(n-l) -\-C 2 53 cos 2a u cos a V ^+^2 53 s i n 2a ^ cos a ^ u a=o a=o a = (n— 1) a=(n— 1) + Ca; S cos a k u cos a p w+# z 53 sin a £ u cos a p u a=o • a=o a=(n-l) m = fc a=(n— 1) =#o 53 cos a V uJ r 53 ^m 53 cos a m u cos a V u a=o m = l a=o m=Z a=(n— 1) + S ^m 2 sinamw cos a p u (287) m = l a=o 201. Examining the limits of (287), it will be noted by a reference to page 63 that k, the maximum value of m for the C terms is ~ when n /n 1 in is even and ~ when n is odd; also, that I has a value of ~ — 1 when w— 1 n is even and — ^— when w is odd. The limits of #>, which is a partic- ular value of m, will, of course, be the same as those of m. a=(n-l) By (273) the quantity 53 cos a V u becomes zero for all the a=o a=(n-l) values of p, and the quantity 53 c °s a m u cos a p u becomes zero .a=o for all values of m and p except when p equals m. By (273), (278) a=(n-l) and (281) the quantity 53 sin a m w cos a p u becomes zero for all a=o values of m and p. Formula (287) may therefore be reduced to the form a=(n— 1) a=(n-l) 23 ^a cos a p u=C p 53 cos2 a V u (288) a=o a=o n For any value of p less than 2 = (n-l) 53 cos2 a V u =i n (277) a=o lb but when p=~, this quantity becomes equal to n (280). TV Therefore for all values of p less than ■= 9 a=(n-l) (7 P =- 5] Kcosapu (289) 1^ a=o Tl but when p is exactly -= 1 a=(n-l) C p =- 53 h & cos apu (290) W a=o Since in tidal work 2? is always taken less than ~ , we are not especially concerned with the latter formula. HARMONIC ANALYSIS' AND PREDICTION OF TIDES 69 202. To obtain the value of any coefficient S, such as S p , multiply each equation of (282) by sin a p u. Sum the resulting equations and obtain a=(n-l) B=(n-1) < 23 h & sin a p u =H 22 sm a V u a=o a=o m = fc a=( n— 1) + 22 a=o L^ a=o J V~2 a=(n-l) m "I + — 23 h & sin a u \ sin 6 t2 a=(n-l) "I — 23 ^a cos 2 a u \ cos 2 W a=o J P2 a=(n-l) -I +| - 2] ^a sin 2 a it sin 2 L^ a=o J r 2 * a^o-i) -l cos A: 6 + - 23 ^a cos A: aw L^ ^o J T2 a=(n-l) -I + - 22 Ksinl aulsinle (294) L^ a=o J 71 12 *If ft is even and k= -77 » this fraction is — instead of — 2 n n 70 U. S. COAST AND GEODETIC SURVEY 204. Although by taking a sufficient number of terms the Fourier series may thus be made to represent a curve which will be exactly satisfied by the n given ordinates, this is, in general, neither necessary nor desirable in tidal work, since it is known that the mean ordinates obtained from the summations of the hourly heights of the tide in- clude many irregularities due to the imperfect elimination of the me- teorological effects and also residual effects of constituents having periods incommensurable with that of the constituent sought. It is desirable to include only the terms of the series which represent the true periodic elements of the constituent. With series of observations of sufficient lengthy the coefficient of the other terms, if sought, will be found to approximate to zero. 205. The short-period constituents as derived from the equilibrium theory are, in general, either diurnal or semidiurnal. If the period of 6 in formula (257) is taken to correspond to the constituent day, the diurnal constituents will be represented by the terms with coefficient C\ and Si, and the semidiurnal constituents by the terms with co- efficients C 2 and S 2 . For the long-period constituents, the period of may be taken to correspond to the constituent month or to the constituent year, in which case the coefficients G{ and Si will refer to the monthly or annual constituents and the coefficients C 2 and S 2 to the semimonthly or semiannual constituents. For most of the constituents the coefficients C x , S u C 2 , and S 2 will be the only ones required, but for the tides depending upon the fourth power of the moon's parallax and for the overtides and the compound tides, other cc efficients will be required. Terms beyond those with coefficients C$ and S 8 , for the overtides of the principal lunar constituent are not generally used in tidal work. 206. When it is known that certain periodic elements exist in a constituent tide and that the mean ordinates obtained from obser- vations include accidental errors that are not periodic, it may be readily shown by the method known as the least square adjustment, using the observational equations represented by (258), that the most probable values of the constant H and the coefficients C p and S p are the same as those given by formulas (285), (289), and (293), respectively. 207. Since in tidal work the value of H , which is the elevation of mean sea level above the datum of observations, is generally deter- mined directly from the original tabulation of hourly heights, formula (285) is unnecessary except for checking purposes. Formulas (289) and (293) are used for obtaining the most probable values of the coefficients C p and S D from the hourly means obtained from the summations. 208. When 24 hourly means are used n=24 and u=15°, and the formulas may be written I a = 23 sm 2 ) FiB cos {**+?+- a ) = Fl FsBco S (bt+f}+ 3 -^) (312) if we put F 2 =— — sin —~- for brevity. 7T p Z 219. Formula (312) represents the mean value of the B ordinate for a particular day of the page record. The average value for the 7 days may be written *™ n gW<+e + ^) n = -3 \ a / =WFJi [cos (bt+0) cos (-3 - 6 -?)- S m (bt+fi) sin (-3 3 -^) + cos (bt+0) cos (-2 ^)-sin (bp + p) sin (-2 3 -^) + cos (W+tf) cos (-1 3 ^)-sin (U+® sin (-1 ^) + cos (bt+Q) cos — sin (bt+(3) sin + cos (bt+® cos (*)-sin (&*+# sin (^) (Formula continued next page) HARMONIC ANALYSIS' AND PREDICTION OF TIDES 75 + cos (bt+(S) cos ( 2 ^)-sin (W+fl sin (2 ^) + cos («+/}) cos (3 ^)-sin (6* +/ 3) sin (3 3 -^)] 1+2 cos 36%> + 2 cos 2 3606£ +2 cog 3 3606£-| ^ (J ft ft ft ! = \F,FM 7* 1 Z 2 J |^2# 360&7? 3 3606?? ' sin 2 £ cos o - a 2a, 2 ^ttttt 1 sin , 3606^ a cos (6*+/3) 126(%T a sin 1806t> cos (bt+0), (313) 220. Replacing the equivalents of F l and jP 2 in (313), the average value of the B ordinate as obtained by the secondary summations may be written 24a . 15ojTir24 m irpb sm 2a _\[_irp' 15p 2 : ] sm 12606?" 7 sin ISObp B cos (bt+ j8) (314) Since the true ordinate of constituent B at any time t is equal to B cos (6#+j8), the reciprocal of the bracketed coefficient will be the augmenting factor necessary to reduce the B ordinate as obtained from the summations to their true values. This augmenting factor may be written ■bp 24a sin Ibby 2a irp 24 sin I5p' 7 sm 1806p sm 12Q0bp (315) The first factor of the above is to be omitted if the primary sum- mations are for constituent S. It will be noted that the middle factor is the same as the augmenting factor that would be used if constituent B had been subjected to the primary summations. PHASE LAG OR EPOCH 221. The phase lag or epoch of a tidal constituent, which is repre- sented by the Greek kappa (k), is the difference between the phase of the observed constituent and the phase of its argument at the same time. This difference remains approximately constant for any con- stituent in a particular locality. The phase of a constituent argument for any time may be obtained from the argument formula in table 2 by making suitable substitutions for the astronomical elements. The argument itself is represented by the general symbol (V-\-u) or E and 76 U. S. 00 AST AND GEODETIC SURVEY its phase or value pertaining to an initial instant of time, such as the beginning of a series of observations, is expressed by (V -\-u). Refer- ring to formula (300), since is reckoned from the beginning of the series, the angular quantity (— f ) is the corresponding phase of the observed constituent at this time. The phase lag may therefore be expressed by the following general formula: K=y +^-(-r)=y +^+r (31 6) 222. Since the argument formulas of all short-period constituents contain some multiple of the hour angle (T) of the mean sun, the arguments themselves will have different values in different longitudes at the same instant of time. If p equals the coefficient of T or the subscript of the constituent and L equals the longitude of the place in degrees reckoned west from Greenwich, L being considered as nega- tive for east longitude, the relation between the local and Greenwich argument for any constituent may be expressed as follows: local (V+u) = Greenwich (V+u) — pL (317) 223. Also, since the absolute time of the beginning of a day or the beginning of a year depends upon the time meridian used in the locality, the initial instant taken for the beginning of a series of obser- vations may differ in different localities even though expressed in the same clock time of the same calendar day. If we let S equal the longitude of the time meridian in degrees, positive for west and nega- tive for east, the same meridian expressed in hours becomes S/15. Letting a equal the speed or hourly rate of change in the constituent argument, the difference in argument due to the difference in the absolute beginning of the series becomes aS/15, and the relation between the local and Greenwich argument due to this difference may be expressed as follows: local (y o +iO = Greenwich (y o +u)—pL+a-S/15 (318) In the above formula the local and Greenwich (V -\-u) pertain to the same clock time but not the same absolute time unless both clocks are set for the meridian of Greenwich. 224. Values of (Vo+u) for the meridian of Greenwich at the beginning of each calendar year 1850 to 2000 are given in table 15 for all constituents represented in the Coast and Geodetic Survey tide-predicting machine. Tables 16 to 18 provide differences for referring the arguments to other days and hours of the year. In the preparation of table 15 that portion of the argument included in the u was treated as a constant with a value pertaining to the middle of the calendar year. If the Greenwich (V -\-u) with its corrections is sub- stituted for the local (V -{-u) in formula (316), we obtain k- Greenwich (V +u)—pL+aS/15 + t (319) 225. The phase lag designated by k is sometimes called the local epoch to distinguish it from certain modified forms which may be used for special purposes. In the preparation of the harmonic constants for predictions it is convenient to combine the longitude and time meridian corrections with the local epoch to form a modified epoch HARMONIC ANALYSIS' AND PREDICTION OF TIDES 77 designated by k' or by the small g. The relation of the modified epoch to the local epoch may then be expressed by the following formula: K f or g=K+pL— a£/15=Greenwich (V +u)+{ (320) 226. The phases of the same tidal constituent in different parts of the world are not directly comparable through their local epochs since these involve the longitude of the locality. For such a comparison it is desirable to have a Greenwich epoch that is independent of both longitude and time meridian. Such an epoch may be designated by the capital G and its relation to the corresponding local epoch ex- pressed as follows: Greenwich epoch (G)=K J r pL= Greenwich (V +u)+aSfl5+f (321) 227. The angle k may be graphically represented by figures 7 and 8. In figure 7, we have a simple representation of a single con- PL J^p(S-L)^ Greenwich V +u I r— I >, i-4 -cl_- ° fc a the phase of constituent A will equal (2n Tr—a—d) + a and the phase of constituent B will equal - (2n 7r-a-0) + Let <£= phase of constituent B— phase of constituent A at this time. Then 0=^ (2n ir-a-0) + p-a (352) 82 U. S. COAST AND GEODETIC SURVEY Substituting the above in (351) — Aa sin 6+Bb sin (— 6) = — Aa sin B-\-Bb sin cos B—Bb cos <£ sin 6 = — (Aa+Bb cos 0) sin 0+56 sin cos 0=0 (353) Then , . Bb sin ^ . tan 6= - A — i pi , (354) Aa-\-Bb cos v ' 239. For the resultant amplitude at the time of this maximum sub- stitute the values of t from (350), in (346), and we have y=A cos (27i 7T— 0)+5 cos - (2n tt— 0— «) + £ = A cos 0+5 cos F^p (277- 7r-0-a) + /3-c*-0 | =^4 cos 0+5 cos (0—0) (355) =Jl cos B-\-B cos cos 0+5 sin sin = (A+B cos 0) cos 0+5 sin cf> sin = V^ 2 +^ 2 +2A5cos ^t + cos cf> In the special cases under consideration the ratio t- is near unity, and the difference between and tan -1 ■ A , „ — — - is therefore very A+B cos (357) The true amplitude of the constituent sought being A, the resultant amplitude must be divided by the factor v l + 5+2fcos0 (358) in order to correct for the influence of the disturbing constituent. 241. The corrections for acceleration and amplitude as indicated by formulas (356) and (358) may to advantage be applied to the con- stants for constituent K x for an approximate elimination of the effects of constituent P x and to the constants for S 2 for an approximate elimination of the effects of constituents K 2 and T 2 . By taking the relations of the theoretical coefficients for the ratios -? and the differ- ences in the equilibrium arguments as the approximate equivalents of the phase differences represented by , tables may be prepared giving the acceleration and resultant amplitudes with the arguments referring to certain solar elements. Thus, from table 2, the following values may be obtained. HARMONIC ANALYSIS AND PREDICTION OF TIDES 83 B A Aa Bb Effect of Pi on Ki Effect of K 2 on S2 r — 0. 33086 0. 27213 0. 05881 3. 03904 3. 66469 17.02813 -2h+p'+\80°. 2/1-2/'. Effect of T2 on S2- -h+pi. Substituting the above in (356) and (358) we have Effect of P t on K x Acceleration = tan sin (2h—v / ) 3.0390-cos (2/*,-/) Resultant amplitude = 0.813 Vl. 6767 — cos (2h—v') Effect of K 2 on S 2 sin (2h-2v") Acceleration == tan" 3.6647 + cos {2h-2v") Resultant amplitude==0.738V1.9734 + cos (2h—2v") Effect of T 2 on S 2 Acceleration = tan -1 -,- - aoo1 , 8^ r 17.0281 + cos (h—pi) Resultant amplitude=0.343V8.5318 + cos (h—p{) (359) (360) (361) (362) (363) (364) 242. The above formulas give the accelerations and resulting amplitudes for any individual high water. For the correction of the constants derived from a series covering many high waters it is necessary to take averages covering the period of observations. Tables 21 to 26 give such average values for different lengths of series, the argument in each case referring to the beginning of the series. In the preceding formulas the mean values of the coefficients were taken to obtain the ratios -^- To take account of the longitude of the moon's node, the node factor should be introduced. If the mean coefficients are indicated by the subscript 0, formulas (356) and (358) may be written Acceleration = tan -1 f(A)A a + COS f(B)B b Resultant ampUtude= A / 1+ (/(||) + 2 /(|| co? , (365) (366) f(A) 243. In the cases under consideration the ratio 77^ will not differ A a greatly from unity, the ratio -js-r will be rather large compared with cos 4>, which can never exceed unity, and the acceleration itself is relatively small. Because of these conditions the following may be taken as the approximate equivalent of (365) : 86 U. S. COAST AND GEODETIC SURVEY A sin a cos a=A' sin a' cos a — 2 F h cos { \{b — a)r+j8} sin a' (381) A cos c/ sin a=A' sin a cos a'— 2 i^ b sin {\{b — cl)t+(3} cos a' (382) Subtracting (382) from (381) A sin («' — a) = 2 F b sin {§(& — a)T+,3— a'} (383) Multiplying (379) and (380) by cos a and sin a , respectively, A cos a' cos <*=4/ cos 2 a' — S i^ b cos {£(& — a)r+0} cos a' (384) 4 sin a' sin a=4' sin 2 a! — 2 i^ sin {|(6 — a)r+/3} sin a' (385) Taking the sum of (384) and (385) A cos (a' — a)=A' — 2 F h cos {±(b-a)r+p-a f } (386) Dividing (383) by (386) 4. tt n 2 F b sin {§(& — a)r+0— a'} /00 _ tan ^-")" J=2 ^ cos {K6-a)r+^ T • (387) From (386) COS (a' — a) ^ ' 249. Substituting the value F b from (376) and the equivalents R>(A), 5.(4), i?CB),-f'(4)-r(4), and -f (5) for 4', 4, 5, a', a, and /3, respectively, we have by (387) and (388) tan[f(4)-r(4i= 5'(4)-2^9 Q J^t-^lR {B ) cos (i(&-a)r-rt5)+r(4)} 2*'(4)-2^? ^|-a)r )T g(g) cos {Ub-a)r-UB) + ^(A)} cos[f(4)-r / (^)] (390) 250. Formula (389) gives an expression for obtaining the difference to be applied to the uneliminated £'(A) in order to obtain the true £(A), and formula (390) gives an expression for obtaining the true amplitude R(A). These formulas cannot, however, be rigorously applied, because the true values of R{B) and f (2?) of the disturbing constituents are, in general, not known, but very satisfactory results may be obtained by using the approximate values of R(B) and $(B) derived from the analysis or by inference. By a series of successive approximations, using each time in the formulas the newly eliminated values for the disturbing constituents, any desired degree of refinement may be obtained, but the first approximation is usually sufficient and all that is justified because of the greater irregularities existing from other causes. 251. Form 245 (fig. 19) provides for the computations necessary in applying formulas (389) and (390). In these formulas the factors represented by — ■ • 1 ( 2 — c-^- r and the angles represented by HARMONIC ANALYSIS' AND PREDICTION OF TIDES 87 ^{b — a)r will depend upon the length of series; but for any given length of series they will be constant for all times and places. Table 29 has been computed to give these quantities for different lengths of series. The factor as directly obtained may be either positive or negative, but for convenience the tabular values are all given as positive, and when the factor as directly obtained is negative the angle has been modified by ±180° in order to compensate for the change of sign in the factor and permit the tabular values to be used directly in formulas (389) and (390). 252. An examination of formulas (389) and (390) will show that the disturbing effect of one constituent upon another will depend largely upon the magnitude of the fraction —tA \ Assuming that b is r te i{o — a)r not equal to a, this fraction and the disturbing effect it represents will 360° approach zero as the length of series r approaches in value ,,_ y or any multiple thereof, or, in other words, as r approaches in length any multiple of the synodic period of constituents A and B. Also, since the numerator of the fraction can never exceed unity, while the denominator may be increased indefinitely, the value of the fraction will, in general, be diminished by increasing the length of series and will approach zero as r approaches infinity. The greater the dif- ference (jb—a) between the speeds of the two constituents the less will be their disburbing effects upon each other. For this reason the effects upon each other of the diurnal and semidiurnal constituents are usually considered as negligible. 253. The quantities R(B) and {(B) of formulas (389) and (390) refer to the true amplitudes and epochs of the disturbing constituents. These true values being in general unknown when the elimination process is to be applied, it is desirable that there should be used in the formulas the closest approximation to such values as are obtainable. If the series of observations covers a period of a year or more, the am- plitudes and epochs as directly obtained from the analysis may be considered sufficiently close approximations for use in the formulas. For short series of observations, however, the values as directly obtained for the amplitudes and epochs of some of the constituents may be so far from the true values that they are entirely unservice- able for use in the formulas. In such cases inferred values for the disturbing constituents should be used. LONG-PERIOD CONSTITUENTS 254. The preceding discussions have been especially applicable to the reduction of the short-period constituents — those having a period of a constituent day or less. They are the constituents that deter- mine the daily or semidaily rise and fall of the tide. Consideration will now be given to the long-period tides which affect the mean level of the water from day to day, but which have practically little or no effect upon the times of the high and low waters. There are five such long-period constituents that are usually treated in works on harmonic analysis — the lunar fortnightly Mf, the lunisolar synodic fortnightly MSf, the lunar monthly Mm, the solar semiannual Ssa, and the solar annual Sa. The first three are usually too small to be of practical importance, but the last two, depending largely upon 88 U. S. COAST AND GEODETIC SURVEY meteorological conditions, often have an appreciable effect upon the mean daily level of the water. 255. To obtain the long-period constituents, methods similar to those adopted for the short-period constituents with certain modifica- tions may be used. For the fortnightly and monthly constituents the constituent month may be divided into 24 equal parts, analogous to the 24 constituent hours of the day. Similarly, for the semiannual and annual constituents the constituent year may be divided into 24 equal parts, although it will often be found more convenient to divide the year into 12 parts to correspond approximately with the 12 calendar months. 256. Instead of distributing the individual hourly heights, as for the short-period constituents, a considerable amount of labor can be saved by using the daily sums of these heights. The mean of each sum is to be considered as applying to the middle instant of the period from hour to 23d hour; that is, at the 11.5 hour of the day. If the constituent month or year is divided into 24 equal parts, the in- stants separating the groups may be numbered consecutively, like the hours, from to 23, with the instant of the first group taken at the exact beginning of the series. A table may now be prepared (table 34) which will show to which division each daily sum, or mean, of the series must be assigned. 257. Letting a=the hourly speed of any constituent, in degrees. p=l when applied to a monthly or an annual constituent, and p=2 when applied to a fortnightly or a semiannual constituent. d=day of series. = solar hour of day. Then and also OClfl 1 constituent period = — solar hours (391) a L constituent month= solar hours (392) 1 constituent year= solar hours (393) Dividing the constituent month or year into 24 equal parts, the length of 1 constituent division = — - solar hours (394) a Therefore, to express the time of any solar hour in units of the con- stituent divisions to which the solar hourly heights are to be assigned, the solar hour should be multiplied by the factor a/15p. Thus, Constituent division =77- (solar hour of series) -lfp|24( 263. This formula is adapted to use on the tide-computing machine. With the constituent cranks set in accordance with the coefficients and initial epochs of the above formula, the machine will indicate the values of y corresponding to successive values of t. The values of y desired for the clearances are those which correspond to t at the 11.5 hour on each day. Thus, the clearance for each successive day of series may be read directly from the dials of the machine. In practice, it may be found more convenient to use the daily sums rather than the daily means for the analysis. In this case the co- efficients of the terms of (402) should be multiplied by the factor 24 before being used in the tide-computing machine. 264. Assuming that all the daily sums are used in the analysis, the augmenting factor represented by formula (308) which is used for the short-period constituent is also applicable to the long-period con- stituents, with p representing the number of constituent periods in a constituent month or year. Thus, for Mm and Sa, p equals 1 , and for Mf, MSf, and Ssa, p equals 2. For the long-period constituents a further correction or augmenting factor is necessary, because the mean or sum of the 24 hourly heights of the day is used to represent the single ordinate at the 11.5 hour of the day. 265. If we let formula (396) be the equation of the long-period constituent sought, formula (400) will give the mean value of the 24 ordinates of the day which, in the grouping for the analysis, is taken as representing the 11.5 hour of the day or the t 6 hour of the series. Since the true constituent ordinate for this hour should be A cos sin fi (at d -\-a), it is evident that an augmenting factor of 24 - — ~- 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. 92 U. S. 00 AST AND GEODETIC SURVEY 266. The complete augmenting factor for the long-period constit- uents, the year or month being represented by 24 means, will be obtained by combining the above factor with that given in formula (308). Thus <• T w24 sin ha , M ™*\ augmenting factor = if^ X s i n \ 2a (403) 24 sin -~~ z If the year or month is represented by only 12 means as when monthly means are used in evaluating Sa and Ssa, the formula becomes augmenting factor= 10 • , c X —■ — tft~ (404) & s 12 sin 15p sin 12a v } Values obtained from these formulas are given in table 20. 267. The following method of reducing the long-period tides, which conforms to the system outlined by Sir George H. Darwin, differs to some extent from that just described. In this discussion it is assumed that a series of 365 days is used. Let the entire tide due to the five long-period constituents already named be represented by the equation y = A cos (at+a)+B cos (bt+(3) + C cos (at+y) (405) +Z>cos (dt+8) + E cos (et+e) 268. For convenience in this discussion let t be reckoned from the 11.5th solar hour of the first day of series instead of the midnight beginning that day. Every value of t to which the daily means refer will then be either or a multiple of 24. Let A', B', C, D', and E' ', equal A cos a, B cos j8, C cos y, D cos 5, and E cos e, respectively, and A", B", C", D", and E", equal —A sin a, —B sin /5, — Csin y, —D sin 5, and — E sin e, respectively. (406) Then formula (405) may be written y=A f cos at-\-B f cos bt-\-C cos ct-\-D' cos dt+E' cos et +A" sin at+B" sin bt+C" sin ct+D" sin dt+E" sin et (407) In the above equation there are 10 unknown quantities, A f , A", B' ', B" , etc., for which values are sought in order to obtain from them the amplitudes and epochs of the five long-period constituents. The most probable values of these quantities may be found by the least square adjustment. 269. Let 2/1, y 2 , . • . . 2/365 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 t to which that mean applies will be 24 (n— 1). By substituting in formula (407) the successive values of y and the values of t to which they correspond, 365 observational equations are formed as follows: HARMONIC ANALYSIS AND PREDICTION OF TIDES 93 yi=A' cos 0+£' cos 0+ ... . + A" sin 0+£" sin 0+ ... . y 2 =A' cos24a+£' cos 246 + .... -\-A" sin 24a+£" sin 246+ .... y 365 =A' cos 24X364~a+5' cos 24X3646 + +^L" sin 24X364a+£" sin 24X3646 + (408) 270. A normal equation is now formed for each unknown quantity by multiplying each observational equation by the coefficient of the unknown quantity in that equation and adding the results. Thus, for the unknown quantity A', we have y x cos 0=A' cos 2 0+J9' cos cos 0+ .... +A" sin cos 0+£" sin cos 0+ ... . y 2 cos 24a=^L' cos 2 24a+i?' cos 246 cos 24a+ • • • + A" sin 24a cos 24a+£" sin 246 cos 24a+ (409) 2/365 cos (24X364a)=A / cos 2 (24X364a) +£' cos (24X3646) cos (24 X 364a) + .... +A" sin (24 X 364a) cos (24 X 364a) +£" sin (24X3646) cos (24 X 364a) + .... Summing n = 385 n=365 X) Vn cos 24(n— l)a=A' S cos 2 24 (ti— l)a n=l n=l n=365 +A" S sin 24(n-l)a cos 24(n-l)a n=l n=365 +B' 2 cos 24(n-l)6 cos 24(n-l)a n=l n = 365 + £" S sin 24 (n— 1)6 cos 24(n— l)a n = l n=365 + 0' Scos24(n-l)c cos24(n-l)a n=l n=365 + C" S sin 24(n-l)c cos 24(n-l)a n=l n=365 +Z>' Scos24(n-l)a 7 cos24(n-l)a n = l n = 365 +Z>" S sin 24(n- l)d cos 24(n— l)a n = l n=305 + E' Scos24(ti-1)€ cos 24(n-l)a n = l n=365 + £"' 2* sin 24(n-l)e cos 24(n-l)a (410) n = 1 which is the normal equation for the unknown quantity A f . 271. In a similar manner we have for the normal equation for the quantity A" 246007—41 7 94 U. S. COAST AND GEODETIC SURVEY 2 y n sin 24 (ti— l)ft =A' 2 cos 24 (n— l)a sin 24(n— l)a+A" 2 sin 2 24(w— l)a +5' 2 cos 24(n-l)6 sin 24(7i-l)ft+£" 2 sin 24 (ti- 1)6 sin 24(ti— l)a -f<7' 2 cos 24(ti-1)c sin 24 (n— l)a+C" 2 sin 24(n— l)c sin 24(n-l)a +#' 2 cos 24(w— l)d sin 24 (n— l)ft+D" 2 sin 24(n— l)d sin 24 (n— l)ft + £"2 cos24(n-l)e sin 24(n—l)a+E" 2 sin24(7i-l)e sin.24(n-l)ft (411) the limits of n being the same as before. Normal equations of forms similar to (410) and (411) are easily obtained for the other unknown quantities. 272. By changing the notation of formulas (265) to (267) the fol- lowing relations may be derived : n^36 5 sin 24na cos 24(ti— l)ft § cos 24 (»-D«=*»+i s^r — - ipoi I i sin 8760a cos 8736a - 182 ' + I sliTm ; (412) n ^f • 2«^/ ,\ i , sin 24na cos 24(?i— l)a S sin' 24(»-l)a=§*-i ^^ >- 10Ol , sin 8760a cos 8736a ,,,„. = 182 2 - ¥ ^^^ (413) n=365 2H cos 24(n—l)6 cos 24 (n— l)ft n=l _- sin 12ti(6— ft) cos 12 (n— 1)(6— a) -^ sin 12(6— a) . n sin \2n{b-\-a) cos 12 (n— l)(6-fft) + " 2 ' sin 12(6+a) ■t sin 4380(6— a) cos 4368(6— a) ~i sin 12(6— a) , i sin 4380 (6+ a) cos 4368(6+ft) , + * sin 12(6+a) (414) n=365 2 sm 24(n—l)6 sin 24(ti— l)ft n=l _! sin 12ti(6— ft) cos 12 (ti— 1)(6— a) ■** sin 12(6— a) i sin 12ti (6+ a) cos 12 (71— l)(6+ft) * sin 12 (6+ a) _i sin 4380(6— ft) cos 4368(6— ft) ~~ ¥ sinl2(6-ft) x si n 4380(6+ft) cos 4368(6+ft) ,. , ~ ¥ sin 12(6+ft) (415} HARMONIC ANALYSIS AND PREDICTION OF TIDES 95 n=365 2H sill 24 (n— 1)6 cos 24 (n— l)a n=l _i sin 12ti(6— a) sin 12 (ti— 1)(6— a) ~ 2 sin 12(6— a) i sin 12n(b+a) sin 12(n— l)(6 + a) + 2 sinl2(6+a) _i sin 4380(6 — a) sin 4368(6— a) ~ 2 sin 12(6— a) . i sin 4380 (6 + a) sin 4368 (6 + a) ' 2 sinl2(6+a) (416) (417a) 273. By substituting in (412) to (416) the numerical values of a, 6, etc., from table 2, the corresponding equivalents for these expressions are obtained. These, in turn, may be substituted in (410), (411), and similar equations for the other unknown quantities to obtain the 10 normal equations given below. In preparing these equations the symbols a, 6, c, d, and e are taken, respectively, as the speeds of constituents Mm, Mf, MSf, Sa, and Ssa. n=365 Sl/n cos 24 (71— \)a = 183.05,4 , + 0.72J9 , + 0.76<7 / +4.88Z) , + 4.96£: / +2.14A"+4.29i?" + 5.04"-0.70£"' n=365 2 ?/n sin 24 (ti— \)a =2.i4A'-4.i5#'-4.90C'+3.8oz? / +3.88 j e;' + 181.95A ,, + 1.01£ , ' + 1.06C , ' + 0.34 J D' , + 0.68£ W n=365 S?/n cos 24(71—1)6 = 0.72A' + 183.17£'-j-0.56<7'-1.50Z>'-1.51£" -4.15^" + 0.88#" + 0.92C"-0.09Z>"-0.18£"' n=365 2 2/n sin 24(71-1)6 =4.29A' + 0.88£' + 0.92C"-f3.05Z>'+3.06£" + 1.0lA" + 181.83£"-0.80C"-0.08#"-0.17.E" n=365 S Vn COS 24(71— \)C = 0.76A' + 0.56J9' + 183. 19(7'- 1.68Z?'- 1.7QE' -4.90^1" + 0.92£"+0.97<7"-0.1lZ>"-0.21 J E"' n=365 S#n sin 24(71— l)c = 5.044' + 0.92JB' + 0.97(7 / + 3.24Z)' + 3.25S' + 1.06^"-0.80£" + 181.81<7"-0.10Z)"-0.20£"' n=365 2 Vrx cos 24 (ti— \)d =4.88A'-1.50B'-l.e>8C' + 182.38D'-0.24E' -f3.80A"+3.05£" + 3.24C" + 0.00Z>" + 0.01£ T " n = 365 S^n sin 24(71-1)^ = -0.34A'— O^JS'-O.llO' + O.OOZJ' + O.OOS' + 0.34^"-0.08£"-0.10<7" + 182.62Z>"-}-0.00£" (417b) (417c) (417d) 96 U. S. COAST AND GEODETIC ' SURVEY n=365 XI Vn cos 24 (ti— l)e =4.96A'- 1.515'- 1.70(7'-0.24Z?' + 182.38#' +3.88A" + 3.06£" + 3.25<7" + 0.00Z>" + 0.00#" n=365 X)y n sin 24(71—1)6 1 = -0.70A'-0. 18B'-0.21<7'+0.01Z>' + 0.00i?' -f-0.68A ,/ -0.175 ,, -0.20(7 // + 0.00Z> /, + 182.62£: // (417e) 274. The numerical value of the first member of each of the above normal equations is obtained from the observations by taking the sum of the product of each daily mean by the cosine or sine of the angle indicated. The solution of the equations give the values of A' , A" , B f ', B" , etc., from which the corresponding values of quantities A and a, B and (3, etc., of formula (405) are readily obtained, since A^Vt^y+C^") 2 and a=tan- A' In calculating the corrected epoch, it must be kept in mind that the t in this reduction is referred to the 11.5 hour of the first day of series instead of the preceding midnight. 275. Before solving equations (417), if the daily means have not already been cleared of the effects of the short-period constituents, it will be necessary to apply corrections to the first member of each of these equations in order to make the clearances. The disturbance in a single daily mean due to the presence of a short-period constituent is represented by equation (398). Intro- ducing the subscript s to distinguish the symbols pertaining to the short-period constituents, the disturbance in the daily mean of the ft th day of series due to the presence of the short-period constituent A a may be written [y*]n= A s — — t — - cos {24:(n— l)a a + 11. 5a s +a a 3 sin |a 3 L v y The disturbances in the products of the daily means by cos 24 (ti— \)cl and sin 2A(n—l)a may therefore be written \y t ] n cos 24(n-l) a sin 12a, ~24 As sin ia a and [?/ 8 ] n sin24(n i [cos {24(n-l) (a 8 +a) + 11. 5a s +a s ] + cos {24(n—l) (a 8 — a) + 11.5a 8 + a g }] \)a (418) (419) "24 8 sin |a s i [sin {24(7i-l) (a B +a) + 11.5a 8 +a 8 } — sin {24(n—l) (a s — a)-f 11.5a s +a 8 }] (420) HARMONIC ANALYSIS AND PREDICTION OF TIDES 97 276. Then, referring to formulas (263) and (264) n=365 23 [yj n cos24(n-l)ft= n=l 1 A sin 12ft 8 Tsui 12X365 (ft B +ft) fiown^/ i \ i h c i i 48^-arWL S inl2( ffis +a) C0S {12X364(a.+«) + ll.o«.+ «.} + Sin tinS!-a7 ffi)cos {12X364(a.-a)+11.5a.+a B }] (421) and n = 365 S [y B ] n sin24(n— l)a= n=l 1 , sin 12a s Tsin 12X365 (a 8 + a) . r -.o W on,i/ i \ i 11 c i 1 48^1SW L sinl2(ft B +ft) ^ Bin {l2X3b4(a.+a) + ll.5a,+ a,} - Sl ^inS-a"7 a) sin {l2X364(a 8 -a) + ll.5a 8 +a 8 }] (422) Now let A' B =-A 8 cos a 8 and (423) A" 8 = — A s sin a 8 then (421) and (422) may be reduced as follows: n=365 2 [l/Jn cos 24 (n— l)a n=l l sin 12a 8 Tsin 12X365 (a s + a) f1 . VOfli , , x , -,-, c •> =T6—- — 1— 5 : — T7T7 — t-S — c °s {12X364(a B +a) + 11.5a B } 48 sin fa s [_ sin 12(ft s +ft) l v ' J , sin 12X365(a 8 — ft) r«r»w«/n / n i -.-• h ^"1 ,«/ + sinl2( as -a) C0S {12X364(a,-a) + 11.5a.}jA'. 1 sir^rsinl2X365( as +a) {12X864fe+a) + 11 ^ } 1 48 sin fa s |_ sin 12(a 8 +a) l v s J 8J sin 12X365(a s — a) . sin 12 (a B — ft) and n = 365 S [y s ]n sin 24(n— l)ft sin {12X364(a B -a) + 11.5a B } Ti" 8 (424) 1 sin 12a 8 r sin 12X365(a 3 +a) sin {12X364(a B +a) + H-5a B } 48 sin |a 8 [_ sm 12(ft 8 +ft) sin 12X365 (a B — a) . Mov/O ~w N , in K ."I*, sin 12(ft B -ft) sm {12X364(ft s -«) + 11.0ft s }J.4 . 1 sin 12ft s rsin 12X365 (ft s + ft) mowocj/ 1 \ 1 n e 1 ~48^1^ L sinl2K+a) C0S {12X364(a 8 +a)+11.5a 8 } - ^sinS-T^ C0S { 12 X364(« s -a) + ll.o« B }]^" 8 (425) 98 U. S. COAST AND GEODETIC SURVEY 277. Formulas (424) and (425) represent the clearances for any long-period constituent A due to any short-period constituent A B . The first must be subtracted from terms corresponding to 2y n cos 24(n—l)a and the latter from terms corresponding to Sy n sin 24(n— \)a of formula (417) before solving the latter. 278. In (424) and (425) the coefficients of A' s and A" s , which for brevity we may designate as C , C" , S', and S", respectively, contain only values that are constant for all series and may therefore be computed once for all. Separate sets of such coefficients must, however, be computed for the effect of each short-period constituent upon each long-period constituent. In the usual reductions in which the effects of 3 short-period constituents upon 5 long-period con- stituents are considered, 15 sets of 4 coefficients each, or 60 coefficients in all, are required. The coefficients are given in the following table : * Long-period constituents Mm Mf MSf Sa Ssa M 2 (C) -0. 0556 -0. 1704 -0. 1708 +0. 0441 -0. 0588 -0. 0776 -0.0206 +0. 1138 -0. 0648 -0.3476 -0.3452 +0. 0405 +0. 0030 -0.0377 +0. 0417 +0. 0105 +0. 0368 -0.2236 -0. 1526 -0.0854 +0. 0166 -0. 0778 +0. 0841 +0. 0338 +5. 739 -2.923 -2.840 -5. 727 +0. 0294 -0. 1938 -0. 1221 -0.0808 +0.0157 -0. 0816 +0. 0875 +0. 0331 -0. 1041 -0. 0752 -0. 0018 +0. 0048 -0. 0176 +0. 0025 +0. 0002 +0. 0001 -0. 1924 -0. 1826 -0.0046 +0. 0090 -0. 1046 (C") -0.0755 (S') . -0. 0035 (SO . +0. 0096 N2 (C) -0.0176 (CO +0. 0025 (SO +0. 0004 (SO +0. 0002 Oi (CO -0. 1934 (CO -0. 1831 (SO --- -0.0093 (SO- +0. 0180 In the above table the sign is so taken that the values are to be applied to the sums directly as indicated. 279. After the clearances have been applied and the normal equa- tions (417) solved and the resulting amplitude and epoch obtained for each of the long-period constituents, the reductions will be completed in accordance with the processes already outlined, but it must be kept in mind that in this reduction the initial value of t is taken to corre- spond to 11:30 a. m. on the first day of series. In obtaining the nu- merical values of such quantities as 2y n cos 24 (n — l)a and 2y n sin 24 (n— l)a, in order to avoid the labor of separate multiplications for each day, the following abbreviations have been proposed by the British authorities. The values of cos 24 (?i— \)a and of sin 24 (w— I) a are divided into 11 groups according as they fall nearest 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, or 1.0. The daily values are then dis- tributed into 11 corresponding groups, so that all values in one group will be multiplied by 0, another group by 0.1, etc. The cos 24(n— l)a and sin 24 (n— I) a include negative as well as positive values. The former are taken into account by changing the sign of the daily mean to which the negative values apply. 280. As a part of the routine reductions of the tidal records from the principal tide stations it is the practice of the office to obtain the mean sea level for each calendar month. It is therefore desirable to *From Scientific Papers by Sir George H. Darwin, Vol. I, p. 64. HARMONIC ANALYSTS AND PREDICTION OF TIDES 99 have a method of using these means directly in the analysis for the annual and semiannual constituents, thus avoiding any special sum- mation for the purpose. The period of the annual constituent is ap- proximately the length of the Julian year, that is, 365.25 days. If this period is divided into 12 equal groups and the mean of the hourly heights for each group taken, these means represent the approximate height of the combined annual and semiannual constituents for the middle of each group, and the middle of the first group will be the initial point from which the zeta (£) as obtained by the usual process is referred. As each group represents 30° of motion for the annual constituent, or 60° for the semiannual constituent, to refer this £ to the actual beginning of the series of observations it will be necessary to apply a correction of 15° for the annual constituent or 30° for the semiannual constituent. 281. In obtaining the monthly means by calendar months the year is divided only approximately into 12 equal groups. The following table shows the difference between the middle of each group repre- senting a calendar month and the middle of the corresponding group obtained by dividing the Julian year into 12 equal parts. It is to be noted that the hourly heights included in a monthly sum extend from hour on the first day of the month to the 23d hour on the last day. The middle of the group as reckoned from the beginning of the month will therefore be 13.98 days, 14.48 days, 14.98 days, or 15.48 days, respectively, according to whether the month has 28, 29, 30, or 31 days. Month Middle of group reckoned from beginning of year Differences Julian year Common year Leap year Common year Leap year January .. . . . . Days 15.22 45.66 76.09 106. 53 136. 97 167. 41 197. 84 228. 28 258. 72 289. 16 319. 59 350. 03 Days 15.48 44.98 74.48 104. 98 135. 48 165. 98 196. 48 227. 48 257. 98 288. 48 318. 98 349. 48 Days 15.48 45.48 75. 48 105. 98 136. 48 166. 98 197. 48 228. 48 258. 98 289. 48 319. 98 350. 48 Days +0. 26 -0.68 -1.61 -1.55 -1.49 -1.43 -1.36 -0.80 -0. 74 -0.68 -0.61 -0.55 Days +0.26 February .. - ... .. -0.18 March. .. -0.61 April .. May ... ... . _. -0.55 -.0. 49 June. ..-. ... ._ -0.43 July -0.36 August . +0.20 September. .. ...... ... +0.26 October .. ... . .. +0.32 November .. +0.39 December. .... ... +0.45 Sums -11. 24 -0.94 -0.74 Means.. -0.06 Speed of Sa constituent per day=0.9856°. Mean differences reduced to degrees of Sa_. . o -0.93 14.07 28.14 -0.06 Correction to f of Sa. .... . ... ... 14.94 Correction to f of Ssa. ... 29.88 282. From the above table it is evident that in the summation for the monthly means for a calendar year the middle of each group of a common year is on an average 0.93° earlier than the middle of the corresponding group when the Julian year is equally subdivided and the middle of each group of a leap year is on an average 0.06° earlier. Subtracting these values from 15°, the interval between the beginning of the observations and the middle of the first group of an equal subdivision, we have 14.07° and 14.94°, for common and leap years, respectively, as a correction to be applied to the f of Sa as 100 U. S. COAST AND GEODETIC SURVEY directly obtained, in order to refer the f to the hour of the 1st day of January. For Ssa the corrections will be twice as great. 283. If the year commences on the first day of any month other than January, the corrections will differ a little from the above. Calculated in a manner similar to that above, the following table gives the correction to be applied to the £ to refer to the first day of any month at which the series commences. The correction to the f of Ssa will be twice the tabular value for Sa. Observations commence — Correction to f of S a to refer to begin- ning of month Observations commence— Correction to f of S a to refer to begin- ning of month Common year Leap year Common year Leap year Jan. L___„_ o 14.07 13.50 15.89 15.31 15.72 15.15 14.94 14.45 15.93 15.43 15.93 15.43 July 1 o 15.56 14.98 14.41 14.82 14.24 14.65 o 15.93 Feb. 1 Aug. 1 _ _. . .__ 15.43 Mar. 1 Sept. 1.- 14.94 Apr. 1 Oct. 1 15.43 Nov. 1 14.94 Dec. 1 15.43 284. If the monthly means extend over many calendar years, it may be convenient to combine them for a single analysis. In this case the (V -\-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 f 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. 285. The augmenting factors to be used for constituents Sa and Ssa when derived from the monthly sea level values are based upon for- mula (404) in paragraph 266 and are as follows: Sa 1.0115, logarithm 0.00497. Ssa 1.0472, logarithm 0.02005. ANALYSIS OF HIGH AND LOW WATERS 286. The automatic tide gage, which furnishes a continuous record of the rise and fall of the tide, now being in general use, it is seldom necessary to rely only upon the high and low waters for an analysis. It may happen, however, that a record of high and low water observa- tions is available for a more or less isolated locality where it has been impractical to secure continuous records. Such records, if they in- clude all the high and low waters for a month or more may be utilized in determining approximate values of the principal harmonic con- stants, but the results are not as satisfactory as those obtained from an analysis of the hourly heights. 287. An elaborate mode of analysis of the high and low waters is contained in volume 1 of Scientific Papers, by Sir George H. Darwin. Other methods are given by Dr. R. A. Harris in his Manual of Tides. The process outlined below follows to some extent one of the methods of Doctor Harris, extending his treatment for the K and O to other constituents. HARMONIC ANALYSIS AND PREDICTION OF TIDES 101 288. The lengths of series may be taken the same as the lengths used as the analysis of the hourly heights (see par. 152). It is some- times convenient to divide a series, whatever its length, into periods of 29 days each. This permits a uniform method of procedure, and a comparison of the results from different series affords a check on the reliableness of the work. 289. The first process in this analysis consists in making the usual high and low water reductions, including the computation of the luni tidal intervals. Form 138 provides for this reduction. The times and heights of the high and low waters, together with the times of the moon's transits, are tabulated. For convenience the standard time of the place of observations may be used for the times of the high and low waters, and the Greenwich mean civil time of the moon's transits over the meridian of Greenwich may be used for the moon's transits. The interval between each transit and the following high and low water is then found, and the mean of all the high water intervals and the mean of all the low water intervals are then obtained separately. The true mean intervals between the time of the moon's transit over the local meridian and the time of the following high and low waters being desired, the means as directly obtained must be corrected to allow for any difference in the kind of time used for the transit of the moon and the time of the tides and also for the difference in time between the transit of the moon over the local meridian and the transit over the meridian to which the tabular values refer. 290. If the tide is of the semidiurnal type, the approximate ampli- tude and epoch for M 2 may be obtained directly from this high and low water reduction. On account of the presence of the other con- stituents the mean range from the high and low waters will always be a little larger than twice the amplitude of M 2 . If the data are available for some other station in the general locality, the ratio of the M 2 amplitude to the mean range of tide at that station may be used in finding the M 2 amplitude from the mean range of tide at the station for which the results are sought. If this ratio cannot be ob- tained for any station in the general locality, the empirical ratio of 0.47 may be used with fairly satisfactory results. After the ampli- tude of M 2 has been thus obtained, it should be corrected for the longitude of the moon's node by factor F from table 12. 291. The epoch of M 2 may be obtained from the corrected high and low water lunitidal intervals HWI, LWI by the following formula: M° 2 =|(£rW r /+XW r /)X28.984 + 90° (426) In the above formula HWI must be greater than LWI, 12.42 hours being added, if necessary, to the HWI as directly obtained from the high and low water reductions. 292. The difference between the average duration of rise and fall of the tide at any place, where the tide is of the semidiurnal type, de- pends largely upon the constituent M 4 . It is possible to obtain from the high and low waters a constituent with the speed of M 4 which, when used in the harmonic prediction of the tides, will cause the mean duration of rise and fall to be the same as that at the station. The effect of M 4 upon the mean duration of rise will depend chiefly upon the relation of its amplitude and epoch to the amplitude and epoch of the principal constituent M 2 . By assuming an M 4 with epoch 102 U. S. COAST AND GEODETIC SURVEY such as to make the constituent symmetrically situated in regard to the maxima and minima of M 2 , the amplitude necessary to account for the mean duration of rise of the tide may be readily calculated. 293. Let DR= duration of rise of tide in hours as obtained from the lunitidal intervals, a=Hourly speed of M 2 . = 28-.°984. M 2 = Amplitude of M 2 . M 2 °=Epoch of M 2 . M 4 = Amplitude of M 4 . M 4 °=Epochof M 4 . Then, for M 4 to be symmetrically situated with respect to the maxima and minima of M 2 M 4 °=2 M 2 °±90° (427) in which the upper or lower sign is to be used according to whether a(DR) is greater or less, respectively, than 180°. Multiples of 360° may be added or rejected to obtain the result as a positive angle less than 360°. The equations of the constituents M 2 and M 4 may be written 2/i =M 2 cos (at+a) (428) y 2 =M 4 cos (2a*+/3) (429) and the resultant curve y=M 2 cos (a#+«)+M 4 cos {2at+p) (430) 294. Values of t which will render (428) a maximum must satisfy the derived equation aM 2 sin (at+a)=0 (431) and for a maximum of (430) t must satisfy the derived equation aM 2 sin {at+ a) +2aM 4 sin (2at+ 0) = (432) For a maximum of (428) a in which n is any integer. a 295. Let - = the acceleration in the high waters of M 2 due to the a ■'■■'... presence of M 4 . With the M 4 wave symmetrically situated with n respect to the M 2 wave, - will also equal the retardation in the low water of M 2 , due to the presence of M 4 , and — will equal the total amount by which the duration of rise of the tide has been diminished by M 4 . If the duration of rise has been increased, 6 will be negative. Then, for a maximum of (430) t JinT-«-9 (434) a and this value of t must satisfy equation (432). HARMONIC ANALYSIS AND PREDICTION OF TIDES 103 296. Substituting in (432), we have aM 2 sin (2nir—0)+2aM A sin {4nir— 2a-f 0— 26) = ,,„k\ -aM 2 sin 0-2aM 4 sin (26+2a-(3) = ^ 60) But 2a-/3=-2M 2 + M 4 (436) From (427) -2M 2 ° + M 4 °=±90 o 180° according to whether the duration of rise is greater or less than > or whether is negative or positive. Then 2a-/3=T90° (437) according to whether is positive or negative. Substituting this in (435) — aM 2 sin 0±2aM A cos 20=0 (438) and M 4== sinfl M 2 ±2 cos 26 ^ 63) the upper or lower sign being used according to whether 6 is positive or negative. As under the assumed conditions 6 must come within the limits ±45°, the ratio of sr^ as derived from (439) will always be positive. 297. The duration of rise of tide due solely to the constituent M 2 is 180° a The duration of rise as modified by the presence of the assumed M 4 is DR =, X -^-- 2 l (440) Therefore 0=f(18O o -aZ?jB) (441) Substituting the above in (439) Ave have M 2 ± 2 cos (lS0°-aDR) 2 cos oDR (44 ^ j TV T _,_ , COS \(lDR , , /^ON ^"^ ^coBataB Ma (443) and M 4 must be positive, and the sign of the above coefficient will depend upon whether aDR is less or greater than 180°. 298. The approximate constants for S 2 , N 2 , K 1} and Oi may be obtained from the observed high and low waters as follows: Add to each low-water height the mean range of tide. Copy the high and modified low water heights into the form for hourly heights (form 362) , always putting the values upon the nearest solar hour. Sum for the desired constituents, using the same stencils as are used for the regular 104 IT. S. COAST AND GEODETIC SURVEY analysis of the hourly heights. Account should be taken of the num- ber of items entering into each sum and the mean for each constituent hour obtained. The 24 hourly means for each constituent are then to be analyzed in the usual manner. 299. The results obtained by this process are, of course, not as dependable as those obtained from a continuous record of hourly heights. The approximate results first obtained can, however, be im- proved by the following treatment if a tide-computing machine is available. Using the approximate constants as determined above for the principal constituents and inferred values for smaller constituents, set the machine for the beginning of the period of observations and find the predicted heights corresponding to the observed times of the high and low waters. Tabulate the differences between the observed and predicted heights for these times, using the hourly height form and entering the values according to the nearest solar hour. These differences are then to be summed and analyzed the same as the original observed heights. In this analysis of the residuals the con- stituent M 2 should be included. The results from the analysis of the residuals are then combined with the constants used for the setting of the predicting machine. 300. In making the combinations the following formulas may be used: Let A' and k' represent the first approximate values of the constants of any constituent. A" and k" , the constants as obtained from the residuals. A and k, the resultant constants sought. Then A=J(A' cos *' + A" cos K r/ ) 2 + {A! sin K '+A" sin *") 2 v444) and , A f sin k'+A" sin k" , AMr . K = tfm A'cosk'+A'cosk" (445) FORMS USED FOR ANALYSIS OF TIDES 301. Forms used by the Coast and Geodetic Survey for the harmonic analysis of tide observations are shown in figures 9 to 19. A series of tide observations at Morro, California, covering the period February 13 to July 25, 1919, is taken as an example to illustrate the detail of the work. 302. Form 362, Hourly heights (fig. 9).— The hourly heights of the tide are first tabulated in form 362. Although the zero of the tide staff is usually taken as the height datum, any other fixed plane will serve this purpose. For practical convenience it is desirable that the datum be low enough to avoid negative tabulations but not so low as to cause the readings to be inconveniently large for summing. 303. The hours refer to mean solar time, which may be either local or standard, astronomical or civil, but standard civil time will generally be the most convenient to use. The series must commence with the zero (0) hour of the adopted time, and all vacancies in the record should be filled by interpolated values in order that each hour of the series may be represented by a tabulated height. It is the general practice to use brackets with interpolated values to distinguish them from the observed heights. The record for successive days of the series must be entered in successive columns of the form, and these HARMONIC ANALYSIS AND PREDICTION OF TTDES 105 columns are to be numbered consecutively, beginning with one (1) for the first day of the series. 304. The series analyzed should be one of the lengths indicated in paragraph 152. Series of observations very nearly equal to one of these standard lengths may be completed by the use of extrapolated hourly heights. If the observations cover a period of several years, the analysis for each year may be made separately, a comparison of the results affording an excellent check on the work. 305. The hourly heights on each page of form 362 are first summed horizontally and vertically. The total of the vertical sums must equal the total of the horizontal sums, and this page sum is entered in the lower right-hand corner of the page. 1 j Form 3C3 | D \*! R :o^o??oo?Z"^ CE TIDES: HOURLY HEIGHTS 1 | Stat I] Chio Morro, California. v cnr . 1919. r«fP«rf V . E. B. Latham. T..t 35° 22' B. T™<,120° 51* W. 120 tf. T de Gauge No. 107 Scale 1:9 Reduced to Stal T n—in 1 i Month 1 snd mo. d. Fab. 13 d. 14 d. 15 d. 16 d. 17 18 d. 19 Hori- zontal Sum. ! Day of : Series. i flour. 1 2 3 4 5 6 7 Feel. Feet. Feet. Feet. Feet Fett. Feet. Feet. 3 9 4.2 4.6 4.5 4.4 4.7 4.6 30.9 1 3.4 3.8 4.2 4 2 4.2 4.9 4.8 29.5 2 30 33 3.5 37 3.8 4.6 4.9 26.8 3 2.8 30 3.0 31 33 4.1 4.5 23.8 4 3 2.8 2.6 2.5 2.7 3.5 3.8 20.9 5 3. 6 3.1 2.5 2.2 2.2 3.0 3.2 19.8 C 4.4 3.6 2.8 2.2 1.9 2.6 2.7 20.2 7 5 1 4-5 3-5 2-6 2 2.5 23 22. 6 8 & 7 5.3 4.3 3.3 2.4 27 2.2 25.9 6. 6.0 4.9 4.1 3.1 3.1 2.4 29.6 10 & 6 6-2 5-4 4-6 3-9 3 6 2-8 32 1 11 46 5.6 5.5 4.9 4.3 4.1 3.2 32.6 Noon. a 9 51 5.1 4.6 4.4 4.5 3.6 31.4 13 | 34 4.3 4.4 4.3 4.2 4.5 3.8 28.9 M 2 6 3.4 3.5 3.6 3.7 4.3 3.8 24.9 ! 15 i X 9 2 6 2.8 2.9 3.1 3.8 3.6 20.7 1G 1.2 2.0 2.2 2.2 2.6 3.2 3.2 16.6 17 1 1.6 1.7 1.6 2.1 2.7 2.8 IS. 5 18 L 3 1.6 1.5 1.3 1.9 2.4 2.5 12.5 10 2 3 2. 2 1.8 1.4 1.9 2.3 2.3 14.2 20 3 2 3. 1 2.6 2.0 2.3 2.5 2.4 18.1 21 4 3.9 3.4 2.8 3.0 3.0 2.9 23.0 22 4 3 4.5 4.1 3.6 3.6 3.6 3.7 27.6 23 45 4.7 4.5 4.1 4.4 4.2 4.* 30.6 | Sum. 84. 9 90. 6 64.4 76.5| 75.6 64.4 60 2 676,6 1 Sum for 29 days, 1 to 29 of = Divisor=696; mean for 29 daye = Figure 106 U. S. COAST AND GEODETIC SURVEY 306. Stencils (figs. 10 and 11). — The first figure is a copy of the M stencil for the even hours of the first 7 days of the series, and the second figure illustrates the application of the same. This stencil being laid over the page of hourly heights shown in figure 9, the heights applying to each of the even constituent hours for this page show through the openings in the stencil, where they appear con- nected by diagonal lines, thus indicating each group to be summed. 307. For each constituent summation, excepting for S, there are provided two stencils for each page of tabulated hourly heights, one for the even constituent hours and the other for the odd constituent hours. ""RsSs&SSF" TIDES: HOURLY HEIGHTS PtAtWi; St«n«ll fftr «nmrrjr»nmant If. \ 'ear: Chief of Party T.or. T Ti, le Gauge No. Scale 1: Reduced to Staf T 11— TM Month and Day. mo. d. d. d. d. d. & d. Hon zonta Sum J Day of 1 Seriea. j| i \ 2 3 4 5 6 7 Hour. <] Feci. | Feet, j reel. 1 ^2 ftrf. | Feet. feet. 1 { .Fe«r. I Feet H^BLgol ■ ^l Hpl. ^^ P 2 ( HBLs sL t , "W^j 1 1 3 4 ( ^H^^ 1 -"4^^^ . ^^P-sJ | ; 5 1 C ( kj ■ ^^■■^2 > Jl 1 |l I . I " *»'. _ j 8 \ WKLe — j , ■ i ^l^> il \ .• TH— •~, rt 1 ► \ 1 ■ f'^JL i^niTfc 11 ■ vB ""J • 1 * «3 i ( • |1 ! 1 13 !»■_ T^r- ^j 1 HR-^ b tr 13r.0._ Lot Yr. Jfc. Do. Br. Kind of time used: 11—6(7 12Q?..AS ... Computed by 2xA&*..A*.Kxiwxil]L,..I>ta*9+..132Q* Omit, e^,. O' i 2 3 * 6 « 7 e 9 to II 1 24.3 20.6 17.9 16.9 21.0 23.0 28.0 31.9 39.2 34.8 31.9 27.4 2 21.8 17.5 14.4 13.6 11.2 12.3 14.6 20.5 21.7 23.9 24.7 24.3 3 19.7 16.9 11.0 9.6 12.2 17.7 26.1 24.9 27.4 27.6 29.8 21.1 4 26.4 18.0 17.3 17.3 22.7 22.6 26.0 34.3 36.5 41.2 33.3 28.6 5 21.5 21.4 17.9 18.2 16.3 19.9 24.9 29.9 37.7 34.8 33.2 29.6 6 20.3 16.8 15.8 12.1 12.5 15.1 21.0 21.4 23.6 24.8 26.0 28.1 7 23.1 16.1 13.3 13.1 15.6 23.7 28.6 33.9 30.1 34.9 27.6 23.6 8 25.5 23.0 21.4 21.6 20.8 23.5 27.2 29.7 43.5 36.4 32.6 27.6 9 20.9 18.5 16.2 12.1 11.3 13.8 18.1 26.3 26.8 28.6 28.0 29.6 10 16.9 13.2 10.2 8.7 11.5 15.5 18.3 21.6 24.4 25.3 28.7 24.3 11 18.6 15.0 12.5 15.7 17.. 2 23.2 29.5 41.1 36,7 34.4 27.8 24.0 12 24.5 25.5. 20.4 20.7 21.0 24.6 32.1 31.7 32.7 36.5 31.6 25.6 13 25.7 17.3 13.2 10.0 11.9 12.5 16.2 20.3 24.3 30.2 30.3 24.4 14 16.7 12.6 8.3 8.7 9.5 14.3 19.0 •23.4 30.2 27.4 27.2 26.2 15 19.0 16.1 16.3 15.7 20.1 26.5 37.7 37.7 40.2 39.3 35.6 29.6 16 29.6 22.8 21.6 22.5 25.7 31.7 31.9 34.9 35.8 38.1 28.3 27.3 17 22.9 18.6 14.5 11.1 10.5 12.3 15.3 19.0 25.4 24.5 24.9 23.6 18 15.4 10.0 6.2 3.2 4.9 10.2 16.0 24.2 24.6 25.1 25.4 27.6 19 16.7 15.4 13.1 15.3 19.8 29.4 31.8 35.8 38.3 37.9 38.7 28.3 20 27.6 21.0 19.8 20.4 28.1 29.9 31.5 36.1 36.4 39.9 28.9 22.8 COM. 437.1 356.3 300.3 286.5 323.8 401.7 495.8 578.5 635.4 645.6 593.5 523.4 P*.. ia ia i* IB te 17 18 10 20 21 22 23 1 22.7 21.1 17.5 13.5 14.5 17.5 17.9 26.2 26.2 27.7 26.9 28.0 576.6 2 25.8 17.6 17.4 18.9 21.5 28.8 32.1 .35.1 36.5 35.9 35.8 26.6 552.5 3 17.5 17.5 14.7 15.4 21.1 23.5 29.2 33,3 35.3 39.6 30.0 24.8 547.8 4 23.2 20.8 12.9 9.0 7.1 7.9 14.0 20.8 22.2 24.9 25.8 25.3 638.0 5 27.5 20.2 16.9 15.5 16.4 23.8 25.3 30.3 27.9 29.0 34.0 25.4 897.6 6 23.8 23.0 20.0 24.2 23.1 25.2 27.5 28.8 39.4 38.2 28.7 24.4 562.8 7 19.5 15.5 17.8 14.6 15.'5 20.2 29.8 31.4 33.7 33.1 30.6 29.0 674.3 8 22.4 19.4 12.3 8.8 7.4 10.7 14.9 19.5 24.1 26.6 32.0 27.3 558.1 9 22.1 18.1 15.4 14.0 17.1 19.1 24.3 29.3 38.0 29.0 28.0 24.6 528.2 10 23.2 22.6 26.3 22.0 24.4 26.9 28.7 33.8 28.1 30.7 26.4 24.6 536.1 11 19.4 18.2 11.8 12.5 13.1 16.8 25.3 26.6 29.1 29.0 30.4 22.9 550.8 12 21.5 13.3 8.4 4.6 4.4 8.7 15.0 21.0 25.5 31.9 28.1 27.0 536.3 13 21.7 18.8 16.5 17.6 17 .'6 21.6 26.9 36.1 35.5 36.4 30.2 27.0 542.2 14 28.6 23.6 27.4 26.4 26.0 28.8 36.8 32.5 31.5 28.2 28.0 25.0 566.3 15 20.3 16.4 12.9 10.6 14.6 21.5 22.0 25.7 27.8 31.1 25.4 22.6 564.6 16 20.9 14.9 10.9 6.1 4.8 9.9 16.9 23.2 32.1 30.8 30.6 28.9 580.2 17 21.3 22.2 17.1 16.8 17.9 26.7 27.6 36.0 33.5 36.5 35.4 32.1 545.7 18 22.2 21.1 21.5 23.6 31.2 31.8 30.4 31.7 31.3 28.2 27.6 21.1 514.4 19 19.6 15.5 12.5 11.3 10.0 12.2 19.9 26.1 28.6 24.1 22.5 19.7 542.6 20 20.4 15.2 7.8 3.6 3.0 7*2 17.1 22.2 27.4 29.8 34.3 27.7 558.1 Rube ^3.6 275.0 316.0 288.9 310.7 388.8 ■.81.6 S69.6 613.7 620.6 590.7 513.9 L1093.0 Figure 12. HARMONIC ANALYSIS AND PREDICTION OF TIDES 109 moon ($), of the lunar perigee (p), of the sun (h), of the solar perigee (pi), and of the moon's ascending node (N), may be obtained from table 4 for the beginning of any year between 1800 and 2000. The values for any year beyond these limits may be readily obtained by taking into account the rate of change in these elements as given in table 1. The corrections necessary in order to refer the elements to any desired month, day, and hour are given in table 5. As the tables refer to Greenwich mean civil time, the argument used in entering them should refer also to this kind of time, and in the lines for the beginning and middle of the series at the head of the form space is therefore provided for entering the equivalent Greenwich hour. Any change in the day may be avoided by using a negative Greenwich hour when necessary. For example,- 1922, January 1, hour, in the standard time of the meridian 15° east of Greenwich, may be written as 1922, January 1,-1 hour in Greenwich time, instead of 1921, December 31, 23 hour, as would otherwise be necessary. If a negative argument is used in table 5, the corresponding tabular value must be taken with its sign reversed. For the middle of the series the nearest integral hour is sufficient. 310. The values of /, v, £, v' ', and 2/' are obtained for the middle of the series from table 6, using N as the argument. If N is between 180° and 360°, each of the last four quantities will be negative, but I Form in ^ESKSSKSHF 1 TIDES: STENCIL SUMS. Station: Morro,..Cali£flrnla._ Lat. : _.3&?..J22!Jff« Component: .'^.....Length o{ series: J-A 3 ... Series begins: 1919 - Peb.-IS-O T,m e .180 O 51* ff« Jfep. Yt lio. Da. Br. -JL20°I*„....„ _ Computed by tod-AJttinnslIj. Deo.9.1920. «"■*». o» 1 a 3 * e e 7 8 , e IO 11 21 25.9 18.1 14.8 14.5 10.6 11.1 17.3 23.8 23.1 24.4 24.1 22.6 22 16.8 14.6 7.7 5.7 6*6 11.1 19.5 23.2 26.5 27.6 30.8 24.9 23 17.8 15.7 15.1 20.1 21.6 30.7 33.3 37.3 39.0 42.8 33.9 28.4 24 7.2 6.8 6.2 6.1 6.5 8.0 9.7 10.9 18.3 12.1 11.0 9.4 Suma-21-24 67.7 55.4 43.8 46.4 45.3 60.9 79.8 95.2 106.9 106.9 99.8 85.3 - 1-20 437,; 356.3 300.3 286,5 323,8 401,7 495,8 578.5 635.4 645.6 593.5 523.4 Sums.- 504.8 411.7 344.1 332.9 369.1 462.6 575.6 673.7 742.3 752.5 693.3 608.7 DlTlsor*.- 164 163 162 165 164 163 163 163 164 165 163 162 Moans.- 3.08 2.53 2.12 2.02 2.25 2.84 3.53 4.13 4.53 4.56 4.25 3.76 c »Mfc ia» IS 14 is is 17 18 is ao at aa as 21 22 23 24 23.3 22.5 23.1 3.3 18.2 20.7 16.3 4.7 17.0 20.2 15.5 3.0 17.3 23.3 26.0 26.9 11.6 11.9 1.7 0.9 24.0 31.7 13.7 0.9 23.7 36.2 19.6 1.7 32.9 35.9 34.0 40.0 25.1 26.6 3.4 5.5 42.1 31.3 26.2 7.0 34.7 26.2 24.0 7.7 31.1 20.5 24.5 7.8 558.8 551.4 573.8 159. fl Sums 21-24 72.2 n 1-20 443.6 59.9 375.0 55.7 518.0 56.6 63.0 2&8.9 310.7 70.3 388.8 86.2 ■481.6 95.4 108.0 569.6 613.7 106.6 620.6 92.6 590.7 83.9 513.9 1843.8 11093.0 Suma.- Dlvisors.- lieana.- 515.8 162 3.18 434.9 163 2.67 373.7 163 2.29 345.5 373.7 459.1 567.8 665.0 721.7 163 162 162 163 163 162 2.12 2.31 2.83 3.48 4.08 4.45 Figure 13. 727.2 683.3 597.8 12936.8 162 163 163 4.49 4.19 3.67 246037—41- 110 U. S. COAST AND GEODETIC STJRVEY is always positive. Although table 6 is computed for the epoch, January 1, 1900, it is applicable without material error for any series of observations. 311. The values of u of L 2 and u of M 1} may be obtained from table 13 for any date between 1900 and 2000, inclusive, using the value of N for interpolation. If the series falls beyond the limits of this table, the following formulas may be used: u of ~L 2 =2£-2v-R (par. 129) (446) u of M^S-H-Q (par. 123). (447) 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. Form :«. w.»r*.r-o rjtXPARTMENT OF COMMERCE. TIDES: V. ■■ COAIT /.HO ttOD.TlO ftUMVKT. S'atitm florro » California Computation of F f-u. 120" 51' W. L5 °...2.2 ' . N. jr^. . 120.85 W,„. ,__ of una 163.*.. Time mer. .120., .Q0._W._3 5 ilyais and Prediction of the Tide. VT. mo. d. hr. nOricnwk),hr.\ Btginniw, of teries. 1919 Feb.. 13 p V ... .8 ) Lcngth : V. mo. i. hr. i\Crttnu-ic'khr\ Middle c/scric, 1919 May. 5 12 1. ...20 / Compute all values to two decimal places. Table* in Harmonic An For the beginning of Beries. For the middle of series. (1)=* (2)-5Tp (3)=n 1 (4)=p, (5)=p (t)=Ar. Tabic 4, to: January 1 or year TiUc 5, correction to 1st of month 2_58____04„| 27.; 41., . 48-47 | .. 3-^5 n 158-12 i .1-34 . 279-60.1 281-55 1 27.-.41- 251.-.21.. i_ 3Q.-.SSJ fl.--.oa. II.- 83 ! - on 13..52..I -Bs.ZEl. n-4p 1 -n- J>l Tabic S, correction to anvnwich hr 4-39 1 n.nd a 7.7 ! »-«* n-nq I _n-n4 .<.,_ 119.02 l (2 - 32,24 i (3) _ 322.32 1 , 4 ,__ 281.55 m . 41.32 i (C)= 245.11 (7)-J(Tab!cG) -21.\7-6_._ (8)- u (Tabic 6) ~....t1Z.:.Q3... (9)~t(TablcG) = ...--ll.\7_l.. (10)-.' (Tabic C) - -a.*-53... (ll)-2v"(Table6) -.lfi.'.SS.. (l2)-P-(I)-(9) - 5S.-.Q3.. (13)-uofL,±lS0-. 170,2.... (Table 13) T01.C. (14)=u of Ki+90*-. — JUiii..n.... (Table 13) (15)-S»-L» . -D."~8JL + (is)..... 138,25.. _<*>- 12 -P8 v.^-2(M-.)= 14Q-52. v.+„=2(M.)= 1.8.7.. 36 wiiw-iMsjr. ! (MK), + Mt . 46.84 .««_ -1.70 + K. _ 60. 00 v.+u- 150. -93.. v.+u = .2.19.07 v.+u- ...106..84. K, + d7)- .....51-47 + Ml .... . 46 .84 S, V.+u-(15)±180* 179.15 (2MK), + M.- 93 - 68 -do)- 8-53 -(!$-_ -86 -78 _ ,r, _ -60.00 v.+u - CO..-0.0.. -39 .94 v.+u-(33) =..558. 30 v+«- 33.68 3.2C 06 . + (20)- 28.2;.9.4. -fin- 16-63 (2N)_ . + N,..- 320-06 V.+u-2(33)-..35.6-..6.a. (MN), + Ml - 46-84 + n. _ 520-06 (17>-(I6)+W ■ .5.1;. 47.. (18)-(l)-(2) . £&-7JL (i9)-(i7)+(is)- 1.38.-.2_5._ (2o)-2(ic>....- 28.2_._94.. (2i)-(i;+(2) ..- 151.:2S~ (22)-(»)-(2i) 132:..6.8.. (23)- (i6)-») - 2.02....45.. (34)-(9)-(S)..~ 97.. (25)-m)+(2*) - 2_Q3« 4.?.. (26)-2(2o)....- .46-.8.4_ on)- (25)+ (26)- .25_Q__2.___ v.+u- J2.9.9.-.57... 233-28 v.+u-3 <33)- -.3.5.4 • .90 . v.+u- 366;.90.. - 6.-.90. L 2 + (2)- 13JL-.fia. + fis>_ 170- fi 0, 334.15 T_ + (33)- 358-30 (MS) 4 + m, - 46. 84 -351.46 _«*_ -40-77 ,.*__ 358.30 v.+u - 302. 28.. v.+u- -17.--3JL. .!..._ 342,69.. v.+u- 317,.53. r+u- 405.14 - 45- .14 M, + (a *. . 202« 45 .i.™- 123-6 + (31)-. + (31)-_ t'.+u - . (00), 334-15 (36)±i8o-- . 180.24. (2SM) 2 + s»..- 3.5.6.-. 60. _«.- -46.84 351-46 -y+u- 326.05 685-61 v.+u..- S3.-.46. v.+u- 309.. 76 . 325-61 r.+u-+(26)-..46.-.fi4.. P, (iw+sto* -...269 .15. + m,..- 46...84. 4-fM). 46-60 Mf V.+u-+(29)-261;46. -(3) -_ V.+u -_ -322.32 MSf + s..- ...3.5.8-..30. _ M . _ -46- 84 v.+u- 311-..46L. (29)-2f» -. 2fiLJdL -53-17 v.+u- 9.3.-44. 306.83 (31)-(16)-(8) - JIM^JUj.. (32)-(3)-(4)..- .4Q.-.77.. (33)-2(U) - -Jl-\-7-Q— <34)-(3)-(l) ..- .?.9.?.'..?P.. (3i)-2(3J) - .4.6'. 60.. O ?A M, V.+U-(27)±1S0- - ZO.-.2J5.. Q. + o,..- .3.42.-.£2.. _n m - -R6-78 v.+u - 255.-.91.. "2 „....- ,...9.3.-.44. + _ 86-78 Mm v.+u- +- ..322. 32 (37)-(3)-(2>..- 290.-.08. (3S)-2<37) - 220... 16. V.+u-. 169.13 v.+«- 47.6 ;07„ -. 116,07. Ssa v.+u- +2 (3) -2.8.4.. 8.4. t Oroenwich hour- original hour+(£ •+15). FlGL • Positive for Wcsl RE 14. onsitudc; negative fo toit Ion nude. HARMONIC ANALYSIS AND PREDICTION OF TIDES 111 312. In finding the difference between the longitude of the time meridian (S) and the longitude of the place (L) consider west longitude as positive and east longitude as negative. In the ordinary use of form 244 it is assumed that civil time has been used in the tabulations of the observations. If, however, the original hourly heights as tabulated in form 362 are in accordance with astronomical time in which the hour represents the noon of the corresponding civil day and the 12th hour the following midnight, form 244 will still be applicable if the longitude of the time meridian (S) is taken equal to the civil time meridian plus 180°. For example, if tabulations have been made in astronomical time for a locality where the civil time is based upon the meridian 15° E., the value for S should be taken as — 15 + 180, or 165°. If tabulations have been in Greenwich astronomical time, S should be taken as 180°. 313. Form 244a, Log F and arguments for elimination (fig. 15). — ■ Items (1) to (11) are compiled here for convenience of reference for Form 844 a DEPARTMENT OF COMMERCE U. S. COAST AND CEOOETIC SURVEY TIDES: Log F and Arguments for Elimination Station MorrQ^.-Califarnia. Length of series 16.5. days. Series .19.19 Feb....l3 0„„ Component Log F Component Log F Component LogF <4dec) (4 dec.) (4 dec.) Ji ...0,Q2pl .0.0160 ...P.:P472 M 8 Nj, 2N .... O, ...9:?726 ....9-.9.932 ...0,0264 MK 2MK MN ...0..3Q.91 K, ...0..GQ23 K ...9..-.986S L,=Log F(M 2 ) + (7) 9.9589 00 p. ...0.-.Q92.9. 0.0000 MS, 2SM . . . Mf ..9.:9.932 M,=Log F (0,)+(8) 9-8856 .0.0596 M, ...... . ...9.-99S2 Qi,2Q . ... . ...00264 MSf 9 3932 M, ...?.,?897 1*2, S„ &,, S„ S,,T, 0.0000 Mm ..9.5772 M, ...9-..9863 *2 W "J . . . . ...9.-.9932 Sa, Ssa . . . . 0.0000 M. JteSBM " l ■ ..P!..0264 (I) = JV T =item (6) from Form 244 = ...245... 11... (2 dec.) <2) = /=item (7) from Form 244 = 21..v7.6 (2 dec.) (3)=P=item (12) from Form 244= ...53_.,03 (2 dec.) (4) = (&-£.>')= item (3)-* item (10), from Form 244 = 327* (0 dec.) {5) = (h -r")= item (3)-}item (11), from Form 244 = 351 (0 dec.) (6) = (fc-p,) =item (3) -item (4), from Form 244 = 41 (0 dec.) (7) =Log /?. from Table 7 = .9.. 9.6.5.7. (4 dec.) (8) =Log Q. from Table 9 = .9...8.5.9.2. (4 dec.) (9) = Natural number from Log F(K.) =...i..0.3.8. (3 dec.) (10) =Log/(K 2 ) = 10 -Log F(K 2 ) = .9...9_528. (4 dec.) (II) = Natural number /(K 2 ) from (10) = ...Q»8.97. (3 dec.) Explanation.— For all tables sec Special Publication No. 98. First fill in items (1) to (8). Then obtain values of log F for all components excepting L 2 and M, from Table 12. Log ^(Lj) =log F(M,) + log #., and log F(M,) =log F(0,) +log <)». Items (9) to (11) are obtained after the rest of the form has been filled out. Figure 15. 112 U. S. COAST AND GEODETIC StJRVEY this and form 452. Items (1) to (6) are obtained from values given in form 244. Item (7) is obtained from table 7, using items (2) and (3) as arguments, and item (8) is obtained from table 9, using item (3) as argument. Items (9) to (11) are obtained after the rest of the form has been filled out. 314. The log i^for each of the listed constituents, except L 2 and M t and those for which the logarithm is given as zero, may be obtained from table 12, using item (2) as the argument. For constituents L 2 and M, Log F(L 2 ) =log ^(M 2 )+item (7) (448) Log F(M!)=log F(Ot) +item (8) (449) If the tidal series analyzed was observed between the years 1900 and 2000, the log F(L 2 ) and log ^(Mi) may be taken directly from Form 194 Department of Commerce coast and geodetic survey TIDES: HARMONIC ANALYSIS Horro, California Lai. _8B!_82UL Series begins™}?..-??* * „ Length of series i§?..„. Time Mcr. ..1? (Davs) a" Hourly Means from Form 142. (1) Hours to 11 (2) IIoursl2to23 ...3:08 3.18 ..?..:53. 2 .67 .2.12. 2 29 ...2.-.02 2. 12 ...2.-.25. 2.31 ...2_-84. 2.83 ...3..-53.. 3 .48 ...4-..1.3 .4. 08 ...4.-.5.S. 4.45 ...4:f.6. 4.49 ....4.2.5 4 19 ...3.-..7.6 3. 67 -0.10 -0.14 -0 .17 -0.10 -0.06 0.01 .05 0. 05 0.08 0.07 | Q 06 0. 09 (4) = Last5valuesof (3) reversed 0.00 0.09 06 0.07 0.08 0.05 0.00 (S) = (l)+(2)~ 6.26 5.20 4 41 4. 14 .4.56 5.67 7 .01 8. 21| 8.98 | 9.05 I 8. 44J 7. 43' (Si-Last half of (5)_. 7.01 8.21 8 -.98 9. 05 8.44 7.43 -0.10 -0. 05 -6". ii -0. 03 0. 02 0.06 "6.05 -0 -013 -6.055 -0-23 -6.23 -0.222 -6 .199 -P.:021 ...0.-P17 ...Q.-.Q58 6.050 -0-17 -0 .120 -0- 035 -6- lie -P.-..Q21 -0;04 . .-0 ,010 0.05 o.o -P-.P42- -0. 05q .ooo -0. 10C .ooo -P.-.i_63 .500 0.0 00 .S6C +P.-.12C i.ow +Q:.iM .860 .±0- .P.2.8 5oo -0.75 -3. 6i -4- 57 -4- ;91 -3- 88 -1. 76 -1.50: -3.958| -i-..?li •_o_'_ 36 Q __ . 500 -67 " +1.52^ ' o o +0.036 -0 .721 ,-..:3-.?5e|.fc,- G025 12.,- -14.612 -T~I7S >2c„ <33)x(3r,) I2e, -3.010 0.00 ...AM 0.00 -1.7601 rO750 0.000 4570 13-27 1.3-..4i 13.39 13.-.Q0 13.10 0-08 6.ii. 0.29 0355 0251 0080 0205 -0145 26.46 26-41 26.49 2287] -zi 9« 26.460 13205 -izzis 0140 *. ..-fl.9.® 0.0T0 i2..= ...ouc -0060 t quadrant when wc have +8 and ■ Component (J8) -log. 12. (39)=log.l2c («0)=(38)-(39)=log. tan. f ........ (41)=f , for beginning of the series.. (42)=local V.+u (From Form 244). (43) = (< 8 -55630. 9.85794 8.69836 1 16474 9. 41162 O 33806 & 39794 177-14 326.05 143 .19 261 -52 46 .84 308 .36 1 01368 9. 78247 9714613" 6. 63634 9. 14615, 8- 77815 6.36798 8:83885. 8.:P6666. 6.83885 275 53 " 70 26 345 .79 76. 99 """93. 68 17067 113.20 .140. 52 253. 72 278-25 ..187.:36 105 . 61 .,.(-,» (44),=log.: (45)=log. cos. r (46)-(3S)-(«),or(39)-(45).. (47)=log. (augmenting factor-M2) (48)=Iog. (reciprocal of 12) for component S . <49) = (46)+(47j,or(40)+tfS)=log.JS' (SO) = log. factor F(From Form 241a) (51)-.(49) + (50) = lo S . //" (52)=natural nuuiberfrom {4 , j)=Ii f fSJ) = nalnral number from (.il)- IV 9 99523 9. 99797 ! 9. 98871 9. 96338 9 . 99548 .•99946. 9:85848. .1.16.951. .9:. 41.365.. 9-. .7.9.3.76.. 9 2 2 C 8.9320 8,78054.. 9.-8856... Q 09530 8 .34569, 8.-73461 9. 9932 9.9897 J8 .-666.14. I 6 060 I 6 046 .Q.0B85 ..8.-33539 1-245 022 1.226 I "6.022 9- 9863 .S.-.72Q93L.. ...0.:0.54.... 0.053 8.-.14917. 9. -.9.7 9.4... 7. 9-9726 8.:.128.57. ...Q.:Q14... 0.013 7 -.8 19.2.9. ..P..0P7.... 0.007 Figure 16. HARMONIC ANALYSIS AND PREDICTION OF TIDES 113 table 13, using the year of observations, together with item (1), as argument. 315. Form 194, Harmonic analysis (fig. 16). — This form is based, primarily, upon formulas (295), (296), (303), and (304) and is designed for the computations of the first approximate values of the epochs (k) and the amplitudes (H) of the harmonic constants. Provisions are made for obtaining the diurnal, semidiurnal, terdiurnal, quarter- diurnal, sixth-diurnal and eighth-diurnal constituents, but only such items need be computed as are necessary for the particular constituents sought. For the principal lunar series M 1; M 2 , M 3 , M 4 , M 6 , and M 8 , compute all items of the form. For the principal solar series Si, S 2 , S 4 , and S 6 , items (14), (16), (33), (35), and (37) may be omitted. For the lunisolar constituents Ki and K 2 , items (14), (16), and (23) to (37) may be omitted. For the diurnal constituents J 1} O x , 00, Pi, Q u 2Q, and p u items (5), (6), and (14) to (34) may be omitted. For the semidiurnal constituents L 2 , N 2 , 2N, R 2 , T 2 , X 2 , /x 2 , v 2 , and 2SM, items (3), (4), (8) to (16), and (23) to (37) may be omitted. For ter- diurnal constituents MK and 2MK, items (5), (6), (9), (12), and (18) to (37) may be omitted. For quarter-diurnal constituents 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 constituent is to be entered at the head of the column or columns indicated by the subscript corresponding to the number of constituent periods in a constituent day, the remaining columns being left blank. 316. The hourly means from form 142 (fig. 13) are entered as items (1) and (2) in regular order, beginning with the mean for 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. 317. 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. Therefore (41) will be in the first quadrant if both s and c 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 nega- tive and c positive. In obtaining (49) use (46) + (47) for all constituents except S, and (46) + (48) for S. The log factor F for item (50) may be obtained from form 244a. 318. Form 194 is designed for use when 24 constituent hourly means have been obtained and all the original hourly heights have been used in the summation. If in the summation for a constituent each constituent 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). 114 U. S. OOAS'T AND GEODETIC SURVEY 319. This form is also adapted for use with the long-period con- stituents. Assuming that the daily means have been cleared of the effects of the short-period constituents (p. 89), and that these means have been assorted into 24 groups to cover the constituent period, the 24 group means may then be entered in form 194 in place of the 24 hourly means used for the short-period constituents. Then, treat- ing the constituents Mm and Sa the same as the diurnal tides and the constituents 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). 320. To obtain Sa and Ssa from the monthly means of sea level, or tide level, the following process may be used: Enter the monthly means beginning with that for January in alternate spaces provided for the hourly means in form 194, placing the value for January in the space for the hour. For convenience consider all the intermediate blank spaces as being filled with zero values and make the computa- tions indicated by (3) to (12) and (18) to (21). Correct the co- efficients of Si and Cj from 12 to 6, at top and foot of columns (9), (12), (19), and (21). In bottom of form enter Sa in column having sub- script 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 f's as obtained from (40) must have the following corrections applied in order to refer them to hour of the first day of January — common years, Sa correction^ + 14.07°, Ssa correction =+28.14°; leap years, Sa correction= + 14.94°, Ssa cor- rection= +29.88°. For convenience in recording the results it is suggested that the f as directly obtained from (40) be entered (in its proper quadrant) in the space just below the logarithm from which it is obtained, 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 y+u, computed to the first day of January, may then be entered immediately under the corrected £ 's and the K'of (43) readily obtained. For (49) the combination (46) + (47) will be used. 321. Form 452, R, k, and $ from analysis and inference (figs. 17 and 18) — This form provides for certain computations preliminary to the regular elimination process. The constants for constituents Ki and S 2 as obtained directly from form 194 may be improved by the appli- cation of corrections from tables 21 to 26; and constants for some of the smaller constituents, which have been poorly determined or not determined at all by the analysis, may be obtained by inference. If the series of observations is very short, the inferred values for the constants of some of the constituents may be better than the un- eliminated values from form 194. 322. Form 452 is based upon paragraphs 229 to 243. It is designed to take account of the diurnal constituent on one side (fig. 17) and the semidiurnal constituents on the other side (fig. 18). 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 x and 2SM may be omitted. 323. For all short series the values in columns (4) and (8) are to be computed in accordance with the equivalents and factors in columns HARMONIC ANALYSIS AND PREDICTION OF TIDES 115 (3) and (7) respectively. If the series is 192 days or more in length, the k of M b Pj, and K 2 for column (4), and the log R of M,, Pi, and K 2 for column (8) may be taken directly from form 194, and if the series is 355 days or more in length the k and log R 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. 324. The tabular values of items (12) and (13) for the diurnal con- stituents and items (14) to (18) for the semidiurnal constituents may 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 acceler- ations may be taken as zero and the resultant amplitude factors as unity. TIDES: /?, «, AND f, FROM ANALYSIS AND INFERENCE. Station SorrOgJfellftari^ •- Length of Series !§> - days. Series begins ....WMp-^MSSLM. DIURNAL COMPONENTS. 3 a c a a 3 From Analysis. 1 3 From Analysis and Inference. 2 a 1 g 3 Fbom Analysis i JJD INFERENCE. R' f « V„+u.* f=(4)-(5) R (1) (2) (3) (4) (5) (6) (?) (8) Ft. (3 dec.) °(2dec.) Equivalent. "(Idee.) °(ldec.) °(Odec.) Factore. (4 dec.) K, M," 0, 00 P, Q. 2Q S, Pi J, M, o, 00 p, Q/i 2Q Si Pi K r+0.5X(14) (Kf)']Ca..8.-Ki2) 116.7 "iio.~9 105.1 150.9 60,0... 326.0 ...326.... 51 139 Ji log. 0.079 log. 7J'(0,) log. F(J,)» + 8.8976 + 9. 7814 - 0. 0201 0.967 48-82 0.060 177.14 0-569 116-61 (ory K r+(14) 99.3 ------ 342.7 325.6 .....11.7.....: 157 log. i?(J,) *(J.) 8,6589 "olig9*~" 0.107 ~182.JS 210.95 K, log. *'(K,) log. (13) + 9. 9852 - 0. 0128 K° 110.9 306.8 164 K, , -1.5X(14) 93.5 255 9 198 log. R(K t ) R(K,) 9- 9724 - 87.7 169.1 279 (S,*/ ... M, log. 0.071 log. R{0,) log. Q* + 8.8513 + 9.7550 -9-8592 K;-1.43X(14)-... 9A.-.3... ...116..1... ..„33.a (9)=P»=. ..53. 03.(2 dec.); (10)=F(K,)*=..l •.Q5JL(3 dec.); (11)= h i/)»- ?.?.7...(0 dec.) log. JI(M,) *(M,) log. R' (0,) i?(0 I )=7?'(0 1 ) 8-7471 9.7550 (12)=acceleration in K, due to P,=.F(K,)XTable 21=..?.»1 (1 dec.) (13)=resultaDt amplitude, K, and P,=1+[F(K,) X (Table 22)]=„.isP3 (2 dec.) (14)=(iq-0n=.-U-?-?— (Idee.) 0! 00 log. 0.043 log. 27/(0,) log. F(OO)* + 8.6335 + 9-7814 -0-0929 Explanation. — Obtain from Form 194 the amplitudes and epochs indicated by the accent ('), and from Form 244 or 244a the quantities indicated by the asterisk (*). If the series is lees than 355 days, omit S,. For all short series, the values in columns (4) and (8) will be computed in accordance with the Equivalents and Factors in columns (3) and (7), respectively; but if the series is 192 days or more in length, the k of M, and Fi for column (4), and the log. R of M, and P, for column (8) may be taken directly from Form 194 ; and if the series is 355 days or more in length, the « and log. R of all the components for which analysis has 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) should be crossed out. The tabular values for (12) and (13) may be obtained from Tables 21 and 22 in Special Publi- cation No. 98 or from plotted curves representing this table. For a series of 355 days or more, (12)=0, and (13)=1. Obtain the k of K, by applying (12) to (K,°)' from Form 194, and use this corrected « in com- puting (14). If the two angles in (14) differ by more than 180°, add 360° to the smaller before taking the difference, which may be either positive or negative. In computing column (8) it will be noted that the corrected log. R (K,) is to be used when inferring P,. log. R{00) R(OO) 8-3220 p, log. 0.331 log. R(K,) log. J\K,)» + 9.5198 + 9.9724 + 0-0160 log. fl(P,) *(P,) 9-5082 Q. log. 0.194 log. J?(0,) + 9.2878 + 9.7550 log- *(©.,) *(Q,) 9.0428 2Q log. 0.026 log. i?(0,) + 8.4150 + 9.7550 log. R(2Q) i?(2Q) 8-1700 Si log. R\S t ) J?(S,)=R'(S,) Pi log. 0.038 log. *(0,) + 8.5798 + 9-7550 log. R( Pl ) *(p.) 8 • 3348 Figure 17. 116 U. S. COAST AND GEODETIC SURVEY 325. The k's of K x and S 2 are to be corrected by the accelerations as indicated before entering in column (4), and in computing item (14) for the diurnal constituents and (21) for the semidiurnal constituents the corrected k's are to be used. If the two angles in item (14) for the diurnal constituents, or in items (20) or (21) for the semidiurnal constituents, differ by more than 180°, the smaller angle 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 x and S 2 are to be used in inferring other constituents depending upon them. 326. Form 2%5, Elimination of component effects (fig. 19). — This form is based upon formulas (389) and (390). One side of the form is designed for the elimination of the effects of the diurnal constitu- ents upon each other and the other side for use with the semidiurnal constituents, the two sides being similar except for the listing of the U. 5. COAST AND GEOPETK SURVEY TIDES: R, *, AND f, FROM ANALYSIS AND INFERENCE. Station ^ro, Cali^nia Length of Series 1.6.3 days. Series begins 1?1?.#.. February 13 SEMIDIURNAL COMPONENTS. From Analysis. (1) Ft. (3 dec.) Idee.) From Analysis and Inference. Equivalent. ' (1 dec.) "(Oder.) 0-.090. .CL.QS5.. .1:245. 0.263 20 63 ...1S..6.7 261 52 326. 84 JLSUL ..306.43 O1OI.5.. 0.010 ..99-.2Q 118- 79 2SM m;+(20) _ WY - (N,y N,°-(20) (S?)'3Q4»&Ki6) s? M,°-f0.464X(21)~ M;-(21) M;-0.866X(20).. (2SM")' ::. ..?04,.2. 32'9. 9 Jtifttfi. .30.8,4. 286.9 .265:4. ...46..-B.. 320.1 _233:.5 .262. 327 ZQA-.Z.. 304:.?.. .3Q4.J2.. 306.5, 312.6 .219..-.1. .358-3 .317*5. ...93,5. ...B5.. .306. 347. ...?3.-4. 180 2 .215 219 .110.. (S)=I»=..J21,.76.__(2 dec.); (10)=P*=...53 A 03_.(2 dec.); (11)=/(K 2 )*...O..l8.97...(3 dec.) (12)=(A-*")*= 33.1 (Odec.); (13)=(>>-Pi)*= .41 (0 dec.) (14)=acceleration in S 2 due to K 2 =Table 23X/(K*)= 1*4 °(1 dec.) (15)=acceleration in S 2 due to T 2 = Table 25 =..~?.?.9_ ° (1 dec.) (16)=(14) + (15) _ _=_r0.t£ °(l dec.) (17)=resultant amplitude, S 3 and K,=l+[Table 24X/(K,)]= JlS.0.1 (18)=resultant amplitude, S,and T 2 =TabIe 26= 2. t 98 (19)=log. (17)+log. (18, 9.9955 _(2 dec.) _(2dec.) (20)=(M;-NJ)=. 21,5 .(4 dec.) ..(1 dec.); (21)-(S,'-M,> .(1 dec.) Explanation. — Obtain from Form 194 the amplitudes and epochs indicated by the accent ('); and from Form 244 or 244a the quantities indicated by the asterisk (*). If the series is less than 355 days, omit 2SM. For all short series, the values in columns (4) and (8) will be computed in accordance with the Equivalents and Factors in columns (3) and (7), respectively; but if the series is 192 days or moro in length, the *• of K, for column (4) and the log. R of K 3 for column (8) may be taken directly from Form 194 ; and if the series is 355 days or more in length, the k and log. R of all the components for which analysis has 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) should be crossed out. The tabular values for (14) to (18) may be obtained from Tables 23 to 26 in Special Publi- cation No. 98, or from plotted curves representing these tables. For a series of 355 dayB or more, (14)=(15)=(16)=0; (17)=(18)=1; and (19)=0- Obtain the k of S, by applying (16) to (S'Y from Form 194, and use this corrected * in com- puting (21). If the two angles in either (20) or (21) differ by more than 180°, add 360" to the smaller before taking the difference, which may be either positive or negative. In computing column (8) it will be noted that the corrected log. R of S 2 is to be used in in- ferring other components depending upon this one. Figure 18. HARMONIC ANALYSIS AND PREDICTION OF TIDES 117 constituents. The symbol A represents the constituent to be cleared, and the symbol B is the general designation for the disturbing con- stituents. The symbol applying to constituent A is to be crossed out in column (1) and entered in column (8). The values for items (9) and (19) are to be taken from columns (1) and (2) of form 452. 327. For obtaining column (2) it will be found convenient to copy the logarithms of the R's of B from column (8) of form 452 on a hori- zontal strip of paper spaced the same as table 29. Applying this strip successively to the upper line of the tabular values for each con- Form 246 DEPARTMENT OF COMMERCE Coast a\T> Geodetic Survey TIDES: ELIMINATION OF COMPONENT EFFECTS Station JA?"X?.*... c .alifornia. Length of series ?-.6?. days. Series begins I.?!?. Feb., .13. a m£,)x Tabic 29 (3) Nat. No. (2) Table 27 Table 29 -HBi) <4) + <19) (ffl <3)Xmo (3) Table 30 (") (3)Xcos (5) Table 30 (8) RESULTS K t L 2 M, N 2 2N R 2 s t T 2 >j Vl 2SM log (4 dec ) ft. (3 dec ) °l do dec ) °(no dec ) (3 dec) (3 dec ) Use 4 dec. for logarithms, 3 dec. for amplitudes, 1 dec. for angles Component A 2 = ..$2... (9) = R'(A 2 ) from analysis »_ 0.090 6 0462 '".0^007.' • 002 | ...i9o..l...3ii... 240.1 261 -.0-..0.05. -. -.002. - 001 -. :037. -. .-..0.02.. +<>..6.Q5. - 002 +...-. 003.. + .-..004.. (10) = (9)-(7) = . 0.080 (ll)=log(6) = 8.672i (12)=log(10) =J. .9031 (13) = (ll)-(12) = logtan« f =..?.:.7690 *(14)=5 f =.. . -30.4 (15)=logcos« r = ..?..• 93 5 6 (16) = (l2)-(l5)=logi?(/l 2 ) =8 •9675 (16a)=logFVl 2 ) =.0.0472 (17) = (16)+logf(yl 2 ) = log//( / 4 2 ) =.9.pi47 (18) = //M 2 ) =...P.:103 (19)=r'(^2) from analysis = 2 Q.6 (20)-(14) + (19)-f (A 2 ) -...350.2 6 5019 7 -2518 8" '5740' 7 -5905 6""-3638' .......6.0.2.. • 037 . .-.004. ...195.. ...253.. ...312.. ..216... ...274... ..333... .5. •5.70.3. 6. .-2 207. ....:«- — ---. ::: .— ...r.r.r... — ■ -— 1... (20a) = (^+t/) = 299.6 (21) = (20) + (K. + u) = «U 2 ) = .4?.?. .-.?..... 1 Sums= -b- 047 t.Q,.Q10.. K 2 L, Mx N 2 2.V Ri s 2 T 2 h "2 2SM 6 .3256 Component Ai = ..r^... (9) = R' (A 2 ) from analysis =. 0-065. ... (10) = (9)- (7) = 0-054 (iD= log (6) =8.bb"bd .____ . _- — — . — - . --- 3 .2443 7" .5542' 6' .651-3" 013 ...... .004 293 '.'..243'.'.' 309" '.'.'.259'.'. -0.014. - -..Q04. +0-011 r r.OO.i.. (12)=log(10) =.8:7324 (13) = (ll)-(12)=log tan8 r = £ » .2676 *(i4)=s r = -1Q.-.5.... (15) = logcos«f =-|-||IX (I0) = (12)-(15)=logft(/l ! ) =8.7397 (16a)=log F(A 2 ) =..?„■ 9589 (17) = (16) + logF(^l 2 )=logW(/l : ) =.8.6986 ()8) = H{A 2 ) =0-050 (19) = f'(/l 2 ) from analysis = 1.5-7 (20) = (14) + (19) = f {A 2 ) = 5.2 . (20a)-(V'„ + u) =. 302.3 6 -1512 '7"J3'74f 7 ".0340" 7 "5292' ":;'oo7" .obi'" 002 76" '135" 5" '151'"' 21" + 007 + 001 - 001 + .■ 002 5 -39S5 6 2592 ""._"-'-"" ...rr-.. ■-:•::•■ ....•..-.-.-.. .„.---- ....:.--- ....---. ...---.. ....!.." '-.. ...... -r-.. 1 Sums = -0. 010 +0.011 (2\) = (20) + (V o +u)=k(A,) = .3.0.7. -5 K, L 2 .1/2 -v 2 2JV R 2 s 2 T, X 2 » 2 2SM 6 .6419 1 | Component yl 2 = ....<>-.- (9) = R' (A 2 ) from analysis = 1-245 (10) = (9)-(7) =1.246 6.r7.592 7"1'5696 6 -673 i "5""J3'47d 7 "."05 30 "6".73'?"6 0.001 137 39 +0 001 +0-001 oc4 " ''""001 Obi 22S 120 i30 22" +.'.•..003. - 001 ° 003 4 .001 + -001 (ii) = log(6) =.7.-6021 ( 1 2) = log ( 10) = - 0955 ..... (13) = (ll)-(12)=logtan« f = 7-5066 *(H)=of = 0;2 (15) = loi?cos« f = 0-0000 (16) = (12)-(15) = logfl(/l 2 ) =0-0055 (16a)=log F(A 2 ) = .'9..-??3'2 (17) = (16)+logF(,4 2 )=log//UU = 0.0887 . (lS) = tf(.4 2 ) = ...1-227 (19) = f'(-4 3 ) from analysis = 2.61-5 (20) = (14) + (19) = r (^2) = Ml-7 (20a) = (V„ + ») = .4.6-8 (21) = (20) + (V.+tt)=«M.) =- 308 .5 6 .3128 ___ ... . — . — 6 02 68 .7.. -.08.0.3. .'.""r.'.Obl. ".".'.".2.2.3.".' 33.0.V. + 001 -. ...001.'. 1 Sums = +Q-. .0.04. -o- 001 t the 2d quadrant wt 1 the 3d quadrant wh 1 the 4th quadrant \v - and (10) is -. - and (10) is + ) = - cd, and (1 Computed by Verified by I. A. A. L.P.D. Feb. 28, 1921 (Date)""* Figure 19. 118 U. S. 00 AST AND GEODETIC SURVEY stituent the logarithms of the resulting products for column (2) may be readily obtained. Similarly, for column (4), the f's of B from column (6) of form 452 may be copied on a strip of paper and applied to the bottom line of the tabular values for each coustituent and the differences obtained. The natural numbers for column (3) correspond- ing to the logarithms in column (2) can usually be obtained most expeditiously from table 27, this table giving the critical logarithm for each change of 0.001 in the corresponding natural number. If the logarithm is less than 6.6990, the natural number will be too small to appear in the third decimal place, and the effects of the corresponding constituent may be considered as nil. The products for columns (6) and (7) may be conveniently obtained from table 30. In column (8) the references to (6) and (7) are to the sums of these columns. The values of log F{A) and (Vq-\-u) for column (8) may be obtained from forms 244 and 244a. 328. In the use of this form it will be noted that the R's and £*'s referring to constituent B are to be the best known values whether derived from the analysis or by inference, but the W and f ' of con- stituent A, entered as items (9) and (19), respectively, must be the unmodified values as obtained directly by form 194. ANALYSIS OF TIDAL CURRENTS 329. Tidal currents are the periodic horizontal movements of the waters of the earth's surface. As they are caused by the same periodic forces that produce the vertical rise and fall of the tide, it is possible to represent these currents by harmonic expressions similar to those used for the tides. Constituents with the same periods as those con- tained in the tides are involved, but the current velocities take the place of the tidal heights. There are two general types of tidal cur- rents, known as the reversing type and the rotary type. 330. In the reversing type the current flows alternately in opposite directions, the velocity increasing from zero at the time of turning to a maximum about 3 hours later and then diminishes to zero again, when it begins to flow in the opposite direction. By considering the velocities as positive in one direction and negative in the opposite direction, such a current may be expressed by a single harmonic series, such as V=Aco$ (at+a)+B cos (bt+ff) + C cos (c*+7) + etc. (450) in which V= velocity of the current in the positive direction at any time t. A, B, C, etc. = maximum velocities of current constituents. a, b, c, etc. = speeds of constituents. a, j8, 7, etc.=initial phases of constituents. 331. 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 HARMONIC ANALYSIS AND PREDICTION OF TIDES 119 representation of such currents it is, therefore, necessary to have two series — one for the north and the other for the east component. 332. For the analysis of either type of current the original hourly velocities or the resolved hourly velocities are tabulated in the same form used for the hourly heights of the tide. To avoid the incon- venience of negative readings in this tabulation, a constant, such as 3 knots, is added to all velocities. These hourly velocities are then summed with the same stencils that are used for the tides, and the hourly mean velocities are analyzed in the same manner as the hourly heights of the tide. The same forms are used for the currents, with the necessary modifications in the headings. The rotary currents will be represented by a double set of constants, one for the north components and the other for the east components. 333. For a 29-day series of observations, it is recommended that the analysis be made for the M series, the S series, and for N 2 , K^ and Oi. For longer series additional constituents may be included. In the analysis of current velocities, the harmonics of the higher degrees such as M 4 and M 6 may be expected to be of relatively greater magnitude than they are in the tides. From theoretical considera- tions it may also be shown that the magnitude of the diurnal constit- uents as compared with the semidiurnal constituents in a simple tidal oscillation is only about one-half as great in the current as in the tide. However, because of the complexity of the tidal and current move- ment, the actual relation between the various constituents as deter- mined by the analysis is subject to wide variations. The constituent Si, which is usually negligible in the tides, may be found to be of ap- preciable magnitude in offshore currents because of the effect of daily periodic land and sea breezes. However, as this constituent has a speed very nearly the same as that of K x it can be separated from the latter only by a long series of observations, preferably a year or more. 334. Form 723 (fig. 20) provides for the determination of harmonic constants from a series of current observations by comparison with corresponding constants from a tidal series covering the same period of time. This comparison is to be used if the series of observations is less than 29 days and may be used for longer series if desired. For the purpose of this comparison the hourly predicted heights at the tide station are usually to be preferred to actual observations since meteor- ological irregularities appearing in observed tides do not necessarily appear in a similar manner in the observed currents. In this work both currents and tides for the simultaneous period are to be summed for constituents M, S, N, K, and O; and the analysis is then carried through form 194 (Tides: Harmonic Analysis) to obtain the values of R f and f for each constituent. The harmonics M 4 , M 6 , and M 8 are to be obtained for the current series, but may be omitted in the tidal series. 335. Enter in Form 723 the accepted H and k of the principal tidal constituents for the reference station and also the values of K/ and f ' obtained from the analyses of the simultaneous series of tides and currents. The necessary calculations in the form are self-explana- tory. The corrected velocity amplitude of each current constituent is obtained by a ratio on the assumption that for each constituent the relation of the corrected amplitude to the uncorrected amplitude is the same for both tide and current. The ratio derived for the con- 120 U. S. 00 AST AND GEODETIC SURVEY y z o or < X I— z UJ a 1 gssa II ;. -o c 3 z o S= J to 00 t4 cc o odj <■ Q. Oi °3 2E Cvj «| o a Us o H J o s > > £ c s-> +J « C « c > •r r- ■P tg a « IH t t « c <* • X t. a tD g H to en H H C g • tt + CO a Ovl < > < t r- > < a o Sz C (3 c c i 1 < 2 a J (/ J 5 ) « ,a c s rn (ID Corrected « at (A) (8)+(9)+(10) - eoi I ! o>: to to <« ri IQJ O; Cxi t-1 Q 6Q «J Q lp *4 it •""• 1 Sj M n' • & r-f. q iHi H H coi wi • • • «i 9 s ! M (5) Corrected H at (A) (3) X (4) •g O: o> § «o o q o UJ O id M Q O d! d •Hi t-j fH c6 9 S S c? A 4 A A 3 "1 + cw o; o 1 o r-i A i-i mi eg ca 10 cm CvJ rH 5* W « • « « oi q q q ■~- aj to II 3« §£ C« CO . I 8 fa < £ CO, rH °! ° CO CM oi c c> d coi 3 «ri ccj a . . . oi oj H H ca » oa Oj d ui oi cq ca to to ca cq a; 4 A A (1) Accepted H at (B) oj ^i (d irt to co' *g td ^J a; 4 to to Component S ; : ; ' i i j i » i Z ; W : ; § • s 1 O ! i5 c~ ! 00 ; cvi «H ^-i. P3 4 i ^J ■P * •H ri a +>i •HI r-i OI ai 9 *H •d S? ^ i 8 -H •H •rl 1 04 M- 5 -H 4 d ^ rH fi> 4 3 Si •H d t» •U 9 Wi u a *4 HARMONIC ANALYSIS AND PREDICTION OF TIDES 121 stituent M 2 is used also for the higher harmonics of M, this being considered more reliable than ratios determined directly from the much smaller amplitudes of these harmonics. The corrected epoch (k) for each current constituent is calculated on the assumption that the difference between the corrected and uncorrected epoch is the same for tide and current. For convenience the zetas (£) rather than the kappas from the simultaneous observations are used in the form and a longitude correction, column (10), is introduced to allow for this fact. Differences in column (9) for the higher harmonics of M 2 are derived from the difference for that constituent because of the uncer- tainty in the determination of epochs of constituents of very small amplitudes. 336. Short series of current observations are frequently taken at half-hourly intervals. As individual observations are somewhat rough, the utilization of the half-hourly observations will add ma- terially to the accuracy of the results obtained from an analysis. Moreover, the closer spacing of the half-hourly values will give a better development of the higher harmonics of M which are of greater relative importance in the currents than in the tides. Special stencils have been prepared for the summation of these observations. Obser- vations taken on the exact hour are tabulated in form 362 as usual, while observations on the half -hour are offset to the right on the intermediate lines. As the series of observations under consideration are short, provisions have been made for obtaining only the diurnal constituents Ki and Oij the semidiurnal constituents M 2 , S 2 , and N 2 ; and the higher harmonics of M. 337. For the diurnal constituents, the special stencils provide for the same distribution, with the inclusion of the half-hourly values, as is obtained with the standard stencils used for the hourly values only. Hourly means for the constituents are obtained and entered in form 194 and all subsequent computations are the same as those based upon the use of the standard stencils. 338. For the semidiurnal constituents M 2 , S 2 , and N 2 , the semi- diurnal period is divided into 24 parts. Special stencils for the con- stituents M 2 and N 2 provide for the distribution of the observed half- hourly velocities into the 24 groups indicated by this division. No stencil is required for the constituent S 2 , the necessary grouping being- accomplished by combining sums for afternoon observations with those for the forenoon observations of corresponding hours. Thus, the noon observations will be included with those taken at midnight, and the observations at 12:30 p. m. with those taken at 0:30 a. m. 339. The resulting means obtained for the semidiurnal constituents by the method described above are in reality half-hourly means, but in adapting form 194 for the analysis, these means may be entered in order in the spaces provided for the hourly means. Then, after doubling all subscripts in the form, the necessary computations may be carried out as indicated. Thus, all computations for the semi- diurnal constituents will be made in the spaces originally designed for the diurnal constituents. The computations for all higher har- monics of even subscripts may be carried out in the same form using the spaces originally designed for the harmonics with subscripts one- half as great. In this adaptation of the form no provision is made for the computation of a harmonic of odd subscript which is here of rela- 122 U. S. COAST AND GEODETIC StrHVEY tively little importance. Other forms which are used in connection with the analysis will not be affected by the use of the special stencils for the half -hourly velocities. 340. Observations on the half-hour may also be analyzed sepa- rately from those on the exact hour, using the standard stencils for the summation. In this case the stencils are moved to the right one column and dropped one line, thus covering the hourly values and exposing those occurring on the half-hour. Allowance must be made for the difference of a half hour in the beginning of the series when computing the (V -\-u)'s in form 244. This may be conveniently done by assuming a time meridian a half -hour or 7)i° westerly from the actual time meridian used so that the first half -hourly observation will correspond to the hour of the assumed time meridian. The difference of 15 minutes for the middle of the series has a negligible effect in the computations and may be disregarded. In other respects the analysis is carried on in the same manner as the analysis for the hourly observations, and the results obtained afford a useful check on the latter. PREDICTION OF TIDES HARMONIC METHOD 341. The methods for the prediction of the tides may be classified as harmonic and nonharmonic. By the harmonic method the ele- mentary constituent tides, represented by harmonic constants, are combined into a composite tide. By the nonharmonic method the 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 har- monic 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 Coast and Geodetic Survey in making predictions for the standard ports of this country. 342. The height of the tide at any time may be represented har- monically by the formula h=H +2fHcos [at+(V +u)-ic] (451) in which h= height of tide at any time t. i7 =mean height of water level above datum used for pre- diction. i?=mean amplitude of any constituent A. j= factor for reducing mean amplitude H to year of pre- diction. a= speed of constituent A. £=time reckoned from some initial epoch such as beginning of year of predictions. (V -\-u)= value of equilibrium argument of constituent A when t=0. k= epoch of constituent A. In the above formula all quantities except h 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. 343. The exact value of t for the times of high and low waters will be roots of the first derivative of formula (451) equated to zero, which may be written — !f=-2 cvfH sin [at+(y o +u)-K] = (452) 123 124 U. S'. COAST AND GEODETIC SURVEY Although formula (452) cannot, in general, be solved by rigorous methods, it may be mechanically solved by a tide-predicting machine of the type used in the office of the Coast and Geodetic Survey. 344. The constant H of formula (451) is the depression of the adopted datum below the mean level of the water at the place of predic- tion. For places on the open coast the mean water level is indentical 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 and Gulf coasts of the United States, including Puerto 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 Hp thus determined is a constant that is available for future predictions at the stations. 345. The amplitude H and the epoch k for each constituent tide to be included in the predictions are the harmonic constants determined by the analysis discussed in the preceding work. 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. 346. The node factor/ (par. 77) is introduced in order to reduce the mean amplitude to the true amplitude depending upon the longitude of the moon's node. The factor/ for any single constituent, therefore, passes through a cycle of values. The change being slow, it is cus- tomary 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 negli- gible. Each constituent has its own set of values for f] but these values are the same for all localities and have been compiled for convenient use in table 14 for the middle of each year from 1850 to 1999. 347. The quantity a represents the angular speed of any constituent per unit of time. In the application of formulas (451) and (452) 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 constituents have been calculated from astronomical data by formulas derived from the development of the tide-producing force which has already been discussed. These speeds have been compiled in table 2 and are essentially constant for all times and places. The quantity (V +u) is the value of the equilib- rium argument of a constituent at the initial instant from which the value of t is reckoned; that is, when t equals zero. In the prediction HARMONIC ANALYSIS AND PREDICTION OF TIDES 125 of tides this initial epoch is usually taken at the midnight beginning the year for which the predictions are to be made. In strictness the V, or uniformily varying portion of the argument alone, refers to the initial epoch, while the u, 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 quantity (Vq-\-u) is different for each constituent and is also different for each initial epoch and for different longitudes on the earth. In table 15 there have been compiled the values of this quantity for the beginning of each year from 1850 to 2000 for the the longitude of Greenwich. The values may be readily modified to adapt them to other initial epochs and other longitudes. 348. Let L=west longitude in degrees of station for which predictions are desired. #=west longitude in degrees of time meridian used at this station. For east longitude, L and S will have negative values. Now let p=0 when referring to the long-period constituents. 1 when referring to the diurnal constituents. 2 when referring to the semidiurnal constituents, etc. then p will be the coefficient of the quantity T in the equilibrium arguments. Now, T 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 observation or of prediction. At any given instant of time the difference between the values of the hour angle Tat two stations will be equal to the difference in longitude of the stations. If, there- fore, the value of the argument (V -\-u) for any constituent 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 T decreases as the west longitude increases. 349. The instant of time to which each of the tabular values of the Greenwich (V G -\-uYs of table 15 refers is the 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 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 S/15 mean solar hours later than the instant to which the tabular Greenwich (Fo+^)'s are referred. 350. In formulas (451) and (452) the symbol a is the general desig- nation of the speed of any constituent; 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 (Vq-^u) of any year in order to obtain the local (Vq+u) for a place in longitude L° west at an initial epoch of hours of time meridian S° west at the beginning of the same calendar year is 246037—41 9 126 U. S. COAST AND GEODETIC STJUVEY aS 15 ~pL. (453) The general expression for the angles of (451) and (452) may now be written a,t-\-(y -\-u)—K=at+ Greenwich (V +u) +jr—pL—K (454) 351. In order to avoid the necessity of applying the corrections for longitude and initial epoch to the Greenwich (F +^) , s for each year, these corrections may be applied once for all to the k's. Let aS T , Yh - V L-K=-K f (455) Then (454) may be written at+{V Q +u)— k= a* + Greenwich (V +u) — k' (456) Thus, by applying the corrections indicated in (455) to the k'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 (Vq-\-u)'8 in determining the actual constituent phases at the beginning of each calendar year. 352. Let Greenwich (V -\-u) — n'—a (457) then formulas (451) and (452) may be written ^=^o+S JH cos (at+a) (458) for height of tide at any time, and 2 ajH sin (at+a)=0 (459) for times of high and low waters. Formula (458) may be easily solved for any single value of t, but for many values of t as are neces- sary in the predictions of the tides for a year at any station the labor involved by an ordinary solution would be very great. Formula (459) can not, in general, be solved by rigorous methods. The in- vention of tide-predicting machines has rendered the solution of both formulas a comparatively simple matter. * TIDE-PREDICTING MACHINE 353. The first tide-predicting machine was designed by Sir William Thomson (afterwards Lord Kelvin) and was made in 1873 under the auspices of the British Association for the Advancement of Science. This was an integrating machine designed to compute the height of the tide in accordance with formula (458). It provided for the sum- mation of 10 of the principal constituents, and the resulting pre- dicted heights were registered by a curve automatically traced by the machine. This machine is described in part I of Thomson and Tait's Natural Philosophy, edition of 1879. Several other tide- predicting machines designed upon the same general principles but providing for an increased number of constituents were afterwards constructed. HARMONIC ANALYSIS AND PREDICTION OF TIDES 127" 354. The first tide-predicting machine used in the United States was designed by William Ferrel, of the U. S. Coast and Geodetic Survey. This machine, which was completed in 1882, was based upon modified! formulas and differed somewhat in design from any other machine that has ever been constructed. No curve was traced, but both the times and heights of the high and low waters were indicated directly by scales on the machine. The intermediate heights of the tide could be obtained only indirectly. A description of this machine is givem in the report of the Coast and Geodetic Survey for the year 1883. 355. The first machine made to compute simultaneously the height of the tide and the times of high and low waters as represented by formulas (458) and (459), respectively, was designed and constructed* in the office of the Coast and Geodetic Survey. It was completed in 1910 and is known as the United States Coast and Geodetic Survey tide-predicting machine No. 2. The machine sums simultaneously the terms of formulas (458) and (459) and registers successive heights of the tide by the movement of a pointer over a dial and also graphi- cally by a curve automatically traced on a moving strip of paper.. The times of high and low waters determined by the values of t which satisfy equation (459) are indicated both by an automatic stopping: of the machine and also by check marks on the graphic record. 356. The general appearance of the machine is illustrated by figure" 21. 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 constituents are made to rotate with angular speeds proportional to the actual speeds of the constituents;: third, a system of cranks and sliding frames for obtaining harmonic motion; fourth, summation chains connecting the individual constitu- ent elements, by means of which the sums of the harmonic terms of formulas (458) and (459) are transmitted to the recording devices;: fifth, a system of dials and pointers for indicating in a convenient man- ner the height of the tide for successive instants of time and also the time of the high and low waters ; sixth, a tide curve or graphic represen- tation of the tide automatically constructed by the machine. The machine is designed to take account of the 37 constituents listed in table 38, including 32 short-period and 5 long-period constituents. 357. The heavy cast-iron base of the machine, which includes the operator's desk, has an extreme length of 11 feet and is 2 feet wide- This forms a very substantial foundation for the superstructure,, increasing its stability and thereby diminishing errors that might result from a lack of rigidity in the fixed parts. On the left side of the desk is located the hand crank for applying the power (1 , fig. 24) r and under the desk are the primary gears for setting in motion the various parts of the machine. The superstructure is in three sections y each consisting of parallel hard-rolled brass plates held from 6 to 7 inches apart by brass bolts. Between these plates are located the shafts and gears that govern the motion of the different parts of the machine. 358. The front section, or dial case, rests upon the desk facing the operator and contains the apparatus for indicating and registering the results obtained by the machine. The middle section rests upon a. depression in the base and contains the mechanism for the harmonic motions for the principal constituents M 2 , S 2 , K lT O l} N*, and M 4 . The 128 U. S. OOAS'T AND GEODETIC SURVEY rear section contains the mechanism for the harmonic motions for the remaining 31 constituents for which the machine provides. 359. The angular motions of the individual constituents, as indicated by the quantity at in formulas (458) and (459), are represented in the machine by the rotation of short horizontal shafts having their bear- ings in the parallel plates. All of these constituent shafts are con- nected by a system of gearing with the hand crank at the left of the dial case and also with the time-registering dials, so that when the machine is in operation the motion of each of these shafts will be proportional to the speed a of the corresponding constituent, and for any interval of time or increment in / as indicated by the time dials the amount of angular motion in any constituent shaft will equal the increment in the product at corresponding to that constituent. 360. Since the corresponding angles in formulas (458) and (459) are identical for all values of t, the motion provided by the gearing will be applicable alike to the solution of both formulas. The mechanism for the summation of the terms of formula (458) is situated on the side of the machine at the left of the operator, and for convenience this side of the machine is called the "height side" (fig. 21), and the mech- anism for the summation of the terms of formula (459) is on the right- hand side of the machine, which is designated as the "time side" (fig- 22). 361. In table 37 are given the details of the general gearing from the hand-operating crank to the main vertical shafts, together with the details of all the gearing in the front section or dial case. It will be noted that S-6 (fig. 25) is the main vertical shaft of the dial case and is connected through the releasable gears to the hour hand, the ;minute hand, and the day dial, respectively. The releasable gears permit the adjustment of these indicators to any time desired. After an original adjustment is made so that the hour and minute hand will f each read at the same instant that the day dial indicates the begin- :ning of a day, further adjustment will, in general, be unnecessary, as tthe gearing itself will cause the indicators to maintain a consistent relation throughout the year, and by use of the hand-operating crank the entire system may be made to indicate any time desired. The period of the hour-hand shaft is 24 dial hours, and the hand moves over a dial graduated accordingly (3, fig. 23). The minute-hand shaft, with a period of 1 dial hour, moves over a dial graduated into m minutes {2, fig. 23). 362. The day dial, which is about 10 inches in diameter, is graduated into 366 parts to represent the 366 days in a leap year. The names of the months and numerals to indicate every fifth day of each month are 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- 23). The progress of the days as the machine is operated is indicated by the rotation of this dial past an index or pointer just below the opening (£, fig. 23). This pointer is secured to a short shaft which carries at its inner end a lever arm with a pin reaching under the lower edge of the day dial, against which it is pressed by a light spring. A portion of the edge of the dial equal to the angular distance from January 1 to February 28 is of a slightly larger radius, so that the pin pressing against it rises and throws the day pointer to the right one day when this portion has passed by. On Special Publication No. 98 Figure 21.— Coast and Geodetic survey Tide-Predicting Machine. Special Publication No. 98 Figure 23 —Tide-Predicting Machine, Recording Devices. Special Publication No. 98 Figure 24.— Tide-Predicting Machine. Driving Gears. Special Publication No. 98 Figure 25.— Tide-Predicting Machine. Dial Case From height side. Special Publication No. Figure 26.— Tide-Predicting Machine, Dial Case From Time Side. Special Publication No. 98 Figure 27.— Tide-Predicting Machine. Vertical Driving Shaft of middle section. Special Publication No. Figure 28— Tide-Predicting Machine, Forward Driving shaft of Rear Section. Special Publication No. 98 Figure 29.— Tide-Predicting Machine, Rear End. Special Publication No. 98 Figure 30.— Tide-Predicting Machine, Details of Releasable Gear. Figure 31.— Tide-Predicting Machine. Details of Constituent Crank. HARMONIC ANALYSIS AND PREDICTION OF TIDES 129 the last day of December this pointer will move back one day to its original position. 363. On the same center with the day pointer there is a smaller index (7, fig. 23) which may be turned either to the right toward a plate in- scribed "Common year," or to the left to a plate inscribed "Leap year." When this smaller index is turned toward the right, the day pointer is free to move in accordance with the change in radius of the edge of the dial. If the smaller index is turned toward the left, the day pointer is locked and must hold a fixed position throughout the year. For the prediction of the tides for two or more common years in succession the day dial must be set forward one day at the close of the year in order that the days of the succeeding year may be cor- rectly registered. The day dial can be released for setting by the nut (5, fig. 23) immediately above the large dial ring. A slower move- ment of the day dial is provided by a releasable gear on the vertical shaft S-6 (fig. 25). 364. There are three main vertical shafts S-13 (fig. 27), S~U (fig. 28), and S-16 (fig. 29), to which are connected the gearing for the individual constituents. 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 shafts and the individual con- stituent crankshafts are, in general, made by two pairs of bevel gears and an intermediate horizontal shaft, except that for the slow moving: constituents 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 releasable so that each crankshaft can be set independently. 365. Main shaft .S-13 in the middle section of the machine drives 9 individual crankshafts representing 6 constituents r 3 of them being^ provided with two crankshafts each. These 6 constituents are M 2> S 2 , Ki, N 2 , M 4 , and Oi, the first three having the double crankshafts. Main shaft S-14 at the front of the rear section of the machine drives-. 16 crankshafts representing one constituent each. These are M 6r MK, S 4 , MN, v 2 , S 6 , jjl 2 , and 2N in the upper range r and MS, M 8 , K 2r 2MK, L 2 , M 3 , 2SM, and Pi in the lower range. Main shaft S-16 at the back of the rear section drives 15 crankshafts. The constituents represented are 00, X 2 , Si, M 1? J 1} Mm, and Ssa, in the upper range, and 2Q, R 2 , T 2 , Q b Pl , Mf, MSf, and Sa in the lower range. 366. For each of the five long-period constituents motion is com- municated from the intermediate shaft by a worm screw and wheel to a small shaft on which is mounted a sliding spur gear. The latter engages a spur gear on the crankshaft, but may be easily discon- nected by drawing out a pin on the time side of the machine, thus permitting the crankshaft to be turned freely when setting the machine. 367. Gear speeds. — The relative angular motion of each constituent crankshaft must correspond as nearly as possible to the theoretical speed of the constituent 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 shafts to the individual crankshafts, the number of teeth in the different gears for each constituent being given in columns I, II, III, and IV. In design- ing the predicting machine it was necessary to find such values for these 130 U. S. COAST AND GEODETIC SURVEY columns as would give gear speeds approximating as closely as possible with the theoretical speeds of the constituents. By comparing the ^ear speeds as obtained with the corresponding theoretical speeds it •will be noted that the accumulated errors of the gears for an entire dial year for all the constituents are negligible in the prediction of the tides. 368- Releasable gears. — Releasable gears (52, fig. 27) on the main ver- tical shafts permit the independent adjustment of the time indicators and individual crankshafts. The details of these gears are illustrated in figure 30. 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 collar 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 ipressed against the edge of the recess a with such force as to make slipping practically impossible. When the nut is loosened, the gear imay be turned independently of the main driving shaft. A small wrench (56, fig. 28) is used for setting these gears. Each of the three main driving shafts is provided with a clamp (55, fig. 28) to secure the shaft from tinning when the nut of the releasable gear is being loosened or tightened. 369,. Constituent cranks. — Secured to the ends of the constituent crank shafts, which projects through the brass plates on both sides of the machine, are brass cranks (4-0, fig. 25) which are provided for the constituent amplitudes. Those on the left or height side of the machine are designated as the constituent height cranks and are used for the coefficients of the cosine terms of formula (458), and those on the right or time side of the machine are designated as the constituent time cranks and are used for the coefficients of the sine terms of formula (459). The time crank on each constituent crank shaft is attached 90° in advance (in the direction of rotation) of the height crank on the same shaft. For the constituents Sa and Ssa no time cranks are provided, as the coefficients of the sine terms corres- ponding to these constituents are too small to be taken into account. The direction of rotation of each constituent crank shaft with its con- stituent cranks is clockwise when viewed from the time side of the machine and counterclockwise when viewed from the height side. The details of a constituent crank are shown in figure 31. 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 b 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 undamped 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 any de- sired position by tightening up on the clamp screw, which, pressing HARMONIC ANALYSIS 1 AND PREDICTION OF TIDES 131 against the small spring at the back, forces the crank-pin block outward against the flanges of the groove with sufficient pressure to prevent any slipping. A milled head wrench B is used for tighten- ing the clamp screw. A small rectangular block e of hardened steel is fitted to turn freely upon the finely polished axle of the crank pin. This block is designed to fit into and slide along the slot of the con- stituent frame. 370. Positive and negative direction. — All the constituent crank 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 constituents in the upper range the positive direction will be upward and the negative direction downward, while for the constituents in the lower range the positive direction will be downward and the negative direction upward. 371. Constituent dials. — To indicate the angular positions of the constituent crank shafts, the pointer (a, fig. 31) moves around a dial (41, fig. 25) which is graduated in degrees. These dials are fastened to the frame of the machine back of the constituent 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 constituent crank 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 constituent the height crank will be in a positive vertical position and the corresponding time crank in a horizontal position. At a reading of 90° the height crank will be horizontal and the time crank in a negative vertical position. 372. With the face of the machine registering the initial epoch, such as January 1, hour, of any year, the value of t then being taken as zero, each constituent crank shaft may be set, by means of its releas- able gear, so that the dial readings will be equal to the a of the corre- sponding constituent as represented in formulas (458) and (459). If the machine is then put in operation, the dial readings will, for succes- sive values of t, continuously correspond to the angle (at-\-a) of the formulas, as the gearing already described will provide for the increment at. 373. Constituent sliding frames. — For each constituent crank there is a light steel frame {42, fig. 25) 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 frame there is a horizontal slot in which the crank pin slides. As the machine is operated the rotation of the crank 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 sliding frame, imparts a vertical harmonic motion to that frame. The frame is in its zero position when the center horizontal line of the slot intersects the axis of the crank shaft. Positive motion is the direction away from the medial horizontal plane of the machine and negative motion is toward the medial plane. The displacement of each constituent height frame from its zero position will always equal the product of the amplitude setting of the crank pin by the cosine 132 U. S. COAST AND GEODETIC STJBVEY of the constituent dial reading, and the displacement of each consti- tuent time frame will always equal the product of the amplitude setting by minus the sine of the constituent dial reading. 374. Constituent 'pulleys. — Each constituent frame is connected with a small movable pulley (43, fig. 25). For all constituents except M 2 , S 2 , N 2 , Ki, Oi, and Sa on the height side and M 2 , S 2 , N 2 , and M 4 on the time side this connection is by a single steel strip, so that the pulley has the same vertical motion as the corresponding frame. 375. Doubling gears. — Because of the very large amplitudes of some of the constituents two methods were used in order to keep the lengths of the cranks within practical limits. For M 2 , S 2 , and K x two sets of shafts and cranks were provided, so that the amplitudes of these constituents 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 large constituents is accomplished by the use of doubling gears between the sliding frame and movable pulley. Two spur gears with the ratio of 1:2 (48, fig. 25) are arranged to turn together on the same axis. The smaller gear engages a rack (46) attached to the sliding frame and the larger gear engages a rack (47) attached to the constituent 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 sliding frame causes a motion twice as great in the constituent pulley. Doubling gears are provided on the height side of the machine for constituents M 2 , S 2 , N 2 , K 1; Oi, and Sa and on the time side for constituents M 2 , S 2 , N 2 , and M 4 . 376. Scales for amplitude settings. — The scales for setting the con- stituent amplitudes are attached to the frame of the machine and are, in general, graduated into units and tenths (44, %• 25). The scales are arranged to read in a negative direction; that is, downward for the constituents of the upper range and upward for the constituents in the lower range. On a small adjustable plate (45) attached to each constituent pulley there is an index line which is set to read zero on the scale when the sliding frame is in its zero position. For setting the crank pins for the constituent 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°. 377. The scales on the height side of the machine, which are used in setting the coefficients of formula (458), 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 constituent which appears in each of the coefficients of formula (459). Dividing the members of this for- mula by m, the speed of constituent M 2 , it becomes S -JH sin (at+ a) = (460) 7/c; 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 the height and time crank for any constituent to be set in accord with the factor jH which is common to the coefficients of both formulas (458) and (459). There are also provided for special use on the time HARMONIC ANALYSIS 1 AND PREDICTION OF TIDES 133 side of the machine unmodified scales graduated uniformly to read in a positive direction. 378. Summation chains. — The summations of the several cosine terms in formula (458) and of the several sine terms in formula (459) are carried on simultaneously by two chains, one (27, fig. 25) on the height side and the other (28, fig. 26) 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. 379. Each of these chains is fastened at one end near the back part of the machine by a pair of adjusting screws (53, fig. 29, and 54, fig- 22). From these adjusting screws each chain passes alternately downward, under a constituent pulley of the lower range and upward over a con- stituent pulley of the upper range, spanning the space between the rear and middle section of the machine by two idler pulleys and con- tinuing until every constituent pulley on each side of the machine is included in the system. The movable pulleys are so arranged that the direction of the chain in passing from one to another is always vertical and parallel to the direction of the motion of the sliding frames. 380. Summation wheels. — The free or movable end of each of the chains is attached to a threaded grooved wheel (29, 30, fig. 25), 12 inches in circumference and threaded to hold more than seven turns of the chain, or about 90 inches in all. These are called the height and time summation wheels. Each is mounted on a shaft that admits a small lateral motion, and by means of a fixed tooth attached to the framework 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 summa- tion wheel; so that the path of the chain as it is wound or unwound from the summation wheel remains unchanged. 381. The height summation wheel (29, fig. 25) is located near the front edge of the middle section of the machine, where it receives the height summation chain directly from the nearest constituent pul- ley. 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 constituent pulley to the summation wheel. Counterpoise weights are connected with the shafts containing the summation wheels in order to keep the summation chains taut. 382. When all of the sliding 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 sliding frame in a positive direction will tend to unwind 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 sliding frames on either side of the machine moving simultaneously, the resultant motion, which is the algebraic sum of all, will be communicated to the sum- mation wheel. The motion of the sliding frame being transmitted to the chain through a movable pulley, the motion of 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 inch to the unit of amplitude, and there- 134 U. S. 00 AST AND GEODETIC SURVEY 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 of amplitude. 383. The zero position of the height summation wheel is indicated by the conjunction of an index line (50, fig. 25) on the arm attached to the wheel and an index line (51, fig. 25) 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 position when all the sliding frames on the height side of machine 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 constituent frames are set at zero. 384. The zero position of the time summation wheel is indicated by the conjunction of an index point (11, fig. 23) attached to the time summation chain and a fixed index (12, fig. 23) in the middle of the horizontal 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 sliding frames on the time side of machine are in their zero positions. 385. Predicted heights of the tide. — When the machine is in operation, the sum of all the cosine terms of formula (458) included in the settings 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. 23) and second by the ordinates of a tide curve that is automatically traced on a roll of paper (15, fig. 23). 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. 386. Height scale. — The height pointer is geared to make one com- plete 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 (458) 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. 23) 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 (458) 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 HARMONIC ANALYSIS 1 AND PREDICTION OF TIDES 135 machine is operated the height pointer will indicate the predicted height of the tide corresponding to the time shown on the time dials. 387. In order to increase the working scale of the machine when pre- dicting 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. 23). 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. 388. Predicted times of the tide.— Simultaneously with the summation of the cosine terms of formula (458) on the height side of the machine,, the summation of the sine terms of formula (460) , which was derived from formula (459), 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. 389. Automatic stopping device. — This device provides for auto- matically 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. 24) operated by an electromagnet (26) mounted under the desk top. The electric circuit for the electromag- net is closed by a contact spring that rests upon a hard-rubber cylinder (31, fig. 25) 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. Thi& 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 136 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. 24) just below the crank the circuit may be held open to prevent the automatic device from operating when so desired. 390. Nonreversing ratchet.-— Upon the crank shaft, close to the bear- ing in the desk frame, there is a small ratchet wheel and above this there is a pawl (2%, fig. 24) 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. 24) just above the crank the pawl is locked so that it cannot engage the ratchet, thus permitting the machine to be turned backward when desired. Pressure on another button releases the pawl. 1391. Tide curve.— The 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 mo vement of a fountain pen (13, fig. 23), 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. 392. Within the dial -case, near the upper right-hand corner, is a mandrel (33, fig. 25), which can be quickly removed and replaced. It is designed to hold the blank roll of paper, the latter being wound upon n 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. 25), 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. 25), then over the feed roller to the receiving roller (36, fig. 25), upon which it is wound. 393. 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 hour. A ratchet and pawl (37, fig. 25) 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. 394. To provide for the winding up of the paper on the receiving roller there is a sprocket wheel (38, fig. 25) 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 1 AND PREDICTION OF TIDES 137 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. 25) 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 tha mandrel on which the blank roll of paper is placed. 395. Marigram gears. — The pen that traces the tide curve is mounted in a carriage which is arranged to slide vertically on a pair of guiding rods and is controlled from a horizontal shaft at the back of the dial case. On this shaft there is mounted a set of three sliding change gears (18, fig. 26), which are designed to mesh, respectively r witri 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. 26), to which is attached one end of the pen-carriage chain (20, fig. 26). 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. 396. Scale of tide curve. — With a working scale of unity, the rotation of the height summation wheel, as transmitted through marigram gear ratio of 1:1' to the curve-line pen, will move the latter vertically 0.1 inch for each unit change in the sum of the harmonic terms and this may be taken as the basic or natural scale of the graphic record. This scale may be enlarged by the factor 3/2 or 2 through the use of one of the other gear ratios and may be further modified to any desired extent by the introduction of an arbitrary working scale factor. Letting G equal the marigram gear ratio (1, 3/2, or 2) and S equal the working scale factor applied to the amplitude settings,- the vertical scale of the graphic record may be expressed as follows: 1 inch of graph represents 10 /GS units of summation (461) 1 summation unit is represented by GS/ 10 inches in graph (462)! The scale ratio of the graph will differ with different units used in the predictions. Thus Graph scale (amplitude settings in feet) = GS/120 (463) Graph scale (amplitude settings in meters) = GS/393.7 (464) Graph scale (amplitude settings in decimeters) = GS/39. 37 (465) 1 397. In selecting the marigram gear ratio and scale factor for the predictions at any station, it is the general aim to secure as large a scale as possible while keeping the graph within the limits of the paper. Some consideration must be given also to the limits of the height dial scale and in some instances to the mechanical limits of the indi- vidual amplitude settings. The marigram gear ratio affects the graph only but the scale factor affects also the amplitude settings and the height dial readings. The extreme amplitude of the graphic 138 U. S. COAST AND GEODETIC STJUVEY record is limited by the width of the paper which extends 3 inches on either side of the medial line, but for mechanical reasons it is desirable in general to keep the record within a band 2% inches on either side of the medial line. The following table suggests suitable scale, dial, and gear combinations for different tidal ranges and different current velocities. The tabular marigram scales are applicable only when the foot or knot has been used as the unit for machine settings. The marigram amplitude limits given in the last column are expressed in the same unit that is used in setting the machine regardless of what unit that may have been. Working scale', height dial, marigram gear, and scale Tidal range limits Current velocity limits Working scale factor Height dial Mari- gram gear Marigram scale Mari- gram ampli- tude limit Settings in feet Settings in knots Feet 0. 0- 2. 5 2. 6- 3. 5 3. 6- 4. 4 1- 6. 6. 1- 8. 8. 1-10. 10. 1-12. 5 12. 6-16. 5 16. 6-20. 20. 1-25. 25. 1-32. 5 32.6- Knots 0. 0- 1. 1.1- 1.5 1. 6- 2. 2. 1- 3. 3.1-4.0 4. 1- 5. 5. 1- 6. 6. 1- 8. 8. 1-10. 10. 1-12. 5 12. 6-16. 16.1- 10 10 10 4 4 4 2 2 2 1 1 1 4 4 4 10 10 10 20 20 20 40 40 40 2:1 3:2 1:1 2:1 3:2 1:1 2:1 3:2 1:1 2:1 3:2 1:1 Ratio 1: 6 1: 8 1:12 1:15 1:20 1:30 1:30 1:40 1:60 1:60 1:80 1:120 Knots per inch 0.50 0.67 1.00 1.25 1.67 2.50 2.50 3.33 5.00 5.00 6.67 10.00 Units 1.5 2.0 3.0 3.7 5.0 7.5 io!o 15.0 15.0 20.0 30.0 When height dial readings are not required, and amplitude settings are in feet, a convenient graph scale of 1: 10 can be obtained by using any one of the following combinations; scale factor 12 with gear ratio 1 :1, rscale factor 8 with gear ratio 3:2, or scale factor 6 with gear ratio 2:1. 398. When the tide-predicting machine is used for the prediction of the tide-producing force, the graph scale to be adopted will depend upon the unit in which the force is to be expressed. Assume that the sum of all terms in the vertical component of the force (par. 79) is desired. Referring to paragraph 43, it will be noted that the extreme rvalue of this component due to the combined action of moon and sun is approximately 0.2 X 10~ 6 with the unit of force taken as g, the mean acceleration of gravity. In this case a convenient scale relation which will bring the graph within the desired limits on the paper is obtained by .adopting a working scale factor of 6X10 7 with the marigram gear ratio of 2 : 1. With this combination 0.1 foot of graph ordinate will represent 10~ 7 g units of force. In practice the scale factor would be combined with the general coefficient common to all terms in the formulas. 399. Pens. — The curve-line pen (13 K fig. 23) 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. HARMONIC ANALYSIS' AND PREDICTION OF TIDES 139 400. Hour-marking device. — The arm for the datum-line pen is se- cured 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. 26). A spring keeps the armatures at equal distances from their re- spective electromagnets. The upper electromagnet is designed for indicating 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. 25). 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 pen downward for an instant, making a corresponding jog in the datum line, 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 double jog for the beginning of each day, the downward stroke of the second jog indicating the zero hour. 401. 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. 23) is in contact with the fixed platinum index (12); that is to say, at the times of high and low waters. When this con- tact 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, &g. 24) just above the hand-crank shaft permits the cutting out of the current from the two electromag- nets. 402. Adjustment oj 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. 403. Height-chain adjustment. — All amplitudes should be set at zero, so that the turning of each constituent 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 constituent 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. 404. 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 constituent 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. 405. 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- 140 U. S. OOAST AND GEODETIC STJBVEY sary that the zero 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. 406. 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 dial case. 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. 407. 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 usually be 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 electromag- net 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 crank. The year index should be set to indicate the kind of year. 408. In the usual operation of the machine a ratchet prevents the operating crank from being turned backward, but this ratchet may be released when desired by pressing on a button in the side of the machine just above the crank. After the face of the machine has been thus set to register the beginning of the predictions the three main vertical shafts should be clamped to prevent them from turning. 409. To set the height amplitudes. — All the constituent cranks on the left or height side of the machine are first turned, by means of the releasable gears on the main vertical shafts, to a vertical position, the cranks of the upper range of constituents pointing downward and those in the lower range upward, in which position all angles will read 180°. For the long-period constituents 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 undamped 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 posi- tion. If no amplitude is given for any constituent, the corresponding crank must be set at zero. 410. 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 constituents Sa and Ssa the amplitudes are set on the height side only. HARMONIC ANALYSIS 1 AND PREDICTION OF TIDES 141 411. To set constituent 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 constituents oa and Ssa, for which there are no dials on the time side, as the readings are the same for both sides. As each constituent 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 undamped to permit them to turn. 412. 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 ring may also be conveniently used as a 4-foot scale. The scale to be used for any station is indicated in Form 445. In removing 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 being 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 pointer be set approximately 45° to the left of its zero position in order to interfere least with the removal or replacement of the scale. 413. The datum or plane oj reference. — The hand-operating crank should be turned forward or backward until the index of the summa- tion 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 position. When in doubt, the operating crank should be turned forward to obtain a number of conjunctions, the corresponding height dial reading for each being noted. The conjunction that corresponds 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 in 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 corresponds to the height of mean sea level above the datum which has been adopted for the predictions, this value being given in Form 445. 414. The marigram gear. — There are three gear combinations, desig- nated 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. 246037—41 10 142 U. S. COAST AND GEODETIC SURVEY 415. In setting up the machine for successive stations there is a mechanical advantage in making the necessary gear changes before 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 jamming the curve pen carriage and throwing the height chain off its pulleys when setting the amplitudes. 416. Inserting paper roll. — To place the paper on the machine, re- move 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 roll 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 chain. After the paper has been secured to the receiving roller these connections should be restored. 417. 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. 418. Verification of machine settings. — Each step in the adjustment and setting of the machine should be carefully checked before pro- ceeding 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 predictions for the same station for the preceding year are available, by turning the machine backward several days and then comparing the predicted tides with those previously obtained. 419. 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 registered on the time dials and the cor- responding height read off of the height dial. 420. If the predicted high and low waters for the year are desired, the operating crank is turned forward until the machine is auto- matically 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 HARMONIC ANALYSIS 1 AND PREDICTION OF TIDES 143 crank more slowly until the machine is stopped as the indexes come 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. FORMS USED WITH TIDE-PREDICTING MACHINE 421. Form 444) standard harmonic constants jor predictions (fig. 32). — This form provides for the compilation of the harmonic con- ,DEf«BTUEK TIDES Wmm STANDARD HARM0Nlc CONSTANTS FOR PREDICTION STATION ..J^ro* California 22 Long.l2Q. -51 Long... 120.85°. N. 1 COMPO- H AMPLITL'I'E Epoch «•-« B ! C | D xri 360°-*' -«' REMARKS M, [...St... .1.227. 0,320 0,260. 308,5. 304.3 .284,5... + 9,8 ..±..1,7. +14,2 ft ■*» 4,91 ..1,28... .1,04. J-.J0*.7 -518.3 . +..54,0.. -5.05., 0. + 61. 3!.. 72.98. 7. Time meridian ..1.2.0 = ..._?.« Extreme range .?.».? Dial 10 i o... h . ! in. 1 K, .1.001.. .0*053.. ..Q..6.0S.. .111,2.. ..17.Q..7.. ...99*1.. ...t.O.S. ...4...00... .+2.48...3I -111.7 .+169,7j.-1.90.,3 +251*6 -1.08,4 Marigrara scale ....l.J.2.0 .... a, 2*4.0 Permanent current The DATUM is a ptmw IWftm- mean I |™"«««-*prlnpr l lower low water ft. .... kn. ....rft. .+19.6 .+..9*3. .0*21... .2*43.. "(MK)*i .0.013. .25.3.7. ..+29.5 P.. 05... +76.8 7283,2 + 1 - s 4 0.009 176.6 + 3.4 0.04 +180.6 -180.0 (MX), + 1 „ 0.065 285,0 +13.6 0.26 + 61.4 -298.6 ".".¥..: 0.006 0.025 148.3 + 5.1 0.02 +206.6 -153.4 1 174.2 +18.0 0.10 +167,81 -192.2 (2N) ? (0,035 .260.5) +18.5 0.14 + 81.0 -279.0 I (00), (0.026.. (.0.009. .0,04.1.. 123.3) .306,6). 132,4 - 8.3 ...+ 6.0. + 4.9 .0.10 .0.04.. +245.0 + 47.4 -115.0 -312.6 "h s, " M, " + — 0.16 . +222.7! -137.3 i J, Mm (.0.048 117.2) -.3.8 .0.19... +246.6 -113.4 + i - Ssa Sa + | - 1 ! MSf ..+ | - ! Mf .+ - i p. (0,023 .....93.9) ..+13*1. ...0*0.9... .. +.2.53.0 -107.0 ! 1 <2Q), .0*107. (.0*013.. (Q.00.3. (0*0.16. ....98*9.. _3Q4*5). .304*1) ...87.1) +13*7 ...+.J2...0. .. + .1*4 .+18.0 .0.43. ..Q.08 . ..+.2.47* 4. -112. ,5 ..-305.5 , .0*01... ..D.Q6... .+..54,5 +254.9.-105,1 ! P, "(2SM)V ... M ' . L, .0,274. ..0,022. ...0*050 .107,7. 345.3 .3.0.7.. .5.. ..+..1.2 ..+14.7 ..+ ..5.5 1.10 +251.1 -103.9 + i ...0..Q9.. ..0.20... +359.5 - 0.5 .7.47,0 -313.0 (2MK), + 1- K 2 ..0...103.. 0.007 .2.69.8.. .105.8. ...+ ..1..0 ...+3.9,3. .0,41.. 0,03 .T..69..2 -290.8 (MS), "\"+ "-' Source of constants £j!Qia.._observ I.9.br.ua.ry...l3.,... 19.19...... ed.. hourly. heights. .for. 163. days beginning. F.J.K., I.'arch . 2.9, ...1.923. 1 ] Compi ed by .....L iP*P.....J Iar.Qh...28 *..1923., Verified by (Dale) Figure 32. 144 U. S. COAST AND GEODETIC SURVEY stants for use in the prediction of the tides and also for certain per- manent 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. 422. The constituents are listed in an order that conforms to the ar- rangement of the corresponding constituent shafts and cranks on the predicting 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. 423. The column of Remarks provides for miscellaneous informa- tion 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. 424. 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 on page 138. 425. The principal hydrographic datums in general use are as fol- lows: Mean low water for the Atlantic and Gulf coasts of the United States and Puerto Rico. Mean lower low water for the Pacific coast of the United States, Canada, and Alaska, and the Hawaiian and Philippine Islands. Approximate low water springs for the rest of the 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 determined from a reduction of the high and low waters. 426. Column A of Form 444 is designed for the differences by which the epochs of the constituents are adapted once for all for use with the unmodified Greenwich (V -\-u) } s of each year. These dif- ferences take account of the longitude of the station and also of the time meridian used for the predictions, and are computed by the formula k'-k=pL-~ (466) in which k —k= adapted epoch— true epoch. p = subscript of constituent, which indicates number of periods in one constituent day. For the long-period constituents Mm, Ssa, Sa, MSf, and Mf, p should be taken as zero. L= longitude of station in degrees ;+ if west, — if east. a = speed of constituent 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 constituent speeds (a) from table 2. 427. Column B is designed for the reduction of the amplitudes to the working scale of the machine. The scale is unity when the 40- HARMONIC ANALYSIS' AND PREDICTION OF TIDES 145 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. 428. Columns C and D are designed to contain the adapted epochs in positive and negative forms which may be used additively with the Greenwich (T 7 +^)'s. It will be found most convenient to compute column D first, by applying the difference in column A to the k 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. DEPARTMENT OF COMM U. S. COAST AND GEOOtTK Hj Form No. 445 TIDES SETTINGS FOR TIDE PREDICTING MACHINE Station Morro, California Year Ifl 23 Compo- nent Aupli- TPDB SETTING DIAL SETTING REMARKS Jan. l.C Feb. I,0 h |r>ec.31,24'' M t ft. -h*r- ...5.*1Q... ..;8.2 t 9_. 54 47*1. .1.85,7 . Time Meridian .A?.P ." Dial 10 W. s 2 1.30 54 54 N 2 1.10 214 133 226 K, --------- . 3.55 ... 0.20 255 252 286 180 255 94 Marigram gear ?.?."" Marigram scale * Ao 2.40 Permanent current ft. kn. 0, .2.00... 287 221 28 .....Ms.... ™(MK)V 0.05 200 93 142 s 4 0.05 180 180 180 (MN), ... •■2 0.25 33 6 323 s, ... M2 "X2NJ"» 0.10 251 179 93 0.15 345 219 26« (00), 0.05 40 167 298 X 2 ...0.0.5... .338 293 249 s, ... . M, .0.2.5:.. 347 329 217 ! Ji 0.15 140 216 228 Mm ... ... ... — Ssa — _ Sa — ... — . MSf ___ ... Mf — ... Pi 0.05 219 162 150 Q. 0.35 35 234 47 T 2 0.10 56 25 56 R» ... ... ....'... ...0*05... .154 358 78. p, 1.10 242 211 242 (2SM), M, ...0*10 ...61 .149... 7 ...158 32 .....??..... .0,20... 338 (2MK) 3 ... K, 6.30 264 325 264 M, 0.05 20 237 63 (MS), • Compu ed by L.P.D. March 28, 1923 v F.J.H, March .29^.1923 Predict* id by Date.. Figure 33. 146 U. S. COAST AND GEODETIC STTRVEY The values for column C may then be obtained by applying 360° to the negative values in column D. 429. Form 44$, settings for tide-predicting machine (fig. 33). — This form is designed for the computations of the settings for the predicting machine for the beginning of each year of predictions. The forms are bound in books, a separate book being used for each year of predictions. This form is used in connection with Form 444, and for convenience the order of arrangement of the constituents is identi- cal in the two forms. The name of the station, the time meridian, the height dial, marigram gear, marigram scale, and datum plane are copied directly from Form 444. 430. For the amplitude settings the amplitudes of column B of Form 444 are multiplied by the factors/ from table 14 for the year for which the predictions are to be made. A convenient way to apply these factors is to prepare a strip of paper with the same vertical spacing as the lines on Form 444 and enter the factors / for the required year on this strip. The strip may then be placed alongside of column B of Form 444 and the multiplication be performed. The same strip will serve for every station for which predictions are to be made for the given year. It has been the recent practice to enter the amplitude settings to the nearest 0.05 foot as being sufficiently close for all practical purposes. 431. For the dial settings for January 1, hour, the Greenwich equilibrium arguments of (F +^)'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 (Vo-f-iO's a strip similar to that used for the factors/ should be pre- pared. The same strip will serve for all stations for the given year. For the dial settings it is customary to use whole degrees, except for constituent M 2 , for which the setting is carried to the first decimal of a degree. 432. The settings for February 1 and December 31 are used for checking purposes to ascertain whether there has been any slipping of the gears during the operation of the machine. To obtain the dial settings for February 1, h , and December 31, 24 h , prepare strips similar to those for the/'s and (F +^)'s. On one enter the angular motion of the constituents from January 1 , h , to February 1 , h ; on a second and a third strip, the angular motion for February 1, h , to December 31, 24 h , for a common and leap year, respectively. For checking purposes a fourth and fifth strip may contain the angular changes for a complete common and a complete leap year, respectively. The values for these strips may be obtained from table 36. These strips will be found more convenient if arranged with two columns each, one column containing the values in a positive form and the other column containing the equivalent negative value which is obtained by subtracting the first from 360°. These strips are good for all years, distinction being made between the common and leap years. By applying the first strip to the dial settings for January 1 the values for February 1 are readily obtained, and by applying the second or third strip to the latter settings those for the end of the year are obtained. The values obtained by applying the fourth or fifth strips to the settings for January 1 should also give the correct setting for the end of the year, and thus serve as a check. The HARMONIC ANALYSIS AND PREDICTION OF TIDES 147 angular changes for computing the settings for any day of the year may be obtained from tables 16 and 17. PREDICTION OF TIDAL CURRENTS 433. Since the tidal current velocities in any locality may be expressed by the sum of a series of harmonic terms involving the same periodic constituents that are found in the tides, the tide- predicting machine may be used for their prediction. For the cur- rents, however, consideration must be given to the direction of flow, and in the use of the machine some particular direction must be assumed. At present the machine is used for the prediction of reversing currents in which the direction of the flood current is taken as positive and the maximum velocity in this direction corre- sponds to the high water of the predicted tide. The ebb current is then considered as having a negative velocity with its maximum corresponding to the low water of the predicted tide. Rotary cur- rents may be predicted by taking the north and east components separately but the labor of obtaining the resultant velocities and directions from these components would be very great without a machine especially designed for the purpose. Predictions can, how- ever, be made along the main axis of a rotary movement without serious difficulties. Formulas for referring the harmonic constants of the north and east components to any desired axis are given in Coast and Geodetic Survey Special Publication No. 215, Manual of Current Observations. 434. The harmonic constants for the prediction of current velocities are derived from current observations by an analysis similar to that used in obtaining the harmonic constants from tide observations. In the current harmonic constants, however, the amplitudes are expressed in a unit of velocity, usually the knot, instead of the linear unit that is used for the tidal harmonic constants. Forms 444 and 445 for the computation of the settings for the tide-predicting machine are applicable for the current predictions and the procedure in filling out these forms is essentially the same as described in paragraphs 421-432 for the tide predictions. The node factors (/) and arguments (Vo+u) are the same as for the tides. The height dial, marigram gear and scale suitable to the current velocity can be obtained from the table on page 138. Instead of a sea level elevation there should be entered in the column of ' 'Remarks" the velocity of any permanent current along the axis in which the predictions are to be made. This velocity should be marked plus (+) or minus (— ) according to whether the permanent current is in the flood or ebb direction. 435. The predicting machine is set with the current harmonic con- stants in the same manner as for the tidal harmonic constants. To take account of the permanent current the height summation wheel should be brought to its zero position and the height hand then set at a dial reading corresponding to the velocity of the permanent current, the hand being set to the right of the scale zero if the per- manent current is in the flood direction and to the left if in the ebb direction. The hand crank should be then turned to bring the height hand to its zero position and the curve-pen set at the medial line of the paper, this line now representing zero velocity or slack water. 148 U. S. COAST AND GEODETIC SURVEY 436. The operation of the machine for the prediction of the cur- rents is similar to that for the prediction of the tides. The machine automatically stops at each maximum flood and ebb velocity and the corresponding times and velocities are then recorded, the flood veloci- ties being read to the right and the ebb velocities to the left of the scale zero. In the prediction of the currents the times of slack water are also desired. These are indicated by the zero position of the recording hand as well as by the intersections of the curve and medial line in the graphic record. The velocity of the current at any inter- mediate time can be read directly from the height dial when the machine has been turned to the time desired and it may be also scaled from the graphic record. 437. Predictions of hydraulic currents in a strait, based upon the difference in the tidal head at the two entrances, may be made by means of harmonic constants derived from the tidal constants for the entrances. Differences in tidal range or in the times of the high and low waters at the two ends of a strait will cause the water surface at one end alternately to rise above and fall below that at the other end, thus creating a periodic reversing current in the strait. Theo- retically, disregarding friction or inertia, the velocity of the current would vary as the square root of the difference in head, being zero when the surface is at the same level at both ends and reaching a maximum when the difference is greatest. Actually they will gen- erally be a lag of some minutes in the response of the current movement to the difference in head which must be determined from observations. 438. Let the two ends of the strait be designated by A and B, with the flow from A toward B considered as flood or positive and the flow in the opposite direction as ebb or negative. With the waterway receiving the tide from two sources, the application of the terms "flood" and "ebb" will be somewhat arbitrary, and care must be taken to indicate clearly the direction assumed for the flood move- ment. In the following discussion tidal constants pertaining to entrances A and B will be distinguished by subscripts a and b, respec- tively, and those pertaining to the difference in tidal head by the subscript d. Since the usual constituent epochs known as "kappas" refer to the local meridian, it will be necessary for the purpose of comparison between places on different meridians to use the Green- wich epochs "6" (par. 226), these being independent of local time and longitude. 439. For any one constituent let T represent time as expressed in degrees of the constituent reckoned from the phase zero of its Greenwich equilibrium argument. Also let Y a and Y b represent the height of the constituent tide for any time T as referred to the mean level at locations A and B, respectively; and let Y d equal the difference (Y a — Y b ). Formulas for heights and difference may now be written Y a =H a cos (T-G a ) for location "A" (467) Y b =H b cos (T-G b ) for location "B" (468) Y d =H a cos (T-G a )-H b cos (T-G b ) = (H a cos G a -H b cos G b ) cos T+{H a sin G a -H b sin G b ) sin T =H d cos (T-G d ) (469) HARMONIC ANALYSIS' AND PREDICTION OF TIDES 149 in which H d =[H*+W-2HJI b cos (G>-Ga)Y G d =tsai ! H a sin G a —H b sin G t H a cos G a —H b cos Gi (470) (471) The proper quadrant for G d is determined by the signs of the numer- ator and denominator of the above fraction, these being the same, respectively, as for the sine and cosine of the angle. Formulas (470) and (471) may be solved graphically (fig. 34) by drawing from any point C a line CD to represent in length and direction H a and G a , respectively; from the point D a line DE to represent in length and direction H b and (G b ±180°), respectively. The connecting line from 90' 180°- C to E will represent by its length the amplitude H d and by its direc- tion the epoch G d . 440. Formulas (470) and (471) may be modified to adapt them for use with tables 41 and 42. From (470) we may obtain or H d IH a =[l J r(H b IH a y+2(H b IH a ) cos (G b -G a ±lS0°)]* HilH b =[l + (H a IH b y+2{H a IH b ) cos (G a -G b ±lS0°W and from (471) we have Tan (G d -G a ) = (H b /H a ) sin «?»- T . Then in the spherical triangle ^Ti, the three sides are A T , v, and (N-£), and the opposite angles are respectively (180°-/), i, and co. 156 U. S. COAST AND GEODETIC SURVEY Therefore we have the following relations which may be used in com- puting the values of /, v, and £ in the table: cos 7= cos i cos co— sin i sin co cos N = 0.91370 — 0.03569' cos N tan i(N-^+v)= COS \ ( : 03 ~ l ), tan £#=1.01883 tan JJV cos f ( l co^-^; tan h(N-£-v)= Sm li"~7 l ), tan £iV=0.64412 tan JiV sin i(co+i) z For the computation of / and 2/', formulas (224) and (232) on pages 45-46 may be used. The tabular values themselves were taken from the preceding edition of this work where they were based upon formulas differing slightly from those given here but any differences arising from the use of the latter may be considered as negligible. Table 7. Values of log R a for amplitude of constituent L 2 . — Values in this table are based upon formula (213) on page 44. Table 8. Values of R for argument of constituent L 2 . — Values in this table are derived from formula (214) on page 44. Table 9. Values of log Q a for amplitude of constituent Mi. — Values in this table are based upon formula (197) on page 41. Table 10. Values of Q for argument of constituent Mi. — Values in this table are derived from formula (203) on page 42. Table 11. Values of u for equilibrium arguments. — This table is based upon the ^-formulas in table 2 and includes values for the principal lunar constituents for each degree of N. The u's of L 2 and Mi, which are functions of both N and P are given separately in table 13 for the years 1900 to 2000. Table 12. Log factor F for each degree of I. — The factor F is the reciprocal of the node factor/ to which references are given in table 2. The values in table 12 are based upon the formulas for these factors and are given for all the lunar constituents used in the tide-predicting machine, excepting values for L 2 and M 2 which are given separately in table 13. Table 13. Values of u and log F for L 2 and M 2 . — From a com- parison of the u's of constituents L 2 , Mi, and M 2 in table 2, it will be noted that the following relations exist: u of L 2 = (u of M 2 ) —R uoi Mi=J(vof M 2 ) + Q Also, the following relations may be derived from formula (215) on page 44 and formula (207) on page 43 since the factor F is the recip- rocal of the node factor /: log i^(L 2 )=log i^(M 2 )+log R a log^(Mi)=logF(Oi)+logQ a The values for table 13 were computed by the above formulas, the component parts being taken from tables 7 to 12, inclusive. The values for log F(Mi) in this table are in accord with Darwin's original HARMONIC ANALYSIS AND PREDICTION OF TTDES 157 formula from which a factor of approximately 1.5 was inadvertently omitted (see page 43). Table 14. Node factor f for middle of each year 1850 to 1999. — The factor/ is the reciprocal of factor F. The values for the years 1850 to 1950 were taken directly from the Manual of Tides, by R. A. Harris, and the values for 1951 to 1999 were derived from tables 12 and 13. Table 15. Equilibrium argument (V -{-u) for beginning of each year 1850 to 2000. — The equilibrium argument is discussed on page 22. The tabular values are computed by the formulas for the argument in table 2, the V referring to the value of V on January 1, hour Greenwich mean civil time, for each year, and the u 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 u 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 month, day, and hour, and are computed from the hourly speeds of the constituents as given in table 2. The differ- ences refer to the uniformly varying portion V of the argument, it being assumed that for practical purposes the portion u is constant for the entire year. The approximate Greenwich (Vo+u) for any desired Greenwich hour may be obtained by applying the appropriate differences from tables 16, 17, and 18 to the value for the first of January of the required year, as given in table 15. To refer this Greenwich (V -\-u) to any local meridian, it is necessary to apply a further correction equal to the product of the longitude in degrees by the subscript of the constituent, which represents the number of periods in a con- stituent day. West longitude is to be considered as positive and east longitude as negative, and the subscripts of the long-period constituents are to be taken as zero. This correction is to be subtracted. The (V -\-u) 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 u 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 19 J/.. — This is a multiplication table especially adapted for use with Form 194, the multipliers being the sines of multiples of 15°. Table 20. Augmenting factors . — A discussion of augmenting factors is given on page 71. The tabular values for the short-period constit- uents are obtained by formulas (308) and (309) on page 72, and those for the long-period constituents by formulas (403) and (404) on page 92. For constituents S x , S 2 , etc. the augmenting factor is unity. Tables 21 to 26. — These tables represent perturbations in K^ and S 2 due to other constituents of nearly equal speeds. They are based upon formulas (359) to (364), inclusive, on page 83. 246037—41 11 158 U. S. COAST AND GEODETIC SURVEY Table 27. Critical logarithms for Form 2^5. — 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. Constituent speed differences. — The constituent speeds as given in table 2 were used in the computation of this table. Table 29. Elimination factors. — These tables provide for certain constant factors in formulas (389) and (390). Separate tables for each length of series and different values for each term of the formulas are required. The tabular values are arranged in groups of three, determined as follows: First value = logarithm of - — .?, — e ■-• tv i(6— a)r Second value = natural number =-X^ — r^— always taken as positive. * K&-a)r Third value =|(6— a)t, if — — ~ — r-^— is positive, or i(6 — a)r± 180, if — -~ — H— is negative. i(b-a)r Table 30. Products for Form 2^5. — This table is designed for ob- taining the products for columns (6) and (7) of Form 245. Table 31. For construction of primary stencils. — This table gives the differences to be applied to the solar hours in order to obtain the constituent hours to which they most nearly coincide. Each differ- ence applies to several successive solar hours, but for brevity only the first solar hour of each group to which the difference applies is given in the table. An asterisk (*) indicates that the solar hour so marked is to be used twice or rejected according to whether the constituent speed is greater or less than \hp, when in the summation it is desired to assign a single solar hour to each successive constituent hour. For the usual summations in which each solar hour height is assigned to the nearest constituent hour no attention need be given to the asterisk. The table is computed by substituting successive integral values for d in formula (243) and reducing the resulting solar hour of series (shs) to the corresponding day and hour. The solar hour to be tabulated is the integral hour that immediately follows the value of (shs) of the formula. If the fractional part of (shs) exceeds 0.5, the tabular solar hour is marked by an asterisk (*). The successive values of d, although used positively in formula (243), are to be considered as negative in the application of the table when the speed of the con- stituent is less than 15p. When the constituent speed is greater than 15p, the difference is to be taken as positive. All tabular differences are brought within the limits +24 hours and —24 hours by rejecting multiples of ±24 hours when necessary, and for convenience in use all differences are given in both positive and negative forms. HARMONIC ANALYSTS AND PREDICTION OF TIDES 159 The following example will illustrate the use of the table: To find constituent 2Q hours corresponding to solar hours 12 to 23 on 16th day of series. By the table we see that solar hour 12 of the 16th day of series is within the group beginning on solar hour 8 of the same day with the tabular difference of +19 or —5 hours, and that the differ- ence changes by —1 hour on solar hours 15 and 21, the latter being marked by an asterisk. Applying the differences indicated, we have for these solar hours on the 16th day of series: Solar hour 12, 13, 14*, 15, 16, 17, 18, 19, 20, 21*, 22, 23 Difference -5 -5 -5 -6 -6 -6 -6 -6 -6 -7 -7 -7 Constituent 2Qhour 7, 8, 9*, 9, 10, 11, 12, 13, 14, 14*, 15, 16 In the results it will be noted that the constituent hours 9 and 14 are each represented by two solar hours. If it should be desired to limit the representation to a single solar hour each, the hours marked with the asterisk should be rejected. To find constituent 00 hours corresponding to solar hours to 18 on the 22d day of series. The hour of the 22d day is in the group beginning on solar hour 14 of the preceding day with the tabular difference of +14 or —10 hours, and changes of +1 hour in the differences occur on solar hours 3 and 17 of the 22d day. It will be noted that the hour 3 is marked by an asterisk. Applying the differences from the table as indicated, we have for the 22d day of series: Solar hours.. 0. 1, 2, 3*, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, W, 16*, 17, 18 Differences -+14, +14, +14, +15, +15, +15, +15, +15, +15, -9, -9, -9, -9, -9, -9, -9, -9, -8, -8 Constituent 00 hours... 14, 15, 16, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 9, 10 In the results it will be noted that constituent hours 17 and 8 are missing. If it is desired to have each of these hours represented also, the solar hours marked by asterisks will be used again. In this table the constituents have been arranged in accordance with the length of the constituent day. Table 32. Divisors jor primary stencil sums. — This table contains the number of solar hourly heights included in each constituent hour group for each of the standard length of series when all the hourly heights have been used in the summation. Table 33. For construction of secondary stencils. — Constituent A is the constituent for which the original primary summations have been made, and constituent B is the constituent for which the sums are to be derived by the secondary stencils. The "Page" refers to the page of the original tabulations of the hourly heights in Form 362. The differences in this table were calculated by formula (252), and the corresponding "Constituent A hours" from formula (250), m being assigned successive values from 1 to 24 for each page of record. Special allowance was made for page 53 of the record to take account of the fact that in a 369-day series this page includes only 5 days of record. The sign of the difference is given at the top of the column. For K-P and R-T the positive sign is to be used for constituents K and R and the negative sign for constituents P and T. For brevity all the 24 constituent hours for every page of record are not directly represented in the table. The difference for the omitted hours for any page should be taken numerically one greater 160 IT. S. COAST AND GEODETIC SURVEY than the difference for the given hours on that page. For an example, take the hours for page 2 for constituent 00 as derived from con- stituent J. According to the table the difference for the constituent hours 10 to 3, inclusive, is 9 hours; therefore the difference for the omitted hours 4 to 9, inclusive, should be taken as 10 hours. For constituent 2Q as derived from constituent O the three differences usually required for each page are given in full. The use of the table may be illustrated from the example above, as follows: Page 2— J-hours 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 Difference +9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 9, 9 OO-hours 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 19, 20 J-hours 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23 Difference +9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9 OO-hours 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8 The period 24 hours should be added or subtracted when necessary in order that the resulting constituent hours may be between and 23. Table 34. For summation of long-period constituents. — This table is designed to show the assignment of the daily page sums of the hourly heights to the constituent divisions to which they most nearly correspond. The table is based upon formula (395). The constituent division to which each day of series is assigned is given in the left-hand column. For Mf, MSf, and Mm there will frequently occur two consecutive days which are to be assigned to the same constituent division. In such cases the day which most nearly corresponds to the constituent division is the only one given in the table, and this is marked by an asterisk (*). The missing day, whether it precedes or follows the one marked by the asterisk, is to be assigned to the same constituent division. For Sa a number of consecutive days of series are assigned to each constituent division. In the table there are given the first and last days of each group. Table 35. Products aS 1 15 for Form 444- — This table contains the products of constituent speeds and time meridian longitudes for formula (466) which is used in obtaining values of (k'-k) for column A of Form 444. Table 36. Angle differences for Form 44/>- — This table gives the differences for obtaining and checking the dial settings for February 1 and December 31, as entered in Form 445. The differences are de- rived from tables 16 and 17. Table 37. Coast and Geodetic Survey tide-predicting machine No. 2 — General gears. — This table gives the details of the general gearing from the hand-operating crank to the main vertical shafts, together with the details of the gearing in the front section or dial case. In this table the gears and shafts are each numbered consecutively for con- venience of reference, the gears being designated by the letter G and the shafts by the letter S. In the second column are given the face of each bevel or spur gear and the diameter of each shaft. The next two columns contain the number of teeth and pitch of each bevel and spur gear. The pitch is the number of teeth per inch of diameter of the gear. The worm screw is equivalent to a gear of one tooth, as it requires a complete revolution of the screw to move the engaged wheel HARMONIC ANALYSIS AND PREDICTION OF TTDES 161 one tooth forward. The period of rotation of each shaft and gear is relative and refers to the time as indicated on the face of the machine, which for convenience is called dial time. Table 38. Coast and Geodetic Survey tide-predicting machine No. 2 — Constituent gears. — This table contains the details of the gearing from the main vertical shafts to the individual constituent cranks. Column I gives the number of teeth in the bevel gear on the main vertical shaft; column II, the number of teeth in the gear on the intermediate shaft that meshes with the gear on the vertical shaft; column III, the number of teeth in the gear on the intermediate shaft that meshes with the gear on the constituent crank shaft; and column IV, the number of teeth in the gear on the crank shaft. For the long-period constituents the worm gear is taken as the equivalent of one tooth. For each of these constituents there is a short secondary shaft on which sliding gears are mounted, but the extra gears do not affect the speed of any of the crank shafts except that for constituent Sa in which case a ratio of 1:2 is introduced. The crank-shaft speed per dial hour for each constituent is equal to 30°X — i r^X — t^ ¥T7- For constituent Sa the product of column II column IV both values appearing in each of the columns II and III is to be taken as the value for the column. The column of "Gear speed per dial hour" contains the speeds as computed by the above formula. For comparison the table contains also the theoretical speed of each of the constituents and the accumulated error per year due to the difference between the theoretical and the gear speeds. For convenience of reference the table includes also the maximum amplitude settings of the constituent cranks. Table 39. Synodic periods of constituents. — This table is derived from table 28, the period represented by 360° being divided by the speed difference and the results reduced to days. Table 40. Day of year corresponding to any date. — This table is convenient for obtaining the difference between any two dates and also in finding the middle of any series. Table 41. Values of h in formula h=(l-\-r 2 -\-2r cos x)K — This table may be used with formulas (472) and (473) on page 149 to obtain constituent amplitudes for the prediction of hydraulic currents. Table 42. Values of k in formula k=tan~ 1 1 —, This table J - 1 -\-r cos x may be used with formulas (474) and (475) on pages 149-150 to obtain constituent epochs for the prediction of hydraulic currents. 162 U. S. COAST AND GEODETIC STJRVEY Table 1. — Fundamental astronomical data Mean distance, earth to sun 92, 897, 416 miles » Mean distance, earth to moon 238, 857 miles ■ Equatorial radius of earth (Hayford's Spheroid of 1909) 3, 963. 34 miles « Polar radius of earth (Hayford's Spheriod of 1909) 3, 949. 99 miles » Mean radius of earth (a), (Intern. Ell.) 6,371,269 meters b = 20,903,071 feet = 3,958. 91 miles Solar parallax (Paris Conference) 8.80" a = 0. 000, 042, 66 radian Lunar equatorial horizontal parallax (Brown) _57' 2.70" a = 0. 016, 59 radian Mean solar parallax in respect to mean radius (a/ci) 0. 000, 042, 61 radian Mean lunar parallax in respect to mean radius (a/c) 0. 016, 57 radian Eccentricity of earth's orbit (e0, epoch Jan. 1, 1900 0. 016, 75 c Eccentricity of moon's orbit (e) 0. 054, 90 d Obliquity of the ecliptic (a>), epoch Jan. 1, 1900 23° 27' 8.26" c = 23. 452° Inclination of moon's orbit to plane of ecliptic (i) 5° 08' 43.3546" d = 5. 145° Ratio of mass of sun to combined mass of earth and moon (Sitter) 327, 932 b Ratio of mass of earth to mass of moon (Hinks) 81. 53 b Mass of sun/mass of earth (S/E) 331, 954 Mass of moon/mass of earth (M/E) 0. 012, 27 Solar coefficient U x = (S/E) (a/c 3 .2569 x 10~ 7 Basic factor U = (M/E) (a/c)* .5582x10-7 Solar factor S' = U x /U 0.4602 In the following formulas for longitude, T represents the number of Julian centuries (36525 days) reckoned from Greenwich mean noon, December 31, 1899 (Gregorian Calendar). Mean longitude of sun (h) = 279° 41' 48.04" + 129,602,768.13" T + 1.089" T 2 « Longitude of solar perigee (p = 281° 13' 15.0" + 6,189.03" T + 1.63" T 2 + 0.012" T 3 ■ Mean longitude of moon (s) = 270° 26' 14.72"+ (1336 rev. + l, 108,411.20") T+ 9.09" T 2 + 0.006,8" T 3 « Longitude of lunar perigee (p) = 334° 19' 40.87" + (11 rev. + 392,515.94") T - 37.24" T 2 - 0.045" T 3 « Longitude of moon's node (N) = 259° 10' 57.12" - (5 rev. + 482,912.63") T + 7.58" T 2 + 0.008" T 3 • Ratio of mean motion of sun to that of moon (m) 0. 074, 804 ■ American Ephemeris and Nautical Almanac for year 1940, p. xx. b Table of astronomical constants by W. de Sitter, published in Bulletin of the Astronomical Institutes of the Netherlands, Vol. VIII, No. 307, July 8, 1938, pp. 230-231. e Astronomical Papers for the American Ephemeris, by Simon New comb; Vol. VI, pp. 9-10, and Vol. IX, pt. 1, p. 224. d The Solar Parallax and Related Constants, by William Harkness, p. 140. HARMONIC ANALYSIS AND PREDICTION OF TIDES 163 Table 1. — Fundamental astronomical data — Continued MEAN LONGITUDE OF SOLAR AND LUNAR ELEMENTS FOR CENTURY YEARS Epoch, Gregorian calendar Greenwich mean civil time Sun h Solar perigee Pi Moon s Lunar perigee P Moon's node AT 1600, Jan. 1, hour. ... O 279. 857 280. 624 280. 407 280. 190 279. 973 o 276. 067 277. 784 279. 502 281. 221 282. 940 99. 725 47. 604 342. 313 277. 026 211. 744 o 7.417 116. 501 225. 453 334. 384 83. 294 o 301. 496 1700, Jan. 1, hour - 167. 343 1800, Jan. 1, hour. . .. 33. 248 1900, Jan. 1, hour. . . 259. 156 2000, Jan. 1, hour _ 125. 069 RATE OF CHANGE IN MEAN LONGITUDE OF SOLAR AND LUNAR ELEMENTS (EPOCH, JAN. 1, 1900) Elements Per Julian cen- tury (36525 days) Per common year (365 days) Per solar day Per solar hour Sun (h) o 100r+ 0.769 1.719 1336r+307. 892 llr+109. 032 -5r-134. 142 o 359. 761, 28 0. 017, 18 13r+129. 384, 82 40. 662, 47 -19.328,19 o 0. 985, 647, 3 0. 000, 047, 1 13. 176, 396. 8 0.111,404,0 -0. 052, 953. 9 o 0. 041, 068, 64 Solar perigee (pi) ... - .- . . 0. 000, 001, 96 Moon («)-.. . 0. 549, 016, 53 Lunar perigee (p) Moon's node (A 7 ") 0, 004, 641. 83 -0,002,206,41 MEAN ASTRONOMICAL PERIODS (Symbols refer to rate of change in mean longitude) Solar days Sidereal day, 360°/ (360° +h) 0.997,270 Lunar day, 360°/(360°+/i-s) 1.035,050 Nodical month, 360°/(s-A) 27.212,220 Tropical month, 360°/s 27.321,582 Anomalistic month, 360°/ (s-p) 27.554,550 Synodical month, 360°/(s-h) 29.530,588 Moon's evectional period, 360°/(s-2h+p) 31,811,939 Eclipse year, 360°/(h-N) 346.620,0 Tropical year, 360°//i 365.242,2 Anomalistic year, S60°/(h-p l ) 365.259,6 Common year 365.000,0 Mean Gregorian year 365.242,5 Mean Julian year 365.250,0 Leap year 366.000,0 Evectional period in moon's parallax, 360°/(/i— p) 411.784,7 Revolution of lunar perigee, 360°/p 8.85 Julian years Revolution of moon's node, 36Q°/N- 18.61 Julian years Revolution of solar perigee, 360°/Pi-- 209 Julian centuries 164 U. S. COAST AND GEODETIC SURVEY Table 2. — Harmonic constituents Ref. Symbol Argument (E) Speed per Coeffi- cient (C) Factor-i No. V u solar hour formula Ai LUNAR LONG-PERIOD Zero (permanent term) .... TERMS, FORMULA (62) o zero 0. 544, 374, 7 1. 088, 749, 4 0. 471, 521, 1 1. 015, 895, 8 1. 098, 033, 1 1.642,407,8 0. 553, 658, 4 2. 186, 782, 5 1. 569, 554, 3 0. 626, 512, 2. 113, 928, 8 0.082,137,3 137) 0. 549. 016, 5 1. 093, 391, 2 0. 004, 641, 8 1. 647, 049, 6 2. 191, 424, 3 85) zero 0. 041, 066, 7 0. 082, 133, 4 0. 082, 137, 3 0. 123, 204, 0. 041, 070, 6 0. 164, 270, 6 0. 5044 0. 0827 0. 0068 0. 0116 0. 0084 0. 1566 0. 0303 0. 0043 0.0040 0. 0043 0. 0006 0. 0025 0. 0001 0. 0399 0. 0065 0. 0022 0. 0032 0. 0009 0. 2340 0.0118 0. 0003 0. 0728 0. 0043 0. 0006 0. 0002 (73) A 2 Mm s—p zero zero zero zero. (73) As 2s-2p (73) At s— 2h+p (73) Ai MSf Mf 2s-2h (73) At 2s -2f (74) Ai . 3s— p. .. -2£ (74) As s+p+180° 4s— 2p___ -2? (74) Ai... -2f -n -2? (74) 3s—2h+p (74) An—. s-f2ft-p+180° (74) An—. 4s-2ft -2£ (74) 2ft —2?. (74) LUNAR LONG-PERIOE S-90° TERMS, FORMULA ( -£ (141) 2s-p-90°_ -£ (141) p-90° -^ (141) 3s-90° -3^... (142) 4s— p— 90° -3£ (142) Bi SOLAR LONG-PERIOD Zero (permanent term) TERMS, FORMULA (] unity B 2 h—pi. ... .. zero zero zero... .. .. . .. unity B 3 2h-2pi unity Be - Ssa 2ft unity B 7 .. 3ft— pi zero. . unity B s . . ft+pi+180° zero. unity B 9 - 4ft-2pi zero unity B 6i Sa 0,.... Qi (Mi).... 2Qi pi SOLAR LONG-PERIOD TERM, PARAGRAPH ft.. - .. _. _. zero . 119 0.041,068,6 unity Au LUNAR DIURNAL 1 T-2s-f-ft+90°- ^ERMS, FORMULA (63 +2%-v ) 13, 943, 035, 6 13, 398, 660, 9 14. 487, 410, 3 12, 854, 286, 2 13, 471, 514, 5 14, 414, 556, 7 12. 927, 139, 8 14, 958, 931, 4 15. 041, 068, 6 14. 496, 693, 9 15. 585, 443, 3 13. 952, 319, 2 16. 129, 818, 14. 569, 547, 6 15. 512, 589, 7 14. 025, 172, 9 16. 056, 964, 4 16. 139, 101, 7 16. 683, 476, 4 15. 594, 727, 17. 227, 851, 1 16. 610, 622, 8 15. 667, 580, 6 17. 154, 997, 5 15. 123, 205, 9 0. 3771 0. 0730 0. 0104 0.0097 0. 0142 0. 0015 0. 0061 0. 0003 0. 3623 0. 0297 0. 0297 0. 0024 0. 0024 0. 0042 0. 0042 0. 0030 0. 0030 0. 0163 0. 0032 0. 0005 0. 0004 0. 0004 0. 0001 0. 0003 (75) :T-3s+ft+;p+90 o — . +2f-y .. (75) T-s+ft-p-90°—_ +2f-v (75) An T-4s-fft+2p+90° +2f-y (75) T-3s+3ft-p+90° T-s-h+p-%° T-4s+3ft+90° ■\-2i-v (75) +2Z-v (75) 30.164,274,6 'Adapted for use with tabular node factors, theoretical value is 0.0317. See p. 43. -2u ■2v -2£-2p. -2$-2*. -2£-&_ -9E-4>„ -2$ -2*. -2£-2j»_ 0.0116 0. 0032 0. 0367 0. 0060 0. 0020 0. 0134 0. 0022 0. 1755 0. 0103 0. 0015 0. 0004 0. 1681 0. 0042 0. 0042 0. 0001 0. 0001 0. 0076 0. 0004 0. 0001 Factor-f formula 0. 0209 0. 5305 0. 9085 0. 1759 0. 0251 0. 0235 0, 0341 0, 0066 0, 0219 0, 0006 0, 0786 0, 0064 0. 0064 0. 0005 0. 0005 0. 0009 0, 0009 0. 0007 0. 0007 0, 0017 0.0003 166 U. S. COAST AND GEODETIC SURVEY Table 2. — Harmonic constituents — Continued Ref. No. An. Symbol ^477 #39 -B40 Bu £42 s 2 T 2 R2 B i7 Ba Bi9 (K 2 )-__ B 50 B51 Bit -B57 - B58 B59 Note 3. Note 4_ ^.82 U K 2 ... Ms Am Asi ^4.88 B82 S3 Argument (E) Speed per solar hour LUNAR SEMIDIURNAL TERMS, FORMULA (139) BT-Ss+Sh+W*.... BT-4s+Bh+p+90*. BT-Bs+Bh-p-90°. BT-s+Bh— 90° BT-Bs+Bh+p-90° B T+s+Bh -90° ._._. +3f-2*_ +3£-2*. +3£-2*. +?-2*. -£-2p. 28. 435, 087, 27. 890, 713, 28. 979, 462, 29. 533, 120, 28. 988, 746, 30. 631, 153, SOLAR SEMIDIURNAL TERMS, FORMULA (187) BT BT-h+ Pl bt+h-pi+iso BT-Bh+Bpi BT+Bh BT+h+pu.. gT+Sh-p!.. BT+Bpi BT+4h-Spi. BT+Ah BT+5h-px BT+3h+pi+m Q zero zero zero zero. zero. zero. zero. zero. zero. zero. zero. zero. 30. 000, 000, I 29. 958, 933, i 30. 041, 066, 29. 917, %6, i 30. 082, 137, : 30.041,070,1 30. 123, 204, i 30. 000, 003, ' 30. 164, 270, I 30. 164, 274, i 30. 205, 341, : 30. 123, 207, ! BT+6h-Bpi | zero | 30.246,407,' COMBINATION SEMIDIURNAL TERMS, FORMULAS (212) AND (230) BT-s+Bh-p+180° I +2£-2*>-R I 29.528,478,! BT+Bh I -2v" | 30.082,137, LUNAR TERDIURNAL TERMS, FORMULA (140) ST-Ss+8h ST-4s+3h+p 3T-2s+3h-p+lS0°. 3T-5s+3h+Bp. 8T-4s+5h-p_. ST-s+Sh 3T-Bs+3h+p. +3£-3v_ +3£-3r_ +3f-3y. +3£-3j>. +3£-3*>. +£-3*-_. +€-3v- 43. 476, 156, 42. 931, 781, 44. 020, 531, 42. 387, 406, 43. 004, 635, 44. 574, 189, 44. 029, 814, 7 SOLAR TERDIURNAL TERM, PARAGRAPH 119 ST.. | zero | 45.000,000,1 Coeffi- cient (C) 0. 0223 0. 0062 0. 0012 0. 0209 0. 0034 0. 0019 0. 4227 0. 0248 0, 0035 0. 0010 0. 0365 0. 0009 0. 0009 0. 0008 0. 0251 0. 1151 0. 0178 0. 0050 0. 0010 0. 0009 0. 0007 0. 0024 0. 0004 Factor-f formula (146) (146) (146) (147) (147) (148) unity unity unity unity unity unity unity unity unity unity unity unity unity (215) (235) (149) (149) (149) (149) (149) (150) (150) unity Note 1— Combines terms An and ^423. Note 2— Combines terms ^22 and £22. Note 3— Combines terms An and ^4.48 Note 4— Combines terms An and B47. HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 2a. — Shallow -water constituents 167 Symbol Argument Speed Factor-/ Origin V u MNS 2 M2+N2-S2 2S2-M2 Semidiurnal 2T-5s+4h+p. +4£-4k -2£+2* 4-2£-2»>-V +4€-4*+i»' degrees per h. 27. 423, 833, 7 31. 015, 895, 8 44. 025, 172, 9 42. 927, 139, 8 45. 041, 068, 6 43. 943, 035, 6 57. 968, 208, 4 58.984,104,2 57. 423. 833, 7 59. 066, 241, 5 60. 000, 000, 86. 952, 312, 7 87. 968, 208, 4 86. 407, 938, 88. 984, 104, 2 87. 423, 833, 7 90. 000, 000, 115.936,416,9 116.952,312,7 117.968,208,4 116. 407, 938, 120. 000, 000, /»(Ms) 2SM 2 ... 2T+2&-M Terdiurnal ZT-2s+3h-90° 3 T- 4s 4-3/1+90° 3T4-A-90 3T-2s+M-90° Quarter diurnal 4T-4s+4/i 4T-2s 4-2/1 4T-5s44ft+p 4T-2S+4A 4T Sixth diurnal 6T-Gs+6h /(Ma) MK3 M2+K1-- /(Ms) x/(Ki) /HM ?X «K„ MKj SK3 2M2-K1 S2+K1.... SO3 S2+Ol-.- 4-2£-k /(Oi) M 4 MS4 2M2 M2+S2 - +4|-4«/ +2f — 2v. . P (Ms) P (Ms) P (Ms) /(Ms)X/(K,) unity P (Ms) MN 4 MK 4 S 4 Me- ---. M2+N2 M2+K2 2S2 3M 2 +4£-4j< +2£-2*-V zero 4-6£-6j/--. 2MSe 2M2+S2 6T-4s4-4ft 4-4£-4j>... P (M s ) 2MNe- 2M2+N2 . 6T-7s+6h+p. 4-6£-6j>. . P (Ms) 2SM 6 2S2+M2 6T-2s+2h.. .. +2£-2k - / (Ms) MSNs— . M2+S2+N2 3S? „ ... QT-5s+4Ji+p. 4-4^-4^. . P (Ms) s 6 6T_. unity M 8 _- 4M 2 3M2+S2 2M2+2S2 2M2+S2+N2 4S 2 Eighth diurnal 8T-8S+8A 8T-6s+6/i +8f-8»' P (Ms) 3MSs 4-6^-6^ P (Ms) 2(MS)g._ 8T-4S4-4/1 -Hf— 4v P (Ms) 2MSN8— 8T-7s+Qh+p +6f-6»'.— P (Ms) Ss 8T zero unity 168 U. S. 00 AST AND GEODETIC SURVEY Table 3. — Latitude factors Yv31 Y Yv30 Y S 30 Y B 32 Y W 42 Yv32 Y W 43 Y 6 31 Yw31 Yw32 Yv40 Yv41 Y 8 40 Yv42 Y S 43 Yv43 Y E 41 Y 8 42 Y W 41 Y 0.500 * 0.000 1.000 1.000 0.000 1.000 * 0.000 0.200 * 0.000 1.000 0.000 0.333 0.200 1 2 3 .500 .498 .496 .035 .070 .105 1.000 0.999 .997 0.999 .998 .995 .017 .035 .052 1.000 0.999 .999 .010 .021 .031 .200 .199 .197 .017 .035 .052 1.000 0.998 .996 .013 .026 .038 .333 .332 .330 .200 .199 .197 1 2 3 4 5 6 .493 .489 .484 .139 .174 .208 .995 .992 .989 .990 .985 .978 .070 .087 .105 .998 .996 .995 .041 .052 .062 .195 .192 .188 .069 .086 .103 .993 .989 .984 .051 .063 .076 .328 .324 .321 .195 • .192 .189 4 5 6 7 8 9 .478 .471 .463 .242 .276 .309 .985 .981 .976 .970 .961 .951 .122 .139 .156 .993 .990 .988 .071 .081 .090 .184 .179 .173 .120 .136 .153 .978 .971 .964 .088 .099 .111 .316 .311 .305 .185 .181 .176 7 8 9 10 11 12 .455 .445 .435 .342 .375 .407 .970 .964 .957 .940 .927 .914 .174 .191 .208 .985 .982 .978 .099 .108 .116 .167 .161 .153 .168 .184 .199 .955 .946 .936 .122 .133 .143 .299 .291 .284 .170 .164 .157 10 11 12 13 14 15 .424 .412 .400 .438 .469 .500 .949 .941 .933 .899 .883 .866 .225 .242 .259 .974 .970 .966 .124 .131 .138 .146 .137 .128 .214 .228 .241 .925 .914 .901 .154 .163 .172 .275 .267 .257 .149 .141 .133 13 14 15 16 17 18 .386 .372 .357 .530 .559 .588 .924 .915 .905 .848 .829 .809 .276 .292 .309 .961 .956 .951 .144 .150 .156 .119 .110 .099 .255 .267 .280 .888 .875 .860 .181 .189 .197 .247 .237 .226 .124 .115 .105 16 17 18 19 20 21 .341 .325 .307 .616 .643 .669 .894 .883 .872 .788 .766 .743 .326 .342 .358 .946 .940 .934 .161 .165 .169 .089 .078 .067 .291 .302 .312 .845 .830 .814 .204 .211 .217 .215 .203 .191 .094 .083 .072 19 20 21 22 23 24 .290 .271 .252 .695 .719 .743 .860 .847 .835 .719 .695 .669 .375 .391 .407 .927 .921 .914 .172 .175 .177 .055 .044 .032 .322 .331 .339 .797 .780 .762 .222 .227 .231 .179 .166 .153 .060 .047 .035 22 23 24 25 26 27 .232 .212 .191 .766 .788 .809 .821 .808 .794 .643 .616 .588 .423 438 .454 .906 .899 .891 .178 .179 .179 .019 .007 -.005 .347 .354 .360 .744 .726 .707 .234 .237 .239 .140 .127 .113 .021 .008 -.006 25 26 27 28 29 30 .169 .147 .125 .829 .848 .866 .780 .765 .750 .559 .530 .500 .469 .485 .500 .883 .875 .866 .178 .177 .175 -.018 -.031 -.043 .366 .371 .375 .688 .671 .650 .241 .242 .242 .100 .086 .072 -.020 -.035 -.050 28 29 30 31 32 33 .102 .079 .055 .883 .899 .914 .735 .719 .703 .469 .438 .407 .515 .530 .545 .857 .848 .839 .172 .169 .165 -.056 -.069 -.081 .378 .381 .383 .630 .610 .590 .241 .240 .238 .058 .045 .031 -.065 -.081 -.097 31 32 33 34 35 36 .031 .007 -.018 .927 .940 .951 .687 .671 .655 .375 .342 .309 .559 .574 .588 .829 .819 .809 .161 .155 .150 -.093 -.106 -.118 .384 .385 .385 .570 .550 .530 .235 .232 .228 .071 .004 -.010 -.113 -.129 -.145 34 35 36 37 38 39 -.043 -.069 -.094 .961 .970 .978 .638 .621 .604 .276 .242 .208 .602 .616 .629 .799 .788 .777 .143 .136 .128 -.130 -.141 -.152 .384 .382 .380 .509 .489 .469 .223 .218 .212 -.023 -.036 -.049 -.162 -.179 -.196 37 38 39 40 41 42 -.120 -.146 -.172 .985 .990 .995 .587 .570 .552 .174 .139 .105 .643 .656 .669 .766 .755 .743 .120 .111 .102 -.163 -.174 -.184 .377 .374 .370 .450 .430 .410 .206 199 .191 -.061 -.073 -.085 -.213 -.230 -.248 40 41 42 43 44 45 -.198 -.224 -.250 .998 .999 1.000 .535 .517 .500 .070 .035 .000 .682 .695 .707 .731 .719 .707 .092 .082 .071 -.194 -.203 -.212 .365 .359 .354 .391 .372 .354 .183 .174 .165 -.096 -.107 -.118 -.265 -.283 -.300 43 44 45 *In these columns reverse signs for south latitude. Other values are applicable to either north or south latitude. HARMONIC ANALYSIS 1 AND PREDICTION OF TIDES' 169 Table 3. — Latitude factors — Continued Y Yv30 Yv31 Ye30 Y 8 32 Y w 42 Yv32 Y w 43 Y 6 31 Y W 31 Y w3 2 Yv40 Yv41 Y.40 Yv42 Y B 43 Yv43 Y 8 41 Y B 42 Y W 41 Y * * * * * 45 -0. 250 1.000 0.500 0.000 0.707 0.707 0.071 -0. 212 0.354 0.354 0.165 -0.118 -0.300 45 46 47 48 -.276 -.302 -.328 0.999 .998 .995 .483 .465 .448 -.035 -.070 -.105 .719 .731 .743 .695 .682 .669 .059 .048 .035 -.221 -.228 -.236 .347 .340 .333 .335 .317 .300 .155 .145 .135 -.128 -.137 -.146 -.317 -.335 -.352 46 47 48 49 50 51 -.354 -.380 -.406 .990 .985 .978 .430 .413 .396 -.139 -.174 -.208 .755 .766 .777 .656 .643 .629 .023 .010 -.003 -.242 -.249 -.254 .325 .317 .308 .282 .266 .249 .124 .112 .101 -.155 -.163 -.170 -.370 -.387 -.404 49 40 51 52 53 54 -.431 -.457 -.482 .970 .961 .951 .379 .362 .345 -.242 -.276 -.309 .788 .799 .809 .616 .602 .588 -.017 -.030 -.044 -.259 -.263 -.267 .299 .289 .280 .233 .218 .203 .089 .076 .064 -.177 -.183 -.189 -.421 -.438 -.455 52 53 54 55 56 57 -.507 -.531 -.555 .940 .927 .914 .329 .313 .297 -.342 -.375 -.407 .819 .829 .839 .574 .559 .545 -.058 -.072 -.087 -.270 -.272 -.274 .269 .259 .249 .189 .175 .162 .051 .038 .025 -.194 -.198 -.202 -.471 -.487 -.503 55 56 57 58 59 60 -.579 -.602 -.625 .899 .883 .866 .281 .265 .250 -.438 -.469 -.500 .848 .857 .866 .530 .515 .500 -.101 -.115 -.130 -.275 -.275 -.275 .238 .227 .217 .149 .137 .125 .012 -.001 -.014 -.204 -.207 -.208 -.519 -.535 -.550 58 59 60 61 62 63 -.647 -.669 -.691 .848 .829 .809 .235 .220 .206 -.530 -.559 -.588 .875 .883 .891 .485 .469 .454 -.144 -.158 -.173 -.274 -.272 -.270 .206 .195 .184 .114 .103 .094 -.028 -.041 -.054 -.209 -.210 -.209 -.565 -.580 -.594 61 62 63 64 65 66 -.712 -.732 -.752 .788 .766 .743 .192 .179 .165 -.616 -.643 -.669 .899 .906 .914 .438 .423 .407 -.187 -.201 -.214 -.266 -.263 -.258 .173 .162 .151 .084 .075 .067 -.067 -.080 -.092 -.208 -.206 -.204 -.608 -.621 -.635 64 65 66 67 68 69 -.771 -.790 -.807 .719 .695 .669 .153 .140 .128 -.695 -.719 -.743 .921 .927 .934 .391 .375 .358 -.228 -.241 -.254 -.253 -.247 -.241 .141 .130 .120 .060 .053 .046 -.105 -.117 -.129 -.201 -.197 -.193 -.647 -.660 -.672 67 68 69 70 71 72 -.825 -.841 -.857 .643 .616 .588 .117 .106 .095 -.766 -.788 -.809 .940 .946 .951 .342 .326 .309 -.266 -.278 -.290 -.234 -.226 -.218 .110 .100 .091 .040 .035 .030 -.141 -.152 -.163 -.188 -.183 -.177 -.683 -.694 -.705 70 71 72 73 74 75 -.872 -.886 -.900 .559 .530 .500 .085 .076 .067 -.829 -.848 -.866 .956 .961 .966 .292 .276 .259 -.301 -.311 -.322 -.209 -.200 -.190 .082 .073 .065 .025 .021 .017 -.173 -.183 -.193 -.170 -.163 -.155 -.715 -.724 -.733 73 74 75 76 77 78 -.912 -.924 -.935 .469 .438 .407 .059 .051 .043 -.883 -.899 -.914 .970 .974 .978 .242 .225 .208 -.331 -.340 -.349 -.179 -.169 -.159 .057 .049 .042 .014 .011 .009 -.202 -.211 -.219 -.147 -.139 -.130 -.741 -.749 -.757 76 77 78 79 80 81 -.945 -.955 -.963 .375 .342 .309 .036 .030 .024 -.927 -.940 -.951 .982 .985 .988 .191 .174 .156 -.357 -.364 -.371 -.146 -.134 -.121 .036 .030 .024 .007 .005 .004 -.226 -.233 -.239 -.120 -.111 -.100 -.764 -.770 -.776 79 80 81 82 83 84 -.971 -.978 -.984 .276 .242 .208 .019 .015 .011 -.961 -.970 -.978 .990 .993 .995 .139 .122 .105 -.377 -.382 -.387 -.109 -.096 -.082 .019 .015 .011 .003 .002 .001 -.245 -.250 -.254 -.090 -.079 -.069 -.781 -.785 -.789 82 83 84 85 86 87 -.989 -.993 -.996 .174 .139 .105 .008 .005 .003 -.985 -.990 -.995 .996 .998 .999 .087 .070 .052 -.391 -.394 -.397 -.069 -.055 -.042 .008 .005 .003 .001 .000 .000 -.258 -.261 -.264 -.057 -.046 -.035 -.792 -.795 -.797 85 86 87 88 89 90 -.998 -1.000 -1.000 .070 .035 .000 .001 .000 .000 -.998 -.999 -1.000 .999 1.000 1.000 .035 .017 .000 -.399 -.400 -.400 -.028 -.014 .000 .001 .000 .000 .000 ..000 .000 -.265 -.266 -.267 -.023 -.012 .000 -.799 -.800 -.800 8S 89 90 *In these columns reverse signs for south latitude. Other values are applicable to either north or south latitude. 170 U. S. COAST AND GEODETIC STJRVEY Table 4. — Mean longitude of lunar and solar elements at Jan. 1, 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 c solai peiigee; N=longitude of moon's node] / sun; p\ =mean longitude Year. s P h Pi N Year. s P h Pi N o o o o o o 1800 1801 1802 1803 342. 31 111.70 241. 08 10.47 225. 45 266. 12 306. 78 347. 44 280. 41 280. 17 279. 93 279. 69 279. 50 279. 52 279. 54 279. 55 33.25 13.92 354. 59 335. 26 1852— 1853-... 1854... . 1855-- 28.44 171.00 330. 38 69.77 181. 24 222. 02 262. 68 303. 34 279. 82 280. 57 280. 33 280. 09 280. 40 280. 41 280. 43 280. 45 107. 55 88.16 68. b4 49.51 1804 1805 1806 1807...,. 139. 85 282. 41 51.80 181. 18 28.10 68.88 109. 54 150. 20 279. 45 280. 20 279. 96 279. 72 279. 57 279. 59 279. 61 279. 62 315. 93 296. 55 277. 23 257. 90 1856— 1857— 1858.... 1859.... 199. 15 341. 72 111. 10 240. 49 344. 00 24.78 65.44 106. 10 279. 85 280. 60 280. 36 280. 12 280. 46 280. 48 280. 50 280. 52 30.18 10.80 351. 47 332. 14 1808 1809 1810 1811 310. 57 93.13 222. 51 351. 90 190. 86 231. 64 272. 30 312. 96 279. 48 280. 23 279. 99 279. 75 279. 64 279. 66 279. 67 279. 69 238. 57 219. 19 199. 86 180. 53 I860— . 1861 1862 1863 9.87 152. 43 281. 82 51.20 146. 77 187. 54 228. 20 268. 87 279. 88 280. 63 280. 39 280. 15 280. 53 280. 55 280. 57 280. 58 312. 81 293. 43 274. 10 254. 78 1812 1813 1814 1815 121. 28 263. 84 33.23 162. 61 353. 63 34.40 75.06 115. 73 279. 51 280. 26 280. 02 279. 78 279. 71 279. 73 279. 74 279. 76 161. 20 141. 82 122. 49 103. 17 1864—. 1865-.. . 1866.--. 1867— 180. 59 323. 15 92.53 221. 92 309. 53 350. 30 30.96 71.63 279. 91 280. 66 280. 42 280. 18 280. 60 280. 62 280. 64 280. 65 235. 45 216. 07 196. 74 177. 41 1816 1817 1818 1819 292. 00 74.56 203. 94 333. 33 156. 39 197. 16 237. 82 278. 49 279. 54 280. 29 280. 05 279. 81 279. 78 279. 79 279. 81 279. 83 83.84 64.46 45.13 25.80 1868— 1869— 1870— 1871— 351. 30 133. 86 263. 25 32.63 112. 29 153. 06 193. 73 234. 39 279. 94 280. 69 280. 45 280. 21 280. 67 280. 69 280. 71 280. 72 158. 08 138. 70 119. 37 100. 04 1820 1821 1822 1823 102. 71 245. 28 14.66 144. 04 319. 15 359. 92 40.59 81.25 279. 57 280. 32 280. 08 279. 84 279. 85 279. 86 279. 88 279. 90 6.47 347. 09 327. 76 308. 43 1872— 1873.— 1874— 1875— 162. 02 304. 58 73.96 203. 35 275. 05 315. 83 356. 49 37.15 279. 97 280. 72 280. 48 280. 24 280. 74 280. 76 280. 77 280. 79 80.72 61.34 42.01 22.68 1824 1825 1826 1827 273 43 55.99 185. 38 314. 76 121. 91 162. 69 203. 35 244. 01 279. 61 280. 35 280. 11 279. 87 279. 91 279. 93 279. 95 279. 97 289. 11 269. 72 250. 40 231. 07 1876— 1877— . 1878— 1879_-__ 332. 73 115. 29 244. 68 14.06 77.81 118. 59 159. 25 199. 91 280. 01 280. 75 280. 51 280. 27 280. 81 280. 83 280. 84 280. 86 3.35 343. 97 324. 64 305. 31 1828 1829 1830 1831 84.15 226. 71 356. 09 125. 48 284. 67 325. 45 6.11 46.77 279. 64 280. 38 280. 14 279. 91 279. 98 280. 00 280. 02 280. 03 211.74 192. 36 173. 03 153. 70 1880— 1881— 1882— 1883- . . 143. 45 286. 01 55.39 184. 78 240. 58 281. 35 322. 01 2.67 280. 04 280. 78 280. 54 280. 31 280. 88 280. 89 280. 91 280. 93 285. 98 266. 60 247. 28 227. 95 1832 1833 1834 1835 254. 86 37.42 166. 81 296. 19 87.43 128. 21 168. 87 209. 53 279. 67 280. 41 280. 18 279. 94 280. 05 280. 07 280. 09 280. 10 134. 37 114. 99 95.66 76.34 1884— 1885.— 1886— 1887— 314. 16 96.72 226. 11 355. 49 43.34 84.11 124. 77 165. 44 280. 07 280. 81 280. 57 280. 34 280. 95 280. 96 280. 98 281. 00 208. 62 189. 24 169. 91 150. 58 1836 1837 1838 1839 65.58 208. 14 337. 52 106. 91 250. 20 290. 97 331. 63 12.30 279. 70 280. 44 280. 21 279. 97 280. 12 280. 14 280. 16 280. 17 57.01 37.63 18.30 358. 97 1888— 1889.— 1890- - 1891.— 124. 88 267. 44 36.82 166. 21 206. 10 246. 87 287. 54 328. 20 280. 10 280. 84 280. 61 280. 37 281. 01 281. 03 281. 05 281. 07 131. 25 111.87 92.54 73.22 1840 1841 1842 1843 236. 29 18.85 148. 24 277. 62 52.96 93.73 134. 39 175. 06 279. 73 280. 48 280. 24 280. 00 280. 19 280. 21 280. 22 280. 24 339. 64 320. 26 300. 93 281. 61 1892- .. . 1893- - 1894— 1895— 295. 59 78.16 207. 54 336. 93 8.86 49.63 90.30 130. 96 280. 13 280. 87 280. 64 280. 40 281. 08 281. 10 281. 12 281. 13 53.89 34.51 15.18 355. 85 1844 1845 1846 1847 47.01 189. 57 318. 95 88.34 215. 72 256. 49 297. 16 337. 82 279. 76 280. 51 280. 27 280.03 280. 26 280. 28 280. 29 280. 31 262. 28 242. 90 223. 57 204. 24 1896--.. 1897.-. 1898— 1899— 106. 31 248. 87 18.26 147. 64 171. 62 212. 40 253. 06 293. 72 280. 16 290. 91 2S0.67 280. 43 281. 15 281. 17 281. 19 281. 20 336. 52 317. 14 297. 81 278. 48 1848 1849 1850 1851 217.72 0.28 129. 67 259. 05 18.48 59.26 99.92 140. 58 279. 79 280. 54 280. 30 280. 06 280. 33 280. 34 280. 36 280. 38 184. 91 165. 53 146. 20 126. 87 HARMONIC ANALYSIS AND PREDICTION OF TIDES 171 Table 4. — Mean longitude of lunar and solar elements at Jan, 1, hour, Greenwich mean civil time, of each year from 1800 to 2000 — Continued Year s P h Pi N Year s P h Pi N o o o o o o o o 1900 1901 1902 1903 277. 03 46.41 175. 80 305. 18 334. 38 15.05 55. 71 96.37 280. 19 279. 95 279. 71 279. 47 281. 22 281. 24 281. 26 281. 27 259. 16 239. 83 220. 50 201. 17 1952.... 1953.... 1954.... 1955— 323. 15 105. 72 235. 10 4.49 290. 16 330. 94 11.60 52.26 279 60 280. 35 280. 11 279. 87 282. 12 282. 13 282. 15 282. 17 333. 45 314.07 294. 75 275. 42 1904 1905 1906 1907 74.57 217. 13 346. 51 115.90 137. 03 177. 81 218. 47 259. 13 279. 23 279. 98 279. 74 279. 50 281. 29 281. 31 281. 32 281. 34 181. 84 162. 46 143. 13 123. 81 1956.... 1957... . 1958.... 1959.... 133. 87 276. 43 45. 82 175. 20 92.92 133. 70 174. 36 215. 02 279. 63 280. 38 280. 14 279. 90 2S2. 18 282. 20 282. 22 282. 24 256. 09 236. 71 217. 38 198. 05 1908 1909 1910 1911 245. 28 27.84 157 23 286. 61 299. 79 340. 57 21.23 61.89 279. 27 280. 01 279. 77 279. 53 281. 36 281. 38 281. 39 281.41 104. 48 85.10 65.77 46.44 I960.... 1961.... 1962.._. 1963 304. 59 87.15 216. 53 345. 92 255.69 296. 46 337. 12 17.78 279. 67 280. 41 280. 17 279.93 282. 25 282. 27 282. 29 282. 30 178. 72 159. 34 140. 01 120. 69 1912 1913 1914 1915 56.00 198. 56 327. 94 97.33 102. 55 143. 33 183.99 224. 65 279. 30 280. 04 279. 80 279. 57 281. 43 281. 44 281. 46 281. 48 27.11 7.73 348. 40 329. 07 1964 1965 1966 1967 115.30 257. 86 27.25 156. 63 58.45 99.22 139. 88 180. 54 279. 70 280. 44 280. 20 279. 97 282. 32 282. 34 282. 36 282. 37 101.36 81.98 62.65 43.32 1916 1917 1918 1919 226. 71 9.27 138.66 268. 04 265. 32 306. 09 346. 75 27.41 279. 33 280. 07 279. 84 279. 60 281.50 281. 51 281. 53 281. 55 309. 75 290. 36 271. 04 251. 71 1968 1969 1970— 1971 286. 02 68.58 197. 96 327. 35 221. 21 261. 98 302.64 343. 31 279. 73 280. 47 280. 24 280. 00 282. 39 282. 41 282. 42 ^82. 44 23.99 4.61 345. 28 325. 95 1920 1921 1922 1923 37.43 179. 99 309. 37 78.76 68.08 108. 85 149. 51 190. 18 279. 36 280. 10 279. 87 279. 63 281. 56 281. 58 281. 60 281. 62 232. 38 213. 00 193. 67 174. 34 1972 1973 1974.... 1975— 96.73 239. 29 8.68 138. 06 23.97 64.74 105. 40 146. 07 279. 76 280. 50 280. 27 280. 03 282. 46 282. 48 282. 49 282. 51 306. 63 287. 24 267. 92 248. 59 1924 1925 1926 1927 208. 14 350. 71 120. 09 249. 47 230. 84 271. 61 312. 27 352. 94 279. 39 280. 14 279. 90 279. 66 281. 63 281. 65 281. 67 281. 69 155. 01 135. 63 116.31 96. 98 1976 1977 1978 1979— 267. 45 50.01 179. 40 308. 78 186. 73 227. 50 268. 17 308. 83 279. 79 280. 54 280.30 280. 06 282. 53 282. 54 282. 56 282. 58 229. 26 209. 88 190. 55 171. 22 1928 1929 1930 1931 18.86 161.42 290. 81 60.19 33.60 74.37 115. 03 155. 70 279. 42 280. 17 279. 93 279. 69 281. 70 281. 72 281. 74 281. 75 77.65 58.27 38.94 19.61 1980— 1981.... 1982.... 1983— 78.16 220. 73 350. 11 119. 50 349. 49 30.26 70.93 111.59 279. 82 280. 57 280. 33 280. 09 282. 60 282. 61 282. 63 282. 65 151. 89 132. 51 113. 19 93.86 1932 1933 1934 1935 189. 58 332. 14 101.52 230. 91 196. 36 237. 13 277. 80 318. 46 279. 45 280. 20 279. 96 279. 72 281. 77 281. 79 281.81 281. 82 0.28 340.90 321. 57 302. 25 1984— 1985.... 1986 1987 248. 88 31.44 160. 83 290. 21 152. 25 193. 02 233. 69 274. 35 279. 85 280. 60 280. 36 280. 12 282. 67 282. 68 282. 70 282. 72 74.53 55.15 35.82 16.49 1936 1937 1938 1939 0.29 142. 85 272. 24 41.62 359. 12 39.89 80.56 121. 22 279. 48 280. 23 279. 99 279. 75 281. 84 281. 86 281. 87 281. 89 282. 92 263. 54 244. 21 224. 88 1988 1989 3990 1991 59.60 202. 16 331. 54 100. 93 315. 01 355. 79 36.45 77.11 279. 88 280. 63 280. 39 280. 15 282. 73 282. 75 282. 77 282. 79 357. 16 337. 78 318. 45 299. 13 1940 1941 1942 1943 171. 01 313. 57 82.95 212. 34 161. 88 202. 65 243. 32 283. 98 279. 51 280. 26 280. 02 279. 78 281. 91 281. 93 281. 94 281.96 205. 55 186. 17 166. 84 147. 51 1992— 1993 1994 1995 230. 31 12.87 142. 26 271. 64 117. 77 158. 55 199. 21 239. 87 279. 91 280. 66 280. 42 280. 18 282. 80 282. 82 282. 84 282. 85 279. 80 260.42 241. 09 221. 76 1944 1945 1946 1947 341. 72 124. 28 253. 67 23.05 324. 64 5.42 46.08 86.74 279. 54 280. 29 280. 05 279. 81 281. 98 281.99 282. 01 282. 03 128. 19 108. 80 89.48 70.15 1996— 1997— 1998.... 1999.... 41.03 193. 59 312. 97 82.36 280. 53 321. 31 1.97 42.63 279. 94 280. 69 280. 45 280. 21 282. 87 282. 89 282.91 282. 92 202. 43 183. 05 163. 72 144. 39 1948 1949 1950 1951 152. 44 295. 00 64.38 193.77 127. 40 168. 18 208. 04 249. 50 279. 57 280. 32 280. 08 279. 84 282. 05 282. 06 282. 08 282. 10 50. 82 31.44 12.11 352. 78 2000— 211. 74 83.29 279. 97 282.94 125.07 172 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 s P h Pi N Month s P h Pi N o o o o o o o Jan. 1 Feb. 1 Mar. 1 0.00 48.47 57.41 0.00 3.45 6.57 0.00 30.56 58.15 0.00 0.00 0.00 0.00 -1.64 -3.12 July 1 Aug. 1 Sept. 1 224. 93 273. 40 321. 86 20.16 23.62 27.07 178. 40 208. 96 239. 51 0.01 0.01 0.01 -9.58 -11.23 -12. 87 Apr. 1 May 1 June 1 105. 88 141. 17 189. 64 10.03 13.37 16.82 88.71 118. 28 148. 83 0.00 0.01 0.01 -4.77 -6.35 -8.00 Oct. 1 Nov. 1 Dec. 1 357. 16 45.62 80.92 30.41 33.87 37.21 269. 08 299. 64 329. 21 0.01 0.01 0.02 -14. 46 -16. 10 -17. 69 DIFFERENCES TO BEGINNING OF EACH DAY OF MONTH FOR COMMON YEARS Day S P h Pi N Day s P h Pi N o o o o o o 1 2 3 0.00 13.18 26.35 0.00 0.11 0.22 0.00 0.99 1.97 0.00 0.00 0.00 0.00 -0.05 -0.11 17 18. 19 210. 82 224. 00 237. 18 1.78 1.89 2.01 15.77 16.76 17.74 0.00 0.00 0.00 -0.85 -0.90 -0.95 4 5 6 39.53 52.71 65.88 0.33 0.45 0.56 2.96 3.94 4.93 0.00 0.00 0.00 -0.16 -0. 21 -0.26 20 21 22 250. 35 263. 53 276. 70 2.12 2.23 2.34 18.73 19.71 20.70 0.00 0.00 0.00 -1.01 -1.06 -1.11 7 8 9 79.06 92.23 105. 41 0.67 0.78 0.89 5.91 6.90 7.89 0.00 0.00 0.00 -0.32 -0.37 -0.42 23 24 25 289. 88 303. 06 316. 23 2.45 2.56 2.67 21.68 22.67 23.66 0.00 0.00 0.00 -1.16 -1.22 -1.27 10 11 12 118. 59 131. 76 144. 94 1.00 1.11 1.23 8.87 9.86 10.84 0.00 0.00 0.00 -0.48 -0.53 -0.58 26 27 28 329. 41 342. 59 355. 76 2.79 2.90 3.01 24.64 25.63 26.61 0.00 0.00 0.00 -1.32 -1.38 -1.43 13 14 15 16 158. 12 171. 29 184. 47 197. 65 1.34 1.45 1.56 1.67 11.83 12.81 13.80 14.78 0.00 0.00 0.00 0.00 -0.64 -0^74 -0.79 29 30 31 32 8.94 22.12 35.29 48.47 3.12 3.23 3.34 3.45 27.60 28.58 29.57 30.56 0.00 0.00 0.00 0.00 -1.48 -1.54 -1.59 -1.64 DIFFERENCES TO BEGINNING OF EACH HOUR OF DAY, GRENWICH CIVIL TIME Hour s P h Pi N Hour s P h Pi N o o o o o 1 2 0.00 0.55 1.10 0.00 0.00 0.01 0.00 0.04 0.08 0.00 0.00 0.00 0.00 0.00 0.00 12 13 14 6.59 7.14 7.69 0.06 0.06 0.06 0.49 0.53 0.57 0.00 0.00 0.00 -0.03 -0.03 -0.03 3 4 5 1.65 2.20 2.75 0.01 0.02 0.02 0.12 0.16 0.21 0.00 0.00 0.00 -0.01 -0.01 -0.01 15 16 17 8.24 8.78 9.33 0.07 0.07 0.08 0.62 0.66 0.70 0.00 0.00 0.00 -0.03 -0.04 -0.04 6 7 8 3.29 3.84 4.39 0.03 0.03 0.04 0.25 0.29 0.33 0.00 0.00 0.00 -0.01 -0.02 -0.02 18 19 20 9.88 10.43 10.98 0.08 0.09 0.09 0.74 0.78 0.82 0.00 0.00 0.00 -0.04 -0.04 -0.04 9_.„ 10 11 4.94 5.49 6.04 0.04 0.05 0.05 0.37 0.41 0.45 0.00 0.00 0.00 -0.02 -0.02 -0.02 21 22. 23 11.53 12.08 12.63 0.10 0.10 0.11 0.86 0.90 0.94 0.00 0.00 0.00 -0.05 -0.05 -0.05 i The table may also be used directly for dates between Jan. 1 and Feb. 29, inclusive, of leap years; but if the required date falls between Mar. 1 and Dec. 31, inclusive, of a leap year, the day of month should be increased by one before entering the table. HARMONIC ANALYSIS AND PREDICTION OF TTDES 173 Table 6. — Values of /, v, £, v' , and 2v" for each degree of N. Positive always 28.60 28.60 28.59 28.58 28.58 28.57 28.56 28.55 28.53 28.52 28.50 28.47 28.45 28.43 28.41 28.39 28.36 28.34 28.31 28.29 28.26 28.23 28.20 28.16 28.13 28.09 28.06 28.02 27.98 27.94 27.90 27.86 27.82 27.77 27.73 27.68 27.63 27.58 27.53 27.48 27.43 27.38 27.32 Diff. 1 1 1 1 1 2 1 2 1 2 2 2 2 2 3 2 3 2 3 3 3 Positive when N is between and 180°; negative when N is between 180 and 360° 0.00 0.19 0.38 0.56 0.75 0.94 1.12 1.31 1.50 1.68 1.87 2.05 2.24 2.42 2.61 2.79 2.98 3.16 3.34 3.52 3.70 3.88 4.06 4.24 4.42 4.60 4.78 4.95 5.13 5.30 5.48 5.65 5.82 5.99 6.16 6.33 6.50 6.83 6.99 7.15 7.31 7.47 7.63 7.79 7.94 Diff. 0.00 0.17 0.34 0.51 0.67 0.84 1.01 85 1. 1. 1. 1. I. 2.02 2.18 2.35 2.51 2.68 2.84 3.01 3.17 3.34 3.50 3.66 3.82 3.98 4.14 4.30 4.46 4.62 4.78 4.94 5.10 5.25 5.41 5.56 5.71 5.86 6.01 6.16 6.31 6.46 6.61 6.75 6.90 7.04 7.18 Diff. 17 17 17 16 17 17 17 17 16 17 17 17 16 17 16 17 16 17 16 17 16 16 16 16 16 16 16 16 16 16 16 15 16 15 15 15 15 15 15 15 15 14 15 14 14 0.00 0.13 0.27 0.40 0.54 0.67 0.80 0.94 1.07 1.20 1.34 1.47 1.60 1.73 1.86 1.99 2.12 2.25 2.38 2.51 2.64 2.77 2.90 3.03 3.15 3.28 3.40 3.53 3.65 3.78 3.90 4.02 4.14 4.26 4.38 4.50 4.62 4.74 4.85 4.97 5.08 5.19 5.30 5.41 5.52 5.63 Diff. 13 14 13 14 13 13 14 13 13 14 13 13 13 13 13 13 13 13 13 13 13 13 13 12 13 12 13 12 13 12 12 12 12 12 12 12 12 11 12 11 11 11 11 11 11 2v" 0.00 0.28 0.57 0.85 1.14 1.42 1.70 1.99 2.27 2.55 4.51 4.78 5.06 5.33 5.60 5.87 6.14 6.41 6.68 6.94 7.21 7.47 7.73 7.99 8.25 8.50 8.75 9.00 9.25 9.50 9.74 9.98 10.22 10.46 10.69 10.93 11.16 11.38 11.60 11.82 Diff. 28 29 28 29 28 28 29 28 28 28 28 28 28 28 28 28 27 28 27 27 27 27 27 27 26 27 26 26 246037—41- -12 174 U. S. COAST AND GEODETIC SURVEY Table 6. — Values of I, v, £, p', and 2v" for each degree of N— Continued N Positive always 27.32 27.27 27.21 27.15 27.09 27.03 26.97 26.91 26.85 26.78 26.72 26.65 26.52 26.45 26.38 26.31 26.24 26.17 26.10 26.03 25.95 25.88 26.80 25.72 25.65 25.57 25.49 25.41 25.33 25.25 25.17 25.09 25.01 24.92 24.84 24.76 24.67 24.59 24.50 24.42 24.33 24.24 24.16 24.07 23.98 Diff. 5 Positive when N is between and 180°; negative when Ni between 180 and 360° 8.10 8.25 8.40 8.55 8.69 8.84 9.12 9.26 9.40 9.54 9.67 9.81 9.94 10.07 10.19 10.32 10.44 10.56 10.68 10.79 10.90 11.01 11.12 11.23 11.33 11.43 11.53 11.63 11.72 11.81 11.89 11.98 12.06 12.14 12.21 12.28 12.35 12.42 12.48 12.54 12.60 12.65 12.70 12.75 Diff. 16 15 15 15 14 15 14 14 14 14 14 13 14 13 13 12 13 12 12 12 11 11 11 11 11 10 10 10 7.18 7.32 7.46 7.60 7.73 7.87 8.00 8.14 8.27 8.40 8.52 8.65 8.77 8.90 9.02 9.14 9.25 9.37 9.48 9.70 9.81 9.92 10.02 10.12 10.22 10.32 10.41 10.50 10.59 10.68 10.77 10. 85 10.93 11.01 11.08 11.15 11.22 11.29 11.36 11.42 11.47 11.53 11.58 11.63 11.68 Diff. 14 14 14 13 14 13 14 13 13 12 13 12 13 12 12 11 12 11 11 11 11 11 10 10 10 10 9 5.74 5.84 5.95 6.05 6.15 6.25 6.35 6.45 6.54 6.64 6.73 6.82 6.91 7.00 7.09 7.17 7.26 7.34 7.42 7.49 7.57 7.64 7.72 7.79 8.05 8.11 8.17 8.34 8.39 8.44 8.48 8.53 8.57 8.61 8.64 8.68 8.71 8.74 8.76 8.79 Diff. 11 10 11 10 10 10 10 10 9 10 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 Diff. 22 22 21 21 20 20 20 20 19 19 19 18 17 18 17 17 16 16 15 15 15 14 13 14 13 12 12 11 11 11 10 10 HARMONIC ANALYSIS AND PREDICTION OF TTDES 175 Table 6. — Values of I, v, £, v' , and 2v" for each degree of N — Continued N Positive always Positive when JV is between and 180° ; negative when N is bet 180 and 360° sveen N V £ v' 2v" 90 23.98 23.89 23.80 23.72 23.63 23.54 23.45 23.36 23.27 23.18 23.09 23. 00 22.91 22.82 22.73 22.64 22.55 22.46 22.37 22.28 22.20 22.11 22.02 21.93 21.84 21.75 21.67 21.58 21.50 21.41 21.32 21.24 21.15 21.07 20.99 20.91 20.82 20.74 20.66 20.58 20.51 20.43 20.35 20.28 20.20 20.13 Diff. 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 8 9 9 9 9 9 8 9 8 9 9 8 9 8 8 8 9 8 8 8 7 8 8 7 8 7 12.75 12.79 12.83 12.87 12.90 12.93 12.95 12.97 12.99 13.01 13.02 13.02 13.02 13.02 13.01 13.00 12.99 12.97 12.95 12.92 12.89 12.85 12.81 12.77 12.72 12.67 12.61 12. 55 12.48 12.41 12.33 12.25 12.17 12.08 11.98 11.88 11.78 11.67 11.55 11.43 11.31 11.18 11.05 10.91 10.77 10.62 Diff. 4 4 4 3 3 2 2 2 2 1 1 1 1 2 2 3 3 4 4 4 5 5 6 6 7 7 8 8 8 9 10 10 10 11 12 12 12 13 13 14 14 15 o 11.68 11.72 11.76 11.80 11.83 11.86 11.89 11.92 11.94 11.95 11.96 11.97 11.98 11.98 11.98 11.97 11.96 11.95 11.93 11.91 11.89 11.86 11.83 11.79 11.75 11.70 11.65 11.60 11.54 11.48 11.41 11.34 11.26 11.18 11.10 11.01 10.92 10.82 10.72 10.61 10.50 10.38 10.26 10.13 10.00 9.87 Diff. 4 4 4 3 3 3 3 2 1 1 1 1 1 1 1 2 2 2 3 3 4 4 5 5 5 6 6 7 7 8 8 8 9 9 10 10 11 11 12 12 13 13 13 o 8.79 8.81 8.83 8.85 8.86 8.87 8.88 8.89 8.90 8.90 8.89 8.89 8.88 8.87 8.86 8.84 8.82 8.80 8.78 8.75 8.72 8.69 8.65 8.61 8.57 8.52 8.48 8.43 8.37 8.31 8.25 8.19 8.13 8.06 7.99 7.91 7.83 7.75 7.67 7.58 7.49 7.40 7.30 7.20 7.10 7.00 Diff. 2 2 2 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 4 4 4 5 4 5 6 6 6 6 6 7 7 8 8 8 8 9 9 9 10 10 10 10 o 17.77 17.79 17.81 17.82 17.83 17.83 17.82 17.81 17.79 17.77 17.74 17.71 17.67 17.62 17.57 17.51 17.45 17.38 17.30 17.22 17.14 17.05 16.95 16.84 16.73 16.62 16.50 16.37 16.24 16.10 15.96 15.81 15.66 15.50 15.33 15.16 14.99 14.81 14.62 14.43 14.23 14.03 13.83 13.62 13.40 13.18 Diff. 2 2 1 1 1 1 2 2 3 3 4 5 5 6 6 7 8 8 8 9 10 11 11 11 12 13 13 14 14 15 15 16 17 17 17 18 19 19 20 20 20 21 22 22 o 270 91 .. 269 92 268 93 267 94 266 95 265 96 264 97. 263 98 262 99 261 100 260 101 259 102 258 103 257 104 256 105 255 106 254 107 253 108 :„_. 252 109 251 110 250 111 249 112 248 113 247 114 246 115 245 116 244 117 243 118 242 119 241 120 240 121 . 239 122. 238 123 237 124 236 125 235 126 234 127. 233 128 232 129 231 130... 230 131 229 132 228 133 227 134.. 226 135 225 176 Table 6. U. S. COAST AND GEODETIC SURVEY ■Values of I, v, £, v' , and 2v" for each degree of N— Continued A r Positive always 20.13 20.05 19.98 19.91 19.84 19.77 19.71 19.64 19.58 19.51 19.45 19.39 19.33 19.27 19.22 19.16 19.11 19.05 19.00 18.95 18.91 18.86 18.82 18.78 18.74 18.70 18.66 18.62 18.59 18.56 18.53 18.50 18.47 18.45 18.43 18.41 18.39 18.37 18.36 18.34 18.33 18.32 18.32 18.31 18.31 18.31 Diff. Positive when iVis between and 180°; negative when N is between 180 and 360° 10.62 10.47 10.31 10.15 9.81 9.63 ).45 ).27 8.89 8.69 8.49 8.28 8.07 7.85 7.63 7.41 7.18 6.95 6.72 6.48 6.24 5.99 5.74 4.72 4.46 4.19 3.92 3.65 3.38 3.10 2.83 2.55 2.27 1.99 1.71 1.42 1.14 0.86 0.57 0.29 0.00 Diff. 15 16 16 17 17 18 18 18 19 19 20 20 21 21 22 22 22 23 23 23 24 24 25 25 25 25 26 26 26 27 27 27 27 28 27 28 28 29 9.87 9.73 9.59 9.44 9.29 9.13 8.97 8.80 8.63 8.10 7.91 7.72 7.52 7.32 7.12 6.91 6.70 6.49 6.27 6.05 5.82 5.59 5.36 5.13 4.89 4.65 4.41 4.16 3.91 3.66 3.41 3.16 2.90 2.64 2.38 2.12 1.86 1.60 1.33 1.07 0.80 0.54 0.27 0.00 Diff. 14 14 15 15 16 16 17 17 17 18 18 19 19 20 20 20 21 21 21 22 22 23 7.00 6.55 6.43 6.31 6.18 6.06 5.93 5.80 5.66 5.52 5.38 5.24 5.09 4.95 4.80 4.65 4.50 4.34 4.19 4.03 3.87 3.70 3.54 3.37 3.20 3.03 2.86 2.69 2.52 2.34 2.17 1.99 1.81 1.63 1.45 1.27 1.09 0.91 0.73 0.55 0.37 0.18 0.00 Diff. 11 11 12 11 12 12 13 12 13 13 14 14 14 14 15 14 15 15 15 16 15 16 16 17 16 17 17 17 17 17 17 18 17 18 18 18 18 18 18 18 18 18 18 19 18 13.18 12.96 12.73 12 49 12.25 12.01 11.76 11.51 11.26 11.00 10.74 10.48 10.21 9.94 9.10 8.81 8.52 8.23 7.94 7.64 7.34 7.04 6.74 6.43 6.12 5.81 5.50 5.19 4.87 4.55 4.23 3.91 3.59 3.27 2.94 2.62 2.29 1.97 1.64 1.31 0.99 0.66 0.33 0.00 Diff. 22 23 24 24 24 25 25 25 26 26 26 27 27 28 28 28 29 29 29 29 30 30 30 N 225 224 223 222 221 220 219 218 217 216 215 214 213 212 211 210 208 207 206 205 204 202 201 200 199 198 197 196 195 194 193 192 191 190 187 186 185 184 183 182 181 180 HARMONIC ANALYSIS AND PREDICTION OF TIDES 177 o I © 3 o co »o »o lOOlO oioo CO CM CM CO CO CO CO 23g CO CO co IO o >o os os oo CM CM CM o >o o 00 t^- t»- CNCNCN o>oo oo r^ t-«. iooio CO CD lO ooo IO -HH -HH ■ o CO COCN CN lOOW 7— 1 1— 1 O o>oo ooos o (35 CM 00 >o CO NiOiO CN r-lO NINH odd g28 CO CO OS odd 1^- CM CO OS co >o >C CM OS ooo> odd CO OS OS o>o — i CO —I CO •HH CN O t^COt^ N>CO 00 tv CO oooooo OlOOiO 00-HHCO 00 00 00 o> ai Oi O 00 CN t- tO 05 NtDOl 82£ ooo CO CO o IO CM O HHO odd 00 00 OS IO CM OS OOOS odd OS 05 CO 00 CM III ■HH -rtH CN IO CO rJH OS 00 t^. oooooo CN CO O tH •hh o "# oof- to ooo t^ l-» CO r-H -HH IO 1* r-H CM CM CNt-I odd 00 <* 1^ OS O0 CO ooo odd OS OS-* <* ■>* IO -HH CN O ooo odd 1 OS d O 00 "* O •>* H t-^ io -# OS O O d os o OS IO i-H OS o CO CN CM r-H o t~- t- 1^ -HH CO ooo o CM CM i-HOSt-H t-hOO odd CO CM CO O CO OS os r~ io ooo odd 00 00 o 3 CNO ooo odd OS OO OS d CO O CO r- IO -HH CDNO CO CN CN lOCNr-H odd O CM H00 O00H 0005 HOO odd O CO CO 00 CO IO ooo odd 00 IO -HH 00 CM CO CO CM O ooo odd o OS OS d IO -HH O0 COCO— I t— CD IO OCOO HCON ■HH CO CN •HH CD CO CM O 00 O O OS o O t- 00 "* Of^CN 00 00 00 OOO OS 00 00 CO CN OS b- CO ■* ooo t^ CM CD «3HO CO CM O OOO CO CN CS OS OS t-^ CD b- CD IO CO CD CO t~ OS CO -HH CO CO CM COt^ 111 ooo ooo ooo OS o o O CO CO O 00 CO ooo odd CM iO CN CO CO »o co io "* ooo odd b. f^ CD CM OS CO CO— I o ooo odd o r-H OS os d 823 00 t^ CO CO "HH O0 CN IO OS IO T)H CO 00 Tf CO IO CO CM o 00 oo »o •>* 001>0 l>- co co ooo odd OCON OS oo IO U3 -* 1 OOO odd tr n >o OoO odd d CO CO CO ■HH -HI IO O0 t^ CD "oooo NHiO IO IO -HH r- < OS r-H CN OS OS -/ ' a, oioo OO 00 OS >o o >c 288 228 CM CM CM CN o «o o 8 S3 Si IO O IO ■HH >0 IO CM CMCN ooo CO CD t^ CM CNCN ooo o »-< 2£c1 CO CO ■"*! »o ■HH U5CO o>oo 178 U. S. 00 AST AND GEODETIC SURVEY m e <^ a 53 § v. o* O ^ 05 ! 1 § .3 iS ? 00 £ II \ B, © >o© O CO CO CO >o © iO ■*•* CO CO CM CM m © IO I-H 1—1 © O iO© ©©© CO CM CM JOOio CM CM CM © CM ©>o© oo t-- t^. io © »o © © >o §Tt<-*t< IO © IO CO CO CM 822 IO © iO ©o© o © O s O © ■* i-( ■* »o IO © © © CO CO ^H t-CM lOt^© © © ©©CN o t^ © CM i-H CM © -* CM© t^lOCM O ©©© lONH CM ©Tt< t^ © CM TtHO ■* i-H oo-* © © ©•*oo o CM >0 t^ 00 00 00 f~ © IO CO --I © t^ ■* CM o 8 OHOS CM 00 © t^1-l© rt< ->*i i-H IO 00 CO S»OM © © ©■*i> ^mio ©l^t^ © io ■* .i-H © -tfl rH CMt-t^ CMCO.-H t^©CM COCM>-< © © OMN ©CM-* IO IO >o >o Tf CO r-l©00 © Tt< CM o CO CM ococo © CM 00 oo coco ©CMt-H 00 CO © 00©O © © O CO i© o ©—1 CM CO CO (N ©©l^ IO CO CM O CM CM ©©© o© >o IO © ^H OOr-lCM ©©.-c -# !>. O0 © © © CM iO o 00O--H CM CO CO CMCNi-H O00N lOCOrt CM ©©o ^H©CO CO 00 © © r-l CO CM © IO © Tj< t^ © o © CM »0 o t-00© lH r-H© © t^© •OtOH o 8 OCOtH *#©CM i-l©00 © i-t ■* r*co© ©1-1© © © ©CM-* © 00 © ©©© O©© 00 t^© ■*«H o © OOOS O CM CO o ©.-ICM »o t^ 00 omN ■OHIO ©"©00 © © >o t^ © IO CM 00-* •*CM*rH © © o 00 OX'* o ©CM CO Tj< ©l^ CON 00 00 00 ©CMf- 00 00 t^ ©©© © © iO 00 © CO COCM^H © © y / 00 00 © »o © to 288 © IO © ,-h-icni CM CM CM IO © IO §388 © »o o >o© IO © CM co© o ■—! £88 © IO © IO © »o •* iO iO © IO © © © r- iO © »o t-00 00 O © HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 9. — Log Q a for amplitude of constituent Mi 179 p LogQa DifT. P P LogQ. Diff. P o o o o o o 180 9. 7133 180 360 45 225 9. 8182 47 135 315 1 2 3 181 182 183 9. 7133 9. 7135 9. 7137 2 2 4 179 178 177 359 358 357 46 47 48 226 227 228 9. 8229 9. 8278 9. 8328 49 50 51 134 133 132 314 313 312 4 5 6 184 185 186 9. 7141 9. 7145 9. 7151 4 6 7 176 1 175 174 356 355 354 49 50 51 229 230 231 9. 8379 9. 8430 9. 8482 51 52 54 131 130 129 311 310 309 7 8 9 187 188 189 9. 7158 9. 7165 9. 7174 7 9 10 173 172 171 353 352 351 52 53 54 232 233 234 9. 8536 9. 8590 9. 8645 54 55 56 128 127 126 308 307 306 10 11 12 190 191 192 9. 7184 9. 7194 9. 7206 10 12 13 170 169 168 350 349 348 55 56 57 235 236 237 9. 8701 9. 8757 9. 8814 56 57 58 125 124 123 305 304 303 13 14 15 193 194 195 9. 7219 9. 7232 9. 7247 13 15 16 167 166 165 347 346 345 58 59 60 238 239 240 9. 8872 9. 8931 9. 8990 59 59 59 122 121 120 302 301 300 16 17 18 196 197 198 9. 7263 9. 7280 9. 7298 17 18 19 164 163 162 344 343 342 61 62 63 241 242 243 9. 9049 9. 9109 9. 9169 60 60 60 119 118 117 299 298 297 19 20 21 199 200 201 9. 7317 9. 7337 9. 7358 20 21 22 161 160 159 341 340 339 64 65 66 244 245 246 9. 9229 9. 9289 9. 9349 60 60 59 116 115 114 296 295 294 22 23 24 202 203 204 9. 7380 9. 7403 9. 7427 23 24 25 158 157 156 338 337 336 67 68 69 247 248 249 9. 9408 9. 9468 9. 9527 60 59 58 113 112 111 293 292 291 25 26 27 205 206 207 9. 7452 9. 7479 9. 7506 27 27 28 155 154 153 335 334 333 70 71 72 250 251 252 9. 9585 9. 9642 9. 9698 57 56 55 110 109 108 290 289 288 28 29 30 208 209 210 9. 7534 9. 7564 9. 7595 30 31 31 152 151 150 332 331 330 73 74 75 253 254 255 9. 9753 9. 9807 9. 9859 54 52 50 107 106 105 287 286 285 31 32 33 211 212 213 9. 7626 9. 7659 9. 7693 33 34 35 149 148 147 329 328 327 76 77 78 256 257 258 9. 9909 9. 9957 0. 0002 48 45 43 104 103 102 284 283 282 34 35 36 214 215 216 9. 7728 9. 7764 9. 7801 36 37 38 146 145 144 326 325 324 79 80 81 259 260 261 0. 0045 0. 0085 0. 0122 40 37 34 101 100 99 281 280 279 37 38 39 217 218 219 9. 7839 9. 7878 9. 7918 39 40 42 143 142 141 323 322 321 82 83 84 262 263 264 0. 0156 0. 0186 0. 0213 30 27 23 98 97 96 278 277 276 40 41 42 220 221 222 9. 7960 9.8002 9.8045 42 43 45 140 139 138 320 319 318 85 86 87 265 266 267 0. 0236 0. 0255 0. 0271 19 16 11 95 94 93 275 274 273 43 44 45 223 224 225 9. 8090 9. 8136 9. 8182 46 46 137 136 135 317 316 315 88 89 90 268 269 270 0. 0282 0. 0288 0. 0290 6 2 92 91 90 272 271 270 180 IT. S. 00 AST AND GEODETIC SURVEY Table 10. — Values of Q for argument of constituent Mx p Q Diff. P Q Diff. P Q Diff. P Q Diff. o o o o o o o 0.0 0.5 45 25.8 0.8 90 90.0 2.1 135 154.2 0.8 1 2 3 0.5 1.0 1.5 0.5 0.5 0.4 46 47 48 26.6 27.4 28.2 0.8 0.8 0.9 91 92 93 92.1 94.1 96.2 2.0 2.1 2.0 136 137 138 155.0 155.8 156.5 0.8 0.7 0.7 4 5 6 1.9 2.4 2.9 0.5 0.5 0.5 49 50 51 29.1 29.9 30.8 0.8 0.9 0.9 94 95 96 98.2 100.3 102.3 2.1 2.0 2.0 139 140 141 157.2 157.9 158.6 0.7 0.7 0.7 7 8 9 3.4 3.9 4.4 0.5 0.5 0.5 52 53 54 31.7 32.7 33.6 1.0 0.9 1.0 97 98 99 104.3 106.2 108. 2 1.9 2.0 1.9 142 143 144 159.3 160.0 160.7 0.7 0.7 0.6 10 11 12 4.9 5.4 5.9 0.5 0.5 0.5 55 56 57 34.6 35.6 36.6 1.0 1.0 1.1 100 101 102 110. 1 111.9 113.8 1.8 1.9 1.7 145 146 147 161.3 162.0 162.6 0.7 0.6 0.6 13 14 15 6.4 6.9 7.4 0.5 0.5 0.5 58 59 60 37.7 38.8 39.9 1.1 1.1 1.2 103 104 105 115.5 117.3 119.0 1.8 1.7 1.7 148 149 150 163.2 163.8 164.4 0.6 0.6 0.6 16 17 18 7.9 8.4 8.9 0.5 0.5 0.5 61 62 63 41.1 42.3 43.5 1.2 1.2 1.2 106 107 108 120.7 122.3 123.9 1.6 1.6 1.6 151 152 153 165.0 165.6 166.2 0.6 0.6 0.5 19 20 21 9.4 10.0 10.5 0.6 0.5 0.5 64 65 66 44.7 46.0 47.3 1.3 1.3 1.4 109 110 111 125.5 127.0 128.5 1.5 1.5 1.4 154 155 156 166.7 167.3 167.9 0.6 0.6 0.5 22 23 24 11.0 11.6 12.1 0.6 0.5 0.6 67 68 69 48.7 50.1 51.5 1.4 1.4 1.5 112 113 114 129.9 131.3 132.7 1.4 1.4 1.3 157 158 159 168.4 169.0 169.5 0.6 0.5 0.5 25 26 27 12.7 13.3 13.8 0.6 0.5 0.6 70 71 72 53.0 54.5 56.1 1.5 1.6 1.6 115 116 117 134.0 135.3 136.5 1.3 1.2 1.2 160 161 162 170.0 170.6 171.1 0.6 0.5 0.5 28 29 30 14.4 IS. 15.6 0.6 0.6 0.6 73 74 75 57.7 59.3 61.0 1.6 1.7 1.7 118 119 120 137.7 138.9 140.1 1.2 1.2 1.1 163 164 165 171.6 172.1 172.6 0.5 0.5 0.5 31 32 33 16.2 16.8 17.4 0.6 0.6 0.6 76 77 78 62.7 64.5 66.2 1.8 1.7 1.9 121 122 123 141.2 142.3 143.4 1.1 1.1 1.0 166 167 168 173.1 173.6 174.1 0.5 0.5 0.5 34 35 36 18.0 18.7 19.3 0.7 0.6 0.7 79 80 81 68.1 7L8 1.8 1.9 2.0 124 125 126 144.4 145.4 146.4 1.0 1.0 0.9 169 170 171 174.6 175.1 175.6 0.5 0.5 0.5 37 38 39 20.0 20.7 21.4 0.7 0.7 0.7 82 83 84 73.8 75.7 77.7 1.9 2.0 2.0 127 128 129 147.3 148.3 149.2 1.0 0.9 0.9 172 173 174 176.1 176.6 177.1 0.5 0.5 0.5 40 41 42 22.1 22.8 23.5 0.7 0.7 0.7 85 86 87 79.7 81.8 83.8 2.1 2.0 2.1 130 131 132 150.1 150.9 151.8 0.8 0.9 0.8 175 176 177 177.6 178.1 178.5 0.5 0.4 0.5 43 44 45 24.2 25.0 25.8 0.8 0.8 88 89 90 85.9 87.9 90.0 2.0 2.1 133 134 135 152.6 153.4 154.2 0.8 0.8 178 179 180 179.0 179.5 180.0 0.5 0.5 HARMONIC ANALYSIS AND PREDICTION OF TIDES 181 Table 10. — Values of Q for argument of constituent Mi — Continued p Q Diff. P Q Diff. P Q Diff. P Q Diff. o o o 180 180.0 0.5 225 205.8 0.8 270 270.0 2.1 315 334.2 0.8 181 182 183 180.5 181.0 181.5 0.5 0.5 0.4 226 227 228 206.6 207.4 208.2 0.8 0.8 0.9 271 272 273 272.1 274.1 276.2 2.0 2.1 2.0 316 317 313 335.0 335.8 336.5 0.8 0.7 0.7 184 185 186 181.9 182.4 182.9 0.5 0.5 0.5 229 230 231 209.1 209.9 210.8 0.8 0.9 0.9 274 275 276 278.2 280.3 282.3 2.1 2.0 2.0 319 320 321 337.2 337.9 338.6 0.7 0.7 0.7 187 188 189 183.4 183.9 184.4 0.5 0.5 0.5 232 233 234 211.7 212.7 213.6 1.0 0.9 1.0 277 278 279 284.3 286. 2 288.2 1.9 2.0 1.9 322 323 324 339.3 340.0 340.7 0.7 0.7 0.6 190 191 192 184.9 185.4 185.9 0.5 0.5 0.5 235 236 237 214.6 215.6 216.6 1.0 1.0 1.1 280 281 282 290.1 291.9 293.8 1.8 1.9 1.7 325 326 327 341.3 342.0 342.6 0.7 0.6 0.6 193 194 195 186.4 186.9 187.4 0.5 0.5 0.5 238 239 240 217.7 218.8 219.9 1.1 1.1 1.2 283 284 285 295.5 297.3 299.0 1.8 1.7 1.7 328 329 330 343.2 343.8 344.4 0.6 0.6 0.6 196 197 198 187.9 188.4 188.9 0.5 0.5 0.5 241 242 243 221.1 222.3 223.5 1.2 1.2 1.2 286 287 288 300.7 302. 3 303.9 1.6 1.6 ].6 331 332 333 345.0 345.6 346.2 0.6 0.6 0.5 199 200 201 189.4 190.0 190.5 0.6 0.5 0.5 244 245 246 224.7 226.0 227.3 1.3 1.3 1.4 289 290 291 305.5 307.0 308.5 1.5 1.5 1.4 334 335 336 346.7 347.3 347.9 0.6 0.6 0.5 202 203 204 191.0 191.6 192.1 0.6 0.5 0.6 247 248 249 228.7 230.1 231.5 1.4 1.4 1.5 292 293 294 309.9 311.3 312.7 1.4 1.4 1.3 337 338 339 348.4 349.0 349.5 0.6 0.5 0.5 205 206 207 192.7 193.3 193.8 0.6 0.5 0.6 250 251 252 233.0 234.5 236.1 1.5 1.6 1.6 295 296 297 314.0 315.3 316.5 1.3 1.2 1.2 340 341 342 350.0 350.6 351.1 0.6 0.5 0.5 208 209 210 194.4 195.0 195.6 0.6 0.6 0.6 253 254 255 237.7 239.3 241.0 1.6 1.7 1.7 298 299 300 317.7 318.9 320.1 1.2 1.2 1.1 343 344 345 351.6 352.1 352.6 0.5 0.5 0.5 211 212 213 196.2 196.8 197.4 0.6 0.6 0.6 256 257 258 242.7 244.5 246.2 1.8 1.7 1.9 301 302 303 321.2 322.3 323.4 1.1 1.1 1.0 346 347 348 353.1 353.6 354.1 0.5 0.5 0.5 214 215 216 198.0 198.7 199.3 0.7 0.6 0.7 259 260 261 248.1 249.9 251.8 1.8 1.9 2.0 304 305 306 324.4 325.4 326.4 1.0 1.0 0.9 349 350 351 354.6 355.1 355.6 0.5 0.5 0.5 217 218 219 200.0 200.7 201.4 0.7 0.7 0.7 262 263 264 253.8 255.7 257.7 1.9 2.0 2.0 307 308 309 327.3 328.3 329.2 1.0 0.9 0.9 352 353 354 356.1 356.6 357.1 0.5 0.5 0.5 220 221 222 202.1 202.8 203.5 0.7 0.7 0.7 265 266 267 259.7 261.8 263.8 2.1 2.0 2.1 310 311 312 330.1 330.9 331.8 0.8 0.9 0.8 355 356 357 357.6 358.1 358.5 0.5 0.4 0.5 223 224 225 204.2 205.0 205.8 0.8 0.8 268 269 270 265.9 267.9 270.0 2.0 2.1 313 314 315 332.6 333.4 334.2 0.8 0.8 358 359 360 359.0 359.5 360.0 0.5 0.5 182 IT. S. COAST AND GEODETIC SURVEY Table 11. — Values of u for equilibrium arguments [Use sign at head of column when Nis between and 180°, reverse sign when Nis between 180 and I N h K t K 2 M2, N2 2N, MS X, n, v M 3 M 4 ,MN M 6 M 8 Oi,Qi 2Q,p 00 MK 2MK Mf N o o o o + 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 2 3 0.19 0.38 0.56 0.13 0.27 0.40 0.28 0.57 0.85 0.04 0.08 0.11 0.05 0.11 0.17 0.08 0.15 0.23 0.11 0.23 0.34 0.15 0.30 0.45 0.15 0.30 0.45 0.53 1.05 1.57 0.17 0.34 0.52 0.06 0.12 0.17 0.34 0.67 1.01 359 358 357 4 5 6 0.75 0.94 1.12 0.54 0.67 0.80 1.14 1.42 1.70 0.15 0.19 0.23 0.23 0.28 0.34 0.30 0.38 0.45 0.45 0.56 0.68 0.60 0.75 0.90 0.60 0.75 0.90 2.10 2.62 3.14 o!86 1.03 0.23 0.29 0.35 1.35 1.68 2.02 356 355 354 7 8 9 1.31 1.50 1.68 0.94 1.07 1.20 1.99 2.27 2.55 0.26 0.30 0.34 0.40 0.45 0.51 0.53 0.60 0.68 0.79 0.90 1.01 1.05 1.20 1.35 1.05 1.20 1.35 3.67 4.19 4.71 1.20 1.37 1.54 0.41 0.47 0.53 2.36 2.69 3.03 353 352 351 JO li 12 1.87 2.05 2.24 1.34 1.47 1.60 2.83 3.11 3.39 0.37 0.41 0.45 0.56 0.62 0.67 0.75 0.82 0.90 1.12 1.24 1.34 1.49 1.64 1.79 1.49 1.64 1.79 5.23 5.75 6.27 1.71 1.88 2.05 0.59 0.64 0.70 3.36 3.70 4.03 350 349 348 13 14 15 2.42 2.61 2.79 1.73 1.86 1.99 3.67 3.95 4.23 0.48 0.52 0.56 0.73 0.78 0.84 0.97 1.04 1.12 1.45 1.56 1.67 1.94 2.09 2.23 1.94 2.09 2.23 6.79 7.31 7.82 2.21 2.38 2.55 0.76 0.82 0.88 4.36 4.70 5.03 347 346 345 16 17 18 2.98 3.16 3.34 2.12 2.25 2.38 4.51 4.78 5.06 0.60 0.63 0.67 0.89 0.95 1.00 1.19 1.26 1.34 1.79 1.90 2,00 2.38 2.53 2.67 2.38 2.53 2.68 8.34 8.85 9.36 2.72 2.89 3.05 . 0.93 0.99 1.05 5.36 5.69 6.02 344 343 342 19 20 21 3.52 3.71 3.89 ?.51 2.64 2.77 5.33 5.60 5.87 0.70 0.74 0.77 1.06 1.11 1.16 1.41 1.48 1.55 2.11 2.21 2.32 2.81 2.95 3.09 2.82 2.97 3.11 9.87 10.38 10.89 3.22 3.38 3.54 1.11 1.17 1.23 6.35 6.67 7.00 341 340 339 22 23 24 4.07 4.25 4.42 2.90 3.03 3.15 6.14 6.41 6.68 0.81 0.84 0.88 1.21 1.26 1.31 1.62 L75 2.42 2.53 2.63 3.23 3.37 3.51 3.26 3.40 3.55 11.39 11.89 12.39 3.71 3.87 4.03 1.28 1.34 1.40 7.33 7.65 7.97 338 337 336 25 26 27 4.60 4.78 4.96 3.28 3.40 3.53 6.94 7.21 7.47 0.91 0.94 0.98 1.37 1.42 1.47 1.82 1.89 1.96 2.73 2.83 2.94 3.64 3.78 3.92 3^83 3.98 12.89 13.39 13.89 4.19 4.35 4.51 1.46 1.52 1.57 8.29 8.61 8.93 335 334 333 28 29 30 5.13 5.30 5.48 3.65 3.78 3.90 7.73 7.99 8.24 1.01 1.04 1.08 1.52 1.57 1.62 2.02 2.09 2.16 3.04 3.13 3.23 4.05 4.18 4.31 4.12 4.26 4.40 14.38 14.87 15.36 4.67 4.82 4.98 1.63 1.69 1.75 9.25 9.57 9.88 332 331 330 31 32 33 5.65 5.82 5.99 4.02 4.14 4.26 8.50 8.75 9.00 1.11 1.14 1.17 1.67 1.72 1.76 2.22 2.29 2.35 3.33 3.43 3.52 4.45 4.58 4.70 4.54 4.68 4.82 15.84 16.32 16.80 5.13 5.29 5.44 1.80 1.86 1.92 10.19 10.50 10.81 329 328 327 34 35 36 6.16 6.33 6.50 4.38 4.50 4.62 9.25 9.50 9.74 1.20 1.24 1.27 1.81 1.85 1.90 2.41 2.47 2.53 3.61 3.71 3.80 4.82 4.94 5.06 4.96 5.10 5.23 17.28 17.76 18.23 5.59 5.74 5.89 1.97 2.03 2.09 11.12 11.43 11.73 326 325 324 37 38 39 6.66 6.83 6.99 4.74 4.85 4.97 9.98 10.22 10.46 1.30 1.33 1.36 1.94 1.99 2.03 2.59 2.65 2.71 3.89 3.98 4.07 5.18 5.30 5.42 5.37 5.50 5.64 18.69 19.16 19.62 6.03 6.18 6.32 2.15 2.20 2.26 12.03 12. 33 12.63 323 322 321 40 41 42 7.15 7.31 7.47 5.08 5.19 5.30 10.69 10.93 11.16 1.38 1.41 1.44 2.08 2.12 2.16 2.77 2.82 2.88 4.15 4.24 4.32 5.54 5.65 5.76 5.77 5.90 6.03 20.08 20.53 20.98 6.46 6.60 6.74 2.31 2.37 2.42 12.92 13.22 13.51 320 319 318 43 44 45 7.63 7.79 7.94 5.41 5.52 5.63 11.38 11.60 11.82 1.47 1.50 1.52 .2.20 2.24 2.28 2.94 2.99 3.04 4.40 4.49 4.57 5.87 5.98 6.09 6.16 6.29 6.42 21.43 21.87 22.31 6.88 7.02 7.15 2.48 2.53 2.59 13.80 14.08 14.37 317 316 315 Note.— For L 2 and Mi see Table 13; for 2SM and MSf, take u of M2 with sign reversed; for Pi, R2, Si, S 2 , S3, S 4 , T2, Mm, Sa, and Ssa, take u=0. HARMONIC ANALYSIS AND PREDICTION OF TIDES 183 Table 11. — Values of u for equilibrium arguments — Continued [ Use sign at head of column when N is between and 180°, reverse sign when N is between 180 and 860°] N h Ki K 2 M2, N2 2N,MS M 3 M f ,MN M 6 Ms Oi,Qi 2Q,p. OO MK 2MK Mf N o o o o 45 7.94 5.63 11.82 1.52 2.28 3.04 4.57 6.09 + 6.42 22. 31 7.15 + 2.59 14.37 315 46 47 48 8.10 8.25 8.40 5.74 5.84 5.95 12.04 12.26 12.47 1.55 1.57 1.60 2.32 2.36 2.40 3.10 3.15 3.20 4.64 4.72 4.80 6.19 6.30 6.40 6.55 6.68 6.80 22.75 23.18 23.60 7.28 7.41 7.54 2.64 2.69 2.75 14.65 14.93 15.20 314 313 312 49 50 51 8.55 8.70 8.84 6.05 6.15 6.25 12.68 12.88 13.08 1.62 1.65 1.67 2.44 2.47 2.51 3.25 3.30 3.34 4.87 4.94 5.01 6.50 6.59 6.68 6.92 7.05 7.17 24.02 24.44 24.85 7.67 7.80 7.92 2.80 2.85 2.91 15.47 15.74 16.01 311 310 309 52 53 54 8.99 9.13 9.27 6.35 6.45 6.54 13.28 13.48 13.67 L72 1.74 2.54 2.58 2.61 3.39 3.44 3.48 5.08 5.15 5.22 6.78 6.87 6.96 7.29 7.41 7.53 25.26 25.66 26.06 8.04 8.17 8.28 2.96 3.01 3.06 16.28 16.54 16.80 308 307 306 55 56 57 9.41 9.54 9.68 6.64 6.73 6.82 13.86 14.05 14.23 1.76 1.78 1.80 2.64 2.67 2.70 3.52 3.56 3.60 5.28 5.34 5.40 7.04 7.12 7.20 7.65 7.76 7.88 26.46 26.85 27.23 8.40 8.51 8.62 3.12 3.17 3.22 17.05 17.30 17.55 305 304 303 58 59 60 9.81 9.94 10.07 6.91 7.00 7.09 14.40 14.58 14.75 1.82 1.84 1.86 2.73 2.76 2.79 3.64 3.68 3.72 5.46 5.52 5.58 7.28 7.36 7.44 7.99 8.10 8.21 27.61 27.98 28.34 8.73 8.84 8.95 3.27 3.32 3.37 17.80 18.04 18.28 302 301 300 61 62 63 10.19 10.32 10.44 7.17 7.26 7.34 14.92 15.08 15.24 1.88 1.90 1.91 2.82 2.84 2.87 3.76 3.79 3.82 5.63 5.69 5.74 7.51 7.58 7.65 8.32 8.42 8.53 28.70 29.06 29.41 9.05 9.15 9.25 3.42 3.46 3.51 18.51 18.74 18.97 299 298 297 64 65 66 10.56 10.68 10.79 7.42 7.49 7.57 15.39 15.54 15.69 1.93 1.94 1.96 2.89 2.92 2.94 3.86 3.89 3.92 5.78 5.83 5.88 7.71 7.78 7.84 8.63 8.73 8.83 29.75 30.09 30.42 9.35 9.44 9.53 3.56 3.61 3.65 19.19 19.41 19.63 296 295 294 67 68 69 10.91 11.02 11.12 7.64 7.72 7.79 15.83 15.96 16.10 1.98 1.99 2.00 2.96 2.98 3.00 3.95 3.98 4.00 5.93 5.97 6.01 7.90 7.96 8.01 8.93 9.03 9.12 30.74 31.06 31.37 9.62 9.71 9.79 3.69 3.74 3.78 19.84 20.04 20.25 293 292 291 70 71 72 11.23 11.33 11.43 7.86 7.92 7.99 16.23 16.35 16.47 2.02 2.03 2.04 3.02 3.04 3.06 4.03 4.06 4.08 6.05 6.08 6.11 8.06 8.11 8.15 9.22 9.31 9.40 31.68 31.98 32.27 9.87 9.95 10.03 3.83 3.87 3.91 20.45 20.64 20.83 290 289 288 73 74 75 11.53 11.63 11.72 8.05 8.11 8.17 16.58 16.69 16.80 2.05 2.06 2.07 3.08 3.09 3.10 4.10 4.12 4.14 6.15 6.18 6.21 8.20 8.24 8.28 9.48 9.57 9.65 32.55 32.82 33.09 10.10 10.17 10.24 3.95 3.99 4.03 21.01 21.20 21.37 287 286 285 76 77 78 11.81 11.90 11.98 8.23 8.28 8.34 16.90 17.00 17.09 2.08 2.09 2.10 3.12 3.13 3.14 4.16 4.18 4.19 6.24 6.26 6.29 8.32 8.35 8.38 9.73 9.81 9.88 33.35 33.60 33.85 10.31 10.37 10.43 4.07 4.11 4.15 21.54 21.71 21.87 284 283 282 79 80 81 12.06 12.14 12.22 8.39 8.44 8.48 17.17 17. 25 17.33 2.10 2.11 2.11 3.15 3.16 3.17 4.20 4.22 4.23 6.31 6.32 6.34 8.41 8.43 8.46 9.96 10.03 10.10 34.09 34.31 34.53 10.49 10.54 10.60 4.18 4.22 4.25 22.02 22.17 22.32 281 280 279 82 83 84 12.29 12.36 12.42 8.53 8.57 8.61 17.40 17.46 17.52 2.12 2.12 2.13 3.18 3.19 3.19 4.24 4.25 4.26 6.36 6.37 6.38 8.48 8.50 8.51 10.17 10.23 10.30 34. 74 34.95 35.14 10.65 10.69 10.73 4.29 4.32 4.35 22. 46 22.59 22.72 278 277 276 85 86 87 12.49 12.55 12.60 8.64 8.68 8.71 17.58 17.63 17.67 2.13 2.13 2.14 3.20 3.20 3.20 4.26 4.27 4.27 6.39 6.40 6.41 8.52 8.53 8.54 10.36 10. 41 10.47 35. 33 35.50 35.67 10.77 10.81 10.84 4.38 4.41 4.44 22.84 22.96 23.07 275 274 273 88 89 90 12.65 12.70 12.75 8.74 8.76 8.79 17. 71 17.74 17.77 2.14 2.14 2.14 3.20 3.20 3.20 4.27 4.27 4.27 6.41 6.41 6.41 8.54 8.54 8.54 10.52 10.57 10.62 35.83 35.98 36.12 10.87 10.90 10.93 4.47 4.49 4.52 23.17 23.27 23.37 272 271 270 Note.— For Lj and Mi see Table 13; for 2SM and MSf, take u of Mj with sign reversed: for Pi, R., Si, Sj, S3, S 4 , T 2 , Mm, Sa, and Ssa, take u=0. 184 U. S. COAST AND GEODETIC STJBVEY Table 11. — Values of u for equilibrium arguments — Continued [Use sign at head of column when Nis between and 180°, reverse sign when N is between 180 and 380°] N Ji Ki K 2 M2, N2 2N,MS X, ix, v M 3 M ( ,MN M 6 M 8 Oi,Qi 2Q, P i OO MK 2MK Mf N o o o • 90 12.75 8.79 17.77 2.14 3.20 4.27 6.41 8.54 + 10.62 36.12 10.93 + 4.52 23.37 270 91 92 93 12.79 12.83 12.87 8.81 8.83 8.85 17.79 17.81 17.82 2.14 2.13 2.13 3.20 3.20 3.20 4.27 4.27 4.26 6.41 6.40 6.40 8.54 8.54 8.53 10.66 10.70 10.74 36.25 36.37 36.48 10.95 10.96 10.98 4.54 4.56 4.58 23.46 23.54 23.61 269 268 267 94 95 96 12.90 12.93 12. 96 8.86 8.87 8.88 17.83 17.83 17.82 2.13 2.13 2.12 3.20 3.19 3.19 4.26 4.26 4.25 6.39 6.38 6.37 8.52 8.51 8.49 10.77 10.80 10.83 36.58 36.67 36.75 10.99 11.00 11.01 4.60 4.62 4.64 23.67 23.73 23.79 266 265 264 97 98 99 12.98 13.00 13.01 8.89 8.90 8.90 17.81 17.79 17.77 2.12 2.11 2.11 3.18 3.17 3.16 4.24 4.22 4.21 6.35 6.34 6.32 8.47 8.45 8.43 10.86 10.88 10.90 36.82 36.87 36.92 11.01 11.01 11.00 4.65 4.67 4.68 23.84 23.88 23.91 263 262 261 100 101 102 13.02 13.03 13.03 8.89 8.89 8.88 17.74 17.71 17.67 2.10 2.09 2.09 3.15 3.14 3.13 4.20 4.19 4.17 6.30 6.28 6.26 8.41 8.38 8.34 10.92 10.93 10.94 36.95 36.98 36.99 11.00 10.99 10.97 4.69 4.70 4.71 23.93 23.95 23.96 260 259 258 103 104 105 13.02 13.02 13.01 8.87 8.86 8.84 17.62 17.57 17.51 2.08 2.07 2.06 3.11 3.10 3.09 4.15 4.14 4.12 6.23 6.20 6.17 8.30 8.27 8.23 10.95 10.95 10.95 37.00 36.99 36.96 10.95 10.93 10.90 4.72 4.72 4.73 23.97 23.97 23.96 257 256 255 106 107 108 12.99 12.97 12.95 8.82 8.80 8.78 17.45 17.38 17.30 2.05 2.04 2.03 3.07 3.06 3.04 4.10 4.08 4.06 6.14 6.11 6.08 8.19 8.15 8.11 10.94 10.94 10. 9.3 36.93 36.89 36.83 10.87 10.84 10.81 4.73 4.73 4.72 23.94 23.91 23.88 254 253 252 109 110 111 12.93 12.90 12.86 8.75 8.72 8.69 17.22 17.14 17.05 2.02 2.00 1.99 3.02 3.00 2.98 4.03 4.00 3.98 6.05 6.01 5.97 8.06 8.01 7.96 10.91 10.89 10.87 36.76 36.67 36.58 10.77 10.72 10.68 4.72 4.72 4.71 23.84 23.79 23.73 251 250 249 112 113 114 12.82 12.77 12.72 8.65 8.61 8.57 16.95 16.84 16.73 1.98 1.96 1.94 2.96 2.94 2.92 3.95 3.92 3.89 5.93 5.88 5.83 7.90 7.84 7.78 10.84 10.81 10.78 36.48 36.36 36.23 10.63 10.57 10.51 4.70 4! 68 23.66 23.59 23.50 248 247 246 115 116 117 12.67 12.61 12.55 8.52 8.48 8.43 16.62 16.50 16.37 1.93 1.91 1.90 2.89 2.87 2.84 3.86 3.82 3.79 5.78 5.74 5.69 7.71 7.65 7.58 10.74 10.70 10.65 36.09 35.93 35.76 10.45 10.39 10.32 4.67 4.65 4.63 23.41 23.31 23.21 245 244 243 118 119 120 12.48 12.41 12.34 8.37 8.31 8.25 16.24 16.10 15.96 1.88 1.86 1.84 2.82 2.79 2.77 3.76 3.72 3.69 5.64 5.59 5.53 7.52 7.45 7.38 10.60 10.55 10.49 35. 57 35.37 35.16 10.25 10.18 10.10 4.61 4.59 4.57 23.09 22.96 22.83 242 241 240 121 122 123 12.26 12.17 12.08 8.19 8.13 8.06 15.81 15.66 15.50 1.82 1.80 1.78 2.74 2.71 2.68 3.65 3.61 3.57 5.47 5.41 5.35 7.30 7.22 7.14 10.43 10.37 10.30 34.94 34.70 34.49 10.02 9.93 9.84 4.54 4.52 4.49 22.69 22.54 22.37 239 238 237 124 125 126 11.98 11.88 11.78 7.99 7.91 7.83 15.33 15.16 14.99 1.76 1.74 1.72 2.64 2.61 2.58 3.52 3.48 3.44 5.29 5.22 5.15 7.05 6.96 6.87 10.22 10.14 10.06 34.19 33.91 33.62 9.75 9.65 9.55 4.46 4.43 4.40 22.20 22.03 21.84 236 235 234 127 128 129 11.67 11.56 11.44 7.75 7.67 7.58 14.81 14.62 14.43 1.70 1.67 1.65 2.54 2.51 2.48 3.39 3.34 3.30 5.09 5.02 4.95 6.78 6.69 6.60 9.97 9.88 9.79 33.31 32.99 32.66 9.45 9.34 9.23 4.36 4.32 4.28 21.64 21.44 21.23 233 232 231 130 131 132 11.31 11.18 11.05 7.49 7.40 7.30 14.23 14.03 13.83 1.63 1.60 1.58 2.44 2.41 2.37 3.26 3.21 3.16 4.88 4.81 4.74 6.51 6.42 6.32 9.69 9.58 9.47 32.31 31.95 31.58 9.12 9.00 8.88 4.23 4.19 4.14 21.00 20.76 20.52 230 229 228 133 134 135 10.91 10.77 10.62 7.20 7.10 7.00 13. 62 13.40 13.18 1.55 1.53 1.50 2.33 2.29 2.25 3.11 3.06 3.00 4.66 4.58 4.51 6.22 6.11 6.01 9.36 9.24 9.12 31.19 30.79 30.37 8.76 8.63 8.50 4.09 4.04 3.99 20.27 20.01 19.75 227 226 225 Note.— For L2 and Mi see table 13; for 2SM and MSf, take u of M2 with sign reversed; for Pi, B2, Si, S2, S3, S<, Tj, Mm, Sa and Ssa, take w=0. HARMONIC ANALYSIS AND PREDICTION OF TIDES 185 Table 11. — Values of u for equilibrium arguments — Continued [Use sign at head of column when Nis between and 180°, reverse sign when Nis between 180 and 860°] N Jl K, K 2 M2, Nj 2N, MS X, 11, v Mj M 4 ,MN Me Mg 2Q,p OO MK 2MK Mf N o o o _ _ _ — — — — — + — — + — 135 10.62 7.00 13.18 1.50 2.25 3.00 4.51 6.01 9.12 30.37 8.0) 3.99 19.75 225 136 10.47 6.89 12.96 1.48 2.21 2.95 4.43 5.90 9.00 29.94 8.36 3.94 19.47 224 137 10.31 6.78 12.73 1.45 2.17 2.90 4 34 5.79 8.87 29.49 8.22 3.88 19.18 223 138 10.15 6.66 12.50 1.42 2.13 2.84 4.26 5.68 8.73 29.04 8.08 3.82 18.88 222 139 9.98 6.55 12.26 1.39 2.09 2.78 4.18 5.57 8.59 28.56 7.94 3.76 18.58 221 140 9.81 6.43 12.02 1.36 2.05 2.73 4.09 5.46 8.45 28.08 7.79 3.70 18.26 220 141 9.64 6.31 11.77 1.34 2.00 2.67 4.01 5.34 8.30 27.58 7.64 3.64 17.94 219 142 9.46 6.18 11.52 1.31 1.96 2.61 3.92 5.22 8.15 27.07 7.49 3.57 17.61 218 143 9.27 6.06 11.27 1.28 1.91 2.55 3.83 5.10 8.00 26.54 7.33 3.50 17.27 217 144 9.08 5.93 11.01 1.25 1.87 2.49 3.74 4.98 7.84 26.00 7.17 3.43 16.92 216 145 8.89 5.80 10.74 1.22 1.82 2.43 3.65 4.86 7.67 25.45 7.01 3.36 16.56 215 146 8.69 5.66 10.48 1.19 1.78 2.37 3.56 4.74 7.50 24.89 6.84 3.29 16.20 214 147 8.49 5.52 10.21 1.16 1.73 2.31 3.47 4.62 7.33 24.31 6.68 3.21 15.82 213 148 8.28 5.38 9.94 1.12 1.69 2.25 3.37 4.50 7.16 23.72 6.51 3.14 15.44 212 149 8.07 5.24 9.66 1.09 1.64 2.18 3.28 4.37 6.98 23.12 6.33 3.06 15.05 211 150 7.85 5.10 9.38 1.06 1.59 2.12 3.18 4.24 6.80 22.51 6.16 2.98 14.65 210 151 7.63 4.95 9.10 1.03 1.54 2.06 3.08 4.11 6.61 21.88 5.98 2.90 14.24 209 152 7.41 4.80 8.81 1.00 1.49 1.99 2.99 3.98 6.42 21.24 5.80 2.81 13.83 208 153 7.18 4.65 8.52 0.96 1.44 1.92 2.89 3.85 6.22 20.59 5.61 2.73 13.41 207 154 6.95 4.50 8.23 0.93 1.39 1.86 3.78 3.71 6.03 19.93 5.43 2.64 12.98 206 155 6.72 4.34 7.94 0.90 1.34 1.79 2.69 3.58 5.82 19.26 5.24 2.55 12.54 205 156 6.48 4.19 7.64 0.86 1.29 1.72 2.59 3.45 5.62 18.58 5.05 2.46 12.10 204 157 6.24 4.03 7.34 0.83 1.24 1.66 2.48 3.31 5.41 17.89 4.85 2.37 11.65 203 158 5.99 3.87 7.04 0.79 1.19 1.59 2.38 3.18 5.20 17.19 4.66 2.28 11.19 202 159 5.74 3.70 6.74 0.76 1.14 1.52 2.28 3.04 4,99 16.48 4.46 2.18 10.73 201 160 5.49 3.54 6.43 0.72 1.09 1.45 2.17 2.90 4.77 15.75 4.26 2.09 10.26 200 161 5.24 3.37 6.12 0.69 1.04 1.38 2.07 2.76 4.55 15.02 4.06 1.99 9.79 199 162 4.98 3.20 5.81 0.66 0.98 1.31 1.97 2.62 4.33 14.29 3.86 1.89 9.31 198 163 4.72 3.03 5.50 0.62 0.93 1.24 1.86 2.48 4.10 13.54 3.66 1.79 8.82 197 164 4.46 2.86 5.19 0.58 0.88 1.17 1.75 2.34 3.87 12.78 3.45 1.70 8.33 196 165 4.19 2.69 4.87 0.55 0.82 1.10 1.64 2.19 3.64 12.02 3.24 1.60 7.83 195 166 3.92 2.52 4.55 0.51 0.77 1.02 1.54 2.05 3.41 11.25 3.03 1.49 7.33 194 167 3.65 2.34 4.23 0.48 0.71 0.95 1.43 1.90 3.18 10.48 2.82 1.39 6.83 193 168 3.38 2.17 3.91 0.44 0.66 0.88 1.32 1.76 2.94 9.70 2.61 1.29 6.32 192 169 3.10 1.99 3.59 0.40 0.61 0.81 1.21 1.62 2.70 8.91 2.39 1.18 5.80 191 170 2.83 1.81 3.27 0.37 0.55 0.74 1.10 1.47 2.46 8.12 2.18 1.08 5.29 190 171 2.55 1.63 2.94 0.33 0.50 0.66 1.00 1.33 2.22 7.32 1.97 0.97 4.77 189 172 2.27 1.45 2.62 0.30 0.44 0.59 0.89 1.18 1.97 6.51 1.75 0.86 4.24 188 173 1.99 1.27 2.29 0.26 0.39 0.51 0.77 1.03 1.73 5.71 1.53 0.76 3.72 187 174 1.71 1.09 1.97 0.22 0.33 0.44 0.66 0.88 1.49 4.90 1.31 0.65 3.19 186 175 1.42 0.91 1.64 0.18 0.28 0.37 0.55 0.74 1.24 4.09 1.10 0.54 2.66 185 176 1.14 0.73 1.31 0.15 0.22 0.30 0.44 0.59 0.99 3.27 0.88 0.43 2.13 184 177 0.86 0.55 0.99 0.11 0.17 0.22 0.33 0.44 0.75 2.46 0.66 0.33 1.60 183 178 0.57 0.37 0.66 0.07 0.11 0.14 0.22 0.29 0.50 1.64 0.44 0.22 1.07 182 179 0.29 0.18 0.33 0.04 0.05 0.07 0.11 0.14 0.25 0.82 0.22 0.11 0.53 181 180 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 180 Note.— For L2 and Mi see Table 13; for 2SM and MSf, take u of Ma with sign reversed; for Pi, Rj, Si, S3, S 4 , T2, Mm, Sa, and Ssa, take u=0. Sj, 186 U. S. COAST AND GEODETIC SURVEY Table 12. — Log factor F corresponding to every tenth of a degree of I Constituent^^ 18.3° Diff. 18.4° Diff. 18.5° Diff. 18.6° Diff. 18.7° Diff. 18.8° Diff. Ji__ .. ... 0. 0827 0. 0547 0. 1263 9. 9839 9. 9758 9. 9678 9. 9516 9. 9355 0. 0939 0. 3139 0. 0386 0. 0224 0. 2039 9. 9465 -21 -12 -22 +2 +4 +4 +8 +10 -22. -69 -10 -7 -46 +8 0. 0806 0. 0535 0. 1241 9. 9841 9. 9762 9. 9682 9.9524 9. 9365 0. 0917 0. 3070 0. 0376 0. 0217 0. 1993 9. 9473 -20 -12 -21 +3 +4 +5 +7 +10 -21 -70 -9 -6 -45 +8 0. 0786 0. 0523 0. 1220 9. 9844 9. 9766 9. 9687 9. 9531 9. 9375 0. 0896 0. 3000 0. 0367 0. 0211 0. 1948 9. 9481 -20 -11 -21 +2 +3 +5 +7 +10 -22 -69 -9 -7 -45 +8 0. 0766 0.0512 0. 1199 9. 9846 9. 9769 9. 9692 9. 9538 9. 9385 0. 0874 0. 2931 0. 0358 0. 0204 0. 1903 9. 9489 -20 -12 -22 +3 +4 +5 +8 +10 -21 -68 -9 -7 -45 +8 0. 0746 0. 0500 0. 1177 9. 9849 9. 9773 9. 9697 9. 9546 9. 9395 0. 0853 0. 2863 0. 0349 0. 0197 0. 1858 9. 9497 -19 -12 -22 +2 +4 +5 +7 +10 -21 -10 -7 -45 +8 0. 0727 0. 0488 0. 1155 9. 9851 9. 9777 9. 9702 9. 9553 9. 9405 0. 0832 0. 2794 0. 0339 0.0190 0. 1813 9. 9505 —20 Ki K 2 M 2 *, N 2 , 2N___. M 3 M t , MN M 6 -11 -21 +3 +3 +5 +8 M 8 Oi, Qi 2Q, pi... oo__. _.:._..... MK +10 -21 -68 —9 2MK Mf -6 —44 Mm +9 \ I Constituent^^ 18.9° Diff. 19.0° Diff. 19.1° Diff. 19.2° Diff. 19.3° Diff. 19.4° Diff. Ji Ki 0. 0707 0. 0477 0. 1134 9. 9854 9 9780 9. 9707 9. 9561 9. 9415 0. 0811 0. 2726 0. 0330 0. 0184 0. 1769 9. 9514 -20 -12 -22 +2 +4 +5 +8 +10 -21 -67 -9 -7 -44 +8 0. 0687 0. 0465 0.1112 9. 9856 9. 9784 9. 9712 9. 9569 9. 9425 0. 0790 0. 2659 0. 0321 0. 0177 0. 1725 9. 9522 -19 -12 -22 +3 +4 +5 +7 +10 -20 -67 -9 -6 -44 +8 0. 0668 0. 0453 0. 1090 9. 9859 9. 9788 9. 9717 9. 9576 9. 9435 0. 0770 0. 2592 0. 0312 0. 0171 0. 1681 9. 9530 -19 -11 -22 +2 +4 +6 +8 +10 -21 -67 -9 -7 -44 +9 0. 0649 0. 0442 0. 1068 9. 9861 9. 9792 9. 9723 9. 9584 9. 9445 0. 0749 0. 2525 0. 0303 0. 0164 0. 1637 9. 9539 -19 -11 -23 +3 +4 +5 +8 +10 -20 -66 -9 -6 -43 +8 0. 0630 0. 0431 0. 1045 9. 9864 9. 9796 9. 9728 9. 9592 9. 9455 0. 0729 0. 2459 0. 0294 0. 0158 0. 1594 9. 9547 -19 -12 -22 +2 +4 +5 +7 +11 -20 -66 -9 -6 -43 +9 0. 0611 0. 0419 0. 1023 9. 9866 9. 9800 9. 9733 9. 9599 9. 9466 0. 0709 0. 2393 0. 0285 0. 0152 0. 1551 9. 9556 -19 — 11 K 2 ._. -22 M 2 *, N 2 , 2N.._. M 3 M 4 , MN M 6 +3 +4 +5 +8 M 8 Oi, Qi,2Q,p!._. 00 .. +10 -21 -66 MK 2MK .. -8 -6 Mf Mm -43 +8 I Constituent" Ji Ki K 2 M 2 *, N 2 , 2N_. Ms M () MN Me M 8 O,, Qi, 2Q, pi. 00 MK 2MK Mf Mm 19.5° 0. 0592 0. 0408 0. 1001 9. 9738 9. 9607 9. 9476 0. 0688 0. 2327 0. 0277 0. 0146 0. 1508 9. 9564 Diff. -18 -12 -23 +3 +3 +5 +8 +11 -20 -65 -43 +9 0. 0574 0. 0396 0. 0978 9. 9872 9. 9807 9. 9743 9. 9615 9. 9487 0. 0668 0. 2262 0. 0268 0. 0140 0. 1465 9. 9573 Diff. -19 -11 -22 +2 +4 +6 +8 +10 -20 -65 -42 +8 0. 0555 0. 0385 0. 0956 9. 9874 9. 9811 9. 9749 9. 9623 9. 9497 0. 0648 0. 2197 0. 0259 0.0133 0. 1423 9. 9581 Diff. -18 -11 -23 +3 +4 +5 +8 +11 -19 -65 -43 +9 0. 0537 0. 0374 0. 0933 9. 9877 9. 9815 9. 9754 9. 9631 9. 9508 0. 0629 0. 2132 0. 0250 0. 0127 0. 1380 Diff. -19 -12 -22 +3 +4 +5 +8 +10 -20 -8 -6 -42 +9 0. 0518 0. 0362 0. 0911 9. 9819 9. 9759 9. 9639 9. 9518 0. 0609 0. 2068 0. 0242 0.0121 0. 1338 9. 9599 Diff. -18 -11 -23 +2 +4 +5 +8 +11 -20 -64 -41 +9 0. 0500 0. 0351 0. 0888 9. 9764 9. 9647 9. 9529 0. 0589 0. 2004 0. 0233 0.0115 0. 1297 *Log F of X i ,H2, V2, MS, 2SM, and MSf are each equal to log F of Ma. Log F of Pi, R 2 , Si, S 2 , S4, Se, T2, Sa, and Ssa are each zero. For log F of L 2 and Mi see Table 13. HARMONIC ANALYSIS AND PREDICTION OF TIDES 187 Table 12. — Log factor F corresponding to every tenth of a degree of I — Con. ^\ I Constituent"^ Ji K, K 2 M 2 *, N 2 , 2N. . . M 3 M 4 , MN M 6 M 8 Oi.Qi.2Q, pi... 00 MK 2MK Mf Mm Constituent^^ Ji K, K 2 M 2 *. N 2 , 2N.... M 3 M 4 , MN M 6 M 8 Oi, Qi, 2Q,pi„. 00 MK 2MK Mf Mm Constit uent^^ Ji K, K 2 M 2 *, Nj, 2N.... M 3 Mi, MN Me M 8 Oi, Q,,2Q,pi... 00 MK 2MK Mf Mm 20.1° Diff. 20.2° Diff. 20.3° Diff. 20.4° 0. 0482 0. 0340 0. 0864 -18 -11 -23 0. 0464 0. 0329 0. 0841 -17 -11 -23 0. 0447 0. 0318 0. 0818 -18 -11 -23 0. 0429 0. 0307 0. 0795 9. 9885 9. 9827 9. 9770 +3 +4 +5 9. 9888 9. 9831 9. 9775 +2 +4 +6 9. 9890 9. 9835 9. 9781 +3 +5 +5 9. 9893 9. 9840 9. 9786 9. 9655 9. 9540 +8 +10 9. 9663 9. 9550 +8 +11 9. 9671 9. 9561 +8 +11 9. 9679 9. 9572 0. 0570 0. 1940 -19 -63 0. 0551 0. 1877 -20 -63 0. 0531 0. 1814 -19 -63 0. 0512 0. 1751 0. 0225 0. 0109 -8 -5 0.0217 0. 0104 -9 -6 0. 0208 0.0098 -8 -5 0. 0200 0. 0093 0. 1255 9. 9617 -41 +9 0. 1214 9. 9626 -41 +9 0. 1173 9. 9635 -41 +9 0. 1132 9.9644 Diff. 20.5° Diff. 20.1 Diff. -18 -11 -24 +3 +4 +6 +8 +11 -19 -62 -41 +9 0.0411 0. 0296 0. 0771 .9844 .9792 9. 9687 9. 9583 0. 0493 0. 1689 0.0191 0. 0087 0. 1091 9. 9653 -17 -11 -23 +3 +4 +5 +9 +11 -18 -62 -8 -5 -40 +9 0. 0394 0. 0285 0. 0748 J. 9797 9. 9696 9. 9594 0. 0475 0. 1627 0.0183 0. 0082 0. 1051 9. 9662 -17 -11 -23 +2 44 +6 +8 +11 -19 -62 -41 +9 0. 0377 0. 0274 0. 0725 ). 9852 .9704 .9605 0. 0456 0. 1565 0. 0175 0. 0076 0. 1010 9. 9671 Diff. -17 -11 -24 +3 +4 +5 +8 +11 -19 -61 -40 +9 0. 0360 0. 0263 0. 0701 9. 9904 9. 9856 9. 9712 9. 9616 0. 0437 0. 1504 0.0167 0. 0071 0. 0970 Diff. -17 -11 -23 +3 +4 +6 +8 +11 -18 -61 +10 29.0° Diff. 21.0° 0. 0343 0. 0252 0. 0678 -17 -11 -24 0. 0326 0. 0241 0. 0654 9. 9907 9. 9860 9. 9814 +3 +4 +5 9. 9910 9. 9864 9. 9819 9. 9720 9. 9627 +9 +12 9. 9729 9. 9639 0. 0419 0. 1443 -19 -61 0. 0400 0. 1382 0.0159 0. 0065 -8 -5 0.0151 0. 0060 0. 0931 9. 9690 -40 +9 0. 0891 9. 9699 Diff. -17 -11 -24 +2 +5 +6 +8 +11 -18 -61 -8 -5 -39 +10 0. 0309 0. 0230 0. 0630 9. 9912 9. 9869 9. 9825 9. 9737 9. 9650 0. 0382 0. 1321 0. 0143 0. 0055 0. 0852 9. 9709 Diff. -17 -11 -23 +3 +4 +6 +9 +11 -18 -8 -5 -40 +9 0. 0292 0. 0219 0. 0607 9. 9915 9. 9873 9. 9831 9. 9746 0. 0364 0. 1261 0.0135 0. 0050 0. 0812 9. 9718 Diff. -16 -10 -24 +3 +4 +5 +8 +12 -8 -5 -39 +10 21.3° Diff. 21.4° 0. 0276 0. 0209 0. 0583 9. 9877 9. 9836 9. 9754 9. 9673 0. 0346 0. 1201 0.0127 0. 0045 0. 0773 9. 9728 -17 -11 +3 +5 +6 +9 +11 -18 -8 -5 +9 0. 0259 0. 0198 0. 0559 9. 9921 9. 9882 9. 9842 9. 9763 0. 0328 0. 1141 0.0119 0. 0040 0. 0735 9. 9737 Diff. 21. 5 C -16 -11 -25 +3 +4 +6 +9 +12 -18 -59 -8 -5 -39 +10 0. 0243 0. 0187 0. 0534 9. 9924 9. 9886 9. 9848 9. 9772 9. 9696 0. 0310 0. 1082 0.0111 0. 0035 0. 0696 9. 9747 Diff. -10 -24 +3 +4 +6 +8 +11 -18 -59 -38 +10 21.6° 0. 0227 0.0177 0. 0510 9. 9927 9. 9890 9. 9854 9. 9707 0. 0292 0. 1023 0. 0103 0. 0030 0. 0658 9. 9757 Diff. -16 -11 -24 +3 +4 +5 +9 +12 -17 -59 -7 -5 -39 +10 21.7° Diff. 21.8° 0.0211 0. 0166 0. 0486 -16 -10 -24 0.0195 0. 0156 0. 0462 9. 9930 9. 9894 9. 9859 +3 +5 +6 9. 9933 9. 9899 9. 9865 9. 9789 9. 9719 +9 +11 9. 9798 9. 9730 0. 0275 0. 0964 -18 -58 0. 0257 0. 0906 0096 0. 0025 -8 -4 0088 0.0021 0. 0619 9. 9767 -38 +9 0. 0581 9. 9776 Diff. -16 -11 -24 +3 44 +6 +8 +12 -17 -58 -7 -5 -37 +10 *Log F of X 2 , M, vi, SM, 2SM, and MSf are each equal to log F of M 2 . Log F of Pi, R 2 , Si, Sg, S 4 , S«, T 2 , Sa, and Ssa are each zero. For log F of L 2 and Mi see Table 13. 188 U. S. COAST AND GEODETIC SURVEY Table 12. — Log factor F corresponding to every tenth of a degree of I — Con. *Log F of X 2 M2, vi, MS, 2SM and MSf are each equal to log F of M2. Log F of Pi, R2 Si, S2, S4, Se, T 2 Sa. and Ss:- 1 . are each zero. For log F of L 2 and Mi see Table 13. HARMONIC ANALYSIS AND PREDICTION OF TIDES 189 Table 12. — Log factor F corresponding to every tenth of a degree of I — Con. \ I Constituent^^ J. K, K 2 M 2 *, N 2 , 2N-— M 3 M 4 , MN M 6 M 8 O,, Qi,2Q,pi„. 00 MK 2MK ML.. Mm 23.7° ). 9912 ). 9963 9. 9991 9. 9986 9. 9981 9. 9972 9. 9962 9. 9942 9. 9846 9. 9954 9. 9944 9. 9894 9. 9975 Diff. -14 -9 -25 +3 +5 +6 +9 +13 -15 -53 -7 -3 -34 +11 23.8° 9. 9898 9. 9954 9. 9973 9. 9994 9. 9991 9. 9987 .9975 9. 9927 9. 9793 ). 9947 J. 9941 Diff. -14 -10 -25 +3 +4 +7 +10 +13 -16 -53 -6 -4 -35 +11 23.9° 9. 9884 9. 9944 9. 9948 9. 9997 9. 9995 9. 9994 0. 9911 9. 9740 ,9941 9. 9825 9. 9997 Diff. -14 -10 -24 +3 +5 +6 +10 +13 -15 -53 -7 -3 -34 +12 24.0 C 9. 9870 9. 9934 9. 9924 0. 0000 0. 0000 0. 0000 0. 0001 0. 0001 9. 9896 9. 9687 9. 9934 9. 9934 9. 9791 0. 0009 Diff. -13 -10 -25 +3 +5 +7 +9 +12 -16 -53 -7 -3 -34 +11 24. l c 9. 9857 9. 9924 0. 0003 0. 0005 0. 0007 0. 0010 0. 0013 9. 9634 9. 9927 9. 9931 9. 9757 0. 0020 Diff. -14 -9 -25 +4 +5 +6 +10 +13 -15 -52 -33 +11 24. 2 C 9. 9843 9. 9915 9. 9874 0. 0007 0. 0010 0. 0013 0. 0020 0. 0026 9. 9865 9. 9582 9. 9921 9. 9928 9. 9724 0. 0031 \ I Cotstituent^ Ji Ki K 2 M 2 *, N 2 , 2N. M 3 Mi, MN M 6 M 8 Oi. Qi, 2Q, pi OO MK 2MK Mf Mm ^\ I Constituent^ Ji Ki K 2 M 2 *, N 2 , 2N_ M 3 M 4 , MN M 6 M 8 . Oi, Qi, 2Q, pi OO MK 2MK Mf Mm 24.3° Diff. 24.4° Diff. 24.5° Diff. 24.6° Diff. 24.7° 9. 9830 9. 9905 9. 9850 -14 -9 -25 9. 9816 9. 9896 9. 9825 -13 -9 -25 9. 9803 9. 9887 9. 9800 -13 -10 -24 9. 9790 9. 9877 9. 9776 -13 - 9 -25 9. 9777 9. 9868 9. 9751 0. 0010 0. 0015 0. 0020 +3 +5 +6 0. 0013 0. 0020 0. 0026 +3 +5 +7 0. 0016 0. 0025 0. 0033 +4 +5 +6 0. 0020 0. 0030 0. 0039 +3 +5 +7 0. 0023 0. 0035 0. 0046 0. 0030 0. 0039 +9 4-14 0. 0039 0. 0053 +10 +13 0. 0049 0. 0066 +10 +13 0. 0059 0. 0079 +10 +13 0. 0069 0. 0092 9. 9850 9. 9530 -15 -52 9. 9835 9. 9478 -15 -52 9. 9820 9. 9426 -15 -51 9. 9805 9. 9375 -15 -51 9. 9790 9. 9324 9. 9915 9. 9925 -6 -3 9. 9909 9. 9922 -6 -3 9. 9903 9. 9919 -6 -3 9. 9897 9. 9916 -6 -2 9. 9891 9. 9914 9. 9690 0. 0043 -34 +11 9. 9656 0. 0054 -33 +12 9. 9623 0. 0066 -33 +11 9. 9590 0. 0077 -33 +12 9. 9557 0. 0089 Diff. -13 -10 -25 +3 +5 +7 +10 +13 -15 -51 -3 -33 +12 9. 9764 9. 9858 9. 9726 0. 0026 0. 0040 0. 0053 0. 0079 0. 0105 9. 9775 9. 9273 t 9911 9. 9524 0. 0101 24.9 C 9. 9751 9. 9849 9. 9701 0. 0030 0. 0045 0. 0059 0. 0089 0. 0119 9. 9760 9. 9222 9. 9491 0.0112 Diff. -13 -9 -24 +3 +5 +7 +10 +13 -14 -51 -32 +12 25.0° 9738 9840 9677 0033 0050 0. 0099 0.0132 9746 9171 9. 9873 9. 9906 9459 0124 Diff. -12 -9 -25 +3 +5 +7 + 10 +14 -15 -50 -6 -2 -33 +12 25.1 C 9. 9726 9. 9831 9. 9652 0. 0036 0. 0055 0. 0073 0. 0109 0. 0146 9. 9731 9.9121 9. 9867 9. 9904 9. 9426 0. 0136 Diff. -13 -9 -24 4-4 +5 +7 +10 +13 -14 -50 -6 -3 -32 4-12 25.2° 9. 9713 9. 9822 0. 0040 0. 0060 0. 0080 0.0119 0. 0159 9. 9717 9. 9071 9. 9861 9. 9901 9. 9394 0. 0148 Diff. -12 -10 -25 +3 +5 +6 +11 +14 -15 -50 -5 -2 -32 +12 25.3° 9. 9701 9. 9812 9. 9603 0. 0043 0. 0065 0.0086 0. 0130 0.0173 9. 9702 9. 9021 9. 9362 0. 0160 Diff. 25.4 C -13 -9 -24 +4 +5 +7 +10 +13 -14 -50 -32 +12 9. 9688 9. 9803 9. 9579 0. 0047 0. 0070 0. 0093 0. 0140 0. 0186 9. 9688 9. 8971 9. 9850 9. 9897 9. 9330 0. 0172 "Log F of A 2 , p 2 , v 2 , MS, 2SM, and MSf are each equal to log F of M 2 . Log F of Pi, R 2 , Si, S 2 , S4, S 6 , T 2 , Sa, and Ssa are each zero. For log F of L 2 and M x see Table 13. 246037 — 41- -13 190 U. S. COAST AND GEODETIC SURVEY Table 12. — Log factor F corresponding to every tenth of a degree of I — Con. 25.5 C 0. 0050 0. 0075 0. 0100 0.0150 0.0200 9. 9674 9. 9298 0.0185 Diff. -12 -9 -25 +3 +5 +7 +10 +14 -5 -2 -32 +12 9. 9785 9. 9529 0. 0053 0. 0080 0. 0107 0. 0160 0. 0214 9. 8873 9. 9839 9.9266 0.0197 Diff. 25. 7 C -12 -9 -25 +4 +5 +7 +11 +14 -14 -49 -6 -2 -31 +12 9. 9652 9. 9776 9. 9504 0. 0057 0. 0085 0.0114 0.0171 0. 0228 9. 9646 9. 8824 9. 9833 9235 Diff. -13 -8 -24 +3 +6 +7 +10 +13 -14 -49 -5 -2 -32 +13 25.5 9. 9768 9. 9480 0. 0060 0. 0091 0. 0121 0.0181 0. 0241 9. 9632 9. 8775 0. 0222 Difl. -12 -9 -25 +4 +5 +7 +11 +14 -5 -2 -31 +12 25.9° 9. 9627 9. 9759 9. 9455 0. 0064 0. 0096 0. 0128 0. 0192 0. 0255 9. 9618 9. 8726 9. 9823 9. 9172 0. 0234 Difl. -11 -9 -24 +3 +5 +7 +10 +14 -14 -49 -31 +13 26.0° 9. 9750 9. 9431 0. 0067 0. 0101 0. 0135 0. 0202 0. 0269 9. 9604 9. 8677 9.9817 9. 9141 0.0247 26. l c Diff. 9. 9604 9. 9741 9. 9406 0. 0071 0. 0106 0. 0142 0. 0213 0. 0283 9. 9590 9. 8629 9. 9812 9. 9110 0. 0259 -12 -9 -24 +3 +6 +7 +10 +15 -13 -48 -5 -2 -31 +13 26.2 C Diff. 9. 9592 9. 9732 9. 9382 0. 0074 0.0112 0. 0149 0. 0223 0. 0298 9. 9577 9. 8581 9. 9807 9. 9079 0. 0272 -12 -8 -25 +4 +5 +7 +11 +14 -14 -48 -5 -1 -31 +13 26.2 9. 9580 9. 9724 9.9357 0. 0078 0.0117 0.0156 0. 0234 0. 0312 9. 9563 9. 8533 9. 9048 0.0285 Diff. 26.4° Diff. 26.5° Diff. 26.6 C -11 -9 -24 +3 +5 +7 +10 +14 -14 -47 -5 -2 -31 +13 9. 9569 9. 9715 9. 9333 0. 0081 0.0122 0. 0163 0. 0244 0. 0326 9. 9549 }. 9797 3. 9878 9. 9017 0. 0298 -12 -9 -25 +4 +6 +7 +11 +14 -13 -5 -1 +12 9. 9557 9. 9706 9. 9308 0. 0085 0.0128 0. 0170 0.0255 0.0340 9. 9536 9. 8438 9. 9792 9. 9877 0. 0310 -11 -8 -24 +4 +5 +7 +11 +14 -13 -47 -5 -2 -30 +13 9. 9546 9. 9698 9. 9284 0. 0089 0.0133 0. 0177 0. 0266 0. 0354 9. 9523 9. 8391 9. 9787 9. 9875 9. 8957 0.0323 26.7 C 9. 9535 0.0092 0.0138 0.0184 0. 0277 0.0369 9. 9509 9. 8344 9. 9782 9. 9874 9. 8926 0.0336 Diff. 26.8° -11 -8 -25 9. 9524 9. 9681 9. 9235 +4 +6 +8 0.0096 0. 0144 0. 0192 +10 +14 0.0287 0. 0383 -13 -47 9. 9496 9. 8297 -5 -2 9. 9777 9. 9872 -30 +14 9. 8896 0. 0350 | Diff. -12 -9 -24 +3 +5 +7 +11 +15 -13 -47 -5 -1 +13 26.9° Diff. 27.0° Diff. 27.1° Diff. 27.2° 9. 9512 9. 9672 9.9211 -11 -8 -24 9. 9501 9. 9664 9. 9187 -11 -8 -25 9. 9490 9. 9656 9. 9162 -11 -9 -24 9. 9479 9. 9647 9. 9138 0. 0099 0.0149 0. 0199 +4 +6 +7 0.0103 0.0155 0.0206 +4 +5 +7 0.0107 0. 0160 0. 0213 +3 +6 +8 0.0110 0. 0166 0. 0221 0.0298 0.0398 +11 +14 0. 0309 0. 0412 +11 +15 0. 0320 0. 0427 +11 +14 0. 0331 0. 0441 9. 9483 9. 8250 -13 -47 9. 9470 9. 8203 -13 -46 9. 9457 9. 8157 -13 -46 9. 9444 9.8111 9. 9772 9. 9871 -5 -1 9. 9767 9. 9870 -5 -1 9. 9762 9. 9869 -4 -1 9. 9758 9. 9868 9. 8866 0. 0363 -29 +13 9. 8837 0. 0376 -30 +13 9. 8807 0. 0389 -30 +14 9. 8777 0.0403 ♦Log F of X 2 , M2, f 2 , MS, 2SM, and MSf are each equal to log F of Mj. Log F of Pi, R2, Si, S2, S4, Se, T2, Sa, and Ssa are each zero. For log F of L 2 and Mi see Table 13 . HARMONIC ANALYSIS AND PREDICTION OF TIDES 191 Table 12. — Log factor F corresponding to every tenth of a degree of I — Con. Constituent ^^\ 27.3° Diff. 27.4° Diff. 27.5° Diff. 27.6° Diff. 27.7° Diff. J t 9. 9469 9. 9639 9. 9114 0.0114 0.0171 0. 0228 0. 0342 0. 0456 9. 9431 9. 8065 9. 9753 9. 9867 9.8748 0. 0416 -11 -8 -24 +4 +6 +7 +11 +15 -13 -46 -4 -1 -29 +14 9. 9458 9. 9631 9. 9090 0.0118 0.0177 0. 0235 0. 0353 0. 0471 9. 9418 9. 8019 9. 9749 9. 9866 9.8719 0. 0430 -11 -8 -24 +3 +5 +8 +11 +15 -13 -46 -5 -1 -30 +14 9. 9447 9. 9623 9. 9066 0.0121 0.0182 0. 0243 0. 0364 0. 0486 9. 9405 9. 7973 9. 9744 9. 9865 9. 8689 0. 0444 -10 -8 -24 +4 +6 +7 +11 +15 -12 -45 -4 -29 +13 9. 9437 9. 9615 9. 9042 0.0125 0. 0188 0. 0250 0.0375 0.0501 9. 9393 9.7928 9. 9740 9. 9865 9. 8660 0.0457 -11 -8 -24 +4 +5 +8 +12 +14 -13 -45 -5 -1 -29 +14 9. 9426 9. 9607 9. 9018 0. 0129 0.0193 0. 0258 0. 0387 0. 0515 9. 9380 9. 7883 9. 9735 9. 9864 9. 8631 0.0471 -10 Ki -8 K2 -24 M 2 *. N 2 , 2N M 3 M 4 , MN +4 +6 +7 M 6 M 8 +11 +15 Oi, Qi,2Q,pi OO MK -12 -45 -4 2MK Mf. -28 Mm +14 ^\. I Constituent \, 27.8° Difi. 27.9° Diff. 28.0° Diff. 28.1° Diff. 28.2° Diff. J! 9. 9416 9. 9599 9. 8994 0.0133 0.0199 0. 0265 0.0398 0. 0530 9. 9368 9. 7838 9. 9731 9. 9864 9. 8603 0. 0485 -11 -9 -24 43 +6 +8 +11 +15 -13 -45 -4 -1 -29 +14 9. 9405 9. 9590 9. 8970 0. 0136 0.0205 0. 0273 0. 0409 0. 0545 9. 9355 9. 7793 9. 9727 9. 9863 9. 8574 0. 0499 -10 -8 -24 +4 +5 +7 +11 +16 -12 -45 -4 -29 +14 9. 9395 9. 9582 9. 8946 0. 0140 0. 0210 0. 0280 0.0420 0. 0561 9. 9343 9. 7748 9. 9723 9. 9863 9. 8545 0. 0513 -10 -8 -24 +4 +6 +8 +12 +15 -13 -45 -5 -1 -28 +14 9. 9385 9. 9574 9. 8922 0.0144 0.0216 0. 0288 0. 0432 0. 0576 9. 9330 9. 7703 9.9718 9. 9862 9. 8517 0.0527 -10 -7 -24 +4 +6 +7 +11 +15 -12 -44 -4 -28 +15 9. 9375 9. 9567 9. 8898 0. 0148 0. 0222 0. 0295 0.0443 0. 0591 9. 9318 9. 7659 9. 9714 9. 9862 9. 8489 0. 0542 -10 Ki K 2 -8 -24 M 2 *, N 2 , 2N M 3 +4 +5 Mi, MN +8 M 8 — +12 M 8 Oi.Qi, 2Q,pi OO MK +15 -12 -44 -4 2MK Mf -29 Mm +14 Constituent "~~~~-— .. 28.3° Diff. 28.4° Diff. 28.5° Diff. 28.6° J, 9.9365 9. 9559 9. 8874 0. 0152 0. 0227 0. 0303 0. 0455 0. 0606 9. 9306 9. 7615 9. 9710 9. 9862 9. 8460 0. 0556 -10 -8 -24 +3 +6 +8 +11 +15 -12 -44 -4 -28 4-14 9. 9355 9. 9551 9.8850 0.0155 0.0233 0.0311 0.0466 0. 0621 9. 9294 9. 7571 9. 9706 9. 9862 9. 8432 0. 0570 -10 -8 -24 +4 +6 +7 +12 +16 -12 -44 -4 -28 -4-14 9. 9345 9. 9543 9. 8826 0. 0159 0. 0239 0. 0318 0. 0478 0. 0637 9.9282 9. 7527 9. 9702 9. 9862 9. 8404 0- 0584 -10 -8 -23 +4 +6 +8 +11 +15 -12 -44 -4 -28 +15 9. 9335 Ki 9. 9535 K 2 9. 8803 M 2 *, N 2 , 2N 0. 0163 M 3 0. 0245 M 4 , MN 0. 0326 M 6 0. 0489 M 8 0. 0652 Oi, Qi, 2Q, pi 9.9270 OO. . 9.7483 MK. 9. 9698 2MK 9. 9862 Mf . . 9. 8376 Mm . ... 0. 0599 *Log F of X 2> m vi, MS. and MSf are each equal to log F of M 2 . Log Fof Pi, R 2 , Si, S 2 , Si, St, T 2 , Sa, and Ssa are each zero. For log F of L 2 and Mi see Table 13. 192 U. S. 00 AST AND GEODETIC SURVEY Table 13. — Values of u and log F of L 2 and M x for years 1900 to 2000 Year N U Of L/2 1900 1901 1902 1903 1904 1905 1906 1907 1909 1910 1911 1912 1913 1914 260 255 250 245 240 235 230 225 220 215 210 205 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 25 20 15 10 5 355 350 345 340 335 330 +11.4 +5.1 -1.3 -6.2 -9.1 -10.0 -9.4 -7.8 -5.5 -3.0 -0.4 +2.2 +4.5 +6.5 +8.0 +8.8 +8.9 +8.2 +6.4 +3.6 +0.1 -3.7 -7.3 -10.1 -11.8 -12.1 -11.3 -9.4 -6.7 -3.4 +0.3 +4.2 +8.0 +11.2 +13.4 +13.8 +11.3 +5.3 -13.2 -19.4 -21.5 -20.4 -17.0 -12.2 -6.4 -0.2 +6.1 +12.1 +17.4 +21.3 +22.7 +19.9 +11.0 -2.6 -14.5 -20.5 -21.2 -18.7 Diff. 6.3 6.4 4.9 2.9 0.9 0.6 1.6 2.3 2.5 2.6 2.6 2.3 2.0 1.5 0.8 0.1 0.7 1.8 2.8 3.5 3.8 3.6 2.8 1.7 0.3 0.8 1.9 2.7 3.3 3.7 3.9 3.8 3.2 2.2 0.4 2.5 6.0 9.2 9.3 6.2 2.1 1.1 3.4 4.8 5.8 6.2 6.3 6.0 5.3 3.9 1.4 2.8 8.9 13.6 11.9 6.0 0.7 2.5 u of Mi 353.5 358.7 3.7 8.8 14.2 20.4 27.6 36.5 47.9 62.5 80.5 117.6 132.0 143.0 151.6 158.6 164.4 169.5 174.1 178.6 183.1 187.7 192.8 205.6 214.4 225.9 241.3 260.7 281.8 300.8 315.8 327.1 335.8 343. 349.1 354.6 0.0 5.5 11.3 18.0 26.2 36.7 50.8 69.8 92.6 114.8 132.5 145.6 155.5 163.2 169.8 175.7 181.3 187.0 193.1 200.0 208.5 Diff. 5.2 5.0 5.1 5.4 6.2 7.2 8.9 11.4 14.6 18.0 19.4 17.7 14.4 11.0 8.6 7.0 5.8 5.1 4.6 4.5 4.5 4.6 5.1 5.8 7.0 8.8 11.5 15.4 19.4 21.1 19.0 15.0 11.3 8.7 7.2 6.1 5.5 5.4 5.5 5.8 6.7 8.2 10.5 14.1 19.0 22.8 22.2 17.7 13.1 5.9 5.6 5.7 6.1 6.9 8.5 Log F (L 2 ) 0. 0964 0. 1125 0. 1052 0. 0793 0. 0445 0. 0092 9. 9779 9. 9529 9. 9347 9. 9233 9. 9183 9. 9193 9. 9254 9. 9369 9. 9523 9. 9713 0. 0148 0. 0357 0. 0523 0. 0616 0. 0611 0. 0502 0. 0307 0. 0058 9. 9790 9. 9536 9. 9320 9. 9159 9. 9069 9. 9058 9. 9137 9. 9314 9. 9596 0. 0473 0. 1009 0. 1472 0. 1663 0. 1454 0. 0945 0. 0343 9. 9787 9. 9014 9. 8823 9. 8764 9. 8841 9. 9060 9. 9428 9. 9954 0. 0634 0. 1414 0. 2089 0. 2276 0. 1826 0. 1072 0. 0334 9. 9730 Difl. 161 73 259 348 353 313 250 182 114 50 10 61 115 154 190 213 222 209 5 109 195 249 254 216 161 90 11 79 177 282 390 487 536 463 191 209 509 602 556 324 191 59 77 219 368 526 680 780 675 187 450 754 738 604 Log F (Mi) 9. 7295 9. 7260 9. 7338 9. 7527 9. 7824 9. 8228 9. 8734 9. 9330 9. 9977 0. 0586 0. 0998 0. 1056 0. 0780 0. 0283 9. 9760 9. 9275 9. 8861 9. 8525 9. 8269 9. 8091 9. 7991 9. 7970 9. 8033 9. 8185 9. 8433 9. 8782 9. 9228 9. 9744 0. 0238 0. 0521 0. 0397 9. 9195 9. 8512 9. 7921 9. 7449 9. 7103 9. 6883 9. 6789 9. 6822 9. 6986 9. 7285 9. 7722 9. 8921 9. 9461 9. 9626 9. 8640 9. 7959 9. 7374 9. 6924 9. 6616 9. 6447 9. 6414 9. 6515 9. 6755 9. 7133 9. 7651 Diff. 35 78 189 297 404 506 596 647 412 58 276 497 523 485 414 336 256 178 100 21 63 152 248 349 446 516 494 283 124 514 591 472 346 220 94 33 164 299 437 564 635 540 165 343 643 6S1 585 450 169 33 101 240 37S 518 HARMONIC ANALYSIS AND PREDICTION OF TIDES 193 Table 13.— Values of u and log F of L 2 and Mi for years 1900 to 2000— Con. N 330 325 320 315 310 305 300 295 290 285 280 275 270 265 260 255 250 245 240 235 230 225 220 215 210 205 200 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 105 100 95 90 85 u of L<2 -18.7 -14.3 -8.8 -2.9 +3.0 +8.6 +13.6 +17.4 +19.6 +19.4 +16.1 +9.6 +1.7 -5.4 -9.8 -11.6 -11.2 -9.4 -6.8 -3.7 -0.5 +2.6 +5.4 +7.6 +9.2 +10.1 +10.1 +9.0 +7.0 +4.1 +0.6 -2.9 -6.1 -8.5 -9.9 -10.3 -9.8 -8.4 -6.3 -3.7 -0.7 +2.6 +5.8 +8.7 +10.9 +11.8 +10.9 +7.3 +0.9 -7.2 -14.5 -18.9 -20.0 -18.4 -15.0 -10.3 -4.8 +1.1 +7.1 Dill. 5.5 5.9 5.9 5.6 5.0 3.8 2.2 0.2 3.3 6.5 7.9 7.1 4.4 1.8 0.4 1.8 2.6 3.1 2.8 2.2 1.6 0.9 0.0 1.1 2.0 2.9 3.5 3.5 3.2 2.4 1.4 0.4 0.5 1.4 2.1 2.6 3.0 3.3 3.2 2.9 2.2 0.9 0.9 3.6 6.4 8.1 7.3 4.4 1.1 1.6 3.4 4.7 5.5 5.9 6.0 u of Mi 208.5 219.4 234.0 253.5 276.3 297.9 314.9 327.4 336.8 344.3 350.6 356.2 1.4 6.6 12.1 18.1 25.2 33.8 45.0 59.5 77.7 97.9 116.7 131.8 143.3 152.2 159.2 165.1 170.1 174.8 179.2 183.5 188.0 192.9 198.5 205.0 213.2 223.6 237.5 255.1 275.3 294.6 310.5 322.7 332.1 339.7 346.0 351.7 357.0 2.2 7.6 13.6 20.7 29.4 40.8 56.3 76.7 99.7 120.6 Diff. 10.9 14.6 19.5 22.8 21.6 17.0 12.5 9.4 7.5 6.3 5.6 5.2 5.2 5.5 6.0 7.1 8.6 11.2 14.5 18.2 20.2 18.8 15.1 11.5 8.9 7.0 5.9 5.0 4.7 4.4 4.3 4.5 4.9 5.6 6.5 8.2 10.4 13.9 17.6 20.2 19.3 15.9 12.2 9.4 7.6 6.3 5.7 5.3 5.2 5.4 6.0 7.1 8.7 11.4 15.5 20.4 23.0 20.9 Log F (L 2 ) 9. 9730 9. 9283 9. 8988 9. 8835 9.8815 9. 8923 9.9153 9. 9499 0. 0458 0. 0954 0. 1305 0. 1368 0.1141 0.0740 0. 0300 9. 9902 9. 9582 9. 9348 9. 9200 9.9133 9. 9139 9. 9210 9. 9338 9. 9512 9. 9721 9. 9948 0. 0172 0. 0368 0. 0509 0. 0572 0. 0545 0.0432 0. 0253 0. 0035 9. 9804 9. 9584 9. 9393 9. 9245 9. 9151 9. 9121 9. 9163 9. 9284 9. 9491 9. 9787 0.0167 0.0607 0. 1041 0. 1347 0. 1381 0. 1105 0. 0630 0. 0105 9. 9628 9. 9245 9. 8977 9. 8831 9.8811 9. 8923 Diff. 447 295 153 20 108 230 346 449 510 4^96 351 63 227 401 440 398 320 234 148 67 71 128 174 209 227 224 196 141 63 27 113 179 218 231 220 191 148 94 30 42 121 207 296 380 440 434 306 34 276 475 525 477 383 146 2(1 112 LogF(M,) 9. 7651 9. 8294 9. 8993 9. 9567 9. 9741 9. 9425 9. 8844 9.8244 9. 7738 9. 7368 9. 7129 9. 7018 9. 7033 9. 7167 9. 7419 9. 7789 9. 8275 9. 8869 9. 9539 0. 0203 0. 0696 0. 0830 0. 0580 0. 0113 9. 9603 9. 9141 9. 8760 9. 8257 9. 8127 9. 8073 9. 8095 9. 8192 9. 8368 9. 8627 9. 8974 9. 9407 9. 9906 0. 0404 0. 0755 0. 0764 0. 0377 9. 9746 9. 9063 9. 8437 9. 7912 9. 7503 9. 7213 9. 7041 9. 6989 9. 7060 9. 7260 9. 7592 9. 9246 9. 9677 9. 9664 9. 9184 Diff. 643 574 174 316 581 600 506 370 239 Hi 15 134 252 370 594 670 664 493 134 250 467 510 462 381 294 209 130 54 22 97 176 259 347 433 499 498 351 9 387 631 525 409 290 172 52 71 200 332 466 579 431 13 480 194 U. S. 00 AST AND GEODETIC SURVEY Table 13.— Values of u and log F of L 2 and M t for years 1900 to 2000— Con. Year N uof L2 Diff. u of Mi Diff. Log F (L 2 ) Diff. Log F (Mi) Diff. 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 40 35 30 25 20 15 10 5 355 350 345 340 335 330 325 320 315 310 305 300 295 290 285 280 275 270 265 260 255 250 245 240 235 230 225 220 215 210 205 200 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 +7.1 +12.8 +17.6 +20.9 +21.3 +17.1 +6.7 -7.2 -17.8 -22.3 -22.0 -18.7 -13.8 -7.9 -1.6 +4.6 +10.5 +15.6 +19.5 +21.4 +20.3 +15.4 +6.9 -2.5 -9.6 -13.2 -13.5 -11.9 -9.0 -5.4 -1.6 +2.2 +5.6 +8.5 +10.6 +11.8 +11.8 +10.7 +8.3 +5.0 +1.2 -2.4 -5.5 -7.6 -8.8 -9.1 -8.5 -7.2 -5.4 -3.2 —0.6 +2.1 +4.7 +7.1 +9.0 +10.0 +9.7 +7.6 +3.2 5.7 4.8 3.3 0.4 4.2 10.4 13.9 10.6 4.5 0.3 3.3 4.9 5.9 6.3 6.2 5.9 5.1 3.9 1.9 1.1 4.9 8.5 9.4 7.1 3.6 0.3 1.6 2.9 3.6 3.8 3.8 3.4 2.9 2.1 1.2 0.0 1.1 2.4 3.3 3.8 3.6 3.1 2.1 1.2 0.3 0.6 1.3 1.8 2.2 2.6 2.7 2.6 2.4 1.9 1.0 0.3 2.1 4.4 120.6 136.7 148.6 157.6 165.0 171.3 177.0 182.6 188.4 194.7 202.0 211.1 223.1 239.3 260.3 283.5 303.9 319.5 330.9 339.6 346.7 352.8 358.3 3.7 9.1 15.1 21.7 29.8 40.2 53.8 71.5 92.2 112.5 129.3 142.0 151.6 159.0 165.2 170.5 175.2 179.7 184.2 188.7 193.6 199.1 205.6 213.5 223.6 236.6 253.2 272.4 291.3 307.3 319.8 329.6 337.3 343.8 349.4 354.6 16.1 11.9 9.0 7.4 6.3 5.7 5.6 5.8 6.3 7.3 9.1 12.0 16.2 21.0 23.2 20.4 15.6 11.4 8.7 7.1 6.1 5.5 5.4 5.4 6.0 6.6 8.1 10.4 13.6 17.7 20.7 20.3 16.8 12.7 9.6 7.4 6.2 5.3 4.7 4.5 4.5 4.5 4.9 5.5 6.5 7.9 10.1 13.0 16.6 19.2 18.9 16.0 12.5 9.8 7.7 6.5 5.6 5.2 9. 8923 9.9175 9. 9576 0. 0134 0. 0840 0. 1614 0. 2202 0. 2218 0. 1646 0. 0865 0. 0145 9.9569 9.9157 9.8895 9. 8776 9.8794 9. 8945 9.9227 9.9637 0.0160 0. 0752 0. 1302 0. 1618 0.1548 0.1155 0. 0625 0.0129 9.9711 9.9190 9.9087 9. 9063 9.9126 9. 9262 9.9458 9. 9960 0. 0216 0. 0432 0. 0573 0. 0617 0.0561 0. 0422 0. 0228 0.0010 9. 9791 9. 9591 9. 9420 9.9290 9.9207 9. 9176 9. 9204 9. 9295 9. 9455 9. 9685 9. 9982 0. 0331 0. 0696 0. 1002 252 401 558 706 774 588 16 572 781 720 576 412 262 119 18 151 410 523 592 550 316 70 393 530 496 418 313 103 24 63 136 196 240 262 256 216 141 44 56 139 194 218 219 200 171 130 83 31 91 160 230 297 349 365 9. 9184 9. 8498 9. 7828 9. 7270 9. 6850 9. 6569 9. 6428 9. 6421 9. 6550 9.6817 9. 7225 9. 7771 9. 8433 9.9110 9.9578 9. 9572 9.9123 9. 8490 9. 7896 9. 7418 9. 7078 9. 6874 9. 6803 9. 6858 9. 7039 9. 7349 9. 7777 9. 8331 9. 8989 9. 9690 0. 0285 0.0555 0. 0395 9. 9949 9. 9427 9. 8952 9. 8569 9. 8285 9. 8002 9. 7989 9.8057 9. 8202 9. 8425 9. 8726 9.9107 9. 9564 0. 0077 0. 0586 0. 0966 0. 1044 0. 0745 0. 0182 9. 9530 9. 8906 9. 8364 9. 7923 9. 7589 9. 7364 670 558 420 281 141 7 129 267 408 546 662 677 468 6 449 633 594 478 340 204 71 55 181 310 428 554 658 701 595 270 446 522 475 383 284 186 97 13 68 145 223 301 381 457 513 509 380 78 299 563 652 624 542 441 334 225 HARMONIC ANALYSIS AND PREDICTION OF TIDES 195 Table 13.— Values of u and log F of L 2 and M { for years 1900 to 2000— Con. N 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 355 350 345 340 335 330 325 320 315 310 305 300 295 290 285 280 275 270 265 260 255 250 245 240 235 230 225 220 215 210 205 200 195 190 185 180 U Of L2 +3.2 -3.0 -9.6 -14.9 -17.7 -17.9 -16.1 -12.7 -8.2 -3.0 +2.6 +8.1 +13.2 +17.4 +19.8 +19.0 +13.4 +2.0 -11.2 -20.1 -23.1 -22.0 -18.2 -12.9 -0.3 +6.1 +12.1 +17.3 +21.1 +22. 6 +20.5 +13.8 +3.2 -7.2 -13.6 -15.9 -15.0 -12.2 -8.4 -4.0 +0.5 +4.8 +8.5 +11.5 +13.4 +14.0 +13.1 +10.6 +6.7 +2.3 -2.0 -5.4 -7.5 -8.5 -8.5 -7.6 -6.3 -4.4 Diff. 6.2 6.6 5.3 2.8 0.2 1.8 3.4 4.5 5.2 5.6 5.5 5.1 4.2 2.4 0.8 5.6 11.4 13.2 8.9 3.0 1.1 3.8 5.3 6.1 6.5 6.4 6.0 5.2 3.8 1.5 2.1 10.6 10.4 6.4 2.3 0.9 2.8 3.8 4.4 4.5 4.3 3.7 3.0 1.9 0.6 0.9 2.5 3.9 4.4 4.3 3.4 2.1 1.0 0.0 0.9 1.3 1.9 ■uof Mi 354.6 359.7 4.8 10.3 16.5 24.0 33.5 46.1 63.1 84.6 107.2 126.2 140.7 151.4 159.8 166.7 172.7 178.4 184.0 189.8 196.4 204.1 213.9 227.0 244.7 267.0 289.9 309.0 323.2 333.8 342.0 348.7 354.7 0.2 5.7 11.4 17.7 25.1 34.4 46.6 62.7 82.9 104. 7 123.7 138.3 149.2 157.6 164.3 169.9 175.0 179.7 184.3 189.0 194.1 199.7 206.3 214.4 224.6 237.7 Diff. 5.1 5.5 6.2 7.5 9.5 12.6 17.0 21.5 22.6 19.0 14.5 10.7 8.4 6.9 6.0 5.7 5.6 5.8 7.7 9.8 13.1 17.7 22.3 22.9 19.1 14.2 10.6 8.2 6.7 6.0 5.5 5.5 5.7 6.3 7.4 9.3 12.2 16.1 20.2 21.8 19.0 14.6 10.9 8.4 6.7 5.6 5.1 4.7 4.6 4.7 5.1 5.6 6.6 8.1 10.2 13.1 Log. F (Li) 0. 1002 0. 1149 0. 1061 0.0757 .0.0331 9. 9885 9. 9488 9. 9163 9. 8963 9. 8864 9. 8884 9. 9030 9. 9312 9. 9740 0.0319 0. 1025 0. 1749 0. 2196 0. 2046 0. 1407 0. 0643 9. 9963 9. 9431 9. 9054 9. 8827 9.8743 9. 8797 9. 9322 9. 9793 0. 0390 0. 1056 0. 1639 0. 1881 0. 1643 0. 1085 0. 0485 9. 9948 9. 9528 9. 9230 9. 9055 9. 8987 9. 9020 9. 9144 9. 9348 9. 9615 9. 9923 0. 0238 0.0511 0.0693 0. 0746 0. 0669 0. 0492 0. 0260 0.0014 9. 9781 9. 9579 9.9417 9. 9299 Diff. 147 304 426 446 397 325 200 99 20 146 282 428 579 706 724 447 150 639 764 532 377 227 84 54 193 332 583 242 238 558 600 537 420 298 175 68 33 124 204 267 308 315 273 182 53 77 177 232 246 233 202 162 118 Log F (M,) 9. 7364 9. 7249 9. 7246 9. 7363 9. 7603 9. 7973 9. 8467 9. 9050 9. 9603 9. 9882 9. 9677 9. 8382 9. 7734 9. 7208 9. 6820 9. 6570 9. 6455 9. 6475 9. 6629 9. 6924 9. 7361 9. 7933 9. 8603 9. 9237 9. 9572 9. 9398 9. 8843 9. 8191 9. 7612 9. 7162 9. 6852 9. 6642 9. 0733 9. 7307 9. 7791 9. 8399 9. 9102 9. 9776 0. 0225 0. 0237 9. 9858 9. 9320 9. 8804 9. 7866 9. 7769 9. 7770 9. 7862 9. 8042 9. 8308 9. 8658 9. 9090 9. 9596 0.0151 0. 0692 Diff. 115 3 117 240 370 494 583 553 279 205 589 706 526 388 250 115 20 154 295 437 572 670 634 335 174 555 652 579 450 310 172 38 91 225 349 484 608 703 674 449 12 379 538 516 424 312 202 97 1 92 180 266 350 432 506 555 541 196 U. S. COAST AND GEODETIC SURVEY Table 13.— Values of u and log F of L 2 and Mi for years 1900 to 2000— Con, Year N 1959 1961 1962 1963 1964 1965 1966 1967 1968 1970 1971 1972 1973 1974 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 105 100 95 90 85 55 50 45 40 35 30 25 20 15 10 5 355 350 345 340 335 330 325 320 315 310 305 300 295 290 285 280 275 270 265 260 255 250 MOf Ii2 -4.4 -2.3 0.0 +2.3 +4.5 +6.5 +8.1 +8.9 +8.7 +7.2 +4.1 -0.6 -6.0 -11.0 -14.5 -16.0 -15.5 -13.4 -10.1 -5.9 -1.1 +4.0 +9.0 +13.3 +16. 5 +17.9 +15.9 +8.9 Diff. -23.1 -21.3 -17.2 -11.8 -5.6 +0.9 +7.4 +13.5 +18.6 +22.2 +23.1 +19.8 +11.1 -1.1 -11.7 -17.1 -18.0 -15.9 -12.2 -7.5 -2.4 +2.7 +7.3 +11.3 +14.4 +16.0 +15.9 +13.7 wofMi 2.3 2.3 2.2 2.0 1.6 0.8 0.2 1.5 3.1 4.7 5.4 5.0 3.5 1.5 0.5 2.1 3.3 4.2 4.8 5.1 5.0 4.3 3.2 1.4 2.0 7.0 11.5 11.6 7.0 1.9 1.8 4.1 5.4 6.2 6.5 6.5 6.1 5.1 3.6 0.9 3.3 8.7 12.2 10.6 5.4 0.9 2.1 3.7 4.7 5.1 5.1 4.6 4.0 3.1 1.6 0.1 2.2 237.7 254.1 272.7 291.0 306.6 318.9 328.5 336.1 342.5 348.1 353.2 358.0 2.9 8.0 13.6 20.2 28.3 38.8 52.9 71.6 93.7 114.9 131.9 144.7 154.3 162.0 168.5 174.3 179.8 185.4 191.4 198.2 206.4 217.0 231.3 250.5 273.5 295.7 313.4 326.4 336.2 343.9 350.4 356.3 1.8 7.4 13.4 20.2 28.4 38.8 52.8 71.3 93.3 114.9 132.0 144.8 154.5 162.1 168.3 Diff. 16.4 18.6 18.3 15.6 12.3 5.6 5.1 4.8 4.9 5.1 5.6 6.6 8.1 10.5 14.1 18.7 22.1 21.2 17.0 12.8 9.6 7.7 6.5 5.8 5.5 5.6 6.0 6.8 8.2 10.6 14.3 19.2 23.0 22.2 17.7 13.0 7.7 6.5 5.9 5.5 5.6 6.0 8.2 10.4 14.0 18.5 22.0 21.6 17.1 12.8 9.7 7.6 6.2 Log F (L 2 ) 9. 9228 9. 9207 9. 9237 9. 9320 9. 9457 9. 9648 9. 9890 0. 0170 0. 0465 0. 0728 0. 0894 0. 0902 0. 0733 0. 0428 0. 0058 9. 9690 9. 9370 9. 9125 9. 8969 9. 8921 9. 8981 9. 9161 9. 9470 9. 9917 0. 0498 0. 1173 0. 1795 0. 2069 0. 1786 0. 1135 0. 0418 9. 9794 9. 9310 9. 8975 9. 8785 9. 8735 9. 8826 9. 9058 9. 9437 9. 9965 0. 0628 0. 1355 0. 1938 0. 2059 0. 1643 0. 0974 0.0313 9. 9767 9. 9359 9. 8945 9. 8919 9. 9005 9. 9188 9. 9462 9. 9805 0.0185 0. 0549 Diff. 71 21 30 83 137 191 242 295 263 166 169 305 370 368 320 245 156 48 60 180 309 447 581 675 622 274 651 717 624 484 335 190 50 91 232 379 528 663 727 583 121 416 546 408 271 143 26 86 183 274 343 380 364 Log F (Mi) 0.0692 0. 1096 0. 1203 0. 0951 0.0444 9. 9841 9. 9252 9. 8729 9. 8294 9. 7950 9. 7703 9. 7554 9. 7507 9. 7565 9. 7735 9. 8025 9. 8437 9. 8956 9. 9523 0. 0049 9. 9657 9. 7674 9. 7185 9. 6832 9. 6614 9. 6529 9. 6577 9. 6759 9. 7081 9. 7545 9. 8137 9. 8803 9. 9370 9. 9555 9. 9227 9. 7946 9. 7388 9. 6965 9. 6683 9. 6540 9. 6531 9. 6656 9. 6912 9. 7305 9. 7833 9. 8482 9. 9191 9. 9802 0. 0058 9. 9834 9. 9326 9. 8760 9. 8269 9. 7895 9. 7642 Diff. 107 252 507 523 435 344 247 149 47 58 170 290 412 510 567 465 61 392 666 702 615 353 218 85 182 322 464 592 666 567 185 328 624 657 558 423 282 143 125 256 528 649 709 611 256 224 508 491 374 253 HARMONIC ANALYSIS AND PREDICTION OF TIDES 197 Table 13.— Values of uand log F of L 2 and Mi for years 1900 to 2000— Con. A r 250 245 240 235 230 225 220 215 210 205 200 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 105 100 95 90 85 70 65 60 55 50 45 40 35 30 25 20 15 10 5 355 350 345 340 335 330 325 320 u of L2 +13.7 +9.6 +4.2 -1.1 -5.3 -8.0 -9.0 -5.9 -3.7 -1.4 +1.0 +3.2 +5.2 +6.9 +8.1 +8.6 +8.3 +6.9 +4.3 +0.6 -3.7 -8.0 -11.4 -13.4 -13.8 -12.8 -10.5 -7.3 -3.4 +0.9 +5.3 +9.5 +13.0 +15.3 +15.4 +12.0 +4.2 -6.6 -16.2 -21.5 -22.4 -20.1 -15.9 -10.4 -4.2 +2.3 +8.6 +14.5 +19. 4 +22.6 +22.7 +18.1 +7.4 -6.0 -15.8 -19.8 -19.4 Diff. 5.4 5.3 4.2 2.7 1.0 0.2 1.1 1.8 2.2 2.3 2.4 2.2 2.0 1.7 1.2 0.5 0.3 1.4 2.6 3.7 4.3 4.3 3.4 2.0 0.4 1.0 2.3 3.2 3.9 4.3 4.4 4.2 3.5 2.3 0.1 3.4 7.8 10.8 5.3 0.9 2.3 4.2 5.5 6.2 6.5 6.3 5.9 4.9 3.2 0.1 4.6 10.7 13.4 4.0 0.4 u of Mi 168.3 173.8 178.8 183.6 188.5 193.8 199.6 206.4 214.7 225.2 238.7 255.5 274.2 292.3 307.6 319. 5 328.8 336.2 342.4 347.8 352.7 357.3 1.9 6.7 11.9 17.9 25.0 34.0 46.0 62.0 82.1 103.8 122.9 137.6 148.7 157.3 164.3 170.4 176.0 181.4 187.0 193.2 200.3 209.0 220.4 236.1 256.6 280.0 301. 317. 3 329.2 338.3 345. 6 351.9 357.7 3.3 9.0 15.2 22.5 Diff. 5.0 4.8 4.9 5.3 10.5 13.5 16.8 18.7 18.1 15.3 11.9 9.3 7.4 6.2 5.4 4.9 4.6 4.6 4.8 5.2 6.0 7.1 9.0 12.0 16.0 20.1 21.7 19.1 14.7 11.1 8.6 7.0 6.1 5.6 5.4 11.4 15.7 20.5 23.4 21.0 16.3 11.9 9.1 7.3 6.3 5.8 5.6 5.7 6.2 7.3 Log F (L 2 ) 0. 0549 0. 0823 0. 0939 0. 0871 0. 0660 0. 0374 0. 0074 9. 9399 9. 9281 9. 9218 9. 9208 9. 9248 9. 9337 9. 9472 9. 9649 9. 9862 0. 0099 0. 0340 0. 0553 0. 0696 0. 0729 0. 0630 0. 0416 0. 0129 9.9817 9. 9524 9. 9278 9. 9101 9. 9006 9. 9004 9. 9104 9. 9315 9. 9645 0. 0099 0. 0659 0. 1263 0. 1741 0. 1839 0. 1474 0. 0849 0. 0199 9. 9639 9. 9210 9. 8920 9. 8768 9. 8754 9. 8880 9. 9150 9. 9572 0. 0151 0. 0868 0. 1629 0. 2165 0. 2126 0. 1553 0. 0809 0. 0132 Diff. 274 116 68 211 286 300 273 228 174 118 63 10 40 89 135 177 213 237 241 213 143 287 312 293 246 177 95 2 100 211 330 454 560 604 478 365 625 650 560 429 290 152 14 126 270 422 579 717 761 536 Log-F(M,) 573 744 677 9. 7642 9. 7508 9. 7486 9. 7568 9. 7751 9. 8033 9.8411 9. 8882 9. 9436 0. 0042 0. 0631 0. 1070 0. 1199 0. 0971 0. 0502 9. 9949 9. 9413 9. 8936 9. 8538 9. 8221 9. 7842 9. 7783 9. 7816 9. 7947 9. 8184 9. 8533 9. 8990 9. 9523 0. 0031 0. 0307 0. 0146 9. 9597 9. 8894 9. 8218 9. 7646 9. 7201 9. 6889 9. 6706 9. 6652 9. 6729 9. 6941 9. 7291 9. 7781 9. 8395 9. 9035 9. 9506 9. 9524 9. 9068 9. 8400 9. 7750 9. 7215 9. 6820 9. 6565 9. 6451 9. 6469 9.6623 9. 6912 9. 7341 Diff. 134 22 82 183 282 378 471 554 606 589 439 129 228 469 553 53(1 317 232 33 131 237 349 457 533 508 276 161 549 703 676 572 445 312 183 54 77 212 350 490 614 640 471 18 456 668 650 535 395 255 114 18 154 289 429 N 250 245 240 235 230 225 220 215 210 205 200 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 355 350 345 340 335 330 325 320 198 U. S. COAST AND GEODETIC STJTtVEY Table 13. — Values of u and log F of L 2 and Mj for years 1900 to 2000- Con, Year 1990 1991 1992 1993 1994 1995 1997 1999 2000 N 320 315 310 305 300 295 290 285 280 275 270 265 255 250 245 240 235 230 225 220 215 210 205 200 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 u of L2 -19.4 -16.4 -11.8 -6.5 -0.9 +4.6 +9.7 +14.0 +17.0 +18.3 +17.3 +13.6 +7.7 +0.8 -5.0 -8.6 -10.2 -9.9 -8.5 -6.4 -3.8 -1.1 +1.6 +4.1 +6.2 +7.9 +8.9 +9.2 +8.6 +7.0 +4.5 +1.2 -2.5 -6.1 -9.1 -11.1 -11.8 -11.4 -9.9 -7.5 -4.4 -0.9 +2.8 Diff. 3.0 4.6 5.3 5.6 5.5 5.1 4.3 3.0 1.3 1.0 3.7 5.9 5.8 3.6 1.6 0.3 1.4 2.1 2.6 2.7 2.7 2.5 2.1 1.7 1.0 0.3 0.6 1.6 2.5 3.3 3.7 3.6 3.0 2.0 0.7 0.4 1.5 2.4 3.1 3.5 3.7 u of Mi 22.5 31.5 43.2 59.0 79.5 102.2 122.6 138.2 149.7 158.6 165.5 171.5 176.9 182.0 187.0 192.4 198.3 205.2 213.5 224.1 237.9 255.1 274.5 293.1 308.5 320.4 329.7 337. 343.0 348.3 353.0 357.5 1.9 6.4 11.3 16.9 23.4 31.5 42.0 55.9 73.9 94.8 114.7 Diff. 9.0 11.7 15.8 20.5 22.7 20.4 15.6 11.5 6.0 5.4 5.1 5.0 5.4 5.9 6.9 8.3 10.6 13.8 17.2 19.4 18.6 15.4 11.9 9.3 7.3 6.0 5.3 4.7 4.5 4.4 4.5 4.9 5.6 6.5 8.1 10.5 13.9 18.0 20.9 19.9 Log F(L 2 ) 0. 0132 9. 9595 9. 9209 9. 8967 9. 8859 9. 8876 9. 9011 9. 9257 9. 9601 0. 0029 0. 0479 0. 0887 0. 1137 0. 1147 0. 0933 0. 0591 0. 0218 9. 9878 9. 9598 9. 9252 9. 9183 9. 9177 9. 9228 9. 9330 9. 9477 9. 9661 9. 9870 0. 0091 0. 0302 0. 0477 0. 0586 0. 0605 0. 0524 0. 0355 0. 0124 9. 9866 9. 9613 9. 9389 9. 9215 9. 9103 9. 9066 9.9111 Diff. 537 242 108 17 135 246 344 428 450 408 250 10 214 342 373 340 280 209 137 51 102 147 184 209 221 211 175 109 19 81 169 231 258 253 224 174 112 37 45 Log-F(Mi) 9. 7341 9. 7906 9. 8582 9. 9276 9. 9776 9. 9827 9. 9433 9. 8848 9. 8283 9. 7819 9. 7493 9. 7290 9. 7211 9. 7249 9. 7400 9. 7662 9. 8033 9. 8510 9. 9086 9. 9731 0. 0372 0. 0868 0. 1041 0. 0839 0. 0395 9. 9876 9. 9382 9. 8955 9. 8607 9. 8339 9. 8148 9. 8035 9. 8044 9. 8174 9. 8395 9. 8714 9. 9130 9. 9627 0. 0141 0.0516 0. 0540 0. 0142 Diff. N 676 694 500 51 394 585 565 464 326 203 79 38 151 262 371 477 576 645 641 173 202 444 519 494 427 348 191 113 45 130 221 319 416 497 514 375 24 320 315 310 305 300 295 290 285 280 275 270 265 260 255 250 245 240 235 230 225 220 215 210 205 200 195 190 185 180 175 170 165 160 155 150 145 140 135 130 125 120 115 110 HARMONIC ANALYSIS AND PREDICTION OF TIDES 199 Table 14.— Node factor f for middle of each year, 1850 to 1999 Constituent 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 J, 0.892 0.922 0.816 1.163 1.023 1.027 1.042 1.056 1.085 1.114 0.874 0.631 0.948 0.974 0.743 1.094 0.948 0.959 0.887 0.905 1.675 1.017 1.026 1.035 1.053 1.071 0.933 0.786 0.976 0.993 0.856 1.059 1.007 0.999 0.977 0.725 1.974 1.005 1.008 1.011 1.016 1.021 0.998 0.983 1.004 1.010 0.990 1.016 1.061 1.037 1.075 1.055 1.559 0. 993 0.989 0.986 0.978 0.971 1. 059 1.204 1.029 1.022 1.129 0.973 1.105 1.069 1.168 1.263 1.118 0.981 0.972 0.963 0.944 0.927 1.110 1. 422 1.048 1.029 1.257 0.933 1.138 1.092 1.246 0.944 1.860 0.972 0.958 0.944 0.918 0.892 1.150 1.608 1.062 1.032 1.360 0.900 1.158 1.107 1.298 0.469 2.348 0.966 0.949 0.932 0.900 0.869 1.174 1.735 1.069 1.032 1.427 0.879 1.165 1.113 1.317 0.962 1.872 0.963 0.945 0.928 0.894 0.861 1.183 1.783 1.072 1.032 1.452 0.871 1.160 1.108 1.302 1.283 1.177 0.965 0.948 0.931 0.899 0.867 1.176 1.745 1.070 1.032 1.432 0.878 1.141 K, K 2 1.095 1.254 L 2 Mi 1.001 1.776 M2*, Njf 2N, X2, H2, V2- M 3 0.971 0. 957 M 4 , MN... 0.942 M 6 Ms 0.915 0.888 Oi, Qi, 2Q, pi 1.153 OO MK 1.627 1.063 2MK Mf . 1.032 1.370 Mm 0.897 Constituent Ju- Ki. K 2 _ Mi M 2 *, Ns, 2N, X 2 , M2, vi. M 3 M 4 , MN M 6 . M 8 _ Oi, Q,, 2Q, p,_ OO MK.. 2MK. ML. Mm. 1860 1.110 1.072 1.179 0.568 2.227 0.980 0.970 0.960 0.941 0.922 1.116 1.447 1.050 1.029 1.270 0.928 1.066 1.041 1.086 0.924 1.792 0.991 0.987 0.983 0.974 0.966 1.065 1.230 1.032 1.023 1.145 0.968 1862 1.013 1.004 0.988 1.225 1.046 1.004 1.006 1.008 1.011 1.015 1.005 1.008 1.007 1.011 1.007 1.011 1863 0.955 0.964 0.897 1.117 1.260 1.016 1.024 1.032 1.049 1.065 0.941 0.807 0.979 0.995 0.872 1.054 1864 0.898 0.926 0.823 0.865 1.680 1.026 1.040 1.054 1.081 1.110 0.880 0.647 0.951 0.976 0.755 1.091 1865 0.852 0.898 0.773 0.879 1.609 1.034 1.G51 1.069 1.105 1.143 0.832 0.540 0.928 0.960 0.670 1.117 0.829 0. 883 0.749 1.082 1.164 1.038 1.057 1.076 1.117 1.159 0.808 0.489 0.916 0.950 0.629 1.130 1867 0.832 0.885 0.753 1.190 0.812 1.037 1.056 1.075 1.115 1.156 0.812 0.497 0.918 0.952 0.635 1.128 0.863 0.904 0.784 1.091 1.189 1.032 1.049 1.066 1.100 1.135 0.843 0.563 0.933 0.964 0.689 1.111 Constituent 1870 1871 1872 1873 1874 1875 1876 1877 1878 0.971 0.974 0.920 1.028 1.014 1.014 1.079 1.050 1.112 1.120 1.079 1.201 1.147 1.099 1.269 1.162 1.111 1.309 1.164 1.112 1.315 1.154 1.104 1.287 1.130 1.087 1.227 0.828 1.811 1.148 1.300 1.224 1.185 0.816 2.004 0.545 2.286 1.087 1.656 1.270 1.227 0.858 1.998 0.543 2.269 1.013 1.019 1.026 1.000 1.001 1.001 0.988 0.982 0.976 0.977 0.966 0.955 0.969 0.954 0.939 0.964 0.947 0.930 0.963 0.946 0.928 0.967 0.951 0.935 0.974 0.961 0.949 1.039 1.052 1.001 1.002 0.965 0.953 0.933 0.912 0.910 0.881 0.896 0.864 0.894 0.862 0.904 0.874 0.924 0.900 0.958 0.858 1.022 1.067 1.080 1.290 1.127 1.500 1.161 1.666 1.179 1.764 1.182 1.779 1.169 1.708 1.140 1.563 0.987 1.000 1.014 1.015 1.037 1.025 1.054 1.030 1.065 1.032 1.071 1.032 1.072 1.032 1.068 1.032 1.059 1.031 0.907 1.043 1.044 1.000 1.181 0.957 1.300 0.919 1.390 0.891 1.442 0.874 1.450 0.872 1.413 0.884 1.335 0.908 Jl-. Ki. K 2 . Mi M 2 *, N2, 2N, X2, 1x2, V2- M 3 _ Ma, MN. M 6 . M 8 . Oi, Qi,2Q, pi. OO ML. 2MK. ML. Mm. ♦Factor /"of MS, 2SM, and MSf are each equal to factor /of M 2 . Factor /of Pi, R 2 ,, Si, S 2 , S 4 , S 6 , T 2 , Sa, and Ssa are each unity. 200 U. S. COAST AND GEODETIC STJUVEY Table 14. — Node factor f for middle of each year, 1850 to 1999 — Continued Constituent 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 Ji 1,047 1.027 1.048 1.246 1.046 0.996 0.994 0.992 0.988 0.984 1.043 1.144 1.023 1.019 1.092 0.984 0,991 0.988 0.951 1.020 1. 528 1.009 1.013 1.017 1.026 1.035 0.980 0.926 0.997 1.005 0.953 1.028 0.932 0.949 0.866 0.786 1.824 1.020 1.031 1.041 1.062 1.084 0.916 0.739 0.968 0.988 0. 823 1.069 0.878 0.914 0.800 0.944 1.529 1.030 1.045 1.060 1.092 1.124 0.860 0.599 0.941 0.969 0.717 1.102 0.840 0.890 0.760 1.152 0.970 1.036 1.054 1.073 1.111 1.151 0.820 0.513 0.922 0.955 0.649 1.124 0.827 0.882 0.748 1.171 0.877 1.038 1.057 1.077 1.118 1.160 0.806 0.486 0.915 0.950 0.626 1.131 0.841 0.891 0.762 1.006 1.364 1.036 1.054 1.072 1.111 1.150 0.821 0.516 0.922 0.955 0.651 1.123 0.880 0.915 0.803 0.824 1.721 1.029 1.044 1.060 1.091 1.123 0.862 0.604 0.942 0.970 0. 721 1.101 0.934 0.950 0.869 0.945 1.593 1.020 1.030 1.040 1.061 1.082 0.919 0.746 0.969 0.988 0.828 1.067 0.994 Ki 0.990 K 2 0.955 L 2 - ---- 1.205 Mi 1.075 M2*, N2, 2N, X 2 , M2, V2- M 3 1.008 1.012 M 4f MN.... 1.016 M 8 -.._, 1.024 M«„ 1.033 Oi, Qi, 2Q, pi 0.983 OO . 0.936 MK 0.998 2MK 1.006 ML 0.959 Mm . 1.026 Constituent 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 Ji 1.049 1.028 1.052 1.153 1.323 0.996 0.993 0.991 0.987 0.982 1.046 1.153 1.024 1.019 1.098 0.983 1.096 1.062 1.148 0.709 2.091 0.984 0.976 0.968 0.952 0.936 1.100 1.375 1.045 1.028 1.230 0.941 1.132 1.088 1.230 0.683 2.158 0.974 0.961 0.948 0.923 0.898 1.142 1.571 1.059 1.031 1.339 0.907 1.155 1.105 1.289 1.185 1.434 0.967 0.950 0.934 0.903 0.873 1.170 1. 713 1.068 1.032 1.418 0.883 1.165 1.112 1.316 1.219 1.369 0.963 0.946 0.928 0.894 0.861 1.182 1.780 1.072 1.032 1.451 0.872 1.162 1.110 1.308 0.704 2.176 0.964 0.947 0.930 0.896 0.864 1.179 1.761 1.071 1.032 1. 441 0.875 1.146 1.099 1.267 0.607 2.240 0.969 0.954 0.939 0.910 0.882 1.160 1.660 1. 005 1.032 1.387 0.892 1.118 1.078 1.197 1.141 1.471 0.978 0.967 0.956 0.934 0.913 1.125 1.491 1.054 1.030 1.296 0.921 1 077 1.054 1.108 1.229 1.166 0.989 0.983 0.977 0.966 0.955 1.078 1.281 1.036 1.025 1.175 0.958 1.026 Ki_.__ 1.012 K 2 1.010 L 2 0.897 Mi 1.781 M 2 *, N 2 , 2N, X 2 , u2, vi. M 3 1.001 1.002 M 4 , MN 1.002 Mb 1.003 M 8 Oi, Qi, 2Q, pi 1.004 1.020 OO 1.058 MK 1.013 2MK 1.014 ML . 1.038 Mm 1.001 Constituent 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 Ji 0.968 0.973 0.916 0.753 1.902 1.013 1.020 1.027 1,040 1.054 0.956 0. 850 0.986 0.999 0.901 1. 045 0.910 0.934 0.838 1. 030 1.399 1.024 1.036 1.049 1.074 1.100 0.893 0.679 0.957 0.980 0.779 1.083 0.861 0.903 0.782 1.193 0.858 1.032 1.049 1.066 1.101 1.136 0. 812 0.559 0.933 0.963 0.686 1.112 0.832 0.885 0.752 1.117 1.069 1.037 1.056 1.076 1.115 1.157 0.811 0.496 0.918 0.952 0. 634 1.128 0.829 0.883 0.750 0. 925 1.507 1.038 1.057 1.076 1.117 1. 159 0.80S 0.490 0.916 0.951 0.630 1.130 0.854 0.899 0.774 0.858 1.643 1.034 1.051 1.068 1.104 1.142 0.834 0.543 0.929 0.960 0.673 1.116 0.900 0.928 0.825 1.051 1.340 1.026 1.039 1.053 1.080 1.108 0.882 0.652 0.952 0.977 0.759 1.089 0.957 0.965 0.900 1.221 0.946 1.016 1.023 1.031 1.047 1.063 0.944 0.814 0.980 0.996 0.877 1.052 1.016 1.005 0.992 1.062 1.479 1.003 1.005 1.007 1.010 1.013 1.008 1.017 1.008 1.012 1.012 1.010 1.069 Ki 1.042 K 2 1.090 L 2 0.653 Mi... 2.112 M 2 *, N 2 , 2N, X 2 , m, V2. M 3 . M <( MN- 0.991 0.986 0.982 M 6 .__. . 0.973 M 8 Oi, r Q,,2Q,p, OO.. 0.964 1.068 1.240 MK 1.033 2MK 1.023 Mf 1.151 Mm 0.966 *Factor/of MS, 2SM, and MSf are each equal to factor /of Ms. Factor /of Pi, R 2 , Si, S 2 , S4, So, T 2 , Sa, and Ssa are each unity. HARMONIC ANALYSIS AND PREDICTION OF TIDES 201 Table 14.— Node factor f for middle of each year, 1850 to 1999— Continued Constituent 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 J, 1.111 1.073 1.182 0.834 1.972 0.980 0.969 0.959 0.940 0.920 1.118 1.455 1.051 1.029 1.275 0.927 1.142 1.096 1.256 1.246 1.248 0.970 0.956 0.942 0.914 0.887 1.154 1.633 1.063 1.032 1.373 0.896 1.160 1.109 1.303 1.135 1. 557 0.965 0.948 0.931 0.898 0.867 1.176 1.748 1.070 1.032 1.434 0.877 1.165 1. 113 1.317 0.561 2.297 0.963 0.945 0.928 0.894 0.861 1.183 1.783 1.072 1.032 1.452 0.871 1.157 1.107 1.296 0.729 2.146 0.966 0.949 0.933 0.901 0.870 1.173 1.732 1.069 1.032 1.425 0.880 1.137 1.092 1.243 1.221 1.310 0.972 0.958 0.945 0.918 0.893 1.148 1.602 1.061 1.032 1.356 0.902 1.104 1.067 1.165 1.172 1.371 0.982 0.972 0.964 0.946 0.928 1.109 1.414 1.048 1.028 1.252 0.934 1.059 1.035 1.071 0.761 1.993 0.993 0.990 0.987 0.980 0.973 1.056 1.195 1.028 1.021 1.124 0.975 1.004 0.997 0.973 0.780 1.909 1.006 1. 009 1.012 1.017 1.023 0.995 0.974 1.003 1.009 0.985 1.018 0.945 Ki 0.958 K 2 0.884 L,2 1.118 Mi 1.243 M2*, N2, 2N, X2, M2» vi. Mi . 1.018 1.027 M 4 , MN 1.036 M 6 M 8 1.054 1.073 0,, Qi, 2Q, pi 00 MK 0.931 0.779 0.975 0.992 0.851 1.060 Constituent 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 J, . 0.890 0.921 0.813 1.198 0.896 1.028 1.042 1.056 1.086 1.116 0.871 0.626 0.947 0.973 0.739 1.096 0.847 0.894 0.767 1.034 1.308 1.035 1.052 1.071 1.108 1.146 0.827 0.528 0.925 0.958 0.660 1.120 0.827 0.882 0.748 0.870 1.597 1.038 1.057 1. 077 1.118 1.160 0.806 0. 487 0.915 0.950 0.626 1.131 0.836 0.887 0.756 0.932 1.503 1.036 1.055 1.074 1.114 1.154 0.815 0.504 0.920 0.953 0.641 1.126 0.870 0.909 0.791 1.133 1.082 1.031 1.047 1.063 1.096 1.130 0.850 0.579 0.937 0.966 0.701 1.107 0.921 0.942 0.852 1.199 0. 954 1.022 1.034 1. 045 1.068 1.092 0.905 0.710 0.963 0.984 0.801 1.076 0.980 0.981 0.934 0.963 1.619 1.011 1.016 1.022 1.033 1 044 0.968 0.889 0.992 1.002 0.928 1.036 1.037 1.020 1.030 0.669 2.063 0.998 0.998 0.997 0.995 994 1.032 1.102 1.018 1.017 1.066 0.993 1.086 1.055 1.127 0.975 1.739 0.986 0.979 0.973 0.959 946 1. 088 1.325 1.040 1.026 1.201 0.950 1.125 Ki 1. 083 K? . . 1.214 L 2 1.270 Mi 1. 138 Mo*, N2, 2N, X2, (12, V2- M 3 0.976 964 M 4 , MN 952 M 6 929 Mg 906 Oi, Qi, 2Q, pi 1 134 OO 1 530 MK 1 056 2MK . 1 031 Mf 1 317 Mm 914 Constituent 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 Ki".""""™"II 1.150 1.102 1.278 1.022 1.312 0.968 0. 952 0.937 0.907 0.878 1.165 1.686 1.066 1.032 1.402 0.887 1.163 1.112 1.312 0.471 2.353 0.964 0.946 0.929 0.895 0.863 1.181 1.772 1.071 1.032 1.446 0.873 1.164 1. 112 1. 313 0.873 1.992 0.964 0.946 0.929 0.895 0.862 1.181 1.773 1.071 1.032 1.447 0.873 1. 151 1.102 1.279 1.270 1.197 0.968 0.952 0.936 0.906 0.877 1.165 1.690 1.067 1.032 1.403 0.887 1.126 1.083 1.216 1.078 1.614 0.975 0.963 0.951 0.928 0.905 1.134 1.535 1.057 1.031 1.320 0.913 1.088 1.056 1.130 0. 636 2.148 0.986 0. 979 0.972 0.958 0.945 1.090 1.332 1.041 1.026 1.205 0.949 1.038 1.021 1.033 0.859 1. 850 0.998 0.997 0.996 0.994 0.992 1.034 1.108 1.019 1.017 1.070 0.991 0.982 0. 982 0.937 1. 190 1.098 1.011 1.016 1.021 1.032 1.043 0.970 0.894 0.992 1.003 0.931 1.035 0.923 0.943 0.854 1.162 1.079 1.022 1.033 1.044 1.067 1.091 0.907 0.714 0.964 0.985 0.804 1.075 0.871 909 K 2 . 0.792 0.93i 1. 534 1.031 1.047 1.063 1.096 1.130 0.852 0.581 0.938 0.966 0.704 1.107 L 2 M, M2*, N2, 2N, X2, m, vi- M 3 M 4 ,MN Mo M 8 Oi, Qi, 2Q,pi. OO MK 2MK Mf Mm ♦Factor /of MS, 2MS, and MSf are each equal to factor /of M 2 . Factor /of Pi, R 2 , Si, S 2 , S4, S 6 , T 2 , Sa, and Ssa are each unity. 202 U. S. COAST AND GEODETIC SURVEY Table 14. — Node factor f for middle of each year, 1850 to 1999— Continued Constituent Ji- K,. K 2 . U- Mi. M2*, N2, 2N, X2, H2, V2 M3 M 4 ,MN Ms. Mj. Oi, Qi, 2Q, pi. 00 MK 2MK Mf_. Mm. 1940 0.836 0.888 0.757 0.860 1.623 1.036 1.055 1.074 1.113 1.154 0.816 0. 505 0.920 0.953 0.642 1.126 1941 0.827 0.882 0.748 1.021 1.313 1.038 1.057 1. 077 1.118 1.160 0.806 0.486 0.915 0.950 0.626 1.131 1942 0.846 0.894 0.766 1.180 0.879 1.035 1.053 1.071 1.108 1.147 0.826 0.526 0.925 0.957 0.659 1.121 1943 0.888 0.920 0.812 1.144 1.076 1.028 1.042 1.057 1.086 1.117 0.870 0.623 0.946 0.973 0.736 1.096 1944 0.944 0.956 0.882 0.876 1.714 1.018 1.027 1.036 1.055 1.074 0.929 0.774 0.974 0.991 0.848 1.061 1945 1.003 0.996 0.970 0.748 1.944 1.006 1.009 1.012 1.018 1.025 0.994 0.969 1.002 1.008 0.981 1.019 1946 1.057 1.034 1.068 1.091 1.480 0.994 0.990 0.987 9.981 0.975 1.055 1.189 1.028 1.021 1.120 0.976 1947 1.103 1.067 1.162 1.255 1.138 0.982 0.973 0.964 0.947 0.929 1.107 1.408 1.047 1.028 1.249 0.935 1948 1.136 1.091 1.242 0.894 1.927 0.972 0.959 0.945 0.919 0.894 1.147 1.598 1.061 1.032 1.354 0.902 Constituent Ji-- K,. K 2 . L 2 -. Mi. M2*, N2, 2N f X2, M2, v% M 3 M 4 , MN Me. Mg_ Oi, Q,, 2Q, pi. OO MK.. 2MK. Mf__ Mm. 1950 1951 1952 1953 1954 1955 1956 1957 1958 1.165 1.113 1.317 1.160 1.109 1.303 1.143 1.096 1.257 1.112 1.074 1.184 1.070 1.043 1.092 1.002 1.001 0.995 0.959 0.966 0.903 0.901 0.929 0.827 0.855 0.900 0.776 1.074 1.717 1.330 1.120 1.014 1.778 0. 653 2.161 1.001 1.664 1.260 0.964 1. 112. 1.276 0.867 1.656 0.915 1.527 0.963 0.945 0.928 0.965 0.948 0.931 0.970 0.956 0. 941. 0.982 0.969 0.959 0.990 0.986 0.981 1.003 1.004 1.006 1.015 1.023 1.031 1.026 1.039 1.052 1.033 1.051 1.068 0.894 0.861 0.898 0.867 0.914 0.887 0.939 0.920 0.972 0.962 1.009 1.012 1.046 1.062 1.079 1.107 1.104 1.141 1.183 1.784 1.177 1.750 1.155 1.637 1.119 1.459 1.069 1.246 1.010 1.023 0.945 0.819 0.884 0.656 0.835 0.546 1.072 1.032 1.070 1.032 1.063 1.032 1.051 1.029 1.033 1.023 1.009 1.012 0.981 0.996 0.953 0.977 0.930 0.960 1.452 0.872 1.435 0.877 1.375 0.896 1.278 0.926 1.154 0.965 1.016 1.008 0.880 1.051 0.761 1.088 0.675 1.116 Constituent 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 Ji 0.831 0.885 0.752 1. 199 0.767 1.037 1.056 1.076 1.116 1.157 0.810 0.495 0.917 0.952 0.633 1.128 0.860 0.903 0.781 1.081 1.197 1.033 1.049 1.066 1.111. 1.137 0.840 0.557 0.932 0.962 0.684 1.113 0.909 0.934 0.836 0.849 1.690 1.024 1.037 1.050 1.075 1.102 0.891 0.675 0. 956 0.980 0.776 1.084 0.766 0.972 0.914 0.893 1.014 1.020 1.027 1.041 1.055 0. 954 0.845 0.985 0.998 0.898 1.046 1.025 1.011 1.008 1.200 1.166 1.001 1.002 1.003 1.C04 1.005 1.018 1.053 1.013 1.014 1.035 1.003 1.076 1.048 1.106 1.237 1.175 0.989 0.983 0.978 0.967 0.956 1.076 1.276 1.036 1.024 1.172 0.959 1.117 1.077 1.195 0.838 1.976 0.978 0.967 0.956 0. 935. 0.914 1.124 1.487 1.053 1.030 1.293 0.922 1.146 1.098 1.265 0.690 2.175 0.969 0.954 0.940 0.911 0.S83 1.159 1.655 1.064 1.032 1. 385 0.893 1.161 1.110 1.307 1.185 1.503 0.964 0.947 0.930 0.897 0.865 1.178 1.758 1.071 1.032 1.439 0.876 1.165 Ki 1.113 K 2 1.316 L2 1.310 Mi 1.197 M2*, N2, 2N, \2, fl2,V2- M3-. 0.963 0.945 M 4 , MN 0.928 M 6 0.894 Ms- . 0.861 Oi, Qi, 2Q, pi 1.182 OO 1.782 MK 1.072 2MK Mf. 1.032 1.451 Mm. 0.872 ♦Factor /of MS, 2SM, and MSf are Factor /of Pi, R 2 , Si, S 2 , Si, S 6 , T 2 , i ch equal to factor / of M2. and Ssa are each unity. HARMONIC ANALYSIS AND PREDICTION OF TIDES 203 Table 14. — Node factor f for middle of each year, 1850 to 1999 — Continued Constituent 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 J, 1.155 1'. 105 1.289 0.882 1. 987 0.966 0.950 0.934 0.903 0.873 1.170 1.716 1.068 1.032 1.417 0.882 1.132 f.088 1.232 0.668 2. 176 0.973 0.960 0.948 0.922 0.898 1.143 1.575 1.059 1.031 1.341 0.906 1.097 1.063 1.150 1.118 1.503 0.983 0.975 0.967 0.951 0.935 1.101 1.380 1.045 1.028 1.233 0.940 1.051 1.029 1.055 1.270 1.012 0.995 0.993 0.991 0.986 0.981 1.047 1.159 1.024 1.020 1.102 0.982 0.995 0.991 0.957 1.014 1. 535 1.008 1.012 1.016 1.024 1.032 0.984 0.940 0.998 1.006 0.962 1.025 0.936 0.951 0.871 0.808 1.777 1.020 1.029 1.039 1.060 1.081 0.920 0.750 0.970 0.989 0.831 1.067 0.881 0.916 0.804 0.988 1.428 1.029 1.044 1.059 1.090 1.122 0.863 0.607 0.943 0.970 0.723 1.100 0.842 0.891 0.763 1.179 0.870 1.035 1.054 1.072 1.110 1.149 0.822 0.517 0.923 0.956 0.652 1.123 0.827 0.882 0.748 1.169 0.874 1.038 1.057 1.077 1.118 1.160 0.806 0.485 0.915 0.950 0.625 1.131 0.839 Kl 0.890 K 2 0.760 L 2 0-994 Mi 1.361 M 2 *, N2, 2N, X2, fl2, V2_ Ms 1.036 1.054 M 4 , MN 1.073 Me—. 1.112 Ms-.. 1.151 Oi, Qi, 2Q, pi 0.819 OO 0.512 MK 0.922 2MK 0.955 ML... 0.647 Mm 1.124 Constituent 1980 1981 1982 1983 1984 1985 1986 1.153 1.104 1.285 1.263 1.292 0.967 0.951 0.935 0.904 0.874 1.168 1.706 1.068 1.032 1.412 0.884 1987 1988 1989 Ji 0.877 0.913 0.799 0.848 1.656 1.030 1.045 1.061 1.092 1.125 0.858 0.596 0.941 0.969 0.715 1.103 0.930 0.948 0.864 1.001 1.468 1.021 1.031 1.042 1.063 1.085 0.915 0.735 0.967 0.987 0.820 1.070 0.989 0.987 0.949 1.238 0.974 1.009 1.013 1.018 1.027 1.036 0.979 0.921 0.996 1.005 0.949 1.029 1.045 1.026 1.045 1.157 1.323 0.997 0.994 0.993 0.989 0.986 1.041 1.137 1.022 1.019 1.088 0.986 1.093 1.060 1.142 0.745 2.050 0.984 0.977 0.969 0.954 0.939 1.096 1.361 1.043 1.027 1.221 0.944 1.130 1.086 1.226 0.811 2.032 0.974 0.962 0.949 0.924 0.901 1.140 1.560 1.058 1.031 1.333 0.909 1.164 1.112 1.315 1.244 1.367 0.964 0.946 0.928 0.894 0.862 1.182 1.778 1.072 1.032 1.450 0.872 1.163 1.111 1.310 0.749 2.142 0.964 0.947 0.930 0.896 0.864 1.180 1.766 1.071 1.032 1.443 0.874 1.148 Ki 1.100 K 2 1.270 L 2 0.746 Ml 2.122 M 2 *, N 2 , 2N, X 2 , ju2, v2. Ms. - 0.969 0.954 Mi,MN Me 0.939 0.910 Mg. 0.881 Oi, Qi, 2Q,pi 1.161 OO 1.668 MK. 1.065 2MK._ . 1.032 Mf 1.392 Mm 0.891 Constituent 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Ji 1.120 1.079 1.203 1.216 1.334 0.977 0.966 0.955 0.932 0.911 1.128 1.505 1.054 1.030 1.303 0.918 1.080 1.051 1.115 1.248 1.156 0.988 0.982 0.976 0.964 0.952 1.081 1.296 1.038 1.025 1.184 0.956 1.030 1.015 1.016 0.898 1.778 1.000 1.000 1.000 1.000 1.C00 1.024 1.072 1.015 1.015 1.048 0.998 0.972 0.976 0.922 0.801 1.829 1.013 1.019 1.025 1.038 1.051 0.960 0.803 0.988 1.000 0.910 1.042 0.914 0.937 0.842 1.077 1.282 1.024 1.036 1.048 1.072 1.C98 0.897 0.688 0.959 0.982 0.786 1.081 0.864 0.905 0.785 1.208 0.800 1.032 1.048 1.065 1.099 1.134 0.844 0.565 0.934 0.964 0.691 1.110 0.833 0.886 0.754 1.107 1.083 1.037 1.056 1.075 1.115 1.156 0.812 0.498 0.918 0.952 0.636 1.128 0.829 0.883 0.750 0.921 1.487 1.038 1.057 1.076 1.117 1.159 0.808 0.489 0.916 0.951 0.629 1.130 0.852 0.897 0.772 0.893 1.560 1.034 1.051 1.069 1.105 1.143 0.832 0.538 0.928 0.959 0.669 1.117 0.896 Ki 0.926 K 2 0.821 L 2 1.096 Mi .. 1.214 M 2 *, N 2 , 2N, X 2 , p.2, V2. M 3 1.027 1.040 M ( ,MN 1.054 M 6 . 1.082 M 8 1. Ill Oi, Qi, 2Q,pi 0.879 OO 0.643 MK 0.950 2MK 0.976 ML 0.752 Mm 1.091 * Factor /of MS, 2SM, and MSf are each equal to factor /of M2. Factor /of Pi, Rj, Si, S 2 , S 4 , So, T 2 , Sa, and Ssa are each unity. 204 U. S. 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COAST AND GEODETIC STJTtVEY ■^U30H!OHmrt(lflOOO'i"ONK|iOOlOlOOOOOinHlOTfl01«10tDMNOONailClO '*'*0)00'*'nHN00SIN'*!O(»N(DO>OM010)0)'<)NOt)NOO©iHfq^lHH HHCOIMHHIN (NrH ININHHHHHMHWNH(N(NN«H NWCO NN'*!ONMtDOSHSC<3!DteO)lO«IC!»NOOOOMO'HH-*MT)(H»'^OOT)(OI>T(( NMNN^tOONNlO^OlO^NOlOtDCOOOOOfflHNffilMNCOOOaiHOOHamN drad«'dddrtHiNM*H'^u5^du5diodddoioHdHC!D»do'ffloi*dd T)NOoo^o>MooNiocoNONHOioooiooooHcsa)aiOooo)NaiHO>HHaN HN'OIXJI'^OOCONCIlOOlNtOCONNCBINOOOONi-lNlOCOH^aiOOr-ltOHOONlO »ioddHrt(N^io«)H'diosNTiicoddioddddtDidd^oo'drt'(NVNNwdd rH rH IN IN rH rH IN CO M»0(»Hl<:iOOHi«HNN^ „ mddNddMdNri^dNMdddNdddddddNrHdHJNTiiccdddwdd ° © © 00 t^- OO 00 t- CO T« CO CO J> CO 00 © 00 00 1>00 © 00 00 t^ l> CO t^ 00 CO lO ■>* lO 00 rH rH rH IN CO rH CO rH rH rH rH r-i i-i rH rH rH IN OP0H(OH0iO<*iONIN'* ^ido'^CMCtf5C>oc>oi^ ° t>-^OTf-rHTtCOCO»at^lO©©Tt rHrH rHININ (NrHrH rH rH CO rH rH rH rH IN IN IN IN rH T|l«MHOlNrtlONHlCj«l«)NtDTtl0JNfflOOOO'*CqM0C»OHO'*- CO ■*© rH rH lO CO »0 »0 lO CM CO CM T* HHHHIONNIMmN rHrH l-H rH N(»ONOffiiH(N ° ^^O300-*rt<00COt^cOTti000O^^THTtlcOO5'cHO3O01^-O0000000TficO ■*»00i-l'*«!®O(N0!l0rHl0tDNOH®aiOOOOO'*INSX<0MN!0Mf4WU501«) ° CO CO CO CO CO CO CO CM CM i-H i-H IN OClmOOOOONiOC»CqC)H MMINCSMMlNiHH CM CM CM CM CO CO CM CM CM CO CM rH CM CM CM CM CM CM r-H rH i-H CM H(N*SCO'*a)NeiOmO)HtDiO'*O'*N MCMC»rHO!DcOOOHONNOlOOOHTtoONCOOOOiO*HOOHtON005NOOiOrtC»HTtC©^iOTt r-H TH CM <3> CM CO i-H i-H CM CM i-H i-H CM CM i-H CO CM CO CM CM CM CM CM t-H CM i-H CM i-H i-H CM CO CO i-H CM i-H OONcOMOOOCO^NOOlOOCOlOOOOOOHNOOOOOOMNOiNOOHtOto^STllNNW NWNOl'tcO^HiOOOW'^HOMaiOOWOOOOWOO'OCOcOOlMNCq^NHttXM ° MNO)OOHHNO'*Nai®iOO-*IMOC5C»NO)OONOO»lO«OOQO*©OHH CMCM CMCM CMrHCMCM CM CM CM CM i-H CM .-H CM CM CO CM .-H rH ^H 0^lOOlOCnOOI^lOCOi-HCDCOOlOCOOOCOt^OOOOCOOO»OOOOuOOOCOOOCMl^CMCMt^»0 NOH'*05Q0N(D'0MHHlOa»H4i00)'COOOO'*C0'0HOf0iaC»NNC0N©lC)H „ o6dH'MNNioodHV'rioo'dio'iddNddddddddiH'do6»dri>o^>rfH/NdiH' CMCM CMCM CM i-Hi-h rHCMCMrHrH CM i-H i-h rH CM CM r-H HMNN!DO0>0>»00NNT)<(0rtN»H«OOOONC0rtC»iHNH00C<:S <0U3O00'*H/NrtlOnHN(DN00H/HHll3OOOOijlCS'0©HWOlONWNNOlOO - NiOrtM'ododdincodNdNH'dHJ^N'dioddddNMdioiNceddM ° OOMN00 00HOCC1O® 00O0)N!DM»M®0)(N(N rH b- h Oi CM rfl -ooctccoio) OH/0ina>0i00t>«'*N(N^C0ffliC)NN'i)ioOOOi0"*«H(0c<5HO00HrtrtOiJ<0) Nod^MMNHioccHH^Vccad^ddddddmio'NH'wioaNNMdddd ° 00 00 lO 1^ l^ Tji CD CO CM rH -^ CO CD -Jl t~- CO IC 00 lOlOCO'fJICOtDlON'HpHrtH rH i-H CO i-H rH CO i-H CO CO CO CO CO rH rH I-H 1-H rH I-H CO CO CO CO rH rH rH CO CO O ■ J i\4&S%%%%%%%g%oo£ 6 7 8 9 0.50 0.354 1.060 1.768 2.474 3 182 3.888 4.596 6.302 6. 0«) 6.716 .51 --- .52 .361 .368 .375 .382 .389 .396 .403 .410 .417 1.068 1.075 1.082 1.089 1.096 1.103 1.110 1.117 1.124 1.775 1.782 1.789 1.796 1.803 1.810 1.817 1.824 1.831 2.482 2.489 2. 496 2.503 2. 510 2. 517 2. 524 2.531 2.538 3.189 3.196 3.203 3.210 3.217 3.224 3.231 3.238 3.245 3.896 3.903 3.910 3.917 3. 924. 3.931. 3.938 3.945 3.952 4. 603 4.610 4.617 4.624 4.631 4.638 4.645 4652 4.659 5. 310 5.317 5. 324 5. 331 5. 338 5.345 5. 352 5.359 5. 366 6.017 6.024 0.031 6.038 6.045 6.052 6. 059 6.066 6.073 6.724 6.731 ,53 6.738 .54 6.745 .55 _ 6.752 .56 6.759 .57 --- 6.766 .58 6.773 .59 6.780 .60 .424 1.131 1.838 2.545 3.252 3.959 4.666 5.373 6.080 6.787 .61 - .431 .438 .445 .452 .460 .467 .474 .481 .488 1.138 1.145 1.152 1.159 1.167 1.174 1.181 1.188 1.195 1.845 1.852 1.859 1.866 1.874 1.881 1.888 1.895 1.902 2.552 2.559 2. 566 2.573 2.581 2.588 2.595 2.602 2.009 3.259 3.266 3.273 3.280 3.288 3.295 3.302 3. 309 3. 316 3.966 3.973 3.980 3.987 3.995 4.002 4.009 4.016 4.023 4.673 4.680 4.687 4.694 4702 4.709 4.716 4. 723 4.730 5.380 5. 387 5.394 5.401 5.409 5.416 5.423 5.430 5.437 6.087 6.094 6.101 6.108 6.116 6.123 6.130 6.137 6.144 6.794 .62 6.801 .63 6.808 .64 6.815 .65 6.823 .66 6.830 .67 6.837 6.844 .69 6.851 .70 .495 1.202 1.909 2.616 3.323 4.030 4.737 5. 444 6.151 6.858 .71 .502 .509 .516 .523 .530 .537 .544 .551 .559 1.209 1.216 1. 223 1.230 1.237 1.244 1.251 1.258 1.266 1.916 1.923 1.930 1.937 1.944 1.951 1.958 1.965 1.973 2.623 2.630 2.637 2.644 2.651 2. GS8 2.665 2.672 2.680 3. 330 3.337 3.344 3.351 3.358 3 365 3. 372 3.379 3.387 4.037 4.044 4.051 4.058 4.065 4.072 4.079 4.086 4.094 4.744 4.751 4.758 4.765 4.772 4.779 4.786 4.793 4.801 5.451 5.458 5.465 5.472 5. 479 5. 486 5.493 5.500 5. 508 6.158 6.165 6.172 6.179 6.186 6.193 6.200 6.207 6.215 6.865 .72 6.872 .73 6.879 .74 6.886 .75 6.893 .76- 6.900 .77 6.907 .78 6.914 .79 6.922 .80 _ .566 1.273 1.980 2.687 3.394 4.101 4.808 5.515 6.222 6.929 .81 .573 .580 .587 .594 .601 .608 .615 .622 .629 1. 280. 1.287 1.294 1.301 1.308 1.315 1.322 1.329 1.336 1.987 1.994 2.001 2.008 2.015 2.022 2.029 2.036 2.043 2.694 2.701 2.708 2.715 2.722 2.729 2.736 2.743 2.750 3. 401 3.408 3.415 3.422 3.429 3.436 3.443 3.450 3.457 4.108 4.115 4.122 4.129 4.136 4.143 4.150 4.157 4.164 4.815 4.822 4.829 4.836 4.843 4.850 4.857 4.864 4.871 5. 522 5.529 5.536 5.543 5.550 5.557 5.564 5.571 5. 578 6.229 6.236 6.243 6.250 6.257 6.264 6.271 6.278 6.285 6. 936 .82 6. 943 .83 6. 950 .84 6. 957 .85 6.964 .86 6.971 .87 6.978 .88 6. 985 .89 6.992 .90 .636 1.343 2.050 2.757 3. 464 4.171 4.878 5.585 6.292 6.999 .91 .643 .650 .658 .665 .672 .679 .686 .693 .700 1.350 1.357 1.365 1.372 1.379 1.386 1.393 1.400 1.407 2.057 2.064 2.072 2.079 2.086 2.093 2.100 2.107 2.114 2.764 2.771 2.779 2.786 2.793 2.800 2.807 2.814 2.821 3. 471 3.478 3.486 3.493 3.500 3.507 3.514 3.521 3.528 4.178 4.185 4.193 4.200 4.207 4.214 4.221 4.228 4.235 4.885 4.892 4.900 4.907 4.914 4.921 4.928 4.935 4.942 5. 592 5.599 5.607 5.614 5.621 5.628 5.635 5.642 5.649 6.299 6.306 6.314 6.321 6.328 6.335 6.342 6.349 6.356 7.006 .92- 7.013 .93.. 7.021 .94 .95 7.028 7. 035 .96 7.042 .97.-. 7.049 .93 7.056 .99 7.063 1.00 . .. . .707 1.414 2. 121 2.828 3.535 4.242 4.949 5.656 6.363 7.070 1 2 3 4 5 6 7 8 9 224 U. S. COAST AND GEODETIC STtttVEY Table 19.— Products for Form 194— Continued [Multiplier = sin 60° =0.866] 1 2 3 4 5 6 7 8 9 0.00 0.000 0.866 1.732 2.598 3.464 4.330 5.196 6.062 6.928 7.794 .01 .009 .017 .026 .035 .043 .052 .061 .069 .078 .875 .883 .892 .901 .909 .918 .927 .935 .944 1.741 1.749 1.758 1.767 1.775 1.784 1.793 1.801 1.810 2.607 2.615 2.624 2.633 2.641 2.650 2.659 2.667 2.676 3.473 3.481 3.490 3.499 3.507 3.516 3.525 3.533 3.542 4.339 4.347 4.356 4.365 4.373 4.382 4.391 4.399 4.408 5.205 5.213 5.222 5.231 5.239 5.248 5.257 5.265 5.274 6.071 6. 079 6.088 6.097 6.105 6.114 6.123 6.131 6.140 6.937 6.945 6.954 6.963 6.971 6.980 6.989 6.997 7.006 7,803 7.811 7.820 7,829 7,837 7.846 7.855 7 863 .02 .03 .04 .05 .06 .07... .08 :.-. .09 7 872 .10 .. . 087 .953 1.819 2.685 3.551 4.417 5.283 6.149 7.015 7.881 .11 .095 .104 .113 .121 .130 .139 .147 .156 .165 .961 .970 .979 .987 .996 1.005 1.013 1.022 1.031 1.827 1.836 1.845 1.853 1.862 1.871 1.879 1.888 1.897 2.693 2.702 2.711 2.719 2.728 2.737 2.745 2.754 2.763 3.559 3.568 3.577 3.585 3.594 3.603 3.611 3.620 3.629 4.425 4.434 4.443 4.451 4.460 4.469 4.477 4.486 4.495 5.291 5.300 5.309 5.317 5.326 5.335 5.343 5.352 5.361 6.157 6.166 6.175 6.183 6.192 6.201 6.209 6.218 6.227 7.023 7.032 7.041 7.049 7.058 7.067 7.075 7.084 7.093 7 889 .12 .13 .. ------ 7.898 7.907 7.915 7.924 7 933 .14 .15 .16 .17 .18 7.941 7 950 .19 7,959 .20 . 173 1.039 1.905 2.771 3.637 4.503 5.369 6.235 7.101 7.967 .21 .22 .182 .191 .199 .208 .216 .225 .234 .242 .251 1.048 1.057 1.065 1.074 1.082 1.091 1.100 1.108 1.117 1.914 1.923 1.931 1.940 1.948 1.957 1.966 1.974 1.983 2.780 2.789 2.797 2.806 2.814 2.823 2.832 2.840 2.849 3.646 3.655 3.663 3.672 3.680 3.689 3.698 3.706 3.715 4.512 4.521 4.529 4.538 4.546 4.555 4.564 4.572 4.581 5.378 5.387 5.395 5.404 5.412 5.421 5.430 5.438 5.447 6.244 6.253 6.261 6.270 6.278 6.287 6.296 6.304 6.313 7.110 7.119 7.127 7.136 7.144 7.153 7.162 7.170 7.179 7,976 7.985 .23 7.993 .24 .25 .26 8.002 8,010 8.019 .27 8.028 .28 .29 8.036 8.045 .30 .260 1.126 1.992 2.858 3.724 4.590 5.456 6.322 7.188 8.054 .31 A .32 .268 .277 .286 .294 .303 .312 .320 .329 .338 1.134 1.143 1.152 1.160 1.169 1.178 1.186 1.195 1.204 2.000 2.009 2.018 2.026 2.035 2.044 2.052 2.061 2.070 2.866 2.875 2.884 2.892 2.901 2. 910 2.918 2.927 2.936 3.732 3.741 3.750 3.758 3.767 3.776 3.784 3.793 3.802 4.598 4.607 4.616 4.624 4.633 4.642 4.650 4.659 4.668 5.464 5.473 5.482 5.490 5. 499 5.508 5.516 5.525 5.534 6.330 6.339 6.348 6.356 6.365 6.374 6.382 6.391 6.400 7.196 7.205 7.214 7.222 7.231 7.240 7.248 7.257 7.266 8.062 8.071 ,33 8.080 ,34 8.088 .35 8.097 .36 8.106 .37 8.114 .38. ,39 ., 8.123 8.132 .40 .-.- .346 1.212 2.078 2.944 3.810 4.676 5.542 6.408 7.274 8.140 .41. -...-.. .355 . .364 1.221 1.230 1.238 1.247 1.256 1.264 1.273 1.282 1.290 2.087 2.096 2.104 2.113 2.122 2.130 2.139 2.148 2.156 2.953 2.962 2.970 2.979 2.988 2.996 3.005 3.014 3.022 3.819 3.828 3.836 3.845 3.854 3.862 3.871 3.880 3.888 4.685 4.694 5.702 4.711 4.720 4.728 4.737 4.746 4.754 5.551 5.560 5.568 5.577 5.586 5.594 5.603 5.612 5.620 6. 417 6.426 6.434 6.443 6.452 6.460 6.469 6.478 6.486 7.283 7.292 7.300 7.309 7.318 7.326 7.335 7.344 7.352 8.149 .42 8.158 .43 .372 .381 .390 .398 .407 .416 .424 8.166 .44 .45 8. 175 8.184 .46 8.192 .47 . 8.201 .48 8.210 .49 8.218 .50 . A .433 1.299 2.165 3.031 3.897 4.763 5.629 6.495 7.361 8,227 1 2 3 4 5 6 7 8 9 HARMONIC ANALYSIS AND PREDICTION OF TDDES 225 Table 19. — Products for Form 194 — Continued [Multiplier=sin 60° = O.866] » 2 3 4 5 6 7 8 9 0.50 0.433 1.299 2.165 3.031 3.897 4.763 5.629 6.495 7.361 8.227 .51 .442 .450 .459 .468 .476 .485 .494 .502 .511 1.308 1.316 1.325 1.334 1.342 1.351 1.360 1.368 1.377 2.174 2.182 2.191 2.200 2.208 2.217 2.226 2.234 2.243 3.040 3.048 3.057 3.066 3.074 3.083 3.092 3.100 3.109 3.906 3.914 3.923 3.932 3.940 3.949 3.958 3.966 3.975 4.772 4.780 4.789 4.798 4.806 4.815 4.824 4.832 4.841 5.638 5.646 5.655 5.664 5.672 5.681 5.690 5.698 5.707 6.504 6.512 6.521 6.530 6.538 6.547 6.556 6.564 6.573 7.370 7.378 7.387 7.396 7.404 7.413 7.422 7.430 7.439 8 236 .52 8 244 .53 8.253 .54 8 262 .55 8.270 .56 8.279 .57 8.288 .58 8.296 .59... 8.305 .60 .520 1.386 2.252 3.118 3.984 4.850 5.716 6.582 7.448 8.314 .61 .528 .537 .546 .554 .563 .572 .580 .589 .598 1.394 1.403 1.412 1.420 1.429 1.438 1.446 1.455 1.464 2.260 2.269 2.278 2.286 2.295 2.304 2.312 2.321 2.330 3.126 3.135 3.144 3.152 3.161 3.170 3.178 3.187 3.196 3.992 4.001 4.010 4.018 4.027 4.036 4.044 4.053 4.062 4.858 4.867 4.876 4.884 4.893 4.902 4.910 4.919 4.928 5.724 5.733 5.742 5.750 5.759 5.768 5.776 5.785 5.794 6.590 6.599 6.608 6.616 6.625 6.634 6.642 6.651 6.660 7.456 7.465 7.474 7.482 7.491 7.500 7.508 7.517 7.526 8.322 .62 8.331 .63 . 8.340 .64 8.348 .65 .66 .67.. . 8.357 8.366 8.374 .68.... 8.383 .69 8.392 .70 .606 1.472 2.338 3.204 4.070 4.936 5.802 6.668 7.534 8.400 .71 .72 .615 .624 .632 .641 .650 .658 .667 .675 .684 1.481 1.490 1.498 1.507 1.516 1.524 1.533 1. 541 1.550 2.347 2.356 2.364 2.373 2.382 2.390 2.399 2.407 2.416 3.213 3.222 3.230 3.239 3.248 3.256 3.265 3.273 3.282 4.079 4.088 4.096 4.105 4.114 4.122 4.131 4.139 4.148 4.945 4.954 4.962 4.977 4.980 4.988 4.997 5.005 5.014 5.811 5. 820 5.828 5.837 5.846 5.854 5.863 5.871 5.880 6.677 6.686 6.694 6.703 6.712 6.720 6.729 7.737 6.746 7.543 7.552 7.560 7.569 7.578 7.586 7.595 7.603 7.612 8.409 8.418 .73 8.426 .74 8.435 .75 8.444 .76.. 8.452 .77 8.461 .78 8.469 .79 8.478 .80 .693 1.559 2.425 3.291 4.157 5.023 5.889 6.755 7.621 8.487 .81.. . .701 .710 .719 .727 .736 .745 .753 .762 .771 1.567 1.576 1.585 1.593 1.602 1.611 1. 619 1. 628 1.637 2. 433 2.442 2.451 2.459 2.468 2.477 2.485 2.494 2.503 3.299 3.308 3.317 3.325 3.334 3.343 3.351 3.360 3.369 4. 165 4.174 4.183 4.191 4.200 4.209 4.217 4.226 4.235 5.031 5.040 5.049 5.057 5.066 5.075 5.083 5.092 5.101 5.897 5.906 5.915 5. 923 5. 932 5.941 5.949 5.958 5.967 6.763 6.772 6.781 6.789 6.798 6.807 6.815 6.824 6.833 7.629 7.638 7.647 7.655 7. 664 7.673 7.681 7.690 7.699 8.495 .82 .83 8.504 8.513 .84 8.521 .85 8.530 .86 8.539 .87 8.547 .88 .89 8.556 8.565 .90.... .779 1.645 2.511 3.377 4.243 5.109 5.975 6.841 7.707 8.573 .91 .788 .797 .805 .814 .823 .831 .840 .849 .857 1.654 1.663 1.671 1.680 1.689 1.697 1.706 1.715 1.723 2.520 2.529 2.537 2.546 2. 555 2.563 2.572 2.581 2.589 3.386 3.395 3.403 3.412 3.421 3.429 3.438 3.447 3.455 4.252 4.261 4.269 4.278 4.287 4.295 4.304 4.313 4.321 5.118 5.127 5.135 5.144 5. 153 5.161 5.170 5.179 5.187 5.984 5.993 6.001 6.010 6.019 6.027 6.036 6.045 6.053 6.850 6.859 6.867 6.876 6.885 6.893 6.902 6.911 6.919 7.716 7.725 7.733 7.742 7.751 7.759 7.768 7.777 7.785 8.582 .92 . . .93 8.591 8.599 .94 .95 8.608 8.617 .96 8.625 .97 .96 8.634 8.643 .99 8.651 1.00.... .866 1.732 2.598 3.464 4.330 5.196 6.062 6.928 7.794 8.660 1 2 3 4 5 6 7 8 9 226 U. S. COAST AND GEODETIC STJRVEY Table 19.— Products for Form 194— Continued [Multiplier=sin 75°=0.966] 1 2 3 4 5 6 7 8 9 0.00 0.000 0.966 1.932 2.898 3.864 4.830 5.796 6.762 7.728 8.694 .01 .010 .019 .029 .039 .048 .058 .068 .077 .087 .976 .985 .995 1.005 1.014 1.024 1.034 1.043 1.053 1.942 1.951 1.961 1.971 1.980 1.990 2.000 2.009 2.019 2.908 2.917 2.927 2.937 2.946 2. 956 2.966 2.975 2.985 3.874 3.883 3.893 3.903 3.912 3.922 3.932 3.941 3.951 4.840 4.849 4.859 4.869 4.878 4.888 4.898 4.907 4.917 5.806 5.815 5.825 5.835 5.844 5.854 5.864 5.873 5.883 6.772 6.781 6.791 6.801 6.810 6.820 6.830 6.839 6.849 7.738 7.747 7.757 7.767 7.776 7.786 7.796 7.805 7.815 8.704 .02.-. 8.713 .03 .04 8.723 8.733 .05 8.742 .06 8.752 .07 .;. . 8.762 .08 ... 8.771 .09... 8.781 .10 .097 1.063 2.029 2.995 3.961 4.927 5.893 6.859 7.825 8.791 .11 .12 .106 .116 .126 .135 .145 .155 .164 .174 .184 1.072 1.082 1.092 1.101 1.111 1.121 1.130 1.140 1.150 2.038 2.048 2.058 2.067 2.077 2.087 2.096 2.106 2.116 3.004 3.014 3.024 3.033 3.043 3.053 3.062 3.072 3.082 3.970 3.980 3.990 3.999 4.009 4.019 4.028 4.038 4.048 4.936 4.946 4.956 4.965 4.975 4.985 4.994 5.004 5.014 5.902 5,912 5.922 5.931 5.941 5.951 5.960 5.970 5.980 6.868 6.878 6.888 6. 897 6.907 6. 917 6.926 6.936 6.946 7.834 7.844 7.854 7.863 7.873 7.883 7.892 7.902 7.912 8.800 8.810 .13 8.820 .14 8.829 .15. .. 8.839 .16 .17... 8.849 8.858 .18 8.868 .19 8.878 .20 .193 1.159 2.125 3.091 4.057 5.023 5.989 6.955 7.921 8.887 .21 .203 .213 .222 .232 .242 .251 .261 .270 .280 1.169 1.179 1.188 1.198 1.208 1.217 1.227 1.236 1.246 2.135 2.145 2.154 2.164 2.174 2.183 2.193 2.202 2.212 3. 101 3.111 3.120 3.130 3.140 3.149 3.159 3.168 3.178 4.067 4. 077 4.086 4.096 4.106 4.115 4.125 4.134 4.144 5.033 5.043 5.052 5.062 5.072 5.081 5.091 5.100 5.110 5.999 6.009 6.018 6.028 6. 038 6.047 6.057 6.066 6.076 6.965 6.975 6.984 6.994 7.004 7.013 7.023 7.032 7.042 7.931 7.941 7.950 7.960 7.970 7.979 7.989 7.998 8.008 8.897 .22 8.907 .23 .24 .. 8.916 8. 926 .25 8. 936 .26 8.945 .27 .28 8.955 8.964 .29 . 8.974 .30 i .290 1.256 2.222 3.188 4.154 5.120 6.086 7.052 8.018 8.984 .81 .299 .309 .319 .328 .338 .348 .357 .367 .377 1.265 1.275 1.285 1.294 1.304 1.314 1.323 1.333 1.343 2.231 2.241 2.251 2.260 2.270 2.280 2.289 2.299 2.309 3.197 3.207 3.217 3.226 3.236 3.246 3.255 3.265 3.275 4.163 4.173 4.183 4.192 4.202 4.212 4.221 4.231 4.241 5.129 5.139 5.149 5.158 5.168 5.178 5. 187 ' 5.197 5.207 6.095 6.105 6.115 6.124 6.134 6.144 6.153 6.163 6.173 7.061 7.071 7.081 7.090 7.100 7.110 7.119 7.129 7.139 8.027 8.037 8.047 8.056 8.066 8.076 8.085 8.095 8.105 8.993 .32 .33 9.003 9.013 .34 9.022 .85 9.032 .86 9.042 .37 .38 9.051 9.061 .39 9.071 .40 .386 1.352 2.318 3.284 4.250 5.216 6.182 7.148 8.114 9.080 .41 .396 .406 .415 .425 .435 .444 .454 .464 .473 1.362 1.372 1.381 1.391 1.401 1.410 1.420 1.430 1.439 2.328 2.338 2.347 2.357 2. 367 2.376 2.386 2.396 2.405 3.294 3.304 3. 313 3.323 3.333 3. 342 3. 352 3. 362 3.371 4.260 4.270 4.279 4.289 4.299 4.308 4.318 4.328 4.337 5.226 5.236 5.245 5.255 5. 265 5.274 5.284 5. 294 5.303 6.192 6.202 6.211 6.221 6.231 6.240 6.250 6. 260 6.269 7.158 7.168 7.177 7.187 7.197 7.206 7.216 7.226 7.235 8.124 8.134 8.143 8.153 8.163 8.172 8.182 8.192 8.201 9.090 .42 9.100 .43 9.109 .44 9.119 .45 9.129 .46... 9.138 .47 9.148 .48... 9.158 .49... 9.167 .50 .483 1,449 2.415 3.381 4.347 5.313 6. 279 7.245 8.211 9.177 1 2 3 4 5 6 7 8 9 HARMONIC ANALYSIS AND PREDICTION OF TIDES 227 Table 19. — Products for Form 194 — Continued [ Multiplier = sin 75° =0.966] 1 2 3 4 5 6 7 8 9 0.50 - 0.483 1.449 2.415 3.381 4.347 5.313 6.279 7.245 8.211 9.177 .51 .493 .502 .512 .522 .531 .541 .551 .560 .570 1.459 1.468 1.478 1.488 1.497 1.507 1.517 1.526 1.536 2.425 2.434 2.444 2.454 2.463 2.473 2.483 2.492 2.502 3.391 3.400 3.410 3.420 3.429 3.439 3.449 3.458 3.468 4.357 4.366 4.376 4.386 4.395 4.405 4.415 4.424 4.434 5.323 5.332 5.342 5.352 5. 361 5.371 5.381 5.390 5.400 6.289 6.298 6.308 6. 318 6.327 6.337 6.347 6.356 6.366 7.255 7.264 7.274 7.284 7.293 7.303 7.313 7.322 7.332 8.221 8. 230 8.240 8.250 8.259 8.269 8.279 8.288 8.298 9.187 .52 9. 196 .53 9.206 .54 9.216 .55 - 9.225 .56 9.235 .57 9.245 .53 9.254 .59 9.264 .60 .580 1.546 2.512 3.478 4.444 5.410 6.376 7.342 8.308 9.274 .81 .589 .599 .609 .618 .628 .638 .647 .657 .667 1.555 1.565 1.575 1.584 1.594 1.604 1.613 1.623 1.633 2.521 2.531 2.541 2.550 2.560 2.570 2.579 2.589 2.599 3.487 3.497 3.507 3.516 3.526 3.536 3. 545 3.555 3.565 4.453 4.463 4.473 4.482 4.492 4.502 4.511 4.521 4.531 5.419 5.429 5. 439 5.448 5.458 5. 468 5.477 5.487 5.497 6.385 6.395 6.405 6.414 0. 424 6.434 6.443 6.453 6.463 7.351 7.361 7.371 7.380 7.390 7.400 7.409 7.419 7.429 8.317 8.327 8.337 8.346 8.356 8.366 8.375 8.385 8.395 9. 283 .62 9.293 .63 9.303 .64 9.312 .65 9.322 .66 9.332 .67 9.341 .68 9.351 .69 9.361 .70 .676 1.642 2.608 3.574 4.540 5.506 6.472 7.438 8.404 9.370 .71 .686 .696 .705 .715 .724 .734 .744 .753 .763 1.652 1.662 1.671 1.681 1.690 1.700 1.710 1.719 1.729 2.618 2.628 2.637 2.647 2.656 2.666 2.676 2.685 2.695 3.584 3.594 3.603 3.613 3.622 3.632 3.642 3.651 3.661 4.550 4.560 4.569 4.579 4.588 4. 598 4.608 4.617 4.627 5.516 5. 526 5.535 5.545 5.554 5.564 5.574 5.583 5.593 6.482 6.492 6.501 6.511 6.520 6.530 6.540 6.549 6.559 7.448 7.458 7.467 7.477 7.486 7.496 7.506 7.515 7.525 8.414 8.424 8.433 8.443 8.452 8.462 8.472 8.481 8.491 9.380 .72 . 9.390 .73 9.399 .74 9.409 .75.. .76 9.418 9.428 .77 9.438 .78 9.447 .79 9.457 .80 .773 1.739 2.705 3.671 4.637 5. 603 6.569 7.535 8.501 9.467 81. - .782 .792 .802 .811 .821 .831 .840 .850 .860 1.748 1.758 1.768 1.777 1.787 1.797 1.806 1.816 1.826 2.714 2.724 2.734 2.743 2.753 2.763 2.772 2.782 2.792 3.680 3.690 3.700 3.709 3.719 3.729 3.738 3.748 3.758 4.646 4.656 4.666 4.675 4.685 4.695 4.704 4.714 4.724 5.612 5.622 5.632 5.641 5. 651 5.661 5.670 5.680 5.690 6.578 5. 588 6.598 6.607 6.617 6.627 6.636 6.646 6.656 7.544 7. 554 7. 564 7.573 7.583 7.593 7.602 7.612 7.622 8.510 ' 8. 520 8.530 8.539 8.549 8.559 8.568 8.578 8.588 9.476 .82 9.486 .83 9.496 .84 9.505 .85 9.515 .86 9.525 .87.. 9.534 .88.. 9.544 .89 9.554 .90 .869 1.835 2.801 3.767 4.733 5.699 6.665 7.631 8.597 9.563 .91. .879 .889 .898 .908 .918 .927 .937 .947 .956 1.845 1.855 1.864 1.874 1.884 1.893 1.903 1.913 1.922 2.811 2.821 2.830 2.840 2.850 2. 859 2.869 2.879 2.888 3.777 3.787 3.796 3.806 3.816 3.825 3.835 3.845 3.854 4.743 4.753 4.762 4.772 4.782 4.791 4.801 4.811 4.820 5.709 5.719 5.728 5.738 5.748 5.757 5.767 5.777 5.786 6. 675 6.685 6.694 6.704 6.714 6.723 6.733 6.743 6.752 7.641 7.651 7.660 7.670 7.680 7.689 7.699 7.709 7.718 8.607 8.617 8.626 8.636 8.646 8.655 8.665 8.675 8.684 9. 573 .92. .93. 9.583 9.592 .94 9.602 .95 9.612 .96 9.621 .97 .... .... 9.631 .98. . 9.641 .99... 9.650 1.00 .966 1.932 2.898 3. 864 4.830 5.796 6.762 7.728 8.694 9.660 1 2 3 4 5 6 7 8 9 228 U. S. COAST AND GEODETIC SURVEY Table 20. — Augmenting factors SHORT-PERIOD CONSTITUENTS,* FORMULA (308) Augment- ing factor Logarithm Remarks Diurnal Ji, Ki, Mi, Oi, 00, Pi, Qi, 2Q, pi Semidurnal K 2 , L 2 , M 2 , N 2 , 2N, R 2 , T 2 , X 2 , m, f2, 2SM. Terdiurnal M 3 , MK, 2MK... Quarter-diurnal M i} MN, MS Sixth-diurnal Me Eighth-diurnal Ms- 1. 0029 1. 0115 1. 0262 1. 0472 1. 1107 1. 2092 0. 001241 0. 004972 0. 011220 0. 020029 0. 045605 0. 082498 Each tabulated solar hourly height used once and once only in summa- tion; group covers one constituent hour; constituent day represented by 24 means. SHORT-rERIOD CONSTITUENTS,* FORMULA (309) Augment- ing factor Logarithm Augment- ing factor Logarithm Remarks Jl— 1. 0031 0. 00134 Pi- - 1. 0028 0. 00123 Ki 1. 0029 0. 00125 Qi 1. 0023 0. 00099 K 2 1. 0116 0. 00500 2Q 1. 0021 0. 00091 L 2 „ 1. 0112 0. 00482 R 2 1. 011.5 0. 00499 Mi 1. 0027 0. 00116 T 2 1. 0115 0. 00496 Mj 1. 0107 0. 00464 \2 1. 0111 0. 00479 Each constituent hour of observa- tion period receives one and M 3 1. 0244 0. 01047 M2 1. 0100 0. 00432 ( only one of solar hourly heights M 4 1. 0440 0. 01868 VI 1. 0104 0. 00449 in the summation; group covers M« 1. 1028 0. 04251 pi 1. 0023 0. 00100 one solar hour; each constituent day represented by 24 means. M 8 - 1. 1934 0. 07680 MK 1. 0250 0. 01074 Na 1. 0103 0. 00447 2MK— . 1. 0238 0. 01021 2N 1. 0099 0. 00430 MN 1. 0431 0. 01833 Oi. 1. 0025 0. 00107 MS 1. 0456 0. 01935 OO 1. 0033 0. 00144 2SM..__ 1. 0123 0. 00532 LONG-PERIOD CONSTITUENTS, FORMULA (403) Mm ML. MSf Sa__ Augment- ing factor 1. 0050 1. 0205 1. 0192 1. 0029 1.0115 Logarithm 0. 00218 0. 00880 0. 00825 0. 00124 0. 00497 Remarks Daily sums used as units in the summation for the divisional means, and all daily sums tised; constituent month for Mm, Mf, MSf, and constituent year for Sa and Ssa represented by 24 means. ANNUAL AND SEMIANNUAL CONSTITUENTS, FORMULA (404) Augment- ing factor Logarithm Remarks Sa 1. 0115 1.0472 0. 00497 0. 02003 [For analysis of 12 monthly means. Ssa 'For constituents Si, S 2 , S3, etc., augmenting factor is unity. HARMONIC ANALYSIS AND PREDICTION OF TTDES 229 Table 21. — Acceleration in epoch of K t due to Pi [Argument h—\v' refers to beginning of series] \ Series 14 days 29 days 58 days 87 days 105 days 134 days 163 days 192 days 221 days 250 days 279 days 297 days 326 days o o o o o o 180 +6.5 +11.4 +14.6 +12.6 +10.1 +5.1 +0.9 +0.2 +2.4 +3.9 +3.9 +3.3 +1.6 10 20 30 190 200 210 +13.9 +17.9 +19.0 +16.4 +18.3 +17.6 +16.0 +15.3 +12.9 +12.0 +9.9 +6.7 +8.8 +6.4 +3.1 +3.4 +1.0 -1.6 -0.1 -1.0 -1.7 +0.7 +0.9 +1.0 +3.0 +3.2 +2.9 +4.1 +3.7 +3.0 +3.6 +2.8 +1.7 +2.7 +1.7 +0.5 +0.9 0.0 -1.0 40 50 60 220 230 240 +17.6 +14.7 +10.8 +15.2 +11.7 +7.4 +9.4 +5.2 +0.7 +2.8 -1.2 -5.2 -0.5 -4.1 -7.2 -3.8 -5.4 -6.0 -2.1 -2.1 -1.9 +1.0 +0.8 +0.6 +2.4 +1.7 +0.9 +2.0 +0.8 -0.4 +0.3 -1.0 -2.2 -0.8 -2.0 -2.9 -1.7 -2.0 -2.1 70 80 90 250 260 270 +6.4 +1.5 -3.5 +2.7 -2.2 -6.9 -3.9 -8.2 -12.0 -8.7 -11.3 -12.5 -9.3 -10.2 -9.7 -5.8 -5.0 -3.8 -1.5 -1.0 -0.5 +0.4 +0.1 -0.1 +0.1 -0.8 -1.6 -1.5 -2.6 -3.5 -3.2 -3.8 -3.9 -3.4 -3.3 -2.9 -1.9 -1.5 -1.1 100 110 120 280 290 300 -8.2 -12.5 -16.0 -11.3 -14.9 -17.5 -14.7 -16.0 -15.2 -12.1 -10.1 -6.9 -8.1 -5.6 -2.7 -2.3 -0.7 +1.0 0.0 +0.6 +1.1 -0.4 -0.6 -0.8 -2.3 -2.8 -3.1 -4.0 -4.0 -3.5 -3.5 -2.7 -1.6 -2.2 -1.3 -0.4 -0.5 0.0 +0.6 130 140 150 310 320 330 -18.4 -18.9 -16.8 -18.3 -16.7 -12.1 -12.2 -7.2 -1.0 -2.9 +1.3 +5.4 +0.5 +3.6 +6.4 +2.6 +4.1 +5.2 +1.5 +1.9 +2.1 -1.0 -1.0 -0.9 -3.1 -2.5 -1.5 -2.5 -1.1 +0.5 -0.3 +0.9 +2.1 +0.7 +1.6 +2.5 +1.1 +1.6 +1.9 160 170 180 340 350 360 -11.3 -2.8 +6.5 -4.7 +3.9 +11.4 +5.4 +10.9 +14.6 +8.9 +11.5 +12.6 +8.6 +10.0 +10.1 +5.9 +5.9 +5.1 +2.0 +1.7 +0.9 -0.7 -0.3 +0.2 -0.1 +1.2 +2.4 +1.9 +3.1 +3.9 +3.1 +3.8 +3.9 +3.1 +3.4 +3.3 +2.1 +2.0 +1.6 Table 22. — Ratio of increase in amplitude of Ki due to Pi [Argument h— \v' refers to beginning of series] Series •V 180 190 200 210 220 230 240 250 260 270 290 300 310 320 330 340 350 360 14 days -0.31 -0.25 -0.15 -0.04 +0.07 +0.17 +0.25 +0.30 +0.33 +0.32 +0.28 +0.22 +0.13 +0.03 -0 -0.19 -0.28 -0.32 -0.31 days -0.26 -0.17 -0.06 +0.04 +0.14 +0.23 +0.28 +0.31 +0.32 +0.29 +0.23 +0.15 +0.05 -0.06 -0.16 -0.25 -0.30 -0.31 -0.26 -0.12 -0.02 +0.07 +0.16 +0.23 +0.27 +0.29 +0.28 +0.24 +0.18 +0.10 +0.01 -0.09 -0.17 -0.23 -0.26 -0.25 -0. 19 -0.12 87 days +0.01 +0.09 +0.16 +0.20 +0.23 +0.24 +0.22 +0.18 +0.12 +0.04 -0.03 -0.10 -0.15 -0.18 -0.19 -0.17 -0.12 -0; 06 +0.01 105 days +0.06 +0.12 +0.17 +0.20 +0.21 +0.19 +0.15 +0.10 +0.05 -0.02 -0.07 -0.11 -0.14 -0.14 -0.13 -0.10 -0.05 0.00 +0.06 134 days +0.09 +0.12 +0.13 +0.13 +0.12 +0.09 +0.05 +0.02 -0.01 -0.04 -0.06 -0.07 -0.07 -0.06 -0.04 -0.01 +0.03 +0.06 +0.09 163 days +0.06 +0.07 +0.06 +0.05 +0.04 +0.03 +0.02 +0.01 0.00 0.00 -0.01 0.00 0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 192 days +0.01 +0.01 +0.02 +0.02 +0.03 +0.03 +0.04 +0.04 +0.04 +0.04 +0.04 +0.04 +0.03 +0.03 +0.02 +0.02 +0.01 +0.01 +0.01 221 days -0.02 0.00 +0.02 +0.04 +0.05 +0.06 +0.07 +0.08 +0.08 +0.07 +0.06 +0.04 +0.02 0.00 -0.01 -0.02 -0.03 -0.03 -0.02 250 days 0.00 +0.02 +0.05 +0.07 +0.08 +0.09 +0.09 +0.09 +0.08 +0.06 +0.03 +0.01 -0.01 -0.03 -0.04 -0.04 -0.03 -0.02 0.00 279 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 +0.05 326 days +C.05 +0.06 +0.07 +0.06 +0.05 +0.04 +0.03 +0.02 +0.01 0.00 0.00 -0.01 0.00 0.00 +0.01 +0.02 +0.03 +0.04 +0.05 230 IT. S. COAST AND GEODETIC SURVEY Table 23. — Acceleration in epoch of S 2 due to K 2 [Argument h—v" refers to beginning of series] s h—v" Series 15 days 29 days 58 days 87 days 105 days 134 days 163 days 192 days 221 days 250 days 279 days 297 days 326 days o o o o o o o o o 180 +3.2 +5.9 +10.1 +10.4 +8.0 +3.2 +0.4 +0.1 +1.3 +2.9 +3.2 +2.4 +0.9 10 20 30 190 200 210 +7.2 +10.3 +13.7 +9.6 +12.6 +14.6 +12.3 +13.2 +12.5 +10.0 +8.4 +5.7 .+6.7 +4.7 +2.3 +2.0 -0.6 -0.9 0.0 -0.5 -0.9 +0.3 +0.5 +0.7 +1.9 +2.4 +2.6 +3.3 +3.3 +2.9 +2.9 +2.2 +1.3 +1.9 +1.1 +0.3 +0.5 0.0 -0.5 40 50 60 220 230 240 +15.4 +15.4 +13.2 +15.0 +13.5 +9.6 +9.9 +5.8 +0.8 +2.5 -1.1 -4.5 -0.4 -3.0 -5.4 -2.2 -3.4 -4.4 -1.3 -1.6 -1.7 +0.8 +0.8 +0.7 +2.5 +2.0 +1.2 +2.0 +0.9 -0.4 +0.3 -0.8 -1.8 -0.6 -1.4 -2.1 -1.0 -1.3 -1.6 70 80 90 250 260 270 +8.6 +1.9 -5.5 +3.7 -3.0 -9.1 -4.4 -8.8 -11.9 -7.4 -9.5 -10.4 -7.2 -8.3 -8.3 -4.9 -4.9 -4.2 -1.7 -1.3 -0.7 +0.5 +0.2 -0.2 +0.1 -1.0 -1.9 -1.6 -2.6 -3.2 -2.6 -3.1 -3.3 -2.6 -2.8 -2.7 -1.7 -1.6 -1.3 100 110 120 280 290 300 -11.2 -14.6 -15.6 -13.2 -15.0 -14.7 -13.2 -12.7 -10.9 -9.9 -8.2 -5.6 -7.3 -5.2 -2.6 -2.7 -0.8 +1.2 0.0 +0.8 +1.4 -0.5 -0.7 -0.8 -2.4 -2.6 -2.4 -3.4 -3.1 -2.5 -3.0 -2.3 -1.4 -2.2 -1.4 -0.4 -0.7 0.0 +0.8 130 140 150 310 320 330 -14.7 -12.4 -9.1 -12.9 -10.0 -6.4 -8.0 ■ -4.4 -0.6 -2.4 +1.0 +4.4 +0.5 +3.4 +5.9 +3.1 +4.4 +4.9 +1.7 +1.7 +1.6 -0.8 -0.7 -0.5 -2.0 -1.5 -0.8 -1.7 -0.7 +0.3 -0.3 +0.8 +1.9 +0.7 +1.6 +2.4 +1.3 +1.7 +1.7 160 170 180 340 350 360 -5.3 -1.1 +3.2 -2.3 +1.9 +5.9 +3.3 +7.0 +10.1 +7.3 +9.4 +10.4 +7.7 +8.4 +8.0 +4.8 +4.2 +3.2 +1.3 +0.9 +0.4 -0.3 -0.1 +0.1 -0.1 +0.7 +1.3 +1.3 +2.2 +2.9 +2.7 +3.2 +3.2 +2.8 +2.8 +2.4 +1.6 +1.3 +0.9 Table 24. — Ratio of increase in amplitude of S 2 due to K 2 [Argument h—v" refers to beginning of series] h— V" Series 15 days 29 days 58 days 87 days 105 days 134 days 163 days 192 days 221 days 250 days 279 days 297 days 326 days 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 20 30 190 200 210 +0. 23 +0.18 +0.10 +0.19 +0.12 +0.04 +0.08 0.00 -0.08 -0.03 -0.09 -0.13 -0.06 -0.10 -0.12 -0.05 -0.06 -0.06 -0.01 -0.01 0.00 +0.03 +0.03 +0.02 +0.04 +0.03 +0.01 +0.02 0.00 -0.01 0.00 -0.02 -0.03 -0.01 -0.02 -0.03 -0.01 -0.01 -0.01 40 50 60 220 230 240 +0.01 -0.08 -0.17 -0.05 -0.14 -0.21 -0.15 -0.19 -0.21 -0.15 -0.16 -0.14 -0.13 -0.11 -0.09 -0.05 -0. 03 -0.01 0.00 +0.01 +0.02 +0.02 +0.02 +0.01 0.00 -0.01 -0.02 -0.03 -0.04 -0.04 -0.04 -0.03 -0.02 -0.03 -0.02 -0.01 0.00 0.00 +0.01 70 80 90 250 260 270 -0.23 -0.27 -0.25 -0.25 -0.25 -0.21 -0.20 -0.16 -0.10 -0.10 -0.05 +0.01 -0.05 0.00 +0.05 +0.02 +0.05 +0.08 +0.03 +0.04 +0.05 +0.01 0.00 0.00 -0.03 -0.02 -0.01 -0.03 -0.02 0.00 -0.01 0.00 +0.02 0.00 +0.02 +0.04 +0.02 +0.03 +0.04 100 110 120 280 290 300 -0.20 -0.13 -0.03 -0.15 -0.06 +0.03 -0.02 +0.05 +0.13 +0.07 +0.12 +0.17 +0.10 +0.14 +0.16 +0.10 +0.11 +0.11 +0.05 +0. 05 +0.04 +0.01 +0.01 +0.01 0.00 +0.01 +0.03 +0.02 +0.04 +0.05 +0.04 +0.06 +0.07 +0.05 +0.06 +0.07 +0.05 +0.05 +0.05 130 140 150 310 320 330 +0.06 +0.14 +0.21 +0.11 +0.18 +0.23 +0.18 +0.22 +0.24 +0.19 +0.19 +0.18 +0.17 +0.15 +0.13 +0.09 +0.07 +0.04 +0.03 +0.02 +0.01 +0.02 +0.02 +0.03 +0.04 +0. 05 +0.06 +0.07 +0.07 +0.07 +0.08 +0.07 +0.06 +0.07 +0.06 +0.05 +0.04 +0.03 +0.02 160 170 180 340 350 360 +0.25 +0.27 +0.26 +0.26 +0.26 +0.24 +0.23 +0.20 +0.15 +0.14 +0.09 +0.03 +0.08 +0.03 -0. 02 +0.01 -0.02 -0.04 0.00 0.00 -0.01 +0.03 +0.03 +0.03 +0.06 +0.06 +0. 05 +0.07 +0.06 +0.04 +0.05 +0.03 +0.01 +0.03 +0.02 0.00 +0.01 0.00 0.00 HARMONIC ANALYSIS AND PREDICTION OF TIDES 231 Table 25. — Acceleration in epoch of S 2 due to T 3 [Argument h—pi reiers to beginning of series] \Series. \ \ h-pi\ 15 days 29 days 58 days 87 days 105 days 134 days 163 days 102 days 221 days 250 days 279 days 297 days 326 days o o c c o o o -0.4 -0.8 -1.6 -2.0 -2.2 -2.4 -2.3 -2.0 -1.5 -1.1 -0.6 -0.4 -0.1 10 20 30 -1.0 —1.6 -2.0 -1.3 -1.8 -2.2 -1.9 -2.2 -2.7 -2.4 -2.7 -2.9 -2.5 -2.7 -2.9 -2.6 -2.7 —2. 7 -2.4 -2.3 -2.2 -2.0 -1.8 -1.7 -1.4 -1.3 -1.1 -0.9 -0.7 -0.6 -0.5 -0.3 -0.2 -0.3 -0.2 0.0 -0 1 0.0 +0.1 40 50 60 -2.4 -2.7 -3.0 -2.6 -2.9 -3.2 -°..o -3.1 -3.2 -3.0 -3.1 -3.0 -2.9 -2.9 -2.8 -2.6 -2 4 -2.2 -2.0 -1.8 -1.5 -1.4 -1.2 -0.9 -0.8 -0.6 -0.3 -0.4 -0.2 +0.1 0.0 +0.1 +0 3 +0.1 +0.2 +0.3 +0.1 +0.2 +0.2 70 80 90 -3.2 -3.4 -3.4 -3 3 -3.3 -3.3 -3.2 -3.1 -2.9 -2.9 -2.6 -2.3 -2.5 -2.2 -1.9 -1.9 -1.5 -1.1 -1.2 -0.8 -0 4 -0.5 -0.2 +0.2 0.0 +0.3 +0.5 +0.3 +0.5 +0.7 +0.4 +0.6 +0.7 +0.4 +0.5 +0.6 +0.3 +0.3 +0.4 100 110 120 -3 3 -3.1 -2.8 -3.1 -2.8 -2.5 -2.6 -2.2 -1.8 -1.9 -1.4 -0.9 -1.4 -1.0 -0.4 -0.7 -0.3 +0.3 0.0 +0.4 +0.8 +0. 5 +0.8 +1.1 +0.8 +1.0 +1.2 +0.9 +1.0 +1.1 +0.8 +0.9 +0.9 +0.6 +0.7 +0.7 +0.4 +0.4 +0.4 130 140 150 -2.4 -1.9 -1.4 -2.0 -1.5 -0.9 -1.2 -0.7 -0.1 -0.4 +0.2 +0.7 +0.1 +0.6 +1.1 +0.8 + 1.2 +1.6 +1.2 +1.5 +1.8 +1.4 +1.6 +1.8 +1.4 +1.5 +1.6 +1.2 +1.3 +1.3 +0.9 +0.9 +0.9 +0.7 +0.7 +0.7 +0.4 +0.3 +0.3 160 170 180 -0.8 -0.2 +0.5 -0.3 +0.3 +0.9 +0.5 +1.1 +1.6 +1.2 +1.7 +2.2 +1.6 +2.0 +2.3 +1.9 +2.2 +2.5 +2.1 +2.2 +2.3 +2.0 +2.0 +2.0 +1.7 +1.6 + 1.6 +1.3 +1.2 +1.1 +0.8 +0.8 +0.7 +0.6 +0.5 +0.4 +0.3 +0.2 +0.1 190 200 210 +1.1 +1.6 +2.1 +1.5 +2.0 +2.4 +2.1 +2.5 +2.8 +2.5 +2.8 +3.0 +2.6 +2.8 +3.0 +2.6 +2.7 +2.6 -12.4 +2.3 +2.2 +2.0 +1.8 +1.7 +1.5 +1.3 +1.1 +1.0 +0.8 +0.6 +0.5 +0.4 +0.2 +0.3 +0.2 0.0 +0.1 0.0 -0.1 220 230 240 +2.6 +2.9 +3.2 +2.8 +3.1 +3.3 +3.1 +3.2 +3.2 +3.1 +3.0 +3.0 +2.9 +2.9 +2.7 +2.5 +2.4 +2.1 +2.0 +1.8 +1.5 +1.5 +1.2 +0.9 +0.9 +0.6 +0.3 +0.4 +0.2 -0.1 0.0 -0.1 -0.3 -0.1 -0.2 -0.3 -0.1 -0.2 -0.3 250 260 270 +3.3 +3.4 +3.3 +3.3 +3.3 +3.2 +3.2 +3.0 +2.8 +2.8 +2.5 +2.2 +2.5 +2.1 +1.8 +1.8 +1.5 +1.1 +1.2 +0.8 +0.4 +0.5 +0.2 -0.2 0.0 -0.3 -0.6 -0.3 -0.5 -0.7 -0.4 -0.6 -0.7 -0.4 -0.5 -0.6 -0.3 -0.3 -0.4 280 290 300 +3.2 +2.9 +2.6 +3.0 +2.7 +2.3 +2.5 +2.1 +1.6 +1.8 +1.3 +0.9 +1.4 +0.9 +0.4 +0.6 +0.2 -0.3 0.0 -0.4 -0.8 -0.5 -0.8 -1.1 -0.8 -1.1 -1.3 -0.9 -1.1 -1.2 -0.8 -0.9 -0.9 -0.7 -0.7 -0.7 -0.4 -0.4 -0.4 310 320 330 +2.2 +1.7 +1.2 + 1.9 +1.4 +0.8 +1.1 +0. 6 +0.1 +0.4 -0.2 -0.7 -0.1 -0.6 -1.0 -0.7 -1.1 -1.5 -1.2 -1.5 -1.8 -1.4 -1.7 -1.8 -1.4 -1.5 -1.6 -1.3 -1.3 -1.3 -0.9 -0.9 -0.9 -0.7 -0.7 -0.6 -0.4 -0.3 -0.3 340 350 360 +0.7 +0.1 -0.4 +0.3 -0.2 -0.8 -0.5 -1.0 -1.5 -1.1 -1.6 -2.0 -1.5 -1.9 -2.2 -1.9 -2.2 -2.4 -2.0 -2.2 -2.3 -2.0 -2.0 -2.0 -1.7 -1.6 -1.5 -1.3 -1.2 -1.1 -0.8 -0.7 -0.6 -0.6 -0.5 -0.4 -0.2 -0.2 -0.1 232 IT. S. COAST AXD GEODETIC SURVEY Table 26. — Resultant amplitude of S 2 due to T 2 [Argument h—pi refers to beginning of series] \Series \ ft-p\ 15 days 29 days 58 days 87 days 105 days 134 days 163 days 192 days 221 days 250 days 279 days 297 days 326 days 1.06 1.06 1.05 1.04 1.03 1.02 1.01 1.00 0.99 0.99 0.99 0.99 0.99 10 20 30 1.06 1.05 1.05 1.05 1.05 1.04 1.05 1.04 1.03 1.03 1.03 1.02 1.03 1.02 1.01 1.01 1.00 1.00 1.00 0.99 0.99 0.99 0.99 0.98 0.99 0.98 0.98 0.99 0.98 0.98 0.99 0.99 0.98 0.99 0.99 0.99 0.99 0.99 0.99 40 50 60 1.04 1.03 1.02 1.03 1.03 1.02 1.02 1.01 1.00 1.01 1.00 0.99 1.00 0.99 0.98 0.99 0.98 0.97 0.98 0.97 0.97 0.98 0.97 0.97 0.98 0.97 0.97 0.98 0.98 0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.99 1.00 1.00 70 80 90 1.01 1.00 1.00 1.01 1.00 0.99 0.99 0.98 0.97 0.98 0.97 0.96 0.97 0.97 0.96 0.97 0.96 0.96 0.96 0.96 0.96 0.97 0.97 0.97 0.97 0.97 0.97 0.98 0.98 0.98 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00 1.00 100 110 120 0.99 0.98 0.97 0.98 0.97 0.96 0.97 0.96 0.95 0.96 0.95 0.95 0.96 0.95 0.95 0.96 0.95 0.95 0.96 0.96 0.96 0.97 0.97 0.97 0.98 0.98 0.98 0.98 0.99 0.99 0.99 0.99 loo; 1.00 1.00 1.00 1.00 1.00 1.00 130 140 150 0.96 0.95 0.95 0.95 0.95 0.94 0.95 0.94 0.94 0.95 0.95 0.95 0.95 0.95 0.95 0.96 0.96 0.96 9.97 0.97 0.97 0.98 0.98 0.99 0.99 0.99 1.00 0.99 0.99 1.00 1.00 1.00 1.01 1.00 1.00 1.01 1.00 1.00 1.01 160 170 180 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.95 0.95 0.95 0.96 0.96 0.96 0.96 0.97 0.97 0.97 0.98 0.98 0.99 0.99 0.99 1.00 1.00 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 190 200 210 0.95 0.95 0.96 0.95 0.95 0.96 0.96 0.96 0.97 0.97 0.98 0.99 0.98 0.99 0.99 0.99 1.00 1.01 1.00 1.01 1.02 1.01 1.02 1.02 1.01 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.01 1.01 220 230 240 0.96 0.97 0.98 0.97 0.98 0.99 0.98 0.99 1.00 1.00 1.00 1.01 1.00 1.01 1.02 1.01 1.02 1.03 1.02 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.03 1.02 1.02 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.01 1.01 250 260 270 0.99 1.00 1.01 1.00 1.01 1.02 1.01 1.02 1.03 1.02 1.03 1.04 1.03 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.03 1.03 1.03 1.02 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.01 1.01 1.00 1.00 280 290 300 1.02 1.03 1.04 1.03 1.03 1.04 1.04 1.04 1.05 1.04 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.04 1.04 1.04 1.04 1.03 1.03 1.03 1.02 1.02 1.02 1.01 1.01 1.01 1.01 1.00 1.01 1.00 1.00 1.00 1.00 1.00 310 320 330 1.04 1.05 1.06 1.05 1.05 1.06 1.05 1.06 1.06. 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.04 1.04 1.04 1.03 1.03 1.03 1.02 1.02 1.02 1.01 1.01 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 340 350 360 1.06 1.06 1.06 1.06 1.06 1.06 1.06 1.05 1.05 1.05 1.05 1.04 1.05 1.04 1.03 1.03 1.03 1.02 1.02 1.02 1.01 1.01 1.00 1.00 1.00 1.00 0.99 1.00 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00 0.99 HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 27. — Critical logarithms for Form 245 233 Natural Loga- Natural Loga- Natural Loga- Natural Loga- 1 Natural Loga- number rithm number rithm number rithm number rithm number rithm 0.000 0.050 .051 8. 6947 8. 7033 0.100 .101 8. 9979 9. 0022 0.150 .151 9.1747 9.1776 0.200 .201 9. 3000 .001 ~""O990~ 9. 3022 .002 7. 1761 .052 8.7119 .102 9.0065 .152 9.1805 ; .202 9. 3043 .003 7. 3980 .053 8. 7202 .103 9. 0108 .153 9.1833 .203 9. 3065 .004 7. 5441 .054 8. 7284 .104 9.0150 .154 9. 1862 , .204 9.3086 .005 7. 6533 .055 8. 7365 .105 9. 0192 .155 9. 1890 .205 9. 3107 .006 7. 7404 .056 8. 7443 .106 9. 0233 .156 9. 1918 .206 9.3129 .007 7. 8130 .057 8. 7521 .107 9. 0274 .157 9. 1946 .207 9. 3150 .008 7. 8751 .058 8. 7597 .108 9.0315 .158 9. 1973 .208 9.3171 .009 7. 9295 .059 8. 7672 .109 9. 0355 .159 9. 2001 .209 9. 3192 .010 7. 9778 .060 8. 7746 .110 9. 0395 .160 9. 2028 .210 9. 3212 .011 8.0212 .061 8. 7818 .111 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 .175 9. 2419 .225 9. 3513 .026 8. 4066 .076 8. 8780 .126 9. 0987 .176 9. 2443 .226 9. 3532 .027 8. 4233 .077 8. 8837 .127 9. 1021 .177 9. 2468 .227 9. 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. 4G99 .080 8. 9004 .130 9.1123 .180 9. 2541 .230 9. 3608 .031 8. 4843 .081 8. 9059 .131 9. 1157 .181 9. 2565 .231 9. 3627 .032 8.4984 .082 8.9112 .132 9. 1190 .182 9. 2589 .232 9. 3646 .033 8.5119 .083 8. 9165 .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 .136 9. 1320 .186 9. 2684 .236 9. 3720 .037 8. 5623 .087 8. 9371 .137 9. 1352 .187 9. 2707 .237 9. 3739 .038 8. 5741 .088 8. 9421 .138 9. 1384 .188 9. 2731 .238 9. 3757 .039 8. 5855 .089 8. 9470 .139 9. 1415 .189 9. 2754 .239 9. 3775 .040 8. 5967 .090 8. 9519 .140 9. 1446 .190 9. 2777 .240 9. 3794 .041 8. 6075 .091 8. 9567 .141 9. 1477 .191 9. 2799 .241 9. 3812 .042 8. 6181 .092 8. 9615 .142 9. 1508 .192 9. 2822 .242 9. 3830 .043 8. 6284 .093 8. 9662 .143 9. 1539 .193 9. 2845 .243 9. 3848 .044 8. 6385 .094 8. 9709 .144 9. 1569 .194 9. 2867 j ! .244 9. 3866 .045 8. 6484 .095 8. 9755 .145 9. 1599 .195 9.2890 .245 9.3883 .046 8.6581 .096 8.9801 .146 9.1629 .196 9.2912 .246 9.3901 .047 8. 6675 .097 8. 9846 .147 9. 1659 .197 9. 2934 .247 9. 3919 .048 8. 6767 .098 8. 9891 .148 9. 1688 .198 9.2956 .248 9. 3936 .049 8. 6858 .099 8. 9935 .149 9. 1718 .199 9.2978 .249 9.3954 .050 8. 6947 .100 8. 9979 .150 9.1747 .200 9. 3000 .250 9. 3971 234 U. S. COAST AND GEODETIC SURVEY i e e 0) w e 0) a, a > O ION ) lO © (NO CO io»o © lO lOO rH CO CO © NH t- t~ OH OO ■«<© rH CO lO »0 CM Tf CO© -*< -# CO-* CM-* •># CM NCM CNCM N© CO-* lOS -H© ffigi o© tig o© 38 3§s ©© o© ©00 ©'©' 1-H© i-H© ©00 i-H© IM© rH © .1 1 + + 1 + + + + cm to ££ 38 lOOl £& Ttl -* SoS COCO 3$ n »o CM "* CO CO SK &S n"3 ■*,-H 83 lO N Tj «Qr< ©© lO CO © CO -* CO N © CO 00 uo »o II II gg "ON © CM © 00 CM© rH CM © N ©'©* N OS N »0 CO CO CO © ^h m (M oo ©o 00 * >0 CO© ©-* rH CO © ■* rHO ** co coco io cm n n rHCO OO "ON O© ©© «# © CM© rid ©'©' rHO rHO O© I ! + + I + ©■8 rid I 00 00 CO © CO rH O 00 i CM -HrH CO© ©CM "Ct* © O CM lO CO rH O © CM ION ©© rlO rid O© i + + I lOCO §8 &s ■* co - "ON d© CO >0 N © S §© rnS O© ©00 + © "# -*© ton »on 0>0 ION rH CO © IO -* rH CO © O© IOiH CM© 000 rHO + + oo cj "IS II §1 is s^ ^n c€i c^c§ sgcs ss •ON OO ©CM «ON ©N rHCO N-* »ON rH CO NOO rH CO ©© rHO rHO + + 4- CM© CM© O© + + + CM© r? M J. o o o HARMONIC ANALYSIS AND PREDICTION OF TIDES 235 - / / ©© O OS n© oh h© ©go o© n© os n ffiN H CO cm n os * o «o N OO HOJ WIN 00 © 00 00 (SO I £2 a N©" n os co'o* OS os o o o'os HO mM O© OO iOh H © H ©' no co 10 00 N i-i i-i co os co io n co cooc hoo ooi © H lO CO CO'O N© ho coos iflN © H ms oooo © io io CO '»© sS NCO oco h« CI CM t^ |v. (--CO H CO O 00 NO 00 H ior- H CO co £j do oo co h ho <* N OOS I OOS HO HO I I OOS + %s HM lO — I 38 l-H H H CO CN O »0>0 iss Set; N N NO H CO O 00 HO I O OS O 00 I + oco oco NO NO I I N© Tt«TfH ©N TJH H OO OOs I "OOI 00 H (M O lO IO h CO PQ H ON OOS ooo ood 0000 NN lO CO O CO HO COOS O O «7 H IO 00 3j ■§ CO 00 *N H io 00 CO H CO K5N ION OOS + O OS I 8S ION OOS NCO Nh **55 o >o NN COH 0CN H N hO + O OS + ■* OS 00 OS o co h oo CO CO N 00 HO OSH Oh t^ C-1 no m co © H CO H H 00 OH IOH O CO CO 00 COH h »o in H O CO nh io n NO CO CM OOS + HO NO + + OOS OOO O CO ooo OO OOS HO + N H 00H OOS O 00 I OOS + HO + O O <0 OS NO ■>*n> NN CO H O CO H CO CO OS ON IOH H CO + H CO O CO N O + ooo + ■CO 00 o >»0 ION N OS CO io OS© OCO OOO lO N ©OS N CO NH CO 00 H N CO H MO H H IOH O © h CO ON HCO H CO N H © CD ©OS + HO NO + + ©00 © 00 + ©OS + N© HO + + N00 HO © OS CO CO 00 CO COH N OS X- CO SB N© HO I I «0 N ©OS O© N H IO OS OS H H N •># N N N MO CD H lO CD IOH T« CO CO H H CO ON CO H ££ N© I N © I H © I ©OS I ©■SO* 38 H H MJ1 N lO 00 © ON ©S H© H© I I Kg cooo HlO >ooo N© CO 00 © © + © © H CO CO © N N N CD 00 00 © CO © H lO CO CD © H 00 H O KO H IOH O © H CO N N NN NO HO © 00 ©OS ©00 + I CO lO lO 00 CO CO lO H IO N io OO N OS CO OO H IO H CO IO N © 00 H OS 00 00 HN N H 000 N OS H O IO N ■ H HO N N O© + H© HO «0 N © OS io ig NO © 00 NN © CO ©00 + S?3 IO N 00© Ni H IO N CO OiO i H H H 00CO 2! H ' O ON H CO © O i © H C> N ©* - + + N© NH ON H N N OS H N IO© O 00 + ©00 + ©© + ©© + CO >o HN H CO N O + H© co >o CO io N © I SfH ©' ©CO H© I N© 0>0 © © O H s© o© OS H 00 00 ©00 N N 00© CO© I IO H N© O © N CO 00 00 f H IO O N N HO 00N O © rf H -i© HO HO © IO ION IO CO Ui N N N OS IO CO © O © CO N IO H OS © 236 U. S. COAST AND GEODETIC STJK.VEY Table 29. — Elimination factors [ Upper line for each constituent gives the logarithms of the factors; middle line, corresponding natural numbers; lower line, angles in degrees] SERIES 14 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C, etc.) (A) Ji Ki Mi d OO Pi Qi 2Q Si 9. 7607 .576 262 Pi Ji... 9. 7968 .626 269 8. 2015 .016 357 9. 3150 .207 264 9. 7890 .615 93 9. 7203 .525 255 8. 3017 .020 353 9. 0913 .123 261 8. 1357 .014 185 Ki 9. 7968 .626 91 9. 7968 .626 269 8. 3839 .024 356 8. 3839 .024 4 9. 9958 .990 346 9. 3150 .207 264 8. 3017 .020 353 9. 9990 .998 353 9. 3344 .216 276 Mi 8. 2015 .016 3 9. 7968 .626 91 9. 7890 .615 267 9. 3150 .207 96 9. 8578 .721 78 8. 3839 .024 356 9. 3150 .207 264 9. 8290 675 85 8. 6530 .045 188 Oi 9. 3150 .207 96 8. 3839 .024 4 9. 7890 .615 93 8. 3826 .024 9 8. 7358 .054 171 9. 7968 .626 269 8. 2015 .016 357 8. 1361 .014 178 9. 8516 .711 281 00 9. 7890 .615 267 8. 3839 .024 356 9. 3150 .207 264 8. 3826 .024 351 8. 9571 .091 342 9. 0878 .122 260 8. 3320 .021 348 8. 7710 .059 349 9. 1065 .128 272 Pi h 9. 7203 .525 105 9. 9958 .990 14 9. 8578 .721 282 8. 7358 .054 189 8. 9571 .091 18 9. 3355 .217 278 8. 2581 .018 186 9. 9990 .998 7 9. 3331 .215 290 Qi 8. 3017 .020 7 9. 3150 .207 96 8. 3839 .024 4 9. 7968 .626 91 9. 0878 .122 100 9. 3355 .217 82 :::::: 9. 7968 .626 269 9. 3283 .213 89 9. 9967 .992 12 2Q 9. 0913 .123 99 8. 3017 .020 7 9. 3150 .207 96 8. 2015 .016 3 8. 3320 .021 12 8. 2581 .018 174 9. 7968 .626 91 7. 1244 .001 9.7298 .537 104 Si 9. 7607 .576 98 9. 9990 .998 7 9. 8290 .675 275 8. 1361 .014 182 8. 7710 .059 11 9. 9990 .998 353 9. 3283 .213 271 7. 1244 .001 9. 3369 .217 283 pi u 8. 1357 .014 175 9. 3344 .216 84 8. 6530 .045 172 9. 8516 .711 79 9. 1065 .128 88 9. 3331 .215 70 9. 9967 .992 348 9. 7298 .537 256 9. 3369 .217 77 HARMONIC ANALYSIS AND PREDICTION OF TIDES 237 Table 29. — Elimination factors — Continued SERIES 15 DAYS. SEMIDIURNAL CONSTITUENTS Disturbing constituents (B, C, etc.) sought 04) Ka La Ma Na 2N Ra Sa Ta Xa M2 V2 2SM Kj 9. 7534 .567 260 8. 9437 .088 342 9. 2424 .175 244 8. 9063 .081 326 9. 9986 .997 353 9. 9950 .989 345 9. 9892 .975 338 9. 6707 .468 247 8. 7223 .053 339 9. 2966 .198 257 8. 8476 .070 168 L 3 9. 7534 .567 100 9. 7627 .579 262 8. 9055 .080 344 9. 2507 .178 246 9. 7927 .620 92 9. 8276 .672 85 9. 8585 .722 77 9. 9961 .991 347 9. 3018 .200 259 8. 1941 .016 357 9. 3301 .214 88 Ms 8. 9437 .088 18 9. 7627 .579 98 9. 7627 .579 262 8. 9055 .080 344 8. 7291 .054 10 8. 1941 .016 3 8. 4114 .026 175 9. 8276 .672 85 8. 1941 .016 357 9. 8276 .672 275 8. 1935 .016 6 Na. 9. 2424 .175 116 8. 9055 .080 16 9. 7627 .579 98 ::::: 9. 7627 .579 262 9. 2760 .189 108 9. 3018 .200 101 9. 3204 .209 93 8. 1941 .016 3 9. 8276 .672 275 9. 9961 .991 13 9. 0793 .120 104 2N 8. 9063 .081 34 9. 2507 .178 114 8. 9055 .080 16 9. 7627 .579 98 8.8167 .066 26 8. 6888 .049 19 8. 4856 .031 11 9. 3018 .200 101 9. 9961 .991 13 9. 6823 .481 111 8. 5765 .038 22 R 2 9. 9986 .997 7 9. 7927 .620 268 8. 7291 .054 350 9. 2760 .189 252 8. 8167 .066 334 9. 9987 .997 353 9. 9950 .989 345 9. 7195 .524 255 8. 5420 .035 347 9. 3168 .207 265 8.4114 .026 175 Sa 9. 9950 .989 15 9. 8276 .672 275 8. 1941 .016 357 9. 3018 .200 259 8. 6888 .049 341 9. 9987 .997 7 9. 9987 .997 353 9. 7627 .579 262 8. 1935 .016 354 9. 3301 .214 272 8. 1941 .016 3 T 2 9.9892 .975 22 9. 8585 .722 283 8.4114 .026 185 9. 3204 .209 267 8. 4856 .031 349 9. 9950 .989 15 9. 9987 .997 7 9. 8010 .632 269 7. 6684 .005 182 9. 3364 .217 280 8. 7291 .054 10 Xj 9. 6707 .468 113 9.9961 .991 13 9. 8276 .672 275 8. 1941 .016 357 9. 3018 .200 259 9. 7195 .524 105 9. 7627 .579 98 9. 8010 .632 91 9. 3301 .214 272 8. 7786 .060 190 9. 3018 .200 101 *>?---- 8. 7223 .053 21 9.2966 .198 103 9. 3018 .200 101 8. 9141 .016 3 9. 8276 .672 85 9. 9961 .991 347 8. 5420 .035 13 8. 1935 .016 6 7. 6684 .005 178 9. 3301 .214 88 9. 7627 .579 98 8. 1926 .016 9 8. 1941 .016 3 9. 8276 .672 85 9. 9961 .991 347 9. 6823 .481 249 9. 3168 .207 95 9. 3301 .214 88 9. 3364 .217 80 8.7786 .060 170 9. 7627 .579 262 9.1043 .127 91 2SM 8. 8476 .070 192 9. 3301 .214 272 8. 1935 .016 354 9. 0793 .120 256 8. 5765 .038 338 8. 4114 .026 185 8. 1941 .016 357 8. 7291 .054 350 9. 3018 .200 259 8. 1926 .016 351 9.1043 .127 269 ... 246037—41- -16 238 TJ. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 29 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C, etc.) (A) Ji Kl M, Oi OO Pi Qi 2Q Si pi jj 8. 6955 .050 351 8. 6896 .049 341 8. 7199 .052 328 8. 8144 .065 13 9. 2092 .162 322 8. 6937 .049 319 8. 6672 .046 310 9.0538 .113 336 8. 3224 .021 344 Ki -- 8. 6955 .050 9 8. 6955 .050 351 8. 7517 .056 338 8. 7517 .056 22 9.9818 .959 331 8. 7199 .052 328 8. 6937 .049 319 9.9954 .990 346 8. 0542 .011 354 Mi — 8. 6896 .049 19 8. 6955 .050 9 8. 8144 .065 347 8.7199 .052 32 9. 0674 .117 161 8. 7517 .056 338 8. 7199 .052 328 8. 4418 .028 175 7. 9579 .009 183 Oi 8. 7199 .052 32 8. 7517 .056 22 8. 8144 .065 13 .:::: 8. 7185 .052 44 8. 2616 .018 174 8. 6955 .050 351 8. 6896 .049 341 8.3262 .021 8 8. 9810 .096 196 OO -. 8.8144 .065 347 8.7517 .056 338 8. 7199 .052 328 8. 7185 .052 316 9.0332 .108 309 8. 6848 .048 306 8. 6504 .045 297 8.9334 .086 324 8.4666 .029 332 Pi 9. 2092 .162 38 9. 9818 .959 29 9. 0674 .117 199 8. 2616 .018 186 9. 0332 .108 51 7. 7378 .005 357 8.2260 .017 348 9.9954 .990 14 8. 6248 .042 202 Qj 8. 6937 .049 41 8. 7199 .052 32 8. 7517 .056 22 8. 6955 .050 9 8. 6848 .048 54 7. 7378 .005 3 8. 6955 .050 351 8. 4846 .031 17 9. 9857 .968 25 2Q 8. 6672 .046 50 8. 6937 .049 41 8. 7199 .052 32 8. 6896 .049 19 8. 6504 .045 63 8. 2260 .017 12 8. 6955 .050 9 8. 5377 .034 27 9. 1825 .152 35 Si 9.0538 .113 24 9.9954 .990 14 8.4418 .028 185 8. 3262 .021 352 8.9334 .086 36 9.9954 .990 346 8. 4846 .031 343 8. 5377 .034 333 ----- 8. 1807 .015 188 01.. . 8.3224 .021 16 8.0542 .011 6 7. 9579 .009 177 8.9810 .096 164 8.4666 .029 28 8.6248 .042 158 9.9857 .968 335 9. 1825 .152 325 8. 1807 .015 172 '..'.I HARMONIC ANALYSIS AND PREDICTION OF TIDES 239 Table 29. — Elimination factors — Continued SERIES 29 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents ( B, C, etc.) sought (A) K 2 U M 2 N 2 2N Rj s 2 T 2 x 2 P2 VI 2SM K 2 -- 8. 8144 .065 347 8. 7517 .056 338 8. 7199 .052 328 8. 6937 .049 319 9. 9954 .990 346 9. 9818 .959 331 9. 9587 .909 317 9.2092 .162 322 8. 3224 .021 344 8. 0542 .011 354 9.0054 ::.: .101 145 L 3 - 8. 8144 .065 13 8. 6955 .050 351 8. 6896 .049 341 8. 6798 .048 332 7. 9581 .009 178 8. 9810 .096 164 9. 2842 .192 150 9. 9857 .968 335 7. 7378 .005 357 8. 2616 .018 186 8. 6248 .042 158 M 2 - 8. 7517 .056 22 8. 6955 .050 9 ::::: 8. 6955 .050 351 8. 6896 .049 341 8. 3262 .021 8 8. 2616 .018 174 8. 7772 .060 159 8. 9810 .096 164 8. 2616 .018 186 8. 9810 .096 196 8.2588 .018 167 N 2 8. 7199 .052 32 8. 6896 .049 19 8. 6955 .050 9 8. 6955 .050 351 8. 4846 .031 17 7. 7378 .005 3 8. 3278 .021 169 8. 2616 .018 174 8. 9810 .096 196 9. 9857 .968 25 7 5900 .004 177 2N. 8. 6937 .049 41 8. 6798 .048 28 8. 6896 .049 19 8. 6955 .050 9 8. 5377 .034 27 8. 2260 .017 12 7. 4179 .003 178 7. 7378 .005 3 9. 9857 .968 25 9. 1825 .152 35 7. 7379 .005 6 R 2 9. 9954 .990 14 7. 9581 .009 182 8. 3262 .021 352 8. 4846 .031 343 8. 5377 .034 333 9. 9954 .990 346 9. 9818 .959 331 9. 0538 .113 336 7. 2754 .002 359 8. 1807 .015 188 8. 7772 .060 159 S 2 - 9. 9818 .959 29 8. 9810 .096 196 8. 2616 .018 186 7. 7378 .005 357 8. 2260 .017 348 9. 9954 .990 14 9. 9954 .990 346 8. 6955 .050 351 8. 2588 .018 193 8. 6248 .042 202 8. 2616 .018 174 Tj 9. 9587 .909 43 9. 2842 .192 210 8. 7772 .060 201 8. 3278 .021 191 7. 4179 .003 182 9. 9818 .959 29 9. 9954 .990 14 8. 4418 .028 185 8. 5780 .038 207 8.8324 .068 217 8. 3262 .021 8 Xs 9. 2092 .162 38 9. 9857 .968 25 8. 9810 .096 196 8. 2616 .018 186 7. 7378 .005 357 9. 0538 .113 24 8. 6955 .050 9 8. 4418 .028 175 8. 6248 .042 202 8. 9640 .092 212 7. 7378 .005 3 *« - 8. 3224 .021 16 7. 7378 .005 3 8. 2616 .018 174 8. 9810 .096 164 9. 9857 .968 335 7. 2754 .002 1 8. 2588 .018 167 8. 5780 .038 153 8. 6248 .042 158 8. 6955 .050 9 8. 2539 .018 161 ft 8. 0542 .011 6 8. 2616 .018 174 8. 9810 .096 164 9. 9857 .968 335 9. 1825 .152 325 8. 1807 .015 172 8. 6248 .042 158 8. 8324 .068 143 8.9640 .092 148 8. 6955 .050 351 8. 5015 .032 151 2SM 9.0054 .101 215 8. 6248 .042 202 8.2588 .018 193 7.5900 .004 183 7. 7379 .005 354 8. 7772 .060 201 8. 2616 .018 186 8. 3262 .021 352 7. 7378 .005 357 8. 2539 .018 199 8.5015 .032 209 240 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 68 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C. etc.) (A) Ji Ki Mi Oi OO Pi Qi 2Q Si pi Ji 8. 6896 .049 341 8. 6896 .049 19 8. 7185 .052 44 8. 7185 .052 316 9.9254 .842 57 8. 6504 .045 63 8. 5715 .037 82 9.9818 .959 29 8.0520 .011 12 8. 6657 .046 322 8. 6896 .049 341 8.8039 .064 25 8. 6504 .045 297 9. 0427 .110 218 8. 7185 .052 44 8. 6504 .045 63 8.4403 .028 190 7.9572 .009 174 8.6504 .045 297 8. 7185 .052 316 8. 8039 .064 335 8. 5737 .037 272 8. 2588 .018 193 8. 6896 .049 19 8. 6657 .046 38 8. 3224 .021 344 8.9640 .092 148 8. 8039 .064 25 8. 7185 .052 44 8. 6504 .045 63 8. 5737 .037 88 8.8349 .068 101 8. 4575 .029 107 8. 3057 .020 126 8. 8391 .069 73 8.4112 .026 57 9. 1056 .128 284 9.9254 .842 303 9. 0427 .110 142 8. 2588 .018 167 8. 8349 .068 259 7. 7379 .005 6 8. 2155 .016 25 9.9818 .959 331 8. 5907 .039 135 8. 5715 .037 278 8. 6504 .045 297 8. 7185 .052 316 8. 6896 .049 341 8. 4575 .029 253 7. 7379 .005 354 8. 6896 .049 19 8. 4645 .029 325 9.9418 .875 309 8. 4713 .030 259 8. 5715 .037 278 8. 6504 .045 297 8. 6657 .046 322 8. 3057 .020 234 8. 2155 .016 335 8. 6896 .049 341 8.4887 .031 307 9. 0969 .125 290 9.0154 .104 313 9.9818 .959 331 8. 4403 .028 170 8.3224 .021 16 8.8391 .069 287 9.9818 .959 29 8.4645 .029 35 8. 4887 .031 53 8. 1761 .015 164 8. 3059 Ki 8. 6896 .049 19 8. 6657 .046 38 8. 6504 .045 63 8. 8039 .064 335 9. 1056 .128 76 8.5715 .037 82 8. 4713 .030 101 9.0154 .104 47 8. 3059 .020 34 .020 329 8. 0520 Mi— .... .011 348 7. 9572 Oi .009 186 8. 9640 00 .092 212 8. 4112 Pi .026 303 8.5907 Ql .039 225 9.9418 2Q - .875 51 9.0969 Si. .125 70 8. 1761 .015 196 --- HARMONIC ANALYSIS AND PREDICTION OF TTDES 241 Table 29. — Elimination factors — Continued SERIES 58 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents (B, C, etc.) sought (A) K 2 L 2 M 2 N 2 2N Ra s 2 T 2 X, M2 Vi 2SM K 2 8. 8039 .064 335 8. 7185 .052 316 8. 6504 .045 297 8. 5715 .037 278 9. 9818 .959 331 9. 9254 .842 303 9. 8237 .666 274 9. 1056 .128 284 8. 3059 .020 329 8.0520 .011 348 8. 9185 .083 110 U - 8. 8039 .064 25 8. 6896 .049 341 8. 6657 .046 322 8.6244 .042 303 7. 9579 .009 177 8.9640 .092 148 9. 2209 .166 120 9. 9418 .875 309 7. 7379 .005 354 8. 2588 .018 193 8. 5907 .039 135 M 2 . 8. 7185 .052 44 8. 6896 .049 19 8. 6896 .049 341 8. 6657 .046 322 8. 3224 .021 16 8. 2588 .018 167 8. 7480 .056 138 8. 9640 .092 148 8.2588 .018 193 8. 9640 .092 212 8.2475 .018 154 N 2 8. 6504 .045 63 8. 6657 .046 38 8. 6896 .049 19 8. 6896 .049 341 8. 4645 .029 35 7. 7379 .005 6 8. 3193 .021 157 8. 2588 .018 167 8.9640 .092 212 9. 9418 .875 51 7. 5898 .004 173 2N_. 8. 5715 .037 82 8. 6244 .042 57 8. 6657 .046 38 8. 6896 .049 19 8. 4887 .031 53 8. 2155 .016 25 7. 4165 .003 176 7. 7379 .005 6 9.9418 .875 51 9. 0969 .125 70 7.7356 .005 12 R2 9.9818 .959 29 7. 9579 .009 183 8. 3224 .021 344 8. 4645 .029 325 8. 4887 .031 307 — -- 9. 9818 .959 331 9. 9254 .842 303 9. 0154 .104 313 7. 2736 .002 357 8. 1761 .015 196 8. 7480 .056 138 S2- - 9. 9254 .842 57 8. 9640 .092 212 8. 2588 .018 193 7. 7379 .005 354 8. 2155 .016 335 9. 9818 .959 29 ::::: 9. 9818 .959 331 8. 6896 .049 341 8. 2475 .018 206 8. 5907 .039 225 8. 2588 .018 167 T 2 - 9. 8237 .666 86 9. 2209 .166 240 8. 7480 .056 222 8. 3193 .021 203 7. 4165 .003 184 9. 9254 .842 57 9.9818 .959 29 8. 4402 .028 190 8. 5270 .034 234 8. 7366 .055 253 8. 3224 .021 16 M 9. 1056 .128 76 9. 9418 .875 51 8. 9640 .092 212 8. 2588 .018 193 7. 7379 .005 354 9. 0154 .104 47 8. 6896 .049 19 8. 4402 .028 170 ----- 8. 5907 .039 225 8. 8933 .078 244 7.7379 .005 6 m - 8. 3059 .020 31 7. 7379 .005 6 8. 2588 .018 167 8. 9640 .092 148 9. 9418 .875 309 7. 2736 .002 3 8. 2475 .018 154 8. 5270 .034 126 8. 5907 .039 135 8. 6896 .049 19 8. 2286 .017 141 n 8.0520 .011 12 8. 2588 .018 167 8.9640 .092 148 9. 9418 .875 309 9.0969 .125 290 8. 1761 .015 164 8. 5907 .039 135 8. 7366 .055 107 8. 8933 .078 116 8. 6896 .049 341 ----- 8. 4439 .028 122 2SM 8. 9185 .083 250 8. 5907 .039 225 8.2475 .018 206 7. 5898 .004 187 7. 7356 .005 348 8. 7480 .056 222 8. 2588 .018 193 8. 3224 .021 344 7. 7379 .005 354 8.2286 .017 219 8. 4439 .028 238 242 IT. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 87 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) Ki- Mi. Oi-. 00. Pl- Qi- 2Q. Si- Disturbing constituents (B, C, etc.) 8. 6798 .048 8. 6244 .042 57 8. 5225 .033 95 8. 7857 .061 322 8.9030 .080 114 8. 3232 .021 123 7.9841 .010 151 8. 9481 .089 71 8. 2780 .019 47 8. 6798 .048 332 8. 6798 .048 28 8. 6607 .046 8. 6607 .046 294 9. 8237 8. 5225 .033 95 8. 3232 .021 123 9. 9587 .909 43 8. 0476 .011 19 M. 8. 6244 .042 303 8. 6798 .048 332 8. 7857 .061 8. 5225 .033 265 9. 0002 .100 237 8. 6607 .046 8. 5225 .033 95 8.4376 .003 195 7.9556 .009 170 8. 5225 .033 265 8. 6607 .046 294 8. 7857 .061 322 8. 2641 .018 227 8. 2539 .018 199 8. 6798 .048 28 8. 6244 .042 57 8. 3155 .021 337 8. 9351 .086 132 OO 8. 7857 .061 38 8. 6607 .046 66 8. 5225 .033 95 8. 2641 .018 133 8. 3377 .022 152 7. 8138 .007 161 7.4337 .003 9 8. 6579 .045 109 8.3116 .020 85 8.9030 .080 246 9. 8237 .666 274 9.0002 .100 123 8. 2539 .018 161 8. 3377 .022 208 7. 7367 .005 9 8. 1982 .016 37 9.9587 .909 317 8. 5315 .034 113 8. 3232 .021 237 8. 5225 .033 265 8. 6607 .046 294 .048 332 7. 8138 .007 199 7. 7367 .005 351 ,6798 .048 .4303 ,027 308 ,8640 ,731 284 2Q 7.9841 .010 209 8. 3232 .021 237 8. 5225 .033 8. 6244 .042 303 7. 4337 .003 351 8. 1982 .016 323 048 332 8. 4014 .025 280 8. 9351 .086 256 8. 9481 9.9587 .909 317 8. 4376 .027 165 8. 3155 .021 23 8. 6579 .045 251 9. 9587 .909 43 8.4303 .027 52 8.4014 .025 .015 156 8. 2780 .019 313 8.0476 .011 341 7.9556 .009 190 8.9351 .086 3116 020 275 5315 034 247 731 76 .9351 .086 104 015 204 HARMONIC ANALYSIS AND PREDICTION OF TIDES 243 Table 29. — Elimination factors — Continued SERIES 87 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents (B, C, etc.) sought (A) Ka . La M 2 Na 2N Ra s 2 T 2 X 2 M2 VI 2SM K 2 8. 7857 .061 322 8. 6798 .048 28 8.6244 .042 57 8. 5247 .033 85 7.9576 .009 185 8. 9351 .086 228 9. 1055 .127 271 9. 8640 .731 76 7. 7367 .005 9 8.2539 .018 161 8.5315 .034 247 8. 6607 .046 294 8. 6798 .048 332 8. 6798 .048 28 8.6244 .042 57 8.3155 .021 337 8. 2539 .018 199 8. 6976 .050 242 8. 9351 .086 228 8.2539 .018 161 8.9351 .086 132 8.2286 .017 219 8. 5225 .033 265 8. 6244 .042 303 8. 6798 .048 332 8. 6798 .048 28 8. 4303 .027 308 7. 7367 .005 351 8. 3068 .020 214 8. 2539 .018 199 8. 9351 .086 132 9.8640 .731 284 7.5883 .004 190 8. 3232 .021 237 8. 5247 .033 275 8. 6244 .042 303 8. 6798 .048 332 8. 4014 .025 280 8. 1982 .016 323 7. 4165 .003 186 7. 7367 .005 351 9.8640 .731 284 8. 9351 .086 256 7. 7314 .005 342 9. 9587 .909 317 7. 9576 .009 175 8. 3155 .021 23 8. 4303 .027 52 8. 4014 .025 80 9.9587 .909 43 9. 8237 .666 86 8. 9481 .089 71 7. 2740 .002 4 8. 1689 .015 156 8. 6976 .050 242 9. 8237 .666 274 8. 9351 .086 132 8. 2539 .018 161 7. 7367 .005 9 8. 1982 .016 37 9.9587 .909 317 9.9587 .909 43 8. 6798 .048 28 8. 2286 .017 141 8. 5315 .034 113 8. 2539 .018 199 9. 5416 .348 231 9. 1055 .127 89 8. 6976 .050 118 8. 3068 .020 146 7. 4165 .003 174 9.8237 .666 274 9. 9587 .909 317 8. 4376 .027 165 8.4358 .027 98 8. 5521 .036 70 8. 3155 .021 337 8. 9030 .080 246 9. 8640 .731 284 8. 9351 .086 132 8. 2539 .018 161 7. 7367 .005 9 8.9481 .089 289 8. 6798 .048 332 8. 4376 .027 195 8. 5315 .034 113 8. 7629 .058 85 7. 7367 .005 351 8. 2780 .019 313 7. 7367 .005 351 8. 2539 .018 199 8.9351 .086 228 9.8640 .731 76 7.2740 .002 356 8.2286 .017 219 8.4358 .027 262 8. 5315 .034 247 8. 6798 .048 332 8. 1849 .015 238 8. 0476 .011 341 8. 2539 .018 199 8. 9351 .086 228 9.8640 .731 76 8. 9351 .086 104 8. 1689 .015 204 8.5315 .034 247 8. 5521 .036 290 8. 7629 .058 275 8.6798 .048 28 8. 3401 .022 267 8.7538 Lj Mj Nj 8. 7857 .061 38 8. 6607 .046 66 8.5225 .033 95 8.3232 .021 123 9.9587 .909 43 9.8237 .666 86 9. 5416 .348 129 8.9030 .080 114 8. 2780 .019 47 8.0476 .011 19 8. 7538 .057 285 .057 75 8. 5315 .034 113 8.2286 .017 141 7.5883 2N R» Sa .004 170 7. 7314 .005 18 8. 6976 .050 118 8.2539 Ta .018 161 8. 3155 Xa .021 23 7. 7367 .005 9 8. 1849 .015 122 8.3401 2SM .022 93 244 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 105 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) Disturbing constituents (B, C, etc.) Ki- Mi. Oil. 00. Pl- 2Q_ 8. 6704 .047 146 8. 5885 .039 112 8. 4422 8. 4953 .031 202 8. 8322 8. 2332 .017 55 7. 7808 .006 21 8. 3722 .024 18 8. 1065 .013 144 K, 8. 6704 .047 214 8. 6704 .047 146 8. 5381 .035 124 8. 5381 .035 236 9. 7311 .538 103 8. 4422 .028 8. 2332 .017 55 9. 9393 .870 52 7. 0766 .001 178 M. 8. 5885 .039 248 8. 6704 .047 214 8. 4953 .031 158 8. 4422 .028 271 8. 8219 .066 318 8. 5381 .035 124 8. 4422 .028 i. 9548 .090 266 .023 32 8. 4422 .028 271 8. 5381 .035 236 8. 4953 .031 202 .019 293 8. 1856 .015 340 8. 6704 .047 146 8. 5885 .039 112 8.6113 .041 078 54 OO 8. 4953 .031 158 8. 5381 .035 124 8. 4422 .028 8.2803 .019 67 8. 4500 .028 47 7. 9556 .009 33 6. 4362 .000 179 7. 5174 .003 175 8. 1640 .015 121 8.8322 .068 291 9. 7311 .538 257 8. 8219 .066 42 8. 1856 .015 20 8. 4500 .028 313 7. 8500 .007 8. 2067 .016 132 9. 9393 .870 ,4685 .029 74 8. 2332 .017 8. 4422 .028 271 8. 5381 .035 236 8. 6704 .047 214 7.9556 .009 327 7. 8500 .007 194 8. 6704 .047 146 8. 2396 .017 322 9. 7951 .624 268 2Q 7. 7808 .006 339 8. 2332 .017 305 8. 4422 .028 271 8. 5885 .039 248 6. 4362 .000 181 8. 2067 .016 228 8. 6704 .047 214 7. 1241 .001 356 8. 7943 .062 302 8. 3722 .024 342 9. 9393 .870 8. 9548 .090 94 8. 6113 .041 72 7. 5174 .003 185 ,870 52 8. 2396 .017 7. 1241 .001 4 8. 3820 .024 126 8. 1065 .013 216 7. 0766 .001 182 8. 3679 .023 .078 306 8. 1640 .015 239 8. 4685 .029 286 9. 7951 .624 92 8. 7943 .062 58 8. 3820 .024 234 HARMONIC ANALYSIS AND PREDICTION OF TIDES 245 Table 29. — Elimination factors — Continued SERIES 105 DAYS. SEMIDIURNAL CONSTITUENTS Constituent sought (A) Ks. La Ma Nj 2N. Ra Sa. Ta X» 2SM Disturbing constituents (B, C, etc.) K 2 8. 4953 .031 158 8. 5381 .035 124 8.4422 .028 89 8. 2332 .017 55 9.9392 869 52 ,7311 538 103 155 155 8. 8322 ,1065 ,013 144 7.0766 .001 178 1. 6847 .048 263 La 4953 031 202 8. 6704 .047 146 8. 5885 .039 112 1.4347 .027 78 8. 9311 85 254 8. 8929 078 306 7. 6403 .004 358 9. 7951 .624 7. 8500 .007 166 8. 1856 .015 8. 4685 .029 M 2 8. 5381 .035 236 8. 6704 .047 214 8. 6704 047 146 8. 5885 39 112 8. 6113 .041 8. 1856 015 340 ,025 212 .078 306 8. 1856 .015 20 8.8929 .078 54 8. 1585 .014 320 N 2 2N ,4422 ,028 271 8. 5885 039 248 8. 6704 .047 214 8. 6704 .047 8. 2395 017 7.8500 007 194 8. 4362 .027 246 8. 1856 .015 340 8.8929 ,078 54 ,7951 ,624 7. 2638 .002 354 8.2332 .017 305 8. 4347 .027 282 8. 5885 .039 248 8. 6704 .047 214 7. 1241 .001 356 8. 2067 .016 228 8. 3366 .022 280 7. 8500 007 194 9. 7951 .624 8. 7943 062 .007 R 2 9392 869 8. 9311 .085 106 8. 6113 .041 72 8. 2395 .017 38 7. 1241 .001 4 9.9392 69 52 9. 7311 538 103 8. 3722 024 18 8. 3410 022 92 .024 126 .025 212 9.7311 .538 257 .078 54 .1856 .015 20 7.8500 .007 166 8. 2067 .016 132 1 9392 9.9392 69 52 8. 6704 .047 146 .1585 ,014 40 ,4685 ,029 74 8. 1856 .015 340 T 2 1892 155 205 7.6403 .004 2 .025 148 8. 4362 027 114 8. 3366 .022 9. 7311 257 9. 9392 8. 9548 90 94 7. 6654 005 168 8. 0785 .012 22 . 6113 .041 8. 8322 .068 291 9. 7951 .624 268 8. 8929 .078 54 8. 1856 .015 20 7. 8500 .007 166 8. 3722 .024 342 8. 6704 .047 214 8. 9548 .090 266 8. 4685 029 74 .046 7. 8500 .007 194 8. 1065 .013 216 7.8500 .007 194 8. 1856 .015 340 ,078 9. 7951 .624 92 1. 3410 .022 8. 1585 014 320 7. 6654 .005 192 .4685 .029 8. 6704 047 214 8. 1117 .013 7. 0766 .001 182 8. 1856 .015 340 8. 8929 .078 306 9. 7951 624 92 8. 7943 .062 58 8. 3820 24 234 8.4685 .029 8.0785 .012 8. 6609 .046 252 8. 6704 .047 146 8. 2581 .018 266 2SM 8. 6847 .048 97 8. 4685 .029 74 8. 1585 .014 40 7. 2638 .002 6 7.8368 .007 152 8. 3896 .025 148 8. 1856 .015 20 8. 6113 .041 72 7. 8500 .007 166 8. 1117 013 60 8. 2581 .018 246 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 134 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C, etc.) (A) Ji Ki Mi Oi OO Pi Qi 2Q Si Pi Ji 8. 4360 .027 205 8. 3946 .025 229 8. 2695 .019 239 8. 0361 .011 170 8. 7345 .054 253 8. 209:4 .016 264 8. 0930 .012 288 8. 6047 .040 319 7. 7771 .006 201 Kj 8. 4360 .027 155 ::::: 8. 4360 .027 205 8. 2628 .018 214 8. 2628 .018 146 9. 5078 .322 228 8. 2695 .019 239 8. 2094 .016 264 9. 8992 .793 294 7. 1819 .002 356 Mi 8. 3946 .025 131 8. 4360 .027 155 8. 0361 .011 190 8. 2695 .019 121 8. 4838 .030 23 8. 2628 .018 214 8. 2695 .019 239 8. 8500 .071 89 8. 2196 .017 332 Oi 8. 2695 .019 121 8. 2628 .018 146 8. 0361 .011 170 8. 1796 .015 111 7. 9151 .008 14 8. 4360 .027 205 8. 3946 .025 229 8. 5206 .033 80 8. 6697 .047 322 00 8. 0361 .011 190 8. 2628 .018 214 8. 2695 .019 239 8. 1796 .015 249 :::::: 8. 4760 .030 262 8. 1133 .013 273 7. 9812 .010 298 8. 2156 .016 328 7. 8315 .007 211 Pi 8. 7345 .054 107 9. 5078 .322 132 8. 4838 .030 337 7.9151 .008 346 8. 4760 .030 98 7. 6424 .004 191 7. 9951 .010 216 9. 8992 .793 66 8. 2746 .019 308 Qi 8. 2094 .016 96 8. 2695 .019 121 8. 2628 .018 146 8. 4360 .027 155 8. 1133 .013 87 7. 6424 .004 169 8. 4360 .027 205 8. 2605 .018 55 9. 6387 .435 117 2Q 8. 0930 .012 72 8. 2094 .016 96 8. 2695 .019 121 8. 3946 .025 131 7. 9812 .010 62 7. 9951 .010 144 8. 4360 .027 155 7. 9233 .008 30 8. 7610 .058 92 Si. 8. 6047 .040 41 9. 8992 .793 66 8. 8500 .071 271 8. 5206 .033 280 8. 2156 .016 32 9. 8992 .793 294 8. 2605 .018 305 7. 9233 .008 330 8. 3143 .021 242 7. 7771 .006 159 7. 1819 .002 4 8. 2196 .017 28 8. 6697 .047 38 7. 8315 .007 149 8. 2746 .019 52 9. 6387 .435 243 8. 7610 .058 268 8. 3143 .021 118 "::: HARMONIC ANALYSIS AND PREDICTION OF TIDES 247 Table 29. — Elimination factors — Continued SERIES 134 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents (B, C, etc.) sought (A) K 2 L 2 M 2 N 2 2N R2 s 2 T 2 X 2 A»2 ft 2SM Ki ... 8. 0361 .011 190 8. 4360 .027 155 8. 3946 .025 • 131 8.3215 .021 106 8. 8285 .067 256 8.6697 .047 322 8. 5871 .039 208 9. 6387 .435 117 7. 6424 .004 169 7. 9151 .008 14 8.2746 .019 308 8 2628 .018 214 8. 4360 .027 205 8. 4360 .027 155 8. 3946 .025 131 8. 5206 .033 280 7. 9151 .008 346 8. 4622 .029 232 8. 6697 .047 322 7. 9151 .008 14 8. 6697 .047 38 7.9028 .008 333 8. 2695 .019 239 8. 3946 .025 229 8. 4360 .027 205 8. 4360 .027 155 8. 2605 .018 305 7.6424 .004 191 8. 3592 .023 257 7. 9151 .008 346 8. 6697 .047 38 9. 6387 .435 243 6. 7753 .001 358 8. 2094 .016 264 8. 3215 .021 254 8. 3946 .025 229 8. 4360 .027 205 7. 9233 .008 330 7. 9951 .010 216 8. 2280 .017 282 7. 6424 .004 191 9. 6387 .435 243 8. 7610 .058 268 7. 6344 .004 202 9. 8992 .793 294 8. 8285 .067 104 8. 5206 .033 80 8. 2605 .018 55 7. 9233 .008 30 9. 8992 .793 66 9. 5079 .322 132 8. 6047 .040 41 8. 2346 .017 93 8. 3143 .021 118 8. 4622 .029 232 9. 5078 .322 228 8. 6697 .047 38 7.9151 .008 14 7. 6424 .004 169 7. 9951 .010 144 9. 8992 .793 294 9. 8992 .793 66 8. 4360 .027 155 7. 9028 .008 27 8.2746 .019 52 7. 9151 .008 346 8. 9538 .090 342 8. 5871 .039 152 8. 4622 .029 128 8. 3592 .023 103 8. 2280 .017 78 9. 5079 .322 228 9. 8992 .793 294 8.8500 .071 89 8. 0509 .011 141 7. 7834 .006 166 8. 5206 .033 280 8. 7345 .054 253 9. 6387 .435 243 8. 6697 .047 38 7.9151 .008 14 7. 6424 .004 169 8. 6047 .040 319 8. 4360 .027 205 8. 8500 .071 271 8. 2746 .019 52 8. 5650 .037 76 7. 6424 .004 191 7. 7771 .006 201 7. 6424 .004 191 7. 9151 .008 346 8. 6697 .047 322 9. 6387 .435 117 8. 2346 .017 267 7. 9028 .008 333 8. 0509 .011 219 8. 2746 .019 308 8. 4360 .027 205 7. 8820 .008 319 7. 1819 .002 356 7.9151 .008 346 8. 6697 .047 322 9. 6387 .435 117 8. 7610 .058 92 8. 3143 .021 242 8. 2746 .019 308 7. 7834 .006 194 8. 5650 .037 284 8. 4360 .027 155 8. 1118 .053 295 8. 5254 L 2 M 2 N 2 2N 8. 0361 .011 170 8. 2628 .018 146 8. 2695 .019 121 8. 2094 .016 96 9. 8992 .793 66 9. 5078 .322 132 8. 9538 .090 18 8. 7345 .054 107 7. 7771 .006 159 7. 1819 .002 4 8. 5254 .034 299 .034 61 8.2746 .019 52 7. 9208 .008 27 6. 7753 .001 2 7. 6344 R 2 S2 T 2 X2— - M2 - v*-- - 2SM___ .004 158 8. 4622 .029 128 7. 9151 .008 14 8. 5206 .033 80 7. 6424 .004 169 7.8820 .008 41 8.1118 .013 65 248 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 163 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C, etc.) (A) h Ki M, Oi OO Pi Qi 2Q Si Pi Jx 8. 1495 .014 195 8. 1495 .014 165 7. 7528 .006 168 7. 7528 .006 192 9. 0723 .118 161 7. 9150 .008 153 7. 9579 .009 137 9. 8470 .703 80 7. 5128 .003 10 8. 1341 .014 210 8. 1495 .014 195 7. 4365 .003 3 7. 9150 .008 207 7. 6604 .005 356 7. 7528 .006 168 7. 9150 .008 153 8. 7629 .058 276 8. 0859 .012 25 7. 9150 .008 207 7. 7528 .006 192 7. 4365 .003 357 7. 7427 .006 204 7. 5513 .004 353 8. 1495 .014 165 8. 1341 .014 150 8. 4422 .028 273 8. 3724 .024 22 7. 4365 .003 3 7. 7528 .006 168 7. 9150 .008 153 7. 7427 .006 156 8. 1140 .013 148 7. 8343 .007 140 7. 8631 .007 125 8. 3776 .024 68 6. 6248 .000 178 8.4234 .027 215 9. 0723 .118 199 7. 6604 .005 4 7. 5513 .004 7 8. 1140 .013 212 7. 4230 .003 172 7. 7409 .006 157 9. 8470 .703 280 7. 9852 .010 29 7. 9579 .009 223 7. 9150 .008 207 7. 7528 .006 192 8. 1495 .014 195 7. 8343 007 220 7. 4230 .003 188 8. 1495 .014 165 8. 2410 .017 288 9. 3888 .245 218 7. 9582 .009 238 7. 9579 .009 223 7. 9150 .008 207 8. 1341 .014 210 7. 8631 .007 235 7. 7409 .006 203 8. 1495 .014 195 8. 0589 .011 57 8. 5769 .038 233 8. 6570 .045 295 9. 8470 .703 280 S. 7629 .058 84 8. 4422 .028 87 8. 3776 .024 292 9. 8470 .703 80 8. 2410 .017 72 8. 0589 .011 57 8. 2562 .018 110 7. 0948 Ki. — 8. 1495 .014 165 8. 1341 .014 150 7. 9150 .008 153 7. 4365 .003 357 8. 4234 .027 145 7. 9579 .009 137 7. 9582 .009 122 8. 6570 .045 65 7. 0948 .001 175 .001 185 7.5128 Mi .003 350 8. 0859 Oi .012 335 8.3724 OO .024 338 6. 6248 Pi .000 182 7. 9852 Qi .010 331 9. 3888 2Q .245 142 8. 5769 Si .038 127 8. 2562 .018 250 -" HARMONIC ANALYSIS AND PREDICTION OF TIDES 249 Table 29. — Elimination factors — Continued SERIES 163 DAYS. SEMIDIURNAL CONSTITUENTS Constituent sought (A) M 2 . N 2 _ 2N- B 2 . T« 2SM. Disturbing constituents (B, C, etc.) K 2 L 2 7. 4365 .003 3 7. 7528 .006 7. 9150 .008 153 7. 9579 .009 137 9. 8470 .703 9. 0723 .118 161 9. 3179 .208 61 8. 4234 .027 145 7. 0948 .001 175 7.5128 .003 10 8. 1450 .014 334 7. 4365 .003 357 8. 1495 .014 165 8. 1341 .014 150 8. 1078 .013 134 8. 7464 .056 257 8. 3724 .024 338 8. 7614 .058 238 .245 142 7. 4230 .003 172 7. 5513 .004 7 7. 9852 .010 331 M 2 7. 7528 .006 192 8. 1495 .014 195 8. 1495 .014 165 8. 1341 .014 150 8. 4422 .028 273 7. 5513 .004 353 8. 4590 .029 253 8. 3724 .024 338 7. 5513 .004 7 8. 3724 .024 22 7. 5483 004 346 7. 9150 .008 207 8. 1341 .014 210 8. 1495 .014 195 8. 1495 .014 165 8. 2410 .017 288 7.4230 .003 188 8. 2850 .019 268 7. 5513 .004 353 8. 3724 .024 22 9. 3888 .245 218 6. 3062 .000 181 2N R 2 7. 9579 .009 223 8. 1078 .013 226 8. 1341 .014 210 8. 1495 .014 195 8. 0588 .011 303 7. 7409 .006 203 8. 1397 .014 284 7. 4230 .003 188 9. 3888 .245 218 8. 5769 .038 233 7. 4186 .003 .8470 ,703 8. 7464 .056 103 8.4422 .028 87 8. 2410 .017 72 8. 0588 .011 57 9. 8470 703 9. 0725 .118 161 .6570 .045 65 .1488 .014 95 8. 2563 .018 110 .4590 ,029 253 9. 0723 .118 8. 3724 .024 22 7. 5513 .004 7 7. 4230 .003 172 7. 7409 .006 157 9. 8470 .703 280 9. 8470 .703 8. 1495 .014 165 7. 5483 .004 14 7. 9852 .010 29 7. 5513 .004 353 Tj 3179 208 8. 7614 .058 122 8. 4590 .029 107 8. 2850 .019 92 8. 1397 .014 76 9. 0725 .118 199 9. 8470 .703 280 8. 7629 .058 84 8. 1668 .015 114 ,016 129 4422 028 273 X 2 ,4234 ,027 215 .245 218 8.3724 .024 22 7. 5513 .004 7 7. 4230 .003 172 8. 6570 .045 295 8. 1495 .014 195 8. 7629 .058 276 7. 9852 .010 29 8. 3386 .022 45 7. 4230 003 188 7. 0948 .001 185 7. 4230 .003 188 7. 5513 .004 353 8.3724 024 338 9. 3888 .245 142 1. 1488 .014 265 7. 5483 004 346 .015 246 7. 9852 010 331 8. 1495 014 195 7. 5426 003 339 vi 2SM 7.5128 .003 350 7. 5513 .004 353 8. 3724 .024 338 .245 142 8. 5769 .038 127 8. 2563 .018 250 7. 9852 .010 331 8. 1966 .016 231 8. 3386 .022 315 8. 1495 .014 165 7. 8424 .007 324 8. 1450 .014 26 7. 9852 010 29 7. 5483 .004 14 3. 3062 .000 179 7.4186 003 164 8. 4590 .029 107 7. 5513 004 7 8.4422 028 87 7. 4230 003 172 7. 5426 .003 21 7. 8424 .007 36 250 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 192 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C, etc.) (A) Ji K! Mi Oi OO Pi Qi 2Q Si Pi J, 7. 6613 .005 186 7. 6613 .005 174 7. 5891 .004 10 7. 5891 .004 350 8. 6868 .049 9 7. 0355 .001 4 6. 5151 .000 178 9. 7807 .604 95 7. 6468 .004 16 7. 6591 .005 192 7. 6613 .005 186 8. 0828 .012 16 7. 0355 .001 356 8. 1441 .014 195 7. 5891 .004 10 7. 0355 .001 4 8. 6866 .049 280 7. 9586 .009 22 7. 0355 .001 356 7. 5891 .004 350 8. 0828 .012 344 7. 5826 .004 340 6.4230 .000 359 7. 6613 .005 174 7. 6591 .005 168 8. 3698 .023 265 7. 7679 .006 6 8. 0828 .012 16 7. 5891 .004 10 7. 0355 .001 4 7. 5826 .004 20 7. 8388 .007 19 7. 3409 .002 14 7. 0344 .001 8 8. 3250 .021 104 7. 6132 .004 26 7. 3819 .002 357 8. 6868 .049 351 8. 1441 .014 165 6. 4230 .000 1 7. 8388 .007 341 7. 1547 .001 175 7. 3491 .002 169 9.7807 .C04 265 7. 3097 .002 7 6. 5151 .000 182 7. 0355 .001 356 7. 5891 .004 350 7. 6613 .005 186 7. 3409 .002 346 7. 1547 .001 185 7. 6613 .005 174 8. 1911 .016 271 8. 8560 .072 192 7. 0698 .001 187 6. 5151 .000 182 7. 0355 .001 356 7. 6591 .005 192 7. 0344 .001 352 7. 3491 .002 191 7. 6613 .005 186 8. 0615 .012 276 8. 0931 .012 198 8.6281 42 271 9. 7807 .604 265 8. 6866 .049 80 8. 3698 .023 95 8.3250 .021 256 9. 7807 .604 95 8. 1911 .016 89 8.0615 .012 84 8. 2024 .016 102 7 3308 Ki-.- 7. 6613 .005 174 7. 6591 .005 168 7. 0355 .001 4 8. 0828 .012 344 7. 3819 .002 3 6. 5151 .000 178 7. 0698 .001 173 8.6281 .042 89 7. 3308 .002 10 .002 350 7 6468 Mi...... .004 344 7 9586 Oi .009 338 7 7679 00 .006 354 7. 6132 P! .004 334 7. 3097 Qi__ .002 353 8. 8560 2Q .072 168 8. 0931 Si .012 162 8.2024 .016 258 ..:: HARMONIC ANALYSIS AND PREDICTION OF TIDES 251 Table 29. — Elimination factors — Continued SERIES 192 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents (B, C, etc.) sought (^4) K 2 L 2 M 2 N 2 2N K 2 s 2 T 2 x 2 M2 PS 2SM K 2 8.0828 .012 344 7. 5891 .004 350 7. 0355 .001 356 6. 5151 .000 182 9. 7807 .604 265 8. 6868 .049 351 9. 2922 .196 256 7. 3819 .002 357 7. 3308 .002 350 7. 6468 .004 344 7. 6012 .004 171 U - - 8. 0828 .012 16 7. 6613 .005 186 7. 6591 .005 192 7. 6554 .005 197 8. 6778 .048 101 7. 7679 .006 6 8. 7615 .058 92 8. 8560 .072 192 7. 1547 .001 185 6. 4230 .000 359 7. 3097 .002 7 M» -- 7. 5891 .004 10 7. 6613 .005 174 :::::: 7. 6613 .005 186 7. 6591 .005 192 8. 3698 .023 95 6. 4230 .000 1 8. 4057 .025 86 7. 7679 .006 6 6. 4230 .000 359 7. 7679 .006 354 6. 4265 .000 1 N 3 7. 0355 .001 4 7. 6591 .005 168 7. 6613 .005 174 7. 6613 .005 186 8. 1911 .016 89 7. 1547 .001 175 8. 2077 .016 80 6. 4230 .000 1 7. 7679 .006 354 8. 8560 .072 168 6. 8803 .001 175 2N __„, 6. 5151 .000 178 7. 6554 .005 163 7. 6591 .005 168 7. 6613 .005 174 8. 0615 .012 84 7. 3491 .002 169 8. 0649 .012 74 7. 1547 .001 175 8. 8560 .072 168 8. 0931 .012 162 7. 1525 .001 170 R 2 - 9. 7807 .604 95 8. 6778 .048 259 8. 3698 .023 265 8. 1911 .016 271 8. 0615 .012 276 9. 7807 .604 265 8. 6863 .049 351 8. 6281 .042 271 8. 0768 .012 264 8. 2024 .016 258 8. 4057 .025 86 S 2 8. 6868 .049 9 7. 7679 .006 354 6.4230 .000 359 7. 1547 .001 185 7. 3491 .002 191 9. 7807 .604 95 9. 7807 .604 265 7. 6613 .005 186 6. 4265 .000 359 7. 3097 .002 353 6.4230 .000 1 T 2 .._ 9. 2922 .196 104 8. 7615 .058 268 8. 4057 .025 274 8. 2077 .016 280 8. 0649 .012 286 8. 6863 .049 9 9. 7807 .604 95 8. 6866 .049 280 8. 0959 .012 273 8. 2350 .017 268 8. 3698 .023 95 X2 7. 3819. .002 3 8. 8560 .072 168 7. 7679 .006 354 6. 4230 .000 359 7. 1547 .001 185 8. 6281 .042 89 7. 6613 .005 174 8. 6866 .049 80 7. 3097 .002 353 7. 7656 .006 347 7.1547 .001 175 7. 3308 .002 10 7. 1547 .001 175 6. 4230 .000 1 7. 7679 .006 6 8. 8560 .072 192 8. 0768 .012 96 6. 4265 .000 1 8. 0959 .012 87 7. 3097 .002 7 7. 6613 .005 174 6. 4253 .000 2 VI.-..- ....... .- 7. 6468 .004 16 6. 4230 .000 1 7. 7679 .006 6 8. 8560 .072 192 8. 0931 .012 198 8.2024 .016 102 7. 3097 .002 7 8. 2350 .017 92 7. 7656 .006 13 7. 6613 .005 186 7. 1202 .001 8 2SM. 7. 6012 .004 189 7. 3097 .002 353 6. 4265 .000 359 6. 8803 .001 185 7. 1525 .001 190 8. 4057 .025 274 6. 4230 .000 359 8. 3698 .023 265 7. 1547 .001 185 6. 4253 .000 358 7. 1202 .001 352 252 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 221 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C, etc.) (A) Ji K! Mi Oi OO Pi Qi 2Q Si Pi ^ 7. 4061 .003 356 7. 4061 .003 4 8. 0179 .010 32 8. 0179 .010 328 9. 2077 .161 38 7. 8848 .008 36 7. 7969 .006 39 9. 6969 .498 109 7. 7209 .005 22 7. 4052 .003 353 7. 4061 .003 356 8. 2672 .019 28 7. 8848 .008 324 8. 4189 .026 214 8. 0179 .010 32 7.8848 .008 36 8. 6172 .041 285 7. 8313 .007 19 7. 8848 .008 324 8. 0179 .010 328 8. 2672 .019 332 7. 9465 .009 296 7. 3352 .002 186 7. 4061 .003 4 7. 4052 .003 7 8. 2991 .020 257 7.8800 .008 170 8. 2672 .019 28 8. 0179 .010 32 7. 8848 .008 36 7. 9465 .009 64 8. 2351 .017 70 7. 8628 .007 68 7. 7946 .006 71 8. 0778 .012 141 7. 8188 .007 54 8. 3590 .023 318 9. 2077 .161 322 8. 4189 .028 146 7. 3352 .002 174 8.2351 .017 290 6. 7176 .001 178 6. 4350 .000 2 9. 6969 .498 251 7. 5854 .004 165 7. 7969 .006 321 7. 8848 .008 324 8. 0179 .010 328 7. 4061 .003 356 7. 8628 .007 292 6. 7176 .001 182 7. 4061 .003 4 8. 1112 .013 253 8. 8310 .068 347 7. 7322 .005 317 7. 7969 .006 321 7. 8848 .008 324 7. 4052 .003 353 7. 7946 .006 289 6.4350 .000 358 7. 4061 .003 356 7. 9748 .009 250 8. 0073 .010 343 8.5324 .034 247 9. 6969 .498 251 8. 6172 .041 75 8. 2991 .020 103 8. 0778 .012 219 9. 6969 .498 109 8. 1112 .013 107 7.9748 .009 110 8. 1495 .014 94 7. 6535 Ki 7. 4061 .003 4 7. 4052 .003 7 7. 8848 .008 36 8. 2672 .019 332 8. 3590 .023 42 7. 7969 .006 39 7. 7322 .005 43 8. 5324 .034 113 7. 6535 .005 26 .005 334 7. 7209 Mi........ .005 338 7. 8313 Oi .007 341 7. 8800 OO .008 190 7.8188 Pi .007 306 7. 5854 Q,_ .004 195 8. 8310 2Q .068 13 8. 0073 Si .010 17 8. 1495 .014 266 :::: HARMONIC ANALYSIS AND PREDICTION OF TIDES 253 Table 29. — Elimination factors— Continued SERIES 221 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents (B, C, etc.) sought (A) K 2 L 2 M 2 N 2 2N R 2 s 2 T 2 x 2 M2 v% 2SM K 2 8. 2872 .019 332 8. 0179 .010 328 7. 8848 .008 324 7. 7969 .006 321 9. 6969 .498 251 9. 2077 .161 322 8. 9831 .096 213 8. 3590 .023 318 7.6535 .005 334 7. 7209 .005 338 8. 2035 .016 136 U 8. 2672 .019 28 7. 4061 .003 356 7. 4052 .003 353 7. 4037 .003 349 8. 6189 .042 99 7. 8800 .008 170 8. 6448 .044 62 8. 8310 .068 347 6. 7176 .001 182 7. 3352 .002 186 7. 5854 .004 165 Mj._ — 8. 0179 .010 32 7. 4061 .003 4 7. 4061 .003 356 7. 4052 .003 353 8. 2991 .020 103 7. 3352 .002 174 8. 3038 .020 65 7. 8800 .008 170 7. 3352 .002 186 7. 8800 .008 190 7. 3333 .002 168 N 2 7. 8848 .008 36 7. 4052 .003 7 7. 4061 .003 4 - — 7. 4061 .003 356 8. 1112 .013 107 6. 7176 .001 178 8. 1229 .013 69 7. 3352 .002 174 7. 8800 .008 190 8. 8310 .068 13 7. 0677 .001 172 2N 7. 7969 .006 39 7. 4037 .003 11 7. 4052 .003 7 7. 4061 .003 4 — - 7. 9748 .009 110 6. 4350 .000 2 7. 9996 .010 73 6.7176 .001 178 8. 8310 .068 13 8. 0073 .010 17 6. 7183 .001 176 R 2 9. 6969 .498 109 8. 6189 .042 261 8. 2991 .020 257 8. 1112 .013 253 7. 9748 .009 250 — - 9. 6970 .498 251 9. 2076 .161 322 8. 5324 .034 247 8. 0145 .010 263 8. 1495 .014 266 8. 3038 .020 65 S2 9. 2077 .161 38 7. 8800 .008 190 7. 3352 .002 186 6. 7176 .001 182 6. 4350 .000 358 9. 6970 .498 109 9. 6970 .498 251 7. 4061 .003 356 7. 3333 .002 192 7. 5854 .004 195 7. 3352 .002 174 T 2 8. 9831 .096 147 8. 6448 .044 298 8. 3038 .020 295 8. 1229 .013 291 7. 9996 .010 287 9. 2076 .161 38 9. 6970 .498 109 8. 6172 .041 285 7. 9704 .009 301 8. 0914 .012 304 8. 2991 .020 103 X2 8. 3590 .023 42 8. 8310 .068 13 7. 8800 .008 190 7. 3352 .002 186 6. 7176 .001 182 8. 5324 .034 113 7. 4061 .003 4 8. 6172 .041 75 7. 5854 .004 195 7. 8738 .007 199 6. 7176 .001 178 M 7. 6535 .005 26 6. 7176 .001 178 7. 3352 .002 174 7. 8800 .008 170 8. 8310 .068 347 8. 0145 .010 97 7. 3333 .002 168 7.9704 .009 59 7. 5854 .004 165 7. 4061 .003 4 7. 3294 .002 162 Vi 7. 7209 .005 22 7. 3352 .002 174 7. 8800 .008 170 8. 8310 .068 347 8. 0073 .010 343 8. 1495 .014 94 7. 5854 .004 165 8. 0914 .012 56 7. 8738 .007 161 7. 4061 .003 356 7. 4945 .003 159 2SM 8. 2035 .016 224 7. 5854 .004 195 7. 3333 .002 192 7. 0677 .001 188 6. 7183 .001 184 8. 3038 .020 295 7. 3352 .002 186 8. 2991 .020 257 6. 7176 .001 182 7. 3294 .002 198 7. 4945 .003 201 246037—41- -17 254 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 250 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) h K, Mi....: Oi-— 00 Pi Qi 2Q - Si — Disturbing constituents (B, C, etc.) 7. 9011 .008 13 26 8. 0302 .011 67 8. 3544 .023 319 8. 4768 .030 7. 9350 .009 7. 8438 .007 93 8. 3527 .023 136 7. 7796 .006 42 Ki M, Oi OO Pi Qi 2Q Si 7. 9011 .008 347 7. 8898 .008 334 8. 0302 .011 293 8. 3544 .023 41 8. 4768 .030 280 7. 9350 .009 280 7. 8438 .007 267 8. 3527 .023 224 7. 9011 .008 347 8. 1489 .014 306 8. 1489 .014 54 9. 3286 .213 294 8. 0302 .011 293 7. 9350 .009 280 9. 5900 .389 237 7.9011 .008 13 8. 3544 .023 319 8.0302 .011 67 8. 5201 .033 127 8. 1489 .014 306 8. 0302 .011 293 8. 5519 .036 70 8. 1489 .014 54 8. 3544 .023 41 7. 9171 .008 108 7. 6029 .004 168 7. 9011 .008 347 7. 8898 .008 334 8. 2274 .017 111 8. 1489 .014 306 8. 0302 .011 293 7. 9171 .008 252 :..:: 8. 1443 .014 239 7. 7748 .006 239 7. 6181 .004 226 6. 8959 .001 183 9. 3286 .213 66 8. 5201 .033 233 7. 6029 .004 192 8. 1443 .014 121 6. 2382 .000 359 7. 3397 .002 346 9. 5900 .389 123 8.0302 .011 67 8. 1489 .014 54 7. 9011 .008 13 7. 7748 .006 121 6. 2382 .000 1 ::::: 7.9011 .008 347 7. 9950 .010 124 7. 9350 .009 80 8. 0302 .011 67 7. 8898 .008 26 7. 6181 .004 134 7. 3397 .002 14 7. 9011 .008 13 7. 7821 .006 137 9. 5900 .389 123 8. 5519 .036 290 8. 2274 .017 249 6. 8959 .001 177 9. 5900 .389 237 7. 9950 .010 236 7. 7821 .006 223 :...: 7. 7661 .006 29 7. 6982 .005 16 8.2405 .017 155 7. 8514 .007 83 7. 8955 .008 142 9. 2133 .163 321 8. 3852 .024 308 8. 0954 .012 85 7. 7796 .006 318 7. 7661 .006 331 .005 344 8. 2405 .017 205 7. 8514 .007 277 7. 8955 .008 218 9. 2133 .163 39 8. 3852 .024 52 8. 0954 .012 275 HARMONIC ANALYSIS AND PREDICTION OF TIDES 255 Table 29. — Elimination factors — Continued SERIES 250 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents (B, C, etc.) sought (A) K 2 U M 2 N 2 2N R 2 s 2 T 2 x 2 M2 n 2SM Kj— - - 8. 3544 .023 319 8. 1489 .014 306 8. 0302 .011 293 7. 9350 .009 280 9. 5900 .389 237 9. 3286 .213 294 8. 4129 .026 350 8. 4768 .030 280 7. 7796 .006 318 7. 7661 .006 331 8. 3023 .020 101 L 2 8. 3544 .023 41 7. 9011 .008 347 7. 8898 .008 334 7. 8703 .007 321 8. 5672 .037 98 8. 2405 .017 155 8. 3634 .023 31 9. 2133 .163 321 6. 2382 .000 359 7.6029 .004 192 7. 8955 .008 143 M 2 8. 1489 .014 54 7. 9011 .008 13 7.9011 .008 347 7. 8898 .008 334 8. 2274 .017 111 7. 6029 .004 168 8. 1376 .014 44 8.2405 .017 155 7. 6029 .004 192 8.2405 .017 205 7. 5928 .004 155 N 2 8. 0302 .011 67 7. 8898 .008 26 7.9011 .008 13 7.9011 .008 347 7. 9950 .010 124 6. 2382 .000 1 8. 0259 .011 58 7. 6029 .004 168 8. 2405 .017 205 9. 2133 .163 39 7. 1697 .001 168 2N 7. 9350 .009 80 7. 8703 .007 39 7. 8898 .008 26 7. 9011 .008 13 7. 7821 .006 137 7. 3397 .002 14 7. 9414 .009 71 6. 2382 .000 1 9. 2133 .163 39 8. 3852 .024 52 6. 2381 .000 2 R 2 9. 5900 .389 123 8. 5672 .037 262 8. 2274 .017 249 7. 9950 .010 236 7. 7821 .006 223 ::::: 9. 5901 .389 237 9. 3286 .213 294 8. 3528 .023 224 7. 9596 .009 261 8. 0954 .012 275 8 137S .014 44 S 2 9. 3286 .213 66 8.2405 .017 205 7. 6029 .004 192 6. 2382 .000 359 7. 3397 .002 346 9. 5901 .389 123 9. 5901 .389 237 7. 9011 .008 347 7. 5928 .004 205 7.8955 .008 218 7 6029) .004 168 T 2 8. 4129 .026 10 8. 3634 .023 329 8. 1376 .014 316 8. 0259 .011 302 7. 9414 .009 289 9. 3286 .213 66 9. 5901 .389 123 8. 5519 .036 290 7. 7084 .005 328 7. 6345 .004 341 8.2274 .017 111 M 8. 4768 .030 80 9. 2133 .163 39 8. 2405 .017 205 7. 6029 .004 192 6. 2382 .000 359 8. 3528 .023 136 7.9011 .008 13 8. 5519 .036 70 7. 8955 .008 218 8. 1962 .016 231 6.2382 .000 1 m — 7. 7796 .006 42 6.2382 .000 1 7. 6029 .004 168 8.2405 .017 155 9. 2133 .163 321 7. 9596 .009 99 7. 5928 .004 155 7. 7084 .005 32 7. 8955 .008 142 7. 9011 .008 13 7. 5759 .004 143 V2 7. 7661 .006 29 7. 6029 .004 168 8. 2405 .017 155 9. 2133 .163 321 8. 3852 .024 308 8. 0954 .012 85 7. 8955 .008 142 7. 6345 .004 19 8. 1962 .016 129 7.9011 .008 347 — - 7. 7671 .006 130 2SM 8. 3023 .020 259 7. 8955 .008 218 7.5928 .004 205 7. 1697 .001 192 6. 2381 .000 358 8. 1376 .014 316 7. 6029 .004 192 8. 2274 .017 249 6. 2382 .000 359 7. 5759 .004 217 7. 7671 .006 230 ::: 256 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 279 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C, etc.) U) h Kj Mi Oi OO Pi Qi 2Q Si Pi Jx 8. 0816 .012 337 8.0816 .012 23 8. 1800 .015 76 8. 18C0 .015 284 9. 3172 .208 95 8. 0127 .010 99 7. 8250 .007 121 9. 4495 .282 138 7. 7947 .006 35 8. 0469 .011 315 8. 0816 .012 337 8. 3961 .025 54 8. 0127 .010 261 8. 5477 .035 252 8. 1800 .015 76 8. 0127 .010 99 8. 4890 .031 295 7. 5510 .004 12 8. 0127 .010 261 8. 1800 .015 284 8. 3961 .025 306 7. 5571 .004 208 7. 7343 .005 199 8. 0816 .012 23 8. 0469 .011 45 8. 1523 .014 241 8. 3798 .024 139 8. 3961 .025 54 8. 1800 .015 76 8. 0127 .010 99 7. 5571 .004 152 7. 3456 .002 171 6. 7358 .001 175 7.1964 .002 18 7. 9211 .008 34 7. 7771 .006 111 8. 3841 .024 242 9. 3172 .208 265 8. 5477 .035 108 7. 7343 .005 161 7. 3456 .002 189 6. 8592 .001 4 7. 5574 .004 26 9. 4495 .282 222 7. 9990 .010 120 7. 8250 .007 239 8. 0127 .010 261 8. 1800 .015 284 8. 0816 .012 337 6. 7358 .001 185 .6. 8592 .001 356 8. 0816 .012 23 7. 8251 .007 219 9. 3242 .211 296 7. 5672 .004 216 7. 8250 .007 239 8.0127 .010 261 8. 0469 .011 315 7. 1964 .002 342 7. 5574 .004 334 8. 0816 .012 337 7. 3460 .002 196 8.4421 .028 274 7. 9987 .010 200 9. 4495 .282 222 8. 4890 .031 65 8. 1523 .014 119 7. 9211 .008 326 9. 4495 .282 138 7. 8251 .007 141 7. 3460 .002 164 8. 0384 .011 77 7. 8339 K 2 8. 0816 .012 23 8. 0469 .011 45 8. 0127 .010 99 8. 3961 .025 306 8. 3841 .024 118 7. 8250 .007 121 7. 5672 .004 144 7. 9987 .010 160 7. 8339 .007 57 .007 303 7. 7947 Mi - „— ■ .006 325 7. 5510 Oi .004 348 8. 3798 00 .024 221 7. 7771 Pj .006 249 7. 9990 Qj .010 240 9. 3242 2Q .211 64 8. 4421 Si .028 86 8. 0384 .011 283 ".:: HARMONIC ANALYSIS AND PREDICTION OF TIDES 257 Table 29. — Elimination factors — Continued SERIES 279 DAYS. SEMIDIURNAL CONSTITUENTS Constituent sought 04) K2- La- M 2 . Ni_. 2N_ R 2 -. Sa~. Ta_. (12 2SM. Disturbing constituents (P, C, etc.) K 2 8. 3961 .025 54 8. 1800 .015 76 8.0127 .010 7. 8250 .007 121 9. 4494 .281 138 9. 3172 .208 95 9. 0421 .110 52 8. 3841 .024 118 7. 8339 .007 57 7. 7947 .006 35 8. 2246 .017 294 L 2 3961 025 306 ,012 23 Oil 45 .010 68 8. 5211 .033 264 8. 3798 .024 221 6.9057 .001 359 9. 3242 .211 64 6. 8592 .001 4 7. 7343 .005 161 .010 240 M 2 .1800 .015 284 .0816 .012 337 8.0816 .012 23 8. 0469 .011 45 8. 1523 .014 241 7. 7343 .005 199 7. 8491 .007 336 8. 3798 .024 221 7. 7343 .005 161 8. 3798 .024 139 7. 7105 .005 218 8.0127 010 261 8.0469 .011 315 .0816 .012 337 N 2 2N 8. 0816 .012 23 7.8251 .007 219 6. 8592 .001 356 7. 9108 .008 314 7. 7343 .005 8. 3798 .024 139 9. 3242 .211 296 7. 2354 .002 195 7. 8250 .007 239 ,010 292 ,011 315 ,0816 .012 337 7. 3463 .002 196 7. 8721 .007 294 6. 8592 .001 356 9. 3242 .211 296 8. 4421 .028 274 6. 8588 .001 352 Ra .281 222 8. 5211 .033 96 8. 1523 .014 119 7. 8251 .007 141 7.3463 .002 164 9. 4496 .282 137 9.3172 .208 95 7.9987 .010 160 7.9102 .008 100 8.0384 .011 77 7. 8491 .007 336 9.3172 .208 265 8. 3798 .024 139 7. 7343 .005 161 6. 8592 .001 4 7. 8721 .007 1. 4496 .282 223 .282 137 8. 0816 .012 123 7. 7105 .005 142 7.9990 .010 120 7. 7343 .005 199 T 3 1. 0421 .110 6.9057 .001 1 7. 8491 .007 24 7.9108 .008 46 7. 8885 .008 69 9.3172 .208 265 282 223 8.4891 .031 65 6. 8703 .001 5 7. 5541 .004 162 8. 1523 .014 241 8. 3841 .024 242 9. 3242 .211 296 8. 3798 .024 139 7. 7343 .005 161 6. 8592 .001 4 7.9987 .010 200 8. 0816 .012 237 8.4891 .031 295 .010 120 8. 2553 .018 97 6. 8592 .001 356 7. 8339 .007 303 6. 8592 .001 356 7. 7343 • 005 199 8. 3798 024 221 9. 3242 .211 64 7. 9102 • 008 260 7.7105 .005 218 6. 8703 .001 355 7.9990 .010 240 8.0816 .012 337 005 236 7. 7947 .006 325 7. 7343 .005 199 8. 3798 .024 221 9. 3242 .211 64 8. 4421 .028 ,011 283 ,010 240 7. 5541 .004 8. 2553 .018 263 8.0816 .012 23 .007 259 2SM 8. 2246 017 .010 120 7. 7105 .005 142 7. 2354 .002 165 .001 7. 8491 .007 24 7. 7343 .005 161 8. 1523 .014 119 6. 8592 .001 4 .005 124 ,007 101 258 U. S. COAST AND GEODETIC STJRVEY Table 29. — Elimination factors — Continued SERIES 297 DAYS. DIURNAL CONSTITUENTS Constituent sought Disturbing constituents (B, C, etc.) {A) Ji Ki Mi Oi OO Pi Q = 2Q Si Pi J,L 8. 2770 .019 220 8. 2770 .019 140 8. 0269 .011 133 8. 0269 .011 227 9. 2565 .181 113 7. 9899 .010 94 7. 7726 .006 54 9. 3360 .217 146 7. 3907 .002 14 8. 1622 .015 260 8. 2770 .019 220 7. 5338 .003 173 7. 9899 .010 266 8. 2044 .016 333 8. 0269 .011 133 7. 9899 .010 94 7. 5392 .003 186 8. 1019 .013 54 7. 9899 .010 266 8. 0269 .011 227 7. 5338 .003 187 7. 8638 .007 273 7. 7467 .006 339 8. 2770 .019 140 8. 1622 .015 100 7. 5336 .003 193 8. 4724 .030 60 7. 5338 .003 173 8. 0269 .011 133 7. 9899 .010 94 7. 8638 .007 87 8. 0954 .012 66 7. 6320 .004 47 6. 7805 .001 7 8. 1433 .014 100 7. 5129 .003 147 8. 3896 .025 287 9. 2565 .181 247 8. 2044 .016 27 7. 7467 .006 21 8. 0954 .012 294 7. 5300 .003 161 7. 8163 .007 121 9. 3360 .217 214 8. 0286 .011 81 7. 7726 .006 306 7. 9899. .010 266 8. 0269 .011 227 8. 2770 .019 220 7. 6320 .004 313 7. 5300 .003 199 8. 2770 .019 140 7. 9031 .008 233 9. 3366 .217 280 7. 1486 .001 346 7. 7726 .006 306 7. 9899 .010 266 8. 1622 .015 260 6. 7805 .001 353 7. 8163 .007 239 8. 2770 .019 220 7. 8741 .007 273 8. 2220 .017 320 8. 4204 .026 253 9. 3360 .217 214 7. 5392 .003 174 7. 5336 .003 167 8. 1433 .014 260 9. 3360 .217 146 7. 9031 .008 127 7. 8741 .007 87 7. 8897 .008 48 7.5223 Ki 8. 2770 .019 140 8. 1622 .015 100 7. 9899 .010 94 7. 5338 .003 187 8. 3896 .025 73 7. 7726 .006 54 7. 1486 .001 14 8. 4204 .026 107 7. 5223 .003 154 .003 206 7. 3907 Mi .002 346 8. 1019 Cm .013 306 8.4724 OO .030 300 7.5129 Pi .003 213 8. 0286 Qi .011 279 9. 3366 2Q .217 80 8. 2220 Si .017 40 7. 8897 .008 312 .:.: HARMONIC ANALYSIS AND PREDICTION OF TIDES 259 Table 29. — Elimination factors — Continued SERIES 297 DAYS. SEMIDIURNAL CONSTITUENTS Constituent sought 04) Kj M 2 - Na. 2N. R 2 - S 2 . Tj. \2— VI 2SM_ Disturbing constituents (B, C, etc.) 7. 5338 .003 173 .011 133 .010 94 7. 7726 .006 54 9. 3359 .217 146 9. 2565 .181 113 9. 1076 .128 79 8. 3896 .025 73 7. 5223 .003 154 7. 3907 .002 14 8. 2357 .017 272 L 2 7. 5338 .003 187 8. 2770 .019 140 8. 1622 .015 100 .009 .1514 ,014 333 .4724 .030 300 .5711 ,037 9. 3366 .217 80 7. 5300 .003 161 7. 7467 .006 21 8. 0286 .011 279 M 2 N 2 !. 0269 .011 227 !. 2770 .019 220 8. 2770 .019 140 8. 1622 .015 100 7. 5339 .003 193 7. 7467 .006 .013 8.4724 .030 300 7. 7467 .006 21 8.4724 .030 60 7. 7179 .005 319 .010 266 8. 1622 .015 260 8. 2770 .019 220 8. 2770 .019 140 7. 9031 .008 233 7. 5300 .003 199 7. 4214 .003 346 7. 7467 .006 339 8.4724 .030 9.3366 .217 280 6. 2014 .000 359 2N 7. 7726 .006 306 7.9326 .009 300 8. 1622 .015 260 8. 2770 .019 220 7. 8741 .007 273 7. 8163 .007 239 7. 5240 .003 205 7. 5300 .003 199 .217 280 8. 2220 .017 320 7. 5050 .003 218 R 2 9. 3359 .217 214 8. 1514 .014 27 7. 5339 .003 167 7.9031 .008 127 7.8741 .007 87 9. 3362 .217 146 9. 2566 .181 113 8. 4204 .026 107 7.0150 .001 48 ,1268 013 306 S 2 9. 2565 .181 247 8. 4724 .030 60 7. 7467 .006 21 7. 5300 .003 161 7. 8163 .007 121 9. 3362 217 214 1. 3362 .217 146 !. 2770 .019 140 7. 7179 .005 41 .011 81 7. 7467 006 339 T 2 9. 1076 .128 281 8. 5711 .037 94 8. 1268 .013 54 7.4214 .003 14 7. 5240 .003 155 9. 2566 .181 247 9. 3362 .217 214 7. 5385 .003 174 7. 8920 .008 75 8. 0039 .010 115 7. 5339 .003 193 8. 3896 .025 287 ,217 8.4724 .030 60 7. 7467 .006 21 7. 5300 .003 161 8.4204 .026 253 8. 2770 .019 220 7. 5385 .003 8. 0286 .011 81 8. 1647 .015 121 7. 5300 .003 7.5223 .003 206 7.5300 .003 199 7. 7467 .006 339 8.4724 .030 300 9. 3366 .217 7. 0150 .001 352 7. 7179 .005 319 7. 8920 .008 285 Oil 279 8. 2770 .019 220 .005 298 7. 3907 .002 346 7. 7467 .006 339 8. 4724 .030 300 9. 3366 .217 80 8. 2220 .017 40 .008 312 .011 279 8.0039 .010 245 8. 1647 .015 239 8. 2770 .019 140 7. 7984 .006 258 2SM 8.2357 .017 8.0286 .011 81 7. 7179 .005 41 5. 2014 .000 1 7. 5050 003 142 8. 1268 .013 54 7. 7467 .006 21 7. 5339 003 167 7. 5300 003 161 ,005 62 7. 7984 .006 102 260 U. S. COAST AND GEODETIC STJBVEY Table 29. — Elimination factors — Continued SERIES 326 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) Mi. 00. Qi- 2Q. Disturbing constituents (B, C. etc.) 8. 1340 .014 150 .012 119 7.8631 .007 125 7. 4352 .003 354 8. 3392 .022 111 7. 8244 .007 95 7. 6841 .005 64 8.2809 .019 130 7. 0934 .001 170 8. 1340 .014 210 8. 1340 .014 150 7. 7427 .006 156 7. 7427 .006 204 9.0470 .111 141 7. 8631 .007 125 7. 8244 .007 95 9. 0723 .118 161 7. 5061 .003 20 Mi .012 241 8. 1340 .014 210 7. 4352 .003 7.8631 .007 235 7. 6587 .005 352 7. 7427 .006 156 7. 8631 .007 125 7. 7472 .006 191 8.0423 .011 50 7.8631 .007 235 7. 7427 .006 204 7. 4352 .003 354 7. 7018 .005 229 7. 5483 .004 346 8. 1340 .014 150 8. 0698 .012 119 7. 0956 .001 185 8. 3386 .022 45 OO 7. 4352 .003 7. 7427 .006 156 7. 8631 .007 125 7. 7018 .005 131 8. 0443 .011 117 7. 7204 .005 101 7. 6227 .004 70 136 5. 6245 .000 176 8. 3392 .022 249 9.0470 .111 219 7. 6587 .005 7. 5483 .004 14 8.0443 .011 243 .003 164 7. 7040 .005 133 9. 0723 .118 7. 9254 7. 8244 .007 265 7. 8631 .007 235 7. 7427 .006 204 8. 1340 .014 210 7. 7204 .005 259 7. 4186 .003 196 5. 1340 .014 150 r. 7258 .005 216 ). 2882 .194 255 2Q 7. 6841 .005 296 7. 8244 .007 265 7. 8631 .007 235 8.0698 .012 241 7. 6227 .004 290 7. 7040 .005 227 8. 1340 .014 210 7.7948 .006 246 8.3594 .023 285 8. 2809 .019 230 9.0723 .118 7. 7472 .006 169 7.0956 .001 175 7.9496 .009 224 9.0723 .118 161 7. 7258 .005 144 7. 7948 .006 114 7. 7844 .006 7.0934 .001 190 7. 5061 .003 340 8.0423 .011 310 8. 3386 .022 315 6. 6245 , .000 184 7.9254 .008 301 9. 2882 .194 105 8. 3594 .023 75 7. 7844 .006 321 HARMONIC ANALYSIS AND PREDICTION OF TIDES 261 Table 29. — Elimination factors — Continued SERIES 326 DAYS. SEMIDIURNAL CONSTITUENTS Constituent sought (A) K:_ L 2 - Ms N 2 2N R 2 - S 2 T 2 X 2 Vi 2SM. Disturbing constituents (B, C, etc.) K, 7. 4352 .003 6 7. 7427 .006 156 7. 8631 .007 125 7. 8244 .007 95 9. 0720 .118 161 .0470 .111 141 .0037 .101 122 .3392 .022 111 7.0934 001 170 7. 5061 .003 20 .0971 ,013 307 La 7. 4352 .003 354 .1340 .014 150 .012 119 7.9526 .009 8. 0858 .012 335 8. 3386 .022 315 8. 4852 .031 296 9. 2882 .194 105 7.4186 .003 164 7. 5483 .004 14 7. 9254 .008 301 M 2 7. 7427 .006 204 8. 1340 .014 210 8. 1340 .014 150 .012 119 7. 0956 .001 185 7. 5483 .004 346 7.9190 .008 326 8. 3386 .022 315 7. 5483 .004 14 8. 3386 .022 45 7. 5347 .003 332 N 2 7. 8631 .007 235 .012 241 i. 1340 .014 210 8. 1340 .014 150 7. 7259 .005 216 7. 4186 .003 196 6. 7213 .001 357 7. 5483 .004 346 .3386 ,022 45 ,2882 ,194 255 ,3062 000 182 2N 7. 8244 .007 265 7. 9526 .009 271 8. 0698 .012 241 8. 1340 014 210 7. 7948 006 246 7.7040 005 227 7. 5123 .003 207 7.4186 .003 196 .2882 .194 255 .3594 .023 285 7. 4010 003 212 R 2 9. 0720 .118 199 8. 0858 .012 25 7. 0956 .001 175 7. 7259 .005 144 7. 7948 .006 114 ,0725 118 161 9. 0472 .112 141 8. 2809 .019 130 7. 0444 .001 7. 7843 006 39 7. 9190 08 326 9. 0470 .111 219 8. 3386 .022 45 7. 5483 .004 14 7. 4186 .003 164 7. 7040 .005 133 9.0725 .118 199 9. 0725 118 161 8. 1340 014 150 7.4347 003 28 7.9254 008 59 7. 5483 .004 346 Tj 9. 0037 .101 238 8. 4852 .031 64 7.9190 .008 34 6. 7213 .001 3 7. 5123 .003 153 .0472 ,112 219 9. 0725 118 199 7. 7568 006 169 7. 7359 005 48 7. 9960 010 78 7. 0956 001 185 8. 3392 .022 249 9. 2882 .194 255 8. 3386 .022 45 7. 5483 .004 14 .003 164 $. 2809 .019 230 1. 1340 .014 210 7. 7468 006 191 7. 9254 .008 59 .1912 ,016 89 7. 4186 003 196 7. 0934 .001 190 7.4186 .003 196 7. 5483 .004 346 .022 315 .194 105 7. 0444 .001 351 7. 5347 003 332 7. 7359 005 312 7. 9254 m 301 8. 1340 014 210 7. 5118 003 317 7. 5061 .003 340 7. 5483 004 346 8. 3386 .022 315 ,2882 ,194 105 8. 3594 .023 75 7. 7843 006 321 7. 9254 08 301 .010 282 8. 1912 016 271 8. 1340 014 150 7. 7477 006 287 2SM 8. 0971 .013 7.9254 7. 5347 .003 28 6. 3062 .000 178 7. 4010 .003 148 7.9190 .008 34 7. 5483 .004 14 7. 0956 001 175 7. 4186 .003 164 7. 5118 .003 43 7. 7477 006 73 262 U. S. COAST AND GEODETIC STJItVEY Table 29. — Elimination factors — Continued SERIES 355 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) M. 00. Disturbing constituents (B, C, etc.) 7. 9464 .009 159 7. 9167 .008 138 7. 5111 .003 157 341 8. 0444 .011 149 7. 6331 .004 136 7. 6506 .004 115 8. 0032 .010 154 6. 7724 .001 5 7. 9464 .009 201 7. 9464 6. 7064 .001 178 6. 7064 .001 182 8. 4581 .029 170 7. 5111 .003 157 7. 6331 .004 136 8. 4598 .029 175 7. 5794 .004 26 M 7. 9167 .008 222 7. 9464 .009 201 7. 8888 .008 19 7.5111 .003 203 7. 7393 .005 191 6. 7064 .001 178 7. 5111 .003 157 7. 8651 .007 196 7. 9839 .010 47 7. 5111 .003 203 6. 7064 .001 182 7. 8888 .008 341 6. 7060 .001 185 7. 2500 .002 352 7. 9464 .009 159 7. 9167 .008 138 6. 7729 .001 357 8. 1364 .014 29 OO .008 19 6. 7064 .001 178 7. 5111 .003 157 6. 7060 .001 175 7. 3914 .002 168 7. 3284 .002 154 7. 4740 .003 133 7. 1838 .002 173 7. 3105 .002 24 8.0444 .011 211 8. 4581 .029 190 7. 7393 .005 7.2500 .002 8 7. 3914 .002 192 7. 2957 .002 167 7. 5554 .004 146 .029 185 7. 7296 .005 7. 6331 .004 224 7. 5111 .003 203 6. 7064 .001 182 7. 9464 .009 201 7. 3284 .002 7. 2957 .002 193 7. 9464 .009 159 7. 4212 .003 9. 1482 .141 230 2Q 7. 6506 .004 245 7. 6331 .004 224 7. 5111 .003 203 7. 9167 .008 222 7. 4740 .003 227 7. 5554 .004 214 7. 9464 .009 201 7. 5984 .004 219 8. 3129 .021 251 8. 0032 .010 206 8. 4598 .029 185 7. 8651 .007 164 6. 7729 .001 3 .002 187 .029 175 .4212 .003 162 .004 141 6.7724 .001 355 7. 5794 .004 334 7. 9839 .010 313 8. 1364 .014 331 7. 3105 .002 336 7. 7296 .005 324 9. 1482 .141 130 8. 3129 .021 ' 109 7. 6607 .005 329 7. 6607 .005 31 HARMONIC ANALYSIS AND PREDICTION OF TIDES 263 Table 29. — Elimination factors — Continued SERIES 355 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents (B, C, etc.) sought (A) K 2 L 2 M 2 Na 2N Ra s 2 T 2 *a M2 V2 2SM Ka 7.8888 .008 341 6. 7064 .001 182 7.5111 .003 203 7. 6331 .004 224 8.4589 .029 185 8. 4581 .029 190 8.4553 .029 195 8. 0444 .011 211 6. 7724 .001 355 7. 5794 .004 334 7.6440 ::: .004 18 La 7. 8888 .008 19 7.9464 .009 201 7. 9167 .008 222 7. 8652 .007 243 8. 0217 .011 24 8. 1364 .014 29 8. 2393 .017 34 9. 1482 .141 230 7. 2957 .002 193 7. 2500 .002 352 7. 7296 .005 36 M 2 6. 7064 .001 178 7.9464 .009 159 7. 9464 .009 201 7.9167 .008 222 6. 7712 .001 3 7. 2500 .002 8 7.4842 .003 13 8. 1364 .014 29 7. 2500 .002 352 8. 1364 .014 331 7. 2458 .002 15 Na 7.5111 .003 157 7.9167 .008 138 7. 9464 .009 159 7.9464 .009 201 7. 4212 .003 162 7. 2957 .002 167 7. 1008 .001 172 7. 2500 .002 8 8.1364 .014 331 9. 1482 .141 130 6. 7017 .001 174 2N... 7. 6331 .004 136 7. 8652 .007 117 7. 9167 .008 138 7. 9464 .009 159 :::: 7. 5985 .004 141 7. 5554 .004 146 7. 5017 .003 151 7. 2957 .002 167 9. 1482 .141 130 8. 3129 .021 109 7.2840 .002 154 R 2 8. 4589 .029 175 8. 0217 .011 336 6. 7712 .001 357 7. 4212 .003 198 7. 5985 .004 219 8.4598 .029 185 8. 4586 .029 190 8. 0034 .010 206 7. 0678 .001 350 7. 6606 .005 329 7.4842 .003 13 S 2 8. 4581 .029 170 8. 1364 .014 331 7. 2500 .002 352 7. 2957 .002 193 7. 5554 .004 214 8. 4598 .029 175 — - 8. 4598 .029 185 7. 9464 .009 201 7. 2458 .002 345 7. 7296 .005 324 7. 2500 .002 8 Ta 8.4553 .029 165 8. 2393 .017 326 7. 4842 .003 347 7. 1008 .001 188 7. 5017 .003 209 8. 4586 .029 170 8. 4598 .029 175 7. 8648 .007 196 7. 3738 .002 340 7. 7893 .006 319 6. 7712 .001 3 X 2 8.0444 .011 149 9. 1482 .141 130 8. 1364 .014 331 7. 2500 .002 352 7. 2957 .002 193 8. 0034 .010 154 7.9464 .009 159 7. 8648 .007 164 .... 7. 7296 .005 324 8. 0795 .012 303 7. 2957 .002 167 6. 7724 .001 5 7. 2957 .002 167 7. 2500 .002 8 8. 1364 .014 29 9. 1482 .141 230 7. 0678 .001 10 7. 2458 .002 15 7. 3738 .002 20 7. 7296 .005 36 .... 7.9464 .009 159 7. 2393 .002 23 7. 5794 .004 26 7. 2500 .002 8 8. 1364 .014 29 9. 1482 .141 230 8. 3129 .021 251 7. 6606 .005 31 7. 7296 .005 36 7. 7893 .006 41 8. 0795 .012 57 7. 9464 .009 201 :::: 7. 5728 .004 44 2SM 7. 6440 .004 342 7. 7296 .005 324 7. 2458 .002 345 6. 7017 .001 186 7.2840 .002 206 7. 4842 .003 347 7. 2500 .002 352 6. 7712 .001 357 7. 2957 .002 193 7.2393 .002 337 7. 5728 .004 316 — 264 U. S. COAST AND GEODETIC SURVEY Table 29. — Elimination factors — Continued SERIES 369 DAYS. DIURNAL CONSTITUENTS Constituent sought (A) Ki. M, Oi._ 00. Qi- 2Q. Disturbing constituents (B, C, etc ) 8. 3503 .022 70 7. 8740 .007 141 7. 8760 .008 73 8. 3371 .022 248 8. 2982 .020 74 7. 5509 .004 143 7. 4182 .003 34 8. 3235 .021 72 5. 7099 .000 8. 3503 .022 290 8. 3503 .022 70 6. 6332 .000 2 6. 6332 .000 358 8. 0072 .010 4 7. 8760 .008 73 7. 5509 .004 143 8. 0074 .010 2 .008 110 M, 7. 8740 .007 219 8. 3503 .022 290 8. 3371 .022 112 7. 8760 .008 287 8. 4104 .026 293 6. 6332 .000 2 7. 8760 .008 73 8. 3792 .024 291 7. 9045 .008 39 7. 8760 .008 287 6. 6332 .000 358 8. 3371 .022 248 6. 6329 .000 356 6. 5537 .000 182 8. 3503 .022 70 7. 8740 .007 141 5. 7100 .000 ,026 OO 8. 3371 .022 112 6. 6332 .000 2 7. 8760 .008 73 6. 6329 .000 4 7. 0438 .001 7. 6584 .005 75 7. 3523 .002 145 6. 8924 .001 4 7. 6527 .004 112 8.2982 .020 286 8. 0072 .010 356 8. 4104 .026 67 6. 5537 .000 178 7. 0438 .001 354 7. 8885 .008 7.6024 .004 139 8. 0074 .010 358 7. 9217 .008 106 7. 5509 .004 217 7. 8760 .008 287 6. 6332 .000 358 8. 3503 .022 290 7. 6584 .005 285 7. 8885 .008 291 8. 3503 .022 70 7. 8824 .008 289 9. 0329 .108 217 2Q 7. 4182 .003 326 7. 5509 .004 217 7. 8760 .008 287 7. 8740 .007 219 7. 3523 .002 215 7. 6024 .004 221 8. 3503 .022 290 7. 5772 .004 219 8. 0586 .011 327 I 3235 .021 8. 0074 .010 358 8. 3792 .024 5. 7100 .000 6. 8924 .001 356 8. 0074 .010 2 7.8824 .008 71 7. 5772 .004 141 7. 9055 .008 108 5. 7099 .000 7. 8892 .008 250 7. 9045 .008 321 8. 4169 .026 252 7. 6527 .004 248 7. 9217 .008 254 9. 0329 .108 143 8. 0586 .011 7. 9055 .008 252 HABMONIC ANALYSIS AND PREDICTION OF TIDES 265 Table 29. — Elimination factors — Continued SERIES 369 DAYS. SEMIDIURNAL CONSTITUENTS Constituent Disturbing constituents (B, C, etc.) sought (A) K 2 U M 2 N 2 2N R2 s 2 T 2 x 2 M2 l>2 2SM K 2 - 8. 3371 .022 248 6. 6332 .000 358 7. 8760 .008 287 7. 5509 .004 217 8. 0074 .010 358 8. 0072 .010 356 8. 0076 .010 354 8. 2982 .020 286 5. 7099 .000 7.8892 .008 250 7. 1088 .001 175 L 2 8. 3371 .022 112 8. 3503 .022 290 7. 8740 .007 219 7. 6166 .004 329 8. 3758 .024 110 8. 4169 .026 108 8.4607 .029 106 9. 0329 .108 217 7. 8885 .008 291 6. 5537 .000 182 7.9217 .008 106 M2 6. 6332 .000 2 8. 3503 .022 70 8. 3503 .022 290 7. 8740 .007 219 5. 7100 .000 6. 5537 .000 178 6.9049 .001 177 8. 4169 .026 108 6. 5537 .000 182 8. 4169 .026 252 6.5549 .000 177 N2 7. 8760 .008 73 7. 8740 .007 141 8. 3503 .022 70 8. 3503 .022 290 7. 8824 .008 71 7. 8885 .008 69 7. 8944 .008 67 6. 5537 .000 178 8. 4169 .026 252 9. 0329 .108 143 7. 6658 .005 67 2N - 7. 5509 .004 143 7. 6166 .004 31 7. 8740 .007 141 8. 3503 .022 70 7. 5772 .004 141 7. 6024 .004 139 7. 6268 .004 138 7. 8885 .008 69 9. 0329 .108 143 8. 0586 .011 33 7. 4452 .003 138 R 2 - - 8. 0074 .010 2 8. 3758 .024 250 5. 7100 .000 7.8824 .008 289 7. 5772 .004 219 8. 0074 .010 358 8. 0072 .010 356 8. 3235 .021 288 6. 1771 .000 181 7.9055 .008 252 6. 9049 .001 177 S2 8. 0072 .010 4 8.4169 .026 252 6. 5537 .000 182 7. 8885 .008 291 7. 6024 .004 221 8.0074 .010 2 8.0074 .010 358 8. 3503 .022 290 6. 5549 .000 183 7. 9217 .008 254 6. 5537 .000 178 T 2 8. 0076 .010 6 8.4607 .029 254 6. 9049 .001 183 7.8944 .008 293 7. 6268 .004 222 8. 0072 .010 4 8.0074 .010 2 8. 3792 .024 291 6. 7601 .001 185 7.9377 .009 256 5. 7100 .000 Xa — - 8. 2982 .020 74 9. 0329 .108 143 8.4164 .026 252 6. 5537 .000 182 7.8885 .008 291 8. 3235 .021 72 8. 3503 .022 70 8. 3792 .024 69 7.9217 .008 254 7. 9044 .008 324 7. 8885 .008 69 5. 7099 .000 7. 8885 .008 69 6. 5537 .000 178 8.4169 .026 108 9. 0329 .108 217 6. 1771 .000 179 6. 5549 .000 177 6. 7601 .001 175 7.9217 .008 106 8. 3503 .022 70 6. 5542 .000 175 »»2-.__ -- 7. 8892 .008 110 6. 5537 .000 178 8. 4169 .026 108 9.0329 .108 217 8. 0586 .011 327 7. 9055 .008 108 7.9217 .008 106 7.9377 .009 104 7. 9044 .C08 36 8. 3503 .022 290 7. 6990 .005 105 2SM 7. 1088 .001 185 7.9217 .008 254 6. 5549 .000 183 7. 6658 .005 293 7. 4452 .003 222 6.9049 .001 183 6. 5537 .000 182 5. 7100 .000 7. 8885 .008 291 6. 5542 .000 185 7. 6990 .005 255 266 U. S. COAST AND GEODETIC SURVEY Table 30. — Products of amplitudes and angular functions for Form 245 1 2 3 4 5 Sin Cos Sin Cos Sin Cos Sin Cos Sin Cos 0.000 1.000 0.000 2.000 0.000 3.000 0.000 4.000 0.000 5.000 90 1 2 3 .017 .035 .052 1.000 0.999 .999 .035 .070 .105 2.000 1.999 1.997 .052 .105 .157 3.000 2.998 2.996 .070 .140 .209 3.999 3.998 3.995 .087 .174 .262 4.999 4.997 4.993 89 88 87 4 5 6 .070 .087 . .105 .998 .996 .995 .140 .174 .209 1.995 1.992 1.989 .209 .261 .314 2.993 2.989 2.984 .279 .349 .418 3.990 3.985 3.978 .349 .436 .523 4.988 4.981 4.973 86 85 84 7 8 9 .122 .139 .156 .993 .990 .988 .244 .278 .313 1.985 1.981 1. 975 .366 .418 .469 2.978 2.971 2.963 .487 .557 .626 3.970 3.961 3.951 .609 .696 .782 4.963. 4.951 4.938 83 82 81 10 11 12 .174 .191 .208 .985 .982 .978 .347 .382 .416 1.970 1.963 1.956 .521 .572 .624 2.954 2.945 2.934 .695 .763 .832 3.939 3,927 3.913 .868 .954 1.040 4.924 4.908 4.891 80 79 78 13 14 15 .225 .242 .259 .974 .970 .966 .450 .484 .518 1.949 1.941 1.932 .675 .726 .776 2.923 2.911 2.898 .900 .968 1.035 3.897 3.881 3.864 1.125 1.210 1.294 4.872 4.852 4.830 77 76 75 16 17 18 .276 .292 .309 .961 .956 .951 .551 .585 .618 1.923 1.913 1.902 .827 .877 .927 2.884 2.869 2.853 1.103 1.169 1.236 3.845 3.825 3.804 1.378 1.462 1.545 4.806 4.782 4.755 74 73 72 19 20 21 .326 .342 .358 .946 .940 .934 .651 .684 .717 1.891 1.879 1.867 .977 1.026 1.075 2.837 2.819 2.801 1.302 1.368 1.433 3.782 3.759 3.734 1.628 1.710 1.792 4.728 4.698 4.668 71 70 69 22 23 24 .375 .391 .407 .927 .920 .914 .749 .781 .813 1.854 1.841 1.827 1.124 1.172 1.220 2.782 2.762 2.741 1.498 1.563 1.627 3.709 3. 682 3.654 1.873 1.954 2.034 4.636 4.602 4.568 68 67 66 25 26 27 .423 .438 .454 .906 .899 .891 .845 .877 .908 1.813 1.798 1.782 1.268 1.315 1.362 2.719 2.696 2.673 1.690 1.753 1.816 3.625 3.595 3.564 2.113 2.192 2.270 4.532 4.494 4.455 65 64 63 28 29 30 .469 .485 .500 .883 .875 .866 .939 .970 1.000 1.766 1.749 1.732 1.408 1.454 1.500 2.649 2.624 2.598 1.878 1.939 2.000 3.532 3.498 3.464 2.347 2.424 2.500 4.415 4.373 4.330 62 61 60 31 32 33 .515 .530 .545 .857 .848 .839 1.030 1.060 1.089 1.714 1.696 1.677 1.545 1.590 1.634 2.572 2.544 2.516 2.060 2.120 2.179 3.429 3.392 3.355 2.575 2.650 2.723 4.286 4.240 4.193 59 58 57 34 35 36 .559 .574 .588 .829 .819 .809 1.118 1.147 1.176 1.658 1.638 1.618 1.678 1.721 1.763 2.487 2.457 2.427 2.237 2.294 2.351 3.316 3.277 3.236 2.796 2.868 2.939 4.145 4.096 4.045 56 55 54 37 38 39 .602 .616 .629 .799 .788 .777 1.204 1.231 1.259 1.597 1.576 1.554 1.805 1.847 1.888 2.396 2.364 2.331 2.407 2.463 2. 517 3.195 3.152 3.109 3.009 3.078 3.147 3.993 3.940 3.886 53 52 51 40 41 42 .643 .656 .669 .766 .755 .743 1.286 1.312 1.338 1.532 1.509 1.486 1.928 1.968 2.007 2.298 2.264 2.229 2.571 2.624 2.677 3.064 3.019 2.973 3.214 3.280 3.346 3.830 3.774 3.716 50 49 48 43 44 45 .682 .695 0.707 .731 .719 0.707 1.364 1.389 1.414 1.463 1.439 1.414 2.046 2.084 2. 121 2.194 2.158 2.121 2.728 2.779 2.828 2.925 2.877 2.828 3.410 3.473 3.536 3.657 3.597 3.536 47 46 45 Cos Sin Cos Sin Cos Sin ' Cos Sin Cos Sin 1 2 3 4 5 HARMONIC ANALYSIS AND PREDICTION OF TIDES 267 Table 30. — Products of amplitudes and angular functions for Form 245 — Continued 6 7 8 9 o Sin Cos Sin Cos Sin Cos Sin Cos 0.000 6.000 0.000 7.000 0.000 8.000 0.000 9.000 90 L 2 3 .105 .209 .314 5.999 5.996 5.992 .122 .244 .366 6.999 6.996 6.990 .140 .279 .419 7.999 7.995 7.989 .157 .314 .471 8.999 8.995 8.988 89 88 87 4 5 6 .419 .523 .627 5.985 5.977 5.967 .488 .610 .732 6.983 6.973 6.962 .558 .697 .836 7.980 7.970 7.956 .628 .784 .941 8.978 8.966 8.951 86 85 84 7 8 9 .731 .835 .939 5.955 5.942 5.926 .853 .974 1.095 6.948 6.932 6.914 .975 1.113 1.251 7.940 7.922 7.902 1.097 1.253 1.408 8.933 8.912 8.889 83 82 81 10 11 12 1.042 1.145 1.247 5.909 5.890 5.869 1.216 1.336 1.455 6.894 6.871 6.847 1.389 1.526 1.663 7.878 7.853 7.825 1.563 1.717 1.871 8.863 8.835 8.803 80 79 78 13 14 15 1.350 1.452 1.553 5.846 5.822 5.796 1.575 1.693 1.812 6.821 6.792 6.762 1.800 1.935 2.071 7.795 7.762 7.727 2.025 2.177 2.329 8.769 8.733 8.693 77 76 75 16 17 18 1.654 1.754 1.854 5.768 5.738 5.706 1.929 2.047 2.163 6.729 6.694 6.657 2.205 2.339 2.472 7.690 7.650 7.608 2.481 2.631 2.781 8.651 8.607 8.560 74 73 72 19 20 21 1.953 2.052 2.150 5.673 5.638 5.601 2.279 2.394 2.509 6.619 6.578 6.535 2.605 2.736 2.867 7.564 7.518 7.469 2.930 3.078 3.225 8.510 8.457 8.402 71 70 69 22 23 24 2.248 2.344 2.440 5.563 5.523 5.481 2.622 2.735 2.847 6.490 6.444 6.395 2.997 3.126 3.254 7.417 7.364 7.308 3.371 3.517 3.661 8.345 8.284 8.222 68 67 66 25 26 27 2.536 2.630 2.724 5.438 5.393 5.346 2.958 3.069 3.178 6.344 6.292 6.237 3.381 3.507 3.632 7.250 7.190 7.128 3.804 3.945 4.086 8.157 8.089 8.019 65 64 63 28 29 30 2.817 2.909 3.000 5.298 5.248 5.196 3.286 3.394 3.500 6.181 6.122 6.062 3.756 3.878 4.000 7.064 6.997 6.928 4.225 4.363 4.500 7.947 7.872 7.794 62 61 60 31 32 33 3.090 3.180 3.268 5.143 5.088 5.032 3.605 3.709 3.812 6.000 5.936 5.871 4.120 4.239 4.357 6.857 6.784 6.709 4.635 4.769 4.902 7.715 7.632 7.548 59 58 57 34 35 36 3.355 3.441 3.527 4.974 4.915 4.854 3.914 4.015 4.115 5.803 5.734 5.663 4.474 4.589 4.702 6.632 6.553 6.472 5.033 5.162 5.290 7.461 7.372 7.281 56 55 54 37 38 39 3.611 3.694 3.776 4.792 4.728 4.663 4.213 4.310 4.405 5.590 5.516 5.440 4.815 4.925 5.035 6.389 6.304 6.217 5.416 5.541 5.664 7.188 7.092 6.994 53 52 51 40 41 42 3.857 3.936 4.015 4.596 4.528 4.459 4.500 4.592 4.684 5.362 5.283 5.202 5.142 5.248 5.353 6. 128 6.038 5.945 5.785 5.905 6.022 6.894 6.792 6.688 50 49 48 43 44 45 4.092 4.168 4.243 4.388 4.316 4.243 4.774 4.863 4.950 5.119 5.035 4.950 5.456 5.557 5.657 5.851 5.755 5.657 6.138 6.252 6.364 6.582 6.474 6.364 47 46 45 Cos Sin Cos Sin Cos Sin Cos Sin 6 7 8 9 268 U. S. COAST AND GEODETIC STJRVEY Table 31. — For construction of primary stencils Difference Constituent 2Q Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 1 7 21 14 21 21 20* 28 20* 35 20 42 20 49 19* 56 19* 63 19 +23 -1 4 8 4 15 4 22 3* 29 3* 36 3 43 3 50 2* 57 2* 64 2 +22 -2 11 11 11 10* 10* 10 10 9* 9* 9 +21 -3 18 18 18 17* 17* 17 17 16* 16* 16 +20 -4 2 1 9 1 16 1 23 0* 30 0* 37 44 23* 23* 23 +19 -5 8 8 8 7* 7* 7 7 51 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 10 5 17 4* 24 4* 31 4* 38 4 45 4 52 3* 59 3* 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 11 2 18 1* 25 1* 32 1* 39 1 46 1 53 0* 60 0* 67 +12 -12 9 9 8* 8* 8 8 8 7* 7* 7 +11 -13 16 16 15* 15* 15 15 15 14* 14* 14 +10 -14 23 23 22* 22* 22 22 22 21* 21* 21 +9 -15 5 6 12 6 19 5* 26 5* 33 5 40 5 47 5 54 4* 61 4* 68 4 +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 13 3 20 2* 27 2* 34 2 41 2 48 1* 55 1* 62 1* 69 1 +5 -19 10 10 9* 9* 9 9 8* 8* 8* 8 +4 -20 17 17 16* 16* 16 16 15* 15* 15* 15 +3 -21 7 14 23* 23* 23 23 22* 22* 22* 22 +2 -22 7 7 21 6* 28 6* 35 6 42 6 49 5* 56 5* 63 5* 70 5 +1 -23 14 14 13* 13* 13 13 12* 12* 12* 12 Difference Constituent 2Q Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 70 19 77 19 84 18* 91 18* 98 18 105 18 112 17* 119 17* 126 17 133 17 +23 -1 71 2 78 2 85 1* 92 1* 99 1 106 1 113 0* 120 0* 127 134 +22 -2 9 9 8* 8* 8 8 7* 7* 7 7 +21 -3 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* 100 5 107 5 114 4* 121 4* 128 4 135 4 +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 2* 87 2* 94 2* 101 2 108 2 115 1* 122 1* 129 1 136 1 +15 -9 10 9* 9* 9* 9 9 8* 8* 8 8 +14 -10 17 16* 16* 16* 16 16 15* 15* 15 15 +13 -11 74 23* 23* 23 23 23 22* 22* 22 22 +12 -12 7 81 6* 88 6* 95 6 102 6 109 6 116 5* 123 5* 130 5 137 5 +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 3 103 3 110 3 117 2* 124 2* 131 2 138 2 +8 -16 11 10* 10* 10 10 10 9* 9* 9 9 +7 -17 18 17* 17* 17 17 16* 16* 16* 16 16 +6 -18 76 1 83 0* 90 0* 97 104 23* 23* 23* 23 23 +5 -19 8 7* 7* 7 7 111 6* 118 6* 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 77 5 84 4* 91 4* 98 4 105 4 112 3* 119 3* 126 3 133 3 140 3 +1 -23 12 11* 11* 11 11 10* 10* 10 10 10 HARMONIC ANALYSIS AND PREDICTION OF TIDES 269 Table 31. — For construction of primary stencils — Continued Difference Constituent 2Q Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. 140 17 147 16* 154 1(3* 161 10 168 16 175 15* 182 15* 189 15 196 15 203 15 +23 -1 141 23* 23* 23 23 22* 22* 22 22 22 +22 -2 7 148 6* 155 6* 162 6 169 6 176 5* 183 5* 190 5 197 5 204 4* +21 -3 14 13* 13* 13 13 12* 12* 12 12 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 2 205 1* +18 -6 10* 10* 10* 10 10 9* 9* 9 9 8* +17 -7 17* 17* 17* 17 17 16* 16* 16 16 15* +16 -8 143 0* 150 0* 157 0* 164 171 23* 23* 23 23 22* +15 -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 3 200 3 207 2* +11 -13 11* 11* 11 11 11 10* 10* 10 10 9* +10 -14 18* 18* 18 18 18 17* 17* 17 17 16* +9 -15 145 1* 152 1* 159 1 166 1 173 0* 180 0* 187 0* 194 201 23* +8 -16 8* 8* 8 8 7* 7* 7* 7 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 11 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 0* +1 -23 9* 9* 9 9 8* 8* 8 8 8 7* Difference Constituent 2Q Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. 210 14* 217 14* 224 14 231 14 238 13* 245 13* 252 13 259 13 266 13 273 12* +23 -1 21* 21* 21 21 20* 20* 20 20 19* 19* +22 -2 211 4* 218 4* 225 4 232 4 239 3* 246 3* 253 3 260 3 267 2* 274 2* +21 -3 11* 11* 11 11 10* 10* 10 10 9* 9* +20 -4 18* 18* 18 18 17* 17* 17 17 1(3* 16* +19 -5 212 1* 219 1* 226 1 233 1 240 0* 247 0* 254 261 23* 23* +18 -6 8* 8* 8 8 7* 7* 7 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 4 269 3* 276 3* +14 -10 12* 12 12 12 11* 11* 11 11 10* 10* +13 -11 19* 19 19 19 18* 18* 18 18 17* 17* +12 -12 214 2* 221 2 228 2 235 2 242 1* 249 1* 256 1 263 1 270 0* 277 0* +11 -13 9* 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 5* 250 5* 257 5 264 5 271 4* 278 4* +7 -17 13* 13 13 12* 12* 12* 12 12 11* 11* +6 -18 20* 20 20 19* 19* 19* 19 19 18* 18* +5 -19 216 3* 223 3 230 3 237 2* 244 2* 251 2 258 2 265 2 272 1* 279 1* +4 -20 10* 10 10 9* 9* 9 9 9 8* 8* +3 -21 17* 17 17 16* 16* 16 16 16 15* 15* +2 -22 217 0* 224 231 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* 246037—41- -18 270 U. S. COAST AND GEODETIC SURVEY Table 31. — For construction of primary stencils — Continued Difference Constituent 2Q Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 280 12* 287 12 294 12 301 11* 308 11* 315 11 322 11 329 10* 336 10* 343 10* +23 -1 19* 19 19 18* 18* 18 18 17* 17* 17* +22 -2 281 2* 288 2 295 2 302 1* 309 1* 316 1 323 1 330 0* 337 0* 344 0* +21 -3 9* 9 9 8* 8* 8 8 7* 7* 7* +20 -4 16* 16 16 15* 15* 15 15 14* 14* 14* +19 -5 23* 23 23 22* 22* 22 22 22* 21* 21 +18 -6 282 6* 289 6 296 6 303 5* 303 5* 317 5 324 5 331 4* 338 4* 345 4 +17 _7 13 13 13 12* 12* 12 12 11* 11* 11 +16 -8 20 20 20 19* 19* 19 19 18* 18* 18 +15 -9 283 3 290 3 297 3 304 2* 311 2* 318 2 325 2 332 1* 339 1* 346 1 +14 -10 10 10 10 9* 9* 9 9 8* 8* 8 +13 -11 17 17 17 16* 16* 16 16 15* 15* 15 +12 -12 284 291 23* 23* 23* 23 23 22* 22* 22 +11 -13 7 7 298 6* 305 6* 312 6* 319 6 326 6 333 5* 340 5* 347 5 +10 -14 14 14 13* 13* 13* 13 13 12* 12* 12 +9 -15 21 21 20* 20* 20* 20 20 19* 19* 19 +8 -16 285 4 292 4 299 3* 306 3* 313 3* 320 3 327 3 334 2* 341 2* 318 2 +7 -17 11 11 10* 10* 10* 10 10 9* 9* 9 +6 -18 18 18 17* 17* 17 17 17 16* 16* 16 +5 -19 286 1 293 1 300 0* 307 0* 314 321 328 23* 23* 23 +4 -20 8 8 7* 7* 7 7 7 335 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 Constituent 2Q Constituent Q Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 350 10 357 10 364 9* 1 10 5 19 13* 28 22* 38 7* 47 16 57 1 +23 -1 17 17 16* 5* 14 23 29 8 16* 48 1* 10* +22 -2 351 358 23* 15 23* 20 8* 17 39 2 11 19* +21 -3 7 7 365 6* 2 11 9 18 30 2* 11* 20 58 5 +20 -4 14 14 13* 9* 18* 21 3 12 21 49 5* 14* +19 -5 21 21 20* 19 12 3* 12* 21* 40 6 15 59 +18 -6 352 4 359 4 366 3* 3 4* 13 22 31 6* 15* 50 0* 9 +17 -7 11 11 10* 13* 22* 22 7* 16 41 1 9* 18* +16 -8 18 18 17* 23 13 8 16* 32 1* 10* 19 60 4 +15 -9 353 1 360 1 367 0* 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* 5 3 12 20* 33 5* 14* 23 61 8 +12 -12 22 21* 21* 12* 21* 24 6 15 23* 52 8* 17* +11 -13 354 5 361 4* 368 4* 22 15 6* 15* 34 0* 43 9 18 62 2* +10 -14 12 11* 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* 7 2 11 19* 35 4* 13 22 63 7 +7 -17 9 8* 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 27 9 18 46 2* 11* 20* +3 -21 13 12* 9 1 9* 18* 37 3 12 21 65 5* +2 -22 20 19* 10 19 28 4 12* 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 271 Table 31. — For construction of primary stencils — Continued Difference Constituent Q Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 66 9* 75 18* 85 3* 94 12 103 21 113 6 122 14* 131 23* 141 8 150 17 +23 -1 19 76 4 12* 21* 104 6* 15 123 132 9 17* 151 2* +22 -2 67 4* 13* 22 95 7 15* 114 0* 9* 18 142 3 12 +21 -3 14 22* 86 7* 16* 105 1 10 18* 133 3* 12* 21 +20 -4 23 77 8 17 96 1* 10* 19* 124 4 13 21* 152 6* +19 -5 68 8* 17* 87 2 11 20 115 4* 13* 22* 143 7 16 +18 -6 18 78 3 11* 20* 106 5 14 23 134 7* 16* 153 1* +17 -7 69 3* 12 21 97 6 14* 23* 125 8 17 144 2 10* +16 -8 12* 21* 88 6* 15 107 116 8* 17* 135 2* 11 20 +15 -9 22 79 7 15* 98 0* 9* 18 126 3 11* 20* 154 5* +14 -10 70 7* 16 89 1 10 18* 117 3* 12* 21 145 6 14* +13 -11 17 80 1* 10* 19 108 4 13 21* 136 6* 15* 155 +12 -12 71 2 11 20 99 4* 13* 22 127 7 16 146 0* 9* +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 91 100 8* 17* 119 2* 11 20 147 5 13* +8 -16 15* 82 0* 9 18 110 3 11* 20* 138 5* 14 23 +7 -17 73 1 10 18* 101 3* 12 21 129 6 14* 23* 157 8* +6 -18 10* 19 92 4 13 21* 120 6* 15 139 148 9 17* +5 -19 19* 83 4* 13* 22 111 7 16 130 0* 9* 18 158 3 +4 -20 74 5 14 22* 102 7* 16* 121 1 10 19 149 3* 12* +3 -21 14* 23* 93 8 17 112 1* 10* 19* 140 4 13 22 +2 -22 75 84 8* 17* 103 2* 11 20 131 4* 13* 22* 159 7 +1 -23 9 18 94 3 11* 20* 122 5* 14 23 150 7* 16* Difference Constituent Q Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. C 160 2 169 10* 178 19* 188 4* 197 13 206 22 216 6* 225 15* 235 0* 244 9 +23 -1 11 20 179 5 13* 22* 207 7* 16 226 1 9* 18* +22 -2 20* 170 5* 14 23 198 8 16* 217 1* 10* 19 245 4 +21 -3 161 6 15 23* 189 8* 17 208 2 11 19* 236 4* 13* +20 -4 15* 171 180 9 18 199 2* 11* 20 227 5 14 22* +19 -5 162 0* 9* 18* 190 3 12 21 218 5* 14* 23 246 8 +18 -6 10 19 181 3* 12* 21* 209 6 15 228 237 8* 17* +17 -7 19* 172 4 13 22 200 6* 15* 219 0* 9 18 247 2* +16 -8 163 5 13* 22* 191 7 16 210 1 9* 18* 238 3* 12 +15 -9 14 23 182 8 16* 201 1* 10 19 229 4 12* 21* +14 -10 23* 173 8* 17 192 2 11 19 220 4* 13 22 248 7 +13 -11 164 9 17* 183 2* 11* 20 211 5 14 22* 239 7* 16 +12 -12 18* 174 3 12 20* 202 5* 14* 23 230 8 17 249 1* +11 -13 165 3* 12* 21* 193 6 15 23* 221 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 9* 18* 222 3 12 21 250 5* +8 -16 166 7* 16* 185 1* 10 19 213 4 12* 21* 241 6 15 +7 -17 17 176 2 10* 19* 204 4* 13 22 232 7 15* 251 0* +6 -18 167 2* 11* 20 195 5 13* 22* 223 7* 16 242 1 10 +5 -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* 187 196 9 18 215 2* 11* 20* 243 5 14 - +2 -22 16 178 1 9* 18* 206 3 12 21 234 5* 14* 23* +1 -23 169 1* 10 19 197 3* 12* 21* 225 6 15 244 253 8* 272 U. S. COAST AND GEODETIC SURVEY Table 31. — For construction of primary stencils — Continued Difference Constituent Q Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. 253 18 263 3 272 11* 281 20* 291 5 300 14 309 23 319 7* 328 16* 338 1* +23 -1 254 3* 12 21 282 6 14* 23* 310 8 17 329 2 10* +22 -2 12* 21* 273 6* 15 292 301 9 17* 320 2* It 20 +21 -3 22 264 7 15* 283 0* 9* 18 311 3 12 20* 339 5* +20 -4 255 7* 16* 274 1 10 18* 302 3* 12* 21 330 6 15 +19 -5 17 265 1* 10* 19* 293 4 13 21* 321 6* 15* 340 +18 -6 256 2 11 20 284 4* 13* 22 312 7 16 331 0* 9* +17 -7 11* 20* 275 5 14 23 303 7* 16* 322 1 10 19 +16 -8 21 266 5* 14* 23* 294 8 17 313 2 10* 19* 341 4 +15 -9 257 6* 15 276 285 8* 17* 304 2* 11 20 332 5 13* +14 -10 15* 267 0* 9* 18 295 3 11* 20* 323 5* 14 23 +13 -11 258 1 10 18* 286 3* 12* 21 314 6 14* 23* 342 8* +12 -12 10* 19 277 4 13 21* 305 6* 15* 324 333 9 17* +11 -13 20 268 4* 13* 22 296 7 16 315 0* 9* 18* 343 3 +10 -14 259 5 14 23 287 7* 16* 306 1 10 19 334 3* 12* +9 -15 14* 23* 278 8 17 297 1* 10* 19* 325 4 13 22 +s -16 260 269 8* 17* 288 2* 11 20 316 4* 13* 22* 344 7 +7 -17 9 18 279 3 11* 20* 307 5* 14 23 335 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* 15 308 317 9 17* 336 2* 11* +4 -20 13* 22 280 7 16 299 0* 9 18 327 3 12 20* +3 -21 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* +1 -23 17* 272 2 11 20 300 4* 13* 22* 328 7 16 347 0* Difference Constituent Q Constituent P Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. 347 10 356 19 366 3* 1 10 15* 20 11 30 6* 40 2 49 21* 59 17 +23 -1 19* 357 4* 13 5* 11 1 20* 16 12 50 7* 60 3 +22 -2 348 5 13* 22* 15* 11 21 6* 31 2 21* 17 12* +21 -3 14 23 367 8 2 1* 21 16* 12 41 7* 51 3 22* +20 -4 23* 358 8* 17 11 12 6* 22 2 21* 17 12* 61 8* +19 -5 349 9 18 368 2* 21 16* 12 32 7* 42 3 22* 18 +18 -6 18* 359 3 12 3 6* 13 2* 22 17* 13 52 8* 62 4 +17 -7 350 3* 12* 21* 16* 12 23 7* 33 3 22* 18 13* +16 -8 13 22 369 6* 4 2* 22 17* 13 43 8* 53 4 23* +15 -9 22* 360 7 16 12 14 7* 24 3 22* 18* 14 63 9* +14 -10 351 8 16* 370 1* 22 17* 13 34 8* 44 4 23* 19 +13 -11 -12 17 352 2* 361 2 11* 5 8 17* 15 3* 13 23 25 8* 18* 35 4 14 23* 54 9* 19 64 5 +12 15 +11 -13 12 20* 6 3* 23 18* 14 45 9* 55 5 65 0* +10 -14 21 362 6 13 16 9 26 4* 36 19* 15 10* +9 -15 -16 -17 -18 353 6* 16 354 1* 10* 15* 363 1 10 19* 23 7 9 18* 8 4* 18* 17 4* 14 18 14 27 9* 19* 9* 19* 37 5* 15 46 5 15 47 1 10* 56 0* 10* 20* 57 6 20 +8 66 6 +7 16 +6 67 1* +5 -19 -20 -21 -22 -23 20 355 5* 15 356 9* 364 5 14* 23* 365 9 18* 14* 9 10 19* 10 5* 10 19* 19 5* 15* 20 1 28 5* 15 29 1 11 20* 38 1 10* 20* 39 6* 16 20* 48 6 16 49 2 11* 16 58 2 11* 21* 59 7 11* +4 21* +3 68 7 +2 17 +1 69 2* HARMONIC ANALYSIS AND PREDICTION OF TIDES 273 Table 31.— For construction of primary stencils — Continued Difference Constituent p Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 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 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 19* 15 10'' 149 6 159 1* +20 -4 71 4 23* 19 14* 10 120 5* 130 1 20* 10 11* +19 -5 13* 81 9 91 4 * 101 19* 15 11 140 6* 150 2 21* +18 -6 23* 19 14* 10 111 5* 121 1 20* 16 11* 160 7 +17 -7 72 9 82 5 92 0* 20 15* 11 131 6* 141 2 21* 17 +16 -8 19 14* 10 102 5* 112 1 20* 16 11* 151 7* 161 3 +15 -9 73 5 83 0* 20 15* 11 122 6* 132 2 21* 17 12* +14 -10 14* 10 93 5* 103 1* 21 16* 12 142 7* 152 3 22* +13 -11 74 0* 20 15* 11 113 6* 123 2 21* 17 12* 162 8 +12 -12 10* 84 6 94 1* 21 16* 12 133 7* 143 3 22* 18 +11 -13 20 15* 11 104 6* 114 2 21* 17* 13 153 8* 163 4 +10 -14 75 6 85 1* 21 16* 12 124 7* 134 3 22* 18 13* +9 -15 15* 11* 95 7 105 2* 22 17* 13 144 8* 154 4 23* +8 -16 76 1* 21 16* 12 115 7* 125 3 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 77 7 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 0* +3 -21 78 2* 22 17* 13 117 8* 127 4* 137 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 19* 15 10* 167 6 Difference Constituent p Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 167 16 177 11* 187 7 197 2* 206 22 216 17* 226 13 236 8* 246 4 255 23* +23 -1 168 1* 21 16* 12 207 7* 217 3* 23 18* 14 256 9* +22 -2 11* 178 7 188 2* 22 17* 13 227 8* 237 4 23* 19 +21 -3 21 17 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 19* 15 +19 -5 17 12* 189 8 199 3* 23 18* 14 9* 248 5 258 0* +18 -6 170 2* 22 17* 13* 209 9 219 4* 229 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 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 19* 15 10* 250 6 260 1* +13 -11 172 4 23* 19 14* 10 221 5* 231 1 20* 16 11* +12 -12 13* 182 9 192 4* 202 19* 15 10* 241 6* 251 2 21* +11 -13 23* 19 14* 10 212 5* 222 1 20* 16 11* 261 7 +10 -14 173 9 183 4* 193 0* 20 15* 11 232 6* 242 2 21* 17 +9 -15 19 14* 10 203 5* 213 1 20* 16 11* 252 7 262 3 +8 -16 174 5 184 0* 20 15* 11 223 6* 233 2 21* 17 12* +7 -17 14* 10 194 5* 204 1 20* 16* 12 243 7* 253 3 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* 12 225 7* 235 3 22* 18 13* +2 -22 15* 11 196 7 206 2* 22 17* 13 245 8* 255 4 23* +1 -23 177 1* 21 16* 12 216 7* 226 3 22* 18 13* 265 9* 274 U. S. COAST AND GEODETIC STJKVEY Table 31. — For construction of primary stencils — Continued Difference Constituent p Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 1 f 265 19 275 14* 285 10 295 5* 305 1 314 20* 324 16 334 12 344 7* 354 3 +23 -1 266 5 276 0* 20 15* 11 315 6* 325 2 21* 17 12* +22 -2 14* 10 286 6 296 1* 21 16* 12 355 7* 345 3 22* +21 -3 267 0* 20 15* 11 306 6* 316 2 21* 17 12* 355 8* +20 -4 10* 277 '6 287 1* 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* 12 317 7* 327 3 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 0* +11 -13 271 2* 22 17* 13 310 9 320 4* 330 19* 15 10* +10 -14 12* 281 8 291 3* 23 18* 14 9* 340 5 350 0* 20 +9 -15 22* 18 13* 301 9 311 4* 321 19* 15 10* 360 6 +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 19* 15 10* 351 6 361 1* +6 -18 273 3* 23* 19 14* 10 322 5* 332 1 20* 16 11* +5 -19 13* 283 9 293 4* 303 19* 15 10* 342 6 352 2 21* +4 -20 23* 19 14* 10 313 5* 323 1 20* 16 11* 362 7 +3 -21 274 9 284 4* 294 19* 15* 11 333 6* 343 2 21* 17 +2 -22 19 14* 10 304 5* 314 1 20* 16 11* 353 7 363 2* +1 -23 275 5 285 0* 20 15* 11 324 6* 334 2 21* 17 12* Difference P Constituent O Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. ft. d. ft. d. ft. 363 22* 1 14 22* 29 3 43 7* 57 12 71 16* 85 21 100 2 114 6* +23 -1 364 8 8 15 12* 17 21* 58 2 72 7 86 11* 16 20* +22 -2 18 22 16 2* 30 7 44 12 16* 21 87 1* 101 6 115 11 +21 -3 365 4 2 12 17 21* 45 2 59 6* 73 11 16 20* 116 1 +20 -4 13* 3 2* 17 7 31 11* 16 21 74 1* 88 6 102 10* 15 +19 -5 23* 16* 21 32 2 46 6* 60 11 15* 20 103 1 117 5* +18 -6 366 9 4 7 18 11* 16 20* 61 1 75 6 89 10* 15 19* +17 -7 19 21 19 1* 33 6 47 11 15* 20 90 0* 104 5 118 10 +16 -8 367 5 5 11 16 20* 48 1 62 5* 76 10 15 19* 119 +15 -9 14* 6 1* 20 6 34 10* 15 20 77 0* 91 5 105 9* 14 +14 -10 368 0* 15* 20 35 1 49 5* 63 10 14* 19 106 120 4* +13 -11 10* 7 6 21 10* 15 19* 64 78 5 92 9* 14 18* +12 -12 20 20 22 0* 36 5 50 9* 14* 19 23* 107 4 121 8* +11 -13 369 6 8 10 14* 19* 51 65 4* 79 9 93 13* 18* 23 +10 -14 15* 9 0* 23 5 37 9* 14 18* 23* 94 4 108 8* 122 13 +9 -15 370 1* 14* 19 23* 52 4* 66 9 80 13* 18 22* 123 3* +8 -16 10 4* 24 9* 38 14 18* 23 81 3* 95 8* 109 13 17* +7 -17 19 23* 39 4 53 8* 67 13* 18 22* 110 3 124 7* +6 -18 11 9 25 13* 18* 23 68 3* 82 8 96 12* 17* 22 +5 -19 23* 26 4 40 8* 54 13 17* 22* 97 3 111 7* 125 12 +4 -20 12 13* 18 22* 55 3* 69 8 83 12* 17 21* 126 2* +3 -21 13 3* 27 8* 41 13 17* 22 84 2* 98 7* 112 12 16* +2 -22 18 22* 42 3 56 7* 70 12* 17 21* 113 2 127 6* +1 -23 14 8 28 12* 17* 22 71 2* 85 7 99 11* 16 21 HARMONIC ANALYSIS AND PREDICTION OF TIDES 275 Table 31. — For construction of primary stencils — Continued Difference Constituent Hours d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 128 11 142 15* 156 20 171 1 185 5* 199 10 213 14* 227 19 242 256 4* +23 -1 129 1 143 6 157 10* 15 19* 200 214 5 228 9* 14 18* +22 -2 15* 20 158 0* 172 5 186 10 14* 19 23* 243 4 257 9 +21 -3 130 5* 144 10 15 19* 187 201 4* 215 9 229 14 18* 23 +20 -4 20 145 0* 159 5 173 9* 14 19 23* 230 4 244 8* 258 13 +19 -5 131 10 14* 19 174 188 4* 202 9 216 13* 18 23 259 3* +18 -6 132 146 5 160 9* 14 18* 23 217 4 231 8* 245 13 17* +17 -7 14* 19 23* 175 4 189 9 203 13* 18 22* 246 3 260 8 +16 -8 133 4* 147 9 161 14 18* 23 204 3* 218 8 232 13 17* 22 +15 -9 19 23* 162 4 176 8* 190 13 18 22* 233 3 247 7* 261 12 +14 -10 134 9 148 13* 18 22* 191 3* 205 8 219 12* 17 21* 262 2* +13 -11 23 149 3* 163 8* 177 13 17* 22 220 2* 234 7* 248 12 16* +12 -12 135 13* 18 22* 178 3 192 7* 206 12* 17 21* 249 2 263 6* +11 -13 136 3* 150 8 164 12* 17* 22 207 2* 221 7 235 11* 16* 21 +10 -14 17* 22* 165 3 179 7* 193 12 16* 21* 236 2 250 6* 264 11 +9 -15 137 8 151 12* 17 21* 194 2* 208 7 222 11* 16 20* 265 1* +8 -16 22 152 2* 166 7* 180 12 16* 21 223 1* 237 6* 251 11 15* +7 -17 138 12* 17 21* 181 2 195 6* 209 11* 16 20* 252 1 266 5* +6 -18 139 2* 153 7 167 11* 16* 21 210 1* 224 6 238 10* 15* 20 +5 -19 16* 21* 168 2 182 6* 196 11 15* 20* 239 1 253 5* 267 10 +4 -20 140 7 154 11* 16 20* 197 1* 211 6 225 10* 15 19* 268 0* +3 -21 21 155 1* 169 6* 183 11 15* 20 226 0* 240 5 254 10 14* +2 -22 141 11* 16 20* 184 1 198 5* 212 10 15 19* 255 269 4* +1 -23 142 1* 156 6 170 10* 15 20 213 0* 227 5 241 9* 14 19 Difference Constituent O Component 2N Hour +23 +22 +21 +20 +19 +18 +17 +16 +15 +14 +13 +12 +11 +10 +9 +8 +7 +6 +5 +4 +3 +2 +1 -1 -2 -3 -4 -5 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 d. h 270 9 23 271 13* 272 3* 18 273 8 22 274 12* 275 2* 16* 276 7 21 277 11* 278 1* 15* 279 6 20 280 10* 281 0* 14* 282 5 19 283 9 23* d. h. 284 13* 285 4 18 286 8 22* 287 12* 288 3 17 289 7 21* 290 11* 291 1* 16 292 6 20* 293 10* 294 0* 15 295 5 19* 296 9* 23* 297 14 298 4 d. h. 298 18 299 8* 22* 300 13 301 3 17 302 7* 21* 303 11* 304 2 16 305 6* 20* 306 10* 307 1 15 308 5* 19* 310 14 311 4 18* 312 8* d. h. 312 23 313 13 314 3 17* 315 V 22 316 12 317 2 16* 318 6* 20* 319 11 320 1 15* 321 5* 19* 322 10 323 14* 324 4* 18* 325 9 23 326 13 d. h. 327 3 ! 17* 328 8 22 329 12 330 2< 16* 331 7 21 332 11 333 1* 15* 334 5* 20 335 10 336 0* 14* 337 4* 19 338 9 23 339 13* 340 3* 18 d. h. 341 8 22 342 12* 343 2* 17 344 7 21 345 11* 346 1* 15* 347 6* 20 348 10* 349 0* 14* 350 5 19 351 9* 23* 352 13* 353 4 18 354 8 22* d. h. 355 12* 356 3 17 357 7 21* 358 11* 359 2 16 360 6 20* 361 10* 362 0* 15 363 5 19* 364 9* 23* 365 14 366 4 18 367 8* 22* 368 13 369 3 d. h. 369 17 370 T 1 8 22 2 12* 3 2* 17 4 7 21* 5 11* 6 2 16 7 6* 20* 8 11 9 1 15* 10 5* 20 11 10 12 0* 14* 13 5 19 14 9* d. h. 14 23* 15 14 16 4 18* 17 8* 23 18 13 19 3* 18 20 8 22* 21 12* 22 3 17 23 7* 21* 24 12 25 2 16* 26 6* 21 27 11 28 1* 15* 276 U. S. COAST AND GEODETIC STJBVEY Table 31. — For construction of primary stencils — Continued Difference Constituent 2N Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 29 6 43 12 57 18 72 86 6 100 12* 114 18* 129 0* 143 6* 157 12* +23 -1 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 +21 -3 31 0* 45 6* 59 13 19 88 1 102 7 116 13 19* 145 1* 159 7* +20 -4 15 21 60 3 74 9 15 21* 117 3* 131 9* 15* 21* +19 -5 32 5 46 11 17 23* 89 5* 103 11* 17* 132 146 6 160 12 +18 -6 19* 47 1* 61 7* 75 13* 20 104 2 118 8 14 20 161 2* +17 -7 33 9 15* 22 76 4 90 10 16 22 133 4* 147 10* 16* +16 -8 34 48 6 62 12 18 91 0* 105 6* 119 12* 18* 148 0* 162 7 +15 -9 14 20 63 2* 77 8* 14* 20* 120 2* 134 9 15 21 +14 -10 35 4* 49 10* 16* 22* 92 5 106 11 17 23 149 5 163 11* +13 -11 18* 50 0* 64 7 78 13 19 107 1 121 7 135 13* 19* 164 1* +12 -12 36 9 15 21 79 3 93 9* 15* 21* 136 3* 150 9* 16 +11 -13 23 51 5 65 11* 17* 23* 108 5* 122 11* 18 151 165 6 +10 -14 37 13* 19* 66 1* 80 7* 94 14 20 123 2 137 8 14 20* +9 -15 38 3* 52 9* 16 22 95 4 109 10 16 22* 152 4* 166 10* +8 -16 18 53 67 6 81 12 18* 110 0* 124 6* 138 12* 18* 167 1 +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* 23 111 5 125 11 17 23 168 5* +5 -19 40 12* 19 69 1 83 7 97 13 19 126 1* 140 7* 154 13* 19* +4 -20 41 3 55 9 15 21 98 3* 112 9 15* 21* 155 3* 169 10 +3 -21 17 23* 70 5* 84 11* 17* 23 127 6 141 12 18 170 +2 -22 42 7* 56 13* 19* 85 1* 99 8 113 14 20 142 2 156 8 14* +1 -23 21* 57 4 71 10 16 22 114 4 128 10* 16* 22* 171 4* Difference Constituent 2N Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 171 19 186 1 200 7 214 13 228 19 243 1* 257 7* 271 13* 285 19* 300 1* +23 -1 172 9 15 21* 215 3* 229 9* 15* 21* 272 4 286 10 16 +22 -2 23* 187 5* 201 11* 17* 23* 244 6 258 12 18 287 301 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 22 231 4 245 10* 16* 22* 288 4* 302 10* +19 -5 18 189 203 6* 217 12* 18* 246 0* 260 6* 274. 13 19 303 1 +18 -6 175 8* 14* 20* 218 2* 232 9 15 21 275 3 289 9 15* +17 -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 9* 15* 22 291 4 205 10 +14 -10 17* 23* 206 5* 220 11* 18 249 263 6 277 12 18 306 0* +13 -11 178 7* 192 13* 20 221 2 235 8 14 20 278 2* 292 8* 14* +12 -12 22 193 4 207 10 16 22* 250 4* 264 16* 16* 22* 307 5 +11 -13 179 12 18 208 0* 222 6* 236 12* 18* 265 0* 279 7 293 13 19 +10 -14 180 2* 194 8* 14* 20* 237 3 251 9 15 21 294 3 308 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 18 268 282 6 296 12 18* +5 -19 183 1* 197 8 211 14 20 240 2 254 8 14* 20* 297 2* 311 8* +4 -20 16 22 212 4 226 10 16* 22* 269 4* 283 10* 16* 23 +3 -21 184 6 198 12* 18* 227 0* 241 6* 255 12* 19 284 1 298 7 312 13 +2 -22 20* 199 2* 213 8* 14* 21 256 3 270 9 15 21 313 3* +1 -23 185 10* 17 23 228 5 242 11 17 23* 285 5* 299 11* 17» HARMONIC ANALYSIS AND PREDICTION OF TTDES Table 31. — For construction of primary stencils — Continued 277 Difference Constituent 2N Constituent n Hour d.h. 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 15 11* 30 6 45 0* 59 19 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 0* 358 6* 23 17* 31 11* 46 6 61 0* 18* +21 -3 316 2* 330 8* 15 21 2 13* 17 8 32 2* 21 15 76 9* +20 -4 17 23 345 5 359 11 3 4* 23 17 47 11* 62 6 77 0* +19 -5 317 7 331 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* 347 361 6 5 0* 19 34 13* 49 8 64 2 20* +16 -8 319 2 333 8 14 20 15* 20 10 35 4 22* 17 79 11* +15 -9 16 22 348 4* 362 10* 6 6* 21 0* 19 50 13* 65 7* 80 2* +14 -10 320 6* 334 12* 18* 363 0* 21 15* 36 10 51 4 22* 17 +13 -11 20* 335 2* 349 9 15 7 12 22 6 37 0* 19 66 13* 81 7* +12 -12 321 11 17 23 364 5 8 2* 21 15* 52 9* 67 4 22* +11 -13 322 1 336 7 350 13* 19* 17* 23 11* 38 6 53 0* 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 23 17 39 11* 54 6 69 0* 18* +8 -16 20 338 2 352 8 14 10 13* 25 8 40 2* 20* 15 84 9 +7 -17 324 10 16* 22* 367 4* 11 4* 22* 17 55 11* 70 6 85 +6 -18 325 0* 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 0* 19 42 13* 57 8 72 2 20* +3 -21 19 341 1* 355 7* 369 13* 15* 28 10 43 4 22* 17 87 11* +2 -22 327 9* 15* 21* 370 3* 14 6 29 0* 19 58 13* 73 7* 88 2 +1 -23 23* 342 6 356 12 21 15* 44 9* 59 4 22* 17 Difference Constituent m Hour d.h. d.h. d.h. d.h. d.h. d.h. d.h. d.h. d.h. d.h. 89 7* 104 2 118 20* 133 14* 148 9 163 3* 177 22 192 16 207 10* 222 5 +23 -1 22* 16* 119 11 134 5* 149 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* 106 13 121 7* 136 1* 20 165 14* 180 9 195 3 21* 16 +19 -5 92 9* 107 4 22 16* 151 11 166 5 23* 18 210 12* 225 6* +18 -6 93 18* 122 13 137 7* 152 1* 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 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 18* 154 12* 169 7 184 1* 20 213 14 228 8* +13 -11 96 2 20* 14* 140 9 155 3* 22 16 199 10* 214 5 23* +12 -12 17 111 11 126 5* 141 18 170 12* 185 7 200 1* 19* 229 14 +11 -13 97 7* 112 2 20* 14* 156 9 171 3* 21* 16 215 10* 230 5 +10 -14 22* 16* 127 11 142 5* 157 18 186 12* 201 7 216 1 19* +9 -15 98 13 113 7* 128 2 20 14* 172 9 187 3 21* 16 231 10* +8 -16 99 4 22 16* 143 11 158 5* 23* 18 202 12* 217 7 232 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 5 23* 18 218 12* 233 6* +5 -19 101 18* 130 13 145 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 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* 206 5 23* 17* +1 -23 103 11 118 5* 133 18* 162 12* 177 7 192 1* 19* 221 14 236 8* 278 U. S. COAST AND GEODETIC SURVEY Table 31. — For construction of primary stencils — Continued Difference Constituent n Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. ) 236 23 251 17* 266 12 281 6* 296 0* 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* 282 12 297 6 312 0* 19 341 13* 356 7* +21 -3 19* 253 14 268 8* 283 2* 21 15* 327 9* 342 4 22* +20 -4 239 10* 254 4* 23 17* 298 12 313 6 328 0* 19 357 13 +19 -5 240 1 19 269 14 284 8 299 2* 21 15* 343 9* 358 4 +18 -6 16 255 10 270 4* 23 17* 314 11* 329 6 344 0* 19 +17 -7 241 6* 256 1 19* 285 13* 300 8 315 2* 21 15 359 9* +16 -8 21* 15* 271 10 286 4* 23 17 330 11* 345 6 360 0* +15 -9 242 12 257 6* 272 1 19 301 13* 316 8 331 2* 20* 15 +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 19 317 13* 332 8 347 2 20* +12 -12 244 8* 259 3 21 15* 303 10 318 4* 22 17 362 11* +11 -13 23 17* 274 12 289 6* 304 0* 19 333 13* 348 8 363 2 +10 -14 245 14 260 8* 275 2* 21 15* 319 10 334 4 22* 17 +9 -15 246 4* 23 17* 290 12 305 6 320 0* 19 349 13* 364 7* +8 -16 19* 261 14 276 8 291 2* 21 15* 335 9* 350 4 22* +7 -17 247 10 262 4* 23 17* 306 11* 321 6 336 0* 19 365 13 +6 -18 248 1 19* 277 13* 292 8 307 2* 21 15 351 9* 366 4 +5 -19 16 263 10 278 4* 23 17 322 11* 337 6 352 0* 18* +4 -20 249 6* 264 1 19* 293 13* 308 8 323 2* 20* 15 367 9* +3 -21 21* 15* 279 10 294 4* 23 17 338 11* 353 6 368 +2 -22 250 12 265 6* 280 1 19 309 13* 324 8 339 2* 20* 15 +1 -23 251 3 21 15* 295 10 310 4* 22* 17 354 11* 369 6 Difference Constituent N Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. ) 1 19 20* 39 2 58 7* 77 13 96 18* 116 135 5* 154 11 173 16* +23 -1 10* 20 16 21* 59 2* 78 8 97 13* 19 136 0* 155 6 174 11* +22 -2 2 5* 21 11 40 16* 22 79 3* 98 9 117 14* 20 156 1* 175 6* +21 -3 3 1 22 6* 41 11* 60 17 22* 99 4 118 9* 137 15 20* 176 2 +20 -4 20 23 1* 42 7 61 12* 80 18 23* 119 5 138 10* 157 15* 21 +19 -5 4 15* 20* 43 2 62 7* 81 13 100 18* 120 139 5* 158 11 177 16* +18 -6 5 10* 24 16 21* 63 3 82 8* 101 14 19* 140 0* 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 7 1 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* 21 47 2* 66 8 85 13 104 18* 124 143 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* 183 7 +11 -13 11 1 30 6* 49 12 68 17* 23 107 4* 126 10 145 15* 21 184 2 +10 -14 20* 31 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 0* 147 6 166 11 185 16* +8 -16 13 11 32 16 21* 71 3 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* 53 12 72 17 23 111 4* 130 10 149 15* 21 188 2* +5 -19 20* 35 2 54 7* 73 13 92 18* 23* 131 5 150 10* 169 16 21* +4 -20 16 15* 21 55 2* 74 8 93 13* 112 19 132 0* 151 6* 170 11* 189 17 +3 -21 17 11 36 16* 22 75 3* 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 279 Table 31. — For construction of primary stencils — Continued Difference Constituent N Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. 192 21* 212 3 231 8* 250 14 269 19* 289 1 308 6* 327 12 346 17* 365 23 +23 -1 193 17 22* 232 4 251 9* 270 15 20 309 1* 328 7 347 12* 366 18 +22 -2 194 12 213 17* 23 252 4* 271 10 290 15* 21 329 2* 348 8 367 13* +21 -3 195 7* 214 13 233 18* 253 272 5 291 10* 310 16 21* 349 3 368 8* +20 -4 196 2* 215 8 234 13* 19 273 0* 292 6 311 11* 330 17 22* 369 4 +19 -5 22 216 3* 235 9 254 14 19* 293 1 312 6* 331 12 350 17* 23 +18 -6 197 17 22* 236 4 255 9* 274 15 20* 313 2 332 7* 351 13 370 18 +17 -7 198 12* 217 18 23 256 4* 275 10 294 15* 21 333 2* 352 8 +16 -8 199 7* 218 13 237 18* 257 276 5* 295 11 314 16* 21* 353 3 +15 -9 200 3 219 8 238 13* 19 277 0* 296 6 315 11* 334 17 22* +14 -10 22 220 3* 239 9 258 14* 20 297 1* 316 6* 335 12 354 17* +13 -11 201 17 22* 240 4 259 9* 278 15 20* 317 2 336 7* 355 13 +12 -12 202 12* 221 18 23* 260 5 279 10* 298 15* 21 337 2* 356 8 +11 -13 203 7* 222 13 241 18* 261 280 5* 299 11 318 16* 22 357 3* +10 -14 204 3 223 8* 242 14 19* 281 0* 300 6 319 11* 338 17 22* +9 -15 22 224 3* 243 9 262 14* 20 301 1* 320 7 339 12* 358 18 +8 -16 205 17* 23 244 4* 263 9* 282 15 20* 321 2 340 7* 359 13 +7 -17 206 12* 225 18 23* 264 5 283 10* 302 16 21* 341 3 360 8* +6 -18 207 8 226 13* 245 18* 265 284 5* 303 11 322 16* 22 361 3* +5 -19 208 3 227 8* 246 14 19* 285 1 304 6* 323 12 342 17* 22* +4 -20 22* 228 3* 247 9 266 14* 20 305 1* 324 7 343 12* 362 18 +3 -21 209 17* 23 248 4* 267 10 286 15* 21 325 2* 344 7* 363 13 :::::::: +2 -22 210 12* 229 18 23* 268 5 287 10* 306 16 21* 345 3 364 8* +1 -23 211 8 230 13* 249 19 269 0* 288 6 307 11 326 16* 22 365 3* Difference Consti tuent v Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. 1 20 18* 40 23 61 3 81 7 101 11 121 15 141 19 161 23 182 3 +23 -1 11 21 15 41 19 23 82 3 102 7 122 11 142 15* 162 19* 23* +22 -2 2 7 22 11 42 15 62 19 23 103 3* 123 7* 143 11* 163 15* 183 19* +21 -3 3 3 23 7 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* 85 12 105 16 125 20 146 166 4 186 8 +18 -6 5 15* 25 19* 46 66 4 86 8 106 12 126 16 20 167 187 4 +17 -7 6 12 26 16 20 67 87 4 107 8 127 12 147 16* 20* 188 0* +16 -8 7 8 27 12 47 16 20 88 0* 108 4* 128 8* 148 12* 168 16* 20* +15 -9 8 4 28 8 48 12* 68 16* 20* 109 0* 129 4* 149 8* 169 12* 189 16* +14 -10 9 0* 29 4* 49 8* 69 12* 89 16* 20* 130 0* 150 5 170 9 190 13 +13 -11 20* 30 0* 50 4* 70 8* 90 13 110 17 21 151 1 171 5 191 9 +12 -12 10 16* 21 51 1 71 5 91 9 111 13 131 17 21 172 1 192 5 +11 -13 11 13 31 17 21 72 1 92 5 112 9 132 13 152 17* 21* 193 1* +10 -14 12 9 32 13 52 17 21 93 1* 113 5* 133 9* 153 13* 173 17* 21* +9 -15 13 5 33 9* 53 13* 73 17* 21* 114 1* 134 5* 154 9* 174 13* 194 17* +8 -16 14 1* 34 5* 54 9* 74 13* 94 17* 21* 135 1* 155 6 175 10 195 14 +7 -17 21* 35 1* 55 5* 75 9* 95 14 115 18 22 156 2 176 6 196 10 +6 -18 15 17* 22 56 2 76 6 96 10 116 14 136 18 22 177 2 197 6* +5 -19 16 14 36 18 22 77 2 97 6 117 10 137 14 157 18* 22* 198 2* +4 -20 17 10 37 14 57 18 22 98 2* 118 6* 138 10* 158 14* 178 18* 22* +3 -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* 59 10* 79 14* 99 18* 22* 140 3 160 7 180 11 200 15 +1 -23 22* 40 2* 60 6* 80 10* 100 15 120 19 23 161 3 181 7 201 11 280 IT. S. COAST AND GEODETIC STJBVEY Table 31. — For construction of primary stencils — Continued Difference Constituent v Hour +23 +22 +21 +20 +19 +18 +17 +16 +15 +14 +13 +12 +11 +10 +9 +8 +7 +6 +5 +4 +3 +2 +1 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 d. h. 202 7* 203 3* 23* 204 20 205 16 206 12 207 8* 208 4* 209 0* 21 210 17 211 13 212 9* 213 5* 214 1* 22 215 18 216 14 217 10* 218 6* 219 2* 23 220 19 221 15 d. h. 222 11* 223 7* 224 3* 225 20 226 16 227 12* 228 8* 229 4* 230 1 21 231 17 232 13* 233 9* 234 5* 235 2 22 236 18 237 14* 238 10* 239 6* 240 3 23 241 19 d. h. 242 15* 243 11* 244 7* 245 4 246 20 247 16* 248 12* 249 8* 250 5 251 1 21* 252 17* 253 13* 254 10 255 6 256 2 22* 257 18* 258 14* 259 11 260 7 261 3 23* d. h. d. h. d. h. d. h. d. h. 262 19* 282 23* 303 3 323 7* 343 11* 263 15* 283 19* 23 324 4 344 8 264 12 284 16 304 20 325 345 4 265 8 285 12 305 16 20 346 266 4 286 8 306 12* 326 16* 20* 267 0* 287 4* 307 8* 327 12* 347 16* 20* 288 0* 308 4* 328 8* 348 12* 268 16* 20* 309 1 329 5 349 9 269 13 289 17 21 330 1 350 5 270 9 290 13 310 17 21 351 1 271 5 291 9 311 13* 331 17* 21* 272 1* 292 5* 312 9* 332 13* 352 17* 21* 293 1* 313 5* 333 9* 353 13* 273 17* 21* 314 2 334 6 354 10 274 14 294 18 22 335 2 355 6 275 10 295 14 296 10 297 6* 315 18 316 14* 317 10* 22 336 18* 337 14* 356 2 22* 357 18* 276 6 277 2* 22* 298 2* 318 6* 338 10* 358 14* 278 18* 22* 319 3 339 7 359 11 279 15 299 19 23 340 3 360 7 280 11 300 15 320 19 23 361 3 281 7 301 11 321 15* 341 19* 23* 282 3 302 7* 322 11* 342 15* 362 19* d. h. 363 16 364 12 365 8 366 4* 367 0* 20* 368 17 369 13 370 9 Difference Constituent 2 ME Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. C 1 22 7 44 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 0* 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 3 7 25 46 17 68 10 90 3 111 20 133 13 155 6 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 2* 26 19* 48 12* 70 5* 22* 113 15* 135 8* 157 1* 178 18* 200 11* +18 -6 6 27 17 49 10 71 3 92 20 114 13 136 6* 23* 179 16* 201 9* +17 -7 22 28 15 50 8 72 1 93 18 115 11 137 4 158 21 180 14 202 7 +16 -8 7 19* 29 12* 51 5* 22* 94 15* 116 8* 138 1* 159 18* 181 11* 203 4* +15 -9 8 17* 30 10* 52 3* 73 20* 95 13* 117 6* 23* 160 16* 182 9* 204 2* +14 -10 9 15 31 8 53 1 74 18 96 11 118 4 139 21 161 14 183 7 205 +13 -11 10 12* 32 5* 22* 75 16 97 9 119 2 140 19 162 12 184 5 22 +12 -12 11 10* 33 3* 54 20* 76 13* 98 6* 23* 141 16* 163 9* 185 2* 206 19* +11 -13 12 8 34 1 55 18 77 11 99 4 120 21 142 14 164 7 186 0* 207 17* +10 -14 13 6 23 56 16 78 9 100 2 121 19 143 12 165 5 22 208 15 +9 -15 14 3* 35 20* 57 13* 79 6* 23* 122 16* 144 9* 166 2* 187 19* 209 12* +8 -16 15 1 36 18* 58 11* 80 4* 101 21* 123 14* 145 7* 167 0* 188 17* 210 10* +7 -17 23 37 16 59 9 81 2 102 19 124 12 146 5 22 189 15 211 8 +6 -18 16 20* 38 13* 60 6* 23* 103 16* 125 10 147 3 168 20 190 13 212 6 +5 -19 17 18* 39 11* 61 4* 82 21* 104 14* 126 7* 148 0* 169 17* 191 10* 213 3* +4 -20 18 16 40 9 62 2 83 19 105 12 127 5 22 170 15 192 8 214 1 +3 -21 19 14 41 7 63 84 17 106 10 128 3 149 20 171 13 193 6 23 +2 -22 20 11* 42 4* 21* 85 14* 107 7* 129 0* 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* HARMONIC ANALYSIS AND PREDICTION OF TIDES 281 Table 31. — For construction of primary stencils — Continued Difference Hour +23 +22 +21 +20 +19 +18 +17 +16 +15 +14 +13 +12 +11 +10 +9 +8 +7 +6 +5 +4 +3 +2 +1 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 d. h. d. h. 217 16 239 9 218 14 240 7 219 11* 241 4* 220 9 242 2 221 7 243 222 4* 21* 223 2* 244 19* 224 245 17 22 246 15 225 19* 247 12* 226 17 248 10 227 15 249 8 228 12* 250 5* 229 10* 251 3* 230 8 252 1 231 5* 22* 232 3* 253 20* 233 1 254 18 23 255 16 234 20* 256 13* 235 18* 257 11* 236 16 258 9 237 13* 259 6* 238 11* 260 4* Constituent 2MK d. h. 261 2 262 21* 263 19* 264 17 265 14* 266 12* 267 10 268 8 269 5* 270 3 271 1 22* 272 20* 273 18 274 16 275 13* 276 11 277 9 278 6* 279 4* 280 2 23* 281 21* d. h. d. h. 282 19 304 12 283 17 305 10 284 14* 306 7* 285 12* 307 5* 286 10 308 3 287 7* 309 0* 288 5* 22* 289 3 310 20 290 1 311 18 22* 312 15* 291 20 313 13* 292 18 314 11 293 15* 315 8* 294 13* 316 6* 295 11 317 4 296 9 318 2 297 6* 23* 298 4 319 21 299 2 320 19 23* 321 16* 300 21* 322 14* 301 19 323 12 302 16* 324 10 303 14* 325 7* d. h. 326 5 327 3 328 0* 22* 329 20 330 17* 331 15* 332 13 333 11 334 8* 335 6* 336 4 337 1* 23* 338 21 339 19 340 16* 341 14 342 12 343 9* 344 7* 345 5 346 3 347 0* d. h. 347 22 348 20 349 17* 350 15* 351 13 352 10* 353 8* 354 6 355 4 356 1* 23* 357 21 358 18* 359 16* 360 14 361 12 362 9* 363 7* 304 5 365 2* 3fifi 0* 22 367 20 368 17* d. h. 369 15 370 13 Constituent MN 7 8 8 T 9 6' 10 6 11 5' 17 1 18 0< 23* 19 23 20 22 21 21* 22 21 d. h. 23 20 24 19* 25 18* 26 18 27 17 28 16* 29 16 30 15 31 14* 32 13* 33 13 34 12* 35 11* 36 11 37 10 Difference Constituent MN Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. C 47 3 70 10 93 17 117 140 7 163 14 186 21 210 4 233 11 256 18 +23 -1 48 2* 71 9* 94 16* 23 141 6 164 13 187 20 211 3 234 10 257 17 +22 -2 49 1* 72 8* 95 15* 118 22* 142 5* 165 12* 188 19* 212 2* 235 9* 258 16* +21 -3 50 1 73 8 96 15 119 22 143 5 166 12 189 18* 213 1* 236 8* 259 15* +20 -4 51 74 7 97 14 120 21 144 4 167 11 190 18 214 1 237 8 260 15 +19 -5 23* 75 6* 98 13* 121 20* 145 3* 168 10* 191 17* 215 0* 238 7* 261 14 +18 -6 52 23 76 6 99 12* 122 19* 146 2* 169 9* 192 16* 23* 239 6* 262 13* +17 -7 53 22 77 5 100 12 123 19 147 2 170 ,9 193 16 216 23 240 6 263 13 +16 -8 54 21* 78 4* 101 11* 124 18* 148 1 171 8 194 15 217 22 241 5 264 12 +15 -9 55 20* 79 3* 102 10* 125 17* 149 0* 172 7* 195 14* 218 21* 242 4* 265 11* +14 -10 56 20 80 3 103 10 126 17 150 173 7 196 14 219 20* 243 3* 266 10* +13 -11 57 19 81 2 104 9 127 16 23 174 6 197 13 220 20 244 3 267 10 +12 -12 58 18* 82 1* 105 8* 128 15* 151 22* 175 5* 198 12* 221 19* 245 2* 268 9* +11 -13 59 18 83 1 106 8 129 14* 152 21* 176 4* 199 11* 222 18* 246 1* 269 8* +10 -14 60 17 84 107 7 130 14 153 21 177 4 200 11 223 18 247 1 270 8 +9 -15 61 16* 23* 108 6* 131 13* 154 20* 178 3 201 10 224 17 248 271 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 21 111 4 134 11 157 18 181 1 204 8 227 15 250 22 274 5 +5 -19 65 13* 88 20* 112 3* 135 10* 158 17* 182 0* 205 7* 228 14* 251 21* 275 4* +4 -20 66 13 89 20 113 3 136 10 159 16* 23* 206 6* 229 13* 252 20* 276 3* +3 -21 67 12 90 19 114 2 137 9 160 16 183 23 207 6 230 13 253 20 277 3 +2 -22 68 11* 91 18* 115 1* 138 8* 161 15* 184 22* 208 5* 231 12 254 19 278 2 +1 -23 69 10* 92 17 116 0* 139 7* 162 14* 185 21* 209 4* 232 11* 255 18* 279 1* 282 U. S. COAST AND GEODETIC SURVEY Table 31. — For construction of primary stencils — Continued Difference Constituent MN Constituent M H mr d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 3 280 0* 303 7* 326 14* 349 21* 1 29 22* 59 11* 89 118 13 148 1* +23 -1 281 304 7 327 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 11 121 150 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 5 330 12 353 19 5 8 34 21 64 9* 93 22* 123 11 153 +19 -5 284 21 308 4 331 11 354 18 6 13* 36 2* 65 15 95 4 124 16* 154 5* +18 -6 285 20* 309 3* 332 10* 355 17* 7 19 37 8 66 20* 96 9* 125 22 155 11 +17 -7 286 20 310 2* 333 9* 356 16* 9 0* 38 13* 68 2 97 15 127 3* 156 16* +16 -8 287 19 311 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 0* 336 7* 359 14* 12 17* 42 6 71 19 101 7* 130 20 160 9 +13 -11 290 17 314 337 7 360 14 13 23 43 11* 73 0* 102 13 132 2 161 14* +12 -12 291 16 23 338 6 361 13 15 4* 44 17 74 6 103 18* 133 7* 162 20 +11 -.3 292 15* 315 22* 339 5* 362 12* 16 10 45 22* 75 11* 105 134 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 9* 77 22* 107 11 137 166 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 -18 297 12 320 19 344 2 367 9 22 13* 52 2* 81 15 111 3* 140 16* 170 5 +5 -19 298 11* 321 18 345 1 368 8 23 19 53 8 82 20* 112 9* 141 22 171 10* +4 -20 299 10* 322 17* 346 0* 369 7* 25 0* 54 13* 84 2 113 15 143 3* 172 16* +3 -21 300 10 323 17 347 370 7 26 6 55 19 85 7* 114 20* 144 9 173 22 +2 -22 -23 301 9 302 8* 324 16 325 15* 23 348 22* 27 11* 28 17 57 0* 58 6 86 13 87 18* 116 2 117 7* 145 14* 146 20 175 3* +1 176 9 Difference Constituent M Constituent MK He ur d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. C 177 14* 207 3 236 16 266 4* 295 17* 325 6 354 19 1 46 5* 92 9* +23 -1 178 20 208 8* 237 21* 267 10 296 23 326 11* 356 0* 2 48 3* 94 7* +22 -2 180 1* 209 14 239 3 268 15* 298 4* 327 17 357 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 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 0* +18 -6 184 23* 214 12* 244 1 273 14 303 2* 332 15 362 4 11 14* 57 18* 103 22* +17 -7 186 5 215 18 245 6* 274 19* 304 8 333 21 363 9* 13 13 59 16* 105 20* +16 -8 187 10* 216 23* 246 12 276 1 305 13* 335 2* 364 15 15 11 61 15 107 18* +15 -9 188 16 218 5 247 17* 277 6* 306 19 336 8 365 20* 17 9 63 13 109 17 +14 -10 189 21* 219 10* 248 23 278 12 308 0* 337 13* 367 2 19 7* 65 11 111 15 +13 -11 191 3 220 16 250 4* 279 17* 309 6 338 19 368 7* 21 5* 67 9* 113 13 +12 -12 192 9 221 21* 251 10 280 23 310 11* 340 0* 369 13 23 3* 69 7* 115 11* +11 -13 193 14* 223 3 252 16 282 4* 311 17 341 6 370 18* 25 2 71 5* 117 9* +10 -14 194 20 224 8* 253 21* 283 10 312 23 342 11* 27 73 4 119 7* +9 -15 196 1* 225 14 255 3 284 15* 314 4* 343 17 28 22 75 2 121 6 +8 -16 197 7 226 19* 256 8* 285 21 315 10 344 22* 30 20* 77 123 4 +7 -17 198 12* 228 1 257 14 287 2* 316 15* 346 4 32 18* 78 22* 125 2 +6 -18 199. 18 229 6* 258 19* 288 8 317 21 347 9* 34 16* 80 20* 127 0* +5 -19 -20 200 23* 202 5 230 12 231 17* 260 1 261 6* 289 13* 290 19 319 2* 320 8 348 15 349 20* 36 14* 38 13 82 18* 84 17 128 22* +4 130 20* +3 -21 203 10* 232 23* 262 12 292 0* 321 13* 351 2 40 11 86 15 132 19 +2 -22 204 16 234 5 263 17* 293 6* 322 19 352 7* 42 9 88 13 134 17 +1 -23 205 21* 235 10* 264 23 294 12 324 0* 353 13* 44 7* 90 11* 136 15 HARMONIC ANALYSIS AND PREDICTION OF TIDES 283 Table 31. — For construction of primary stencils — Continued Difference Constituent MK Constituent X Hour +23 +22 +21 +20 +19 +18 +17 +16 +15 +14 +13 +12 +11 +10 +9 +8 +7 +6 +5 +4 +3 +2 +1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 d. ft. 138 13* 140 11* 142 9* 144 8 146 6 148 4 150 2* 152 0* 153 22* 155 21 157 19 159 17 161 15* 163 13* 165 11* 167 10 169 8 171 6 173 4* 175 2* 177 0* 178 22* 180 21 182 19 d. ft. 184 17 186 15' 188 13' 190 11" 192 10 194 8 198 4* 200 2* 202 0* 203 23 205 21 207 19 209 17" 211 15* 213 13* 215 12 217 10 219 8 221 6* 223 225 227 4* 2< 1 228 23 d. ft. 230 21 232 19* 234 17* 236 15* 238 14 240 12 242 10 244 8 246 6* 248 4* 250 2* 252 1 253 23 255 21 257 19* 259 17* 261 15* 263 14 265 12 267 10 269 8* 271 6* 273 4* 275 3 d. ft. 277 1 278 23 280 21* 282 19* 284 17* 286 16 288 14 290 12 292 10* 294 8* 296 6* 298 5 300 3 302 1 303 23* 305 21* 307 19* 309 18 311 16 313 14 315 12 317 10* 319 8* 321 6* d. ft. 323 5 325 3 327 1 328 23* 330 21* 332 19* 334 18 336 16 338 14 340 12* 342 10* 344 8* 346 7 348 5 350 3 352 1* 353 23* 355 21* 357 20 359 18 361 16 363 14* 365 12* 367 10* d. ft. 369 9 371 7 d. ft. 1 2 4* 4 11* 6 18* 9 1* 11 8* 13 16 15 23 18 6 20 13 22 20* 25 3* 27 10* 28 17* 32 0* 34 8 36 15 38 22 41 5 43 12* 45 19* 48 2* 50 9* 52 16* d. h. 55 57 7 59 14 61 21 64 4* 66 11* 68 18* 71 1* 73 8' 75 16 77 23 82 13 84 20 87 3* 89 10* 91 17* 94 0* 98 15 100 22 103 5 105 12 107 19* d. ft. 110 2* 112 9* 114 16* 117 119 7 121 14 123 21 126 4 128 11* 130 18* 133 1* 135 8* 137 15* 139 23 142 6 144 13 146 20 149 3* 151 10* 153 17* 156 0* 158 7* 160 15 162 22 d. ft. 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* 195 1* 197 8* 199 15* 201 23 204 6 206 13 208 20 211 3 213 10* 215 17* 218 0* Difference Constituent X Constituent MS Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. 3 220 7* 275 10* 330 13 1 58 20* 117 22 176 23* 236 1 295 2* 354 4 +23 -1 222 15 277 17* 332 20 2 6* 61 7* 120 9 179 10* 238 12 297 13* 356 15 +22 -2 224 22 280 0* 335 3 4 17* 63 19 122 20* 181 21* 240 23 300 0* 359 2 +21 -3 227 5 282 7* 337 10 7 4* 66 6 125 7* 184 9 243 10* 302 11* 361 13 +20 -4 229 12 284 14* 339 17* 9 15* 68 17 127 18* 186 20 245 21* 304 23 364 0* +19 -5 231 19 286 22 342 0* 12 2* 71 4 130 5* 189 7 248 8* 307 10 366 11* +18 -6 234 2* 289 5 344 7* 14 13* 73 15 132 16* 191 18 250 19* 309 21 368 22* +17 -7 236 9* 291 12 346 14* 17 0* 76 2 135 3* 194 5 253 6* 312 8 371 9* +16 -8 238 16* 293 19 348 22 19 11* 78 13 137 14* 196 16 255 17* 314 19 +15 -9 240 23* 296 2* 351 5 21 23 81 140 1* 199 3 258 4* 317 6 +14 -10 243 7 298 9* 353 12 24 10 83 11* 142 13 201 14 260 15* 319 17 +13 -11 245 14 300 16* 355 19 26 21 85 22* 145 204 1* 263 3 322 4 +12 -12 247 21 302 23* 358 2 29 8 88 9* 147 11 206 12* 265 14 324 15* +11 -13 250 4 305 6* 360 9* 31 19 90 20* 149 22 208 23* 268 1 327 2* +10 -14 252 11 307 14 362 16* 34 6 93 7* 152 9 211 10* 270 12 329 13* +9 -15 254 18* 309 21 364 23* 36 17 95 18* 154 20 213 21* 272 23 332 0* +8 -16 257 1* 312 4 367 6* 39 4 98 5* 157 7 216 8* 275 10 334 11* +7 -17 259 8* 314 11 369 14 41 15* 100 16* 159 18 218 19* 277 21 336 22* +6 -18 261 15* 316 18* 371 21 44 2* 103 4 162 5* 221 6* 280 8 339 9* +5 +4 +3 +2 -19 -20 -21 -22 -23 263 22* 266 6 268 13 270 20 273 3 319 1* 321 8* 323 15* 325 22* 328 6 46 13* 49 0* 51 11* 53 22* 56 9* 105 15 108 2 110 13 113 115 11 164 16* 167 3* 169 14* 172 1* 174 12* 223 18 226 5 228 16 231 3 233 14 282 19* 285 6* 287 17* 290 4* 292 15* 341 20* 344 8 346 19 349 6 351 17 +1 284 U. S. COAST AND GEODETIC SURVEY Table 31. — For construction of primary stencils — Continued Difference Constituent L Constituent P Constit- uent T Ho ( +23 +22 ur ) -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 d. h. 1 2 8* 5 7 16 10 7* 12 23 15 14* 18 6* 20 22 23 13* 26 5 28 21 31 12* 34 4 36 19* 39 11* 42 3 44 18* 47 10 50 2 52 17* 55 9 58 0* 60 16* d. h. 63 8 65 23* 68 15 71 7 73 22* 76 14 79 5* 81 21* 84 13 87 4* 89 20 92 12 95 3* 97 19 100 10* 103 2* 105 18 108 9* 111 1 113 17 116 8* 119 121 15* 124 7* d. h. 126 23 129 14* 132 6 134 22 137 13* 140 5 142 20* 145 12* 148 4 150 19* 153 11 156 2* 158 18* 161 10 164 1* 166 17 169 9 172 0* 174 16 177 7* 179 23* 182 15 185 6* 187 22 d. h. 190 14 193 5* 195 21 198 12* 201 4* 203 20 206 11* 209 3 211 19 214 10* 217 2 219 17* 222 9* 225 1 227 16* 230 8 233 235 15* 238 7 240 22* 243 14* 246 6 248 21* 251 13 d. h. 254 5 256 20* 259 12 262 3* 264 19* 267 11 270 2* 272 18 275 10 278 1* 280 17 283 8* 286 0* 288 16 291 7* 293 23 296 15 299 6* 301 22 304 13* 307 5* 309 21 312 12* 315 4 d. h. 317 20 320 11* 323 3 325 18* 328 10* 331 2 333 17* 336 9 339 1 341 16* 344 8 346 23* 349 15* 352 7 354 22* 357 14 360 6 362 21* 365 13 368 4* 370 20* d. h. 1 8 15* 23 20* 39 2 54 7 69 12* 84 17* 99 23 115 4 130 9* 145 14* 160 20 176 1 191 6* 206 11* 221 17 236 22 252 3* 267 8* 282 13* 297 19 313 328 5* 343 10* d. h. 358 16 373 21 d. h. 1 16 6 46 16* +21 77 3 +20 107 13* +19 138 +18 168 10* +17 198 21 +16 229 7* +15 259 18 +14 290 4* +13 +12 +11 320 15 351 1* 381 12 +10 +9 +8 +7 +6 +5 +4 +3 +2 +1 Difference Con stituent R Constituent K Constituent 2SM Hour +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -3 -2 -1 d. h. 1 16 6 46 16* 77 3 107 13* 138 168 10* 198 21 229 7* 259 18 290 4* 320 15 351 1* 381 12 d. h. 1 8 15* 23 20* 39 2 54 7 69 12* 84 17* 99 23 115 4 130 9* 145 14* 160 20 176 1 191 6* 206 11* 221 17 236 22 252 3* 267 8* 282 13* 297 19 313 328 5* 343 10* d. h. 358 16 373 21 d. h. 1 15* 2 21 4 2* 5 8 6 13* 7 19 9 0* 10 6 11 12 12 17* 13 23 15 4* 16 10 17 15* 18 21 20 2* 21 8 22 13* 23 19 25 0* 26 6 27 11* 28 17 d. h. 29 22* 31 4 32 10 33 15* 34 21 36 2* 37 8 38 13* 39 19 41 0* 42 6 43 11* 44 17 45 22* 47 4 48 9* 49 15 50 20* 52 2* 53 8 | 54 13*i 55 19 57 0*i 58 6 d. h. d. h. d. h. d. h. 59 11* 89 118 13 148 1* 60 17 90 5* 119 18* 149 7 61 22* 91 11 121 150 12* 63 4 92 17 122 5* 151 18 64 9* 93 22* 123 11 153 65 15 95 4 124 16* 154 5* 66 20* 96 9* 125 22 155 11 68 2 97 15 127 3* 156 16* 69 7* 98 20* 128 9 157 22 70 13 100 2 129 14* 159 3* 71 19 101 7* 130 20 160 9 73 0* 102 13 132 2 161 14* 74 6 103 18* 133 7* 162 20 75 11* 105 134 13 164 1* 76 17 106 5* 135 18* 165 7 77 22* 107 11 137 166 12* 79 4 108 16* 138 5* 167 18 80 9* 109 22 139 11 168 23* 81 15 111 3* 140 16* 170 5 82 20* 112 9* 141 22 171 10* 84 2 113 15 143 3* 172 16* 85 7* 114 20* 144 9 173 22 86 13 116 2 145 14* 175 3* 87 18* 117 7* 146 20 176 9 HARMONIC ANALYSIS AND PREDICTION OF TIDES 285 Table 31. — For construction of primary stencils — Continued Difference Constituent 2SM Constituent J + 1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 mr -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 d. h. 207 3 208 8* 209 14 210 19* 212 1 213 7 214 12* 215 18 216 23* 218 5 219 10* 220 16 221 21* 223 3 224 8* 225 14 226 19* 228 1 229 6* 230 12 231 17* 232 23* 234 5 235 10* d. h. 236 16 237 21* 239 3 240 8* 241 14 242 19* 244 1 245 6* 246 12 247 17* 248 23 250 4* 251 10 252 16 253 21* 255 3 256 8* 257 14 258 19* 260 1 261 6* 262 12 263 17* 264 23 d. h. 266 4* 267 10 268 15* 269 21 271 2* 272 8 273 14 274 19* 276 1 277 6* 278 12 279 17* 280 23 282 4* 283 10 284 15* 285 21 287 2* 288 8 289 13* 290 19 292 0* 293 6* 294 12 d. h. 295 17* 296 23 298 4* 299 10 300 15* 301 21 303 2* 304 8 305 13* 306 19 308 0* 309 6 310 11* 311 17 312 23 314 4* 315 10 316 15* 317 21 319 2* 320 8 321 13* 322 19 324 0* d. h. 325 6 326 11* 327 17 328 22* 330 4 331 9* 332 15 333 21 335 2* 336 8 337 13* 338 19 340 0* 341 6 342 11* 343 17 344 22* 346 4 347 9* 348 15 349 20* 351 2 352 7* 353 13* d. h. 354 19 356 0* 357 6 358 11* 359 17 360 22* 362 4 363 9* 364 15 365 20* 367 2 368 7* 369 13 370 18* d. h. 1 13* 2 15 3 17 4 18* 5 20 6 21* 7 23* 9 1 10 2* 11 4 12 6 13 7* 14 9 15 10* 16 12* 17 14 18 15* 19 17 20 18* 21 20* 22 22 23 23* 25 1 d. h. 26 3 27 4* 28 6 29 7* 30 9* 31 11 32 12* 33 14 34 16 35 17* 36 19 37 20* 38 22* 40 41 1* 42 3 43 5 44 6* 45 8 46 9* 47 11* 48 13 49 14* 50 16 d. h. 51 18 52 19* 53 21 54 22* 56 0* 57 2 58 3* 59 5 60 6* 61 8* 62 10 63 11* 64 13 65 15 66 16* 67 18 68 19* 69 21* 70 23 72 0* 73 2 74 4 75 5* 76 7 d. h. 77 8* 78 10* 79 12 80 13* 81 15 82 17 83 18* 84 20 85 21* 86 23* 88 1 89 2* 90 4 91 6 92 7* 93 9 94 10* 95 12* +15 + 16 +17 + 18 96 14 97 15* +19 +20 98 17 99 18* +21 +22 100 20* +23 101 22 Difference Constituent J Hour d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. d. h. 102 23* 128 14* 154 5* 179 20* 205 11* 231 2 256 17 282 8 307 23 333 14 + 1 -23 104 1 129 16 155 7 180 22 206 13 232 4 257 18* 283 9* 309 0* 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 13* 185 4* 210 19* 236 10* 262 1 287 16 313 7 338 22 +6 -18 109 9* 135 0* 160 15 186 6 211 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 0* 239 15 265 6 290 21 316 12 342 3 +9 -15 112 14 138 5 163 20 189 11 215 2 240 17 266 7* 291 22* 317 13* 343 4* +10 -14 113 16 139 6* 164 21* 190 12* 216 3* 241 18* 267 9* 293 0* 318 15 344 6 +11 -13 114 17* 140 8* 165 23* 191 14 217 5 242 20 268 11 294 2 319 17 345 7* +12 -12 115 19 141 10 167 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* 270 14 296 5 321 20 347 11 +14 -10 117 22* 143 13 169 4 194 19 220 10 246 1 271 16 297 6* 322 21* 348 12* +15 -9 119 144 15 170 6 195 20* 221 11* 247 2* 272 17* 298 8* 323 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 9 198 223 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 9 277 302 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 -3 125 9* 151 0* 176 15* 202 6* 227 21* 253 12* 279 3 304 18 330 9 356 +22 -2 126 11* 152 2 177 17 203 8 228 23 254 14 280 5 305 19* 331 10* 357 1* +23 -1 127 13 153 4 178 18* 204 9* 230 0* 255 15* 281 6* 306 21* 332 12* 358 3 246037—41 19 286 U. S. COAST AND GEODETIC STJRVEY Table 31. — For construction of primary stencils — Continued Diffe rence ur Con. J Constituent OO He d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. 359 5 1 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 0* 28 4* 41 8* 54 12* 16* 20* 94 1 107 5 +3 -21 362 9* 2 9* 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 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 21 111 1 +10 -14 369 21 6 6 19 10 14 18 22 72 2 85 6 98 10 14 +11 -13 370 22* 19 23 33 3 46 7 59 11 15 19 23* 112 3* +12 -12 7 8 20 12 16* 20* 60 0* 73 4* 86 8* 99 12* 16* +13 -11 21* 21 1* 34 5* 47 9* 13* 17* 21* 100 1* 113 5* +14 -10 8 10* 14* 18* 22* 61 2* 74 6* 87 11 15 19 +15 -9 23* 22 3* 35 8 48 12 16 20 88 101 4 114 8 +16 -8 9 13 17 21 49 1 62 5 75 9 13 17 21 +17 -7 10 2 23 6 36 10 14 18 22 89 2* 102 6* 115 10* +18 -6 15 19 23* 50 3* 63 7* 76 11* 15* 19* 23* +19 -5 11 4* 24 8* 37 12* 16* 20* 77 0* 90 4* 103 8* 116 12* +20 -4 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 +22 -2 20 26 39 4 52 8 65 12 16 20 105 118 4 +23 -1 13 9 13 17 21 66 1 79 5 92 9* 13* 17* Difference Constituent OO Hour d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. d. ft. C 119 6* 132 10* 145 14* 158 18* 171 22* 185 2* 198 6* 211 11 224 15 237 19 +1 -23 19* 23* 146 4 159 8 172 12 16 20 212 225 4 238 8 +2 -22 120 9 133 13 17 21 173 1 186 5 199 9 13 17 21 +3 -21 22 134 2 147 6 160 10 14 18 22 213 2* 226 6* 239 10* +4 -20 121 11 15 19* 23* 174 3* 187 7* 200 11* 15* 19* 23* +5 -19 122 0* 135 4* 148 8* 161 12* 16* 20* 201 0* 214 4* 227 8* 240 12* +6 -18 13* 17* 21* 162 1* 175 5* 188 9* 14 18 22 241 2 +7 -17 123 2* 136 6* 149 11 15 19 23 202 3 215 7 228 11 15 +8 -16 16 20 150 163 4 176 8 189 12 16 20 229 242 4 +9 -15 124 5 137 9 13 17 21 190 1 203 5* 216 9* 13* 17* +10 -14 18 22 151 2* 164 6* 177 10* 14* 18* 22* 230 2* 243 6* +11 -13 125 7* 138 11* 15* 19* 23* 191 3* 204 7* 217 11* 15* 19* +12 -12 20* 139 0* 152 4* 165 8* 178 12* 16* 21 218 1 231 5 244 9 +13 -11 126 9* 13* 18 22 179 2 192 6 205 10 14 18 22 +14 -10 23 140 3 153 7 166 11 15 19 23 219 3 232 7 245 11 +15 -9 127 12 16 20 167 180 4 193 8 206 12* 16* 20* 246 0* +16 -8 128 1 141 5 154 9* 13* 17* 21* 207 1* 220 5* 233 9* 13* +17 -7 14* 18* 22* 168 2* 181 6* 194 10* 14* 18* 22* 247 2* +18 -6 129 3* 142 7* 155 11* 15* 19* 23* 208 4 221 8 234 12 16 +19 -5 16* 20* 156 1 169 5 182 9 195 13 17 21 235 1 248 5 +20 -4 130 6 143 10 14 18 22 196 2 209 6 222 10 14 18 +21 -3 19 23 157 3 170 7 183 11 15 19* 23* 236 3* 249 7* +22 -2 131 8 144 12* 16* 20* 184 0* 197 4* 210 8* 223 12* 16* 20* +23 -1 21* 145 1* 158 5* 171 9* 13* 17* 21* 224 1* 237 5* 250 9* HARMONIC ANALYSIS AND PREDICTION OF TIDES 287 Table 31. — For construction of primary stencils — Continued Difference Hour +1 +2 +3 +4 +5 +6 +7 +8 +9 + 10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +21 +22 +23 -23 -22 -21 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -5 -4 -3 -2 -1 Constituent 00 d. h. 250 23 251 12 252 1 14* 253 3* 16* 254 6 19 255 8 21* 256 10* 23* 257 13 258 2 15* 259 4* 17* 260 7 20 261 9 22* 262 11* 263 0* 14 d. h. 264 3 16 265 5* 18* 266 7* 21 267 10 23 268 12* 269 1* 14* 270 4 17 271 6 19* 272 8* 21* 273 11 274 13 275 2* 15* 276 4* 18 d. h. 277 7 20 278 9* 22* 279 11* 280 1 14 281 3 16* 282 5* 18* 283 8 21 284 10 23* 285 12* 286 1* 15 287 4 17 288 6* 19* 289 8* 22 d. h. 290 11 291 13* 292 2* 15* 293 5 18 294 7 20* 295 9* 22* 296 12 297 1 14 298 3* 16* 299 5* 19 300 8 21 301 10* 23* 302 12* 303 2 d. h. 303 15 304 4 17* 305 6* 19* 306 9 22 307 11 308 0* 13* 309 2* 16 310 5 18 311 7* 20* 312 9* 23 313 12 314 1 14* 315 3* 17 316 6 d. h. 316 19 317 8* 21* 318 10* 319 13 320 2 15* 321 4* 17* 322 7 20 323 9 22* 324 11* 325 0* 14 326 3 16 327 5* 18* 328 7* 21 329 10 d. h. 329 23 330 12* 331 1* 14* 332 4 17 333 6 19* 334 8* 21* 335 11 336 13 337 2* 15* 338 4* 18 339 7 20 340 9* 22* 341 11* 342 1 14 d. h. 343 3 16* 344 5* 18* 345 8 21 346 10 23* 347 12* 348 1* 15 349 4 17 350 6* 19* 351 8* 22 352 11 353 13* 354 2* 15* 355 5 18 d. h. 356 7 20* 357 9* 22* 358 12 359 1 14 360 3* 16* 361 5* 19 362 8 21 363 10* 23* 364 12* 365 2 15 366 4 17* 367 6* 19* 368 9 22 rf. ft. 369 11 370 0* 288 U. S. COAST AND GEODETIC SURVEY Table 32. — Divisors for primary stencil sums Constituent J Series Hour 1 2 3 4 5 6 7 8 9___i... 10 : 11 12 13 14 15 16 17 18 19 20 21 22 23 29 58 87 105 134 163 192 221 250 279 297 326 355 30 59 87 106 134 164 192 221 250 279 298 326 355 31 59 89 106 135 164 193 222 250 280 298 327 356 28 58 86 104 134 162 192 220 250 278 296 326 354 30 59 88 106 135 165 192 222 251 280 299 326 356 29 59 88 104 135 163 193 222 250 280 297 327 355 28 59 87 105 134 163 193 221 251 278 297 326 355 30 57 88 106 134 165 192 222 250 280 298 326 356 28 58 87 104 134 163 193 221 250 279 297 327 354 29 58 88 106 134 164 193 222 251 279 298 326 356 29 57 87 105 134 163 192 222 250 280 297 326 355 28 58 86 104 134 162 193 220 250 278 297 326 354 30 59 88 107 134 164 193 223 251 280 299 327 357 29 57 87 104 134 162 191 221 250 279 296 326 354 28 58 85 104 133 162 191 220 250 278 297 325 354 30 58 88 106 134 164 192 223 250 280 297 327 356 29 58 87 105 135 162 192 220 251 279 296 327 355 28 58 86 105 133 163 191 220 250 279 297 325 355 30 57 87 105 134 163 192 221 250 280 296 326 355 28 58 86 104 134 162 192 220 250 278 296 325 355 29 58 87 106 133 163 191 221 249 280 297 325 356 29 57 87 104 134 162 191 220 249 279 296 326 354 28 58 85 104 133 162 191 219 249 277 296 325 354 30 58 88 106 134 164 192 222 249 279 298 326 356 28 57 86 104 134 161 191 219 249 277 295 325 353 Constituent K Series 14 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour 15 30 59 88 106 135 164 193 221 250 279 297 326 355 369 1 14 30 59 88 106 135 164 193 222 251 279 297 326 355 369 2 14 29 59 88 106 135 164 193 222 251 280 298 327 355 369 3 14 29 59 88 106 135 164 193 222 251 280 298 327 356 370 4 14 29 57 87 105 134 163 192 221 250 279 297 326 355 369 5 14 29 58 88 105 134 163 192 221 250 279 297 326 355 369 6 14 29 58 87 106 135 163 192 221 250 279 297 326 355 369 7 14 29 58 87 105 135 164 193 221 250 279 296 325 354 368 8 14 29 58 87 105 135 164 193 222 251 280 298 327 355 369 9 14 29 58 87 105 134 164 193 222 251 280 298 327 356 370 10 14 29 57 86 104 133 163 192 221 250 279 297 326 355 369 11 14 29 58 87 105 133 162 192 221 250 279 297 326 355 369 12 14 29 58 87 105 134 163 192 221 250 279 297 326 355 369 13 14 29 58 87 105 134 163 192 222 250 279 297 326 355 369 14 14 29 58 87 105 134 163 192 222 251 280 297 326 355 369 15 13 28 57 86 104 133 162 191 220 250 279 297 326 355 368 16 14 29 58 86 104 133 162 191 220 249 279 297 326 355 369 17 14 29 58 87 105 133 162 191 220 249 279 297 326 355 369 18 14 29 58 87 105 134 163 191 220 249 278 297 326 355 369 19 14 29 58 87 105 134 163 192 221 250 278 297 326 355 369 20 14 29 58 87 105 134 163 192 221 250 279 297 326 355 369 21 14 28 57 86 104 133 162 191 220 249 278 296 325 355 369 22 14 29 58 86 104 133 162 191 220 249 278 296 325 355 369 23 14 29 58 87 105 134 162 191 220 249 278 296 325 354 369 HARMONIC ANALYSIS AND PREDICTION OF TIDES 289 Table 32. — Divisors for primary stencil sums — Continued Constituent L Series 29 58 87 105 134 163 192 221 250 279 297 326 355 29 59 87 105 133 163 191 221 250 279 297 326 355 29 59 87 106 134 164 192 222 251 279 297 326 355 29 58 87 106 134 163 192 221 250 280 298 326 356 30 58 87 105 134 163 192 221 250 279 298 326 356 30 58 88 106 135 164 192 222 250 279 297 326 355 29 58 88 106 134 164 192 222 250 280 298 327 356 29 57 86 105 133 163 191 221 249 279 297 325 355 30 59 88 106 135 164 193 222 250 279 298 326 356 30 58 88 105 135 164 193 222 251 280 298 327 357 29 57 87 104 133 163 191 221 250 279 296 326 355 30 58 87 105 134 164 192 221 249 279 296 326 354 29 58 87 105 134 162 192 222 250 280 297 326 355 29 58 87 104 134 162 192 221 250 279 297 326 355 29 58 88 105 135 163 192 220 250 279 296 326 354 29 58 88 105 134 163 193 221 250 280 297 327 355 28 58 86 105 134 163 192 221 250 279 297 327 355 28 58 86 104 134 162 191 220 249 278 296 325 353 28 57 86 104 134 162 192 220 250 278 297 326 355 29 58 87 105 134 162 192 220 250 278 296 326 355 29 58 87 105 135 163 192 221 250 279 297 326 354 28 58 86 105 134 163 192 221 250 279 297 327 355 28 58 86 104 132 162 191 219 249 277 296 324 354 29 58 87 105 134 163 193 221 251 279 297 325 355 29 58 87 105 134 163 193 221 251 279 298 326 355 Hour 1 2 3 4 5 6 7 8 9 10--.---- 11 12 13 14. ..:... 15 .- 16 17 18 19 20 21 22 23 Constituent M Series 15 29 58 87 105 134 163 192 221 250 279 297 326 355 15 29 59 87 105 135 164 192 222 250 279 297 325 355 15 29 57 87 105 134 163 192 221 250 279 296 326 354 15 28 58 86 105 134 162 192 221 250 279 296 325 354 16 29 59 88 107 135 165 193 222 251 281 299 328 357 16 30 58 87 106 135 164 193 222 251 280 297 326 355 15 28 57 86 104 134 163 192 221 250 278 296 325 354 15 29 58 87 106 134 163 192 222 250 280 297 326 355 16 29 58 87 105 134 163 192 221 250 279 296 326 354 16 29 59 87 106 135 164 193 221 251 280 298 326 355 15 29 58 87 106 135 165 193 223 251 280 298 327 357 15 29 57 87 105 134 163 192 221 250 279 296 326 354 15 28 57 86 104 133 162 192 221 250 278 296 325 354 15 29 58 87 105 133 162 191 220 250 280 297 326 355 15 30 59 88 105 134 163 192 221 250 279 298 327 355 15 29 58 87 105 134 163 192 220 250 278 297 326 356 14 29 58 87 104 134 163 192 222 250 279 298 326 356 15 29 57 87 104 133 162 191 220 249 278 296 326 354 15 29 59 87 105 134 162 192 220 250 279 298 326 355 14 29 58 87 105 133 163 191 220 249 278 297 326 355 15 30 58 88 105 135 163 192 221 250 279 297 326 356 14 28 57 86 103 133 162 191 220 249 277 296 325 354 14 29 58 87 105 133 162 192 221 250 280 298 327 356 15 30 59 88 105 134 163 192 221 249 279 298 327 355 15 29 58 87 105 134 163 192 220 250 278 296 325 355 Hour 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 290 U. S. COAST AND GEODETIC SURVEY Table 32. — Divisors for primary stencil sums— Continued Constituent N Series Hour 1 2 3 4 15 29 87 105 134 163 192 221 250 279 297 326 355 87 105 106 105 106 105 106 105 105 107 105 105 106 106 104 104 105 104 104 105 105 104 103 106 104 134 135 133 134 135 134 133 133 135 134 134 135 134 133 133 135 134 134 134 134 133 133 137 133 163 165 162 163 164 164 163 163 164 163 163 165 164 161 161 164 162 162 164 163 161 161 165 162 191 194 191 192 193 192 191 191 194 192 191 193 192 191 191 194 191 191 192 191 192 194 191 220 223 220 221 223 222 221 220 223 221 221 224 220 219 220 222 220 220 222 220 219 220 223 220 250 252 248 249 252 250 249 250 251 252 250 249 250 251 249 249 252 250 249 249 252 249 279 281 278 279 282 279 278 279 281 279 279 281 278 277 279 280 277 278 280 278 277 279 281 277 297 299 296 297 299 297 296 298 299 297 297 299 297 295 297 298 295 296 296 295 297 298 295 327 327 324 326 328 326 324 326 327 325 326 328 325 324 326 327 325 327 327 325 325 326 328 325 356 357 354 355 357 355 354 357 356 354 356 357 355 354 354 355 353 355 356 354 353 354 356 353 Constituent 2N Series Hour 29 58 87 87 87 87 105 105 106 105 106 104 105 106 106 106 104 106 106 106 105 104 105 104 105 104 103 106 104 106 103 134 163 192 221 135 135 134 136 132 134 135 135 135 133 135 135 135 134 133 133 134 134 132 133 134 134 135 131 163 165 164 164 161 163 164 165 163 162 164 164 165 162 161 163 163 163 161 161 164 163 163 161 193 194 193 193 190 192 194 193 192 191 193 194 193 190 191 192 192 192 189 191 193 192 192 189 222 223 222 221 220 222 222 223 220 220 223 222 221 219 220 222 221 220 219 220 222 221 220 219 250 251 252 250 251 249 251 252 251 249 250 251 251 249 248 250 250 251 249 248 249 251 250 249 248 279 281 279 280 278 280 281 280 279 278 280 280 277 278 278 280 279 278 278 278 280 279 278 277 297 299 299 297 295 297 296 296 295 296 297 298 297 296 296 297 299 297 296 295 326 327 329 325 326 325 326 328 325 325 326 327 326 325 325 326 327 325 326 325 326 328 325 326 325 355 357 357 353 356 353 356 356 354 365 354 357 355 354 354 355 356 354 355 354 355 357 354 355 354 HARMONIC ANALYSIS AND PREDICTION OF TIDES 291 Table 32. — Divisors for primary stencil sums — Continued Constituent O 14 29 58 87 105 134 163 192 221 250 279 297 326 355 13 29 58 87 106 135 164 192 222 251 279 298 327 355 14 29 59 88 105 134 164 192 221 251 280 298 327 355 14 28 57 86 105 133 162 192 221 250 279 296 325 354 14 30 57 87 105 134 164 193 221 251 280 297 326 356 14 29 58 87 106 135 163 193 222 250 280 297 325 354 14 29 59 87 105 135 163 192 222 251 280 297 326 355 14 29 58 87 105 134 164 193 222 251 279 297 326 355 14 28 58 87 105 135 164 192 222 251 280 297 327 355 14 29 58 88 106 134 164 193 221 250 278 296 325 355 15 30 58 87 106 134 163 193 221 250 279 297 326 355 14 29 58 87 105 135 164 193 221 249 279 297 326 356 14 30 59 88 107 136 164 192 222 251 280 299 327 356 14 29 59 87 105 135 163 192 221 250 279 297 327 355 13 28 57 87 104 132 161 189 219 248 276 295 324 253 14 29 58 87 105 133 163 192 220 250 279 297 327 356 14 29 58 87 105 133 162 192 221 250 280 297 326 355 14 30 58 87 104 134 163 192 221 250 279 298 325 355 14 29 58 86 104 133 161 191 220 248 277 296 325 354 13 28 58 87 104 134 163 191 221 250 279 297 327 355 14 29 58 88 104 133 163 192 221 250 278 297 326 355 15 29 58 87 105 134 163 192 220 250 279 296 326 355 14 29 58 87 105 133 163 192 220 249 279 297 326 356 15 30 57 86 105 134 162 191 221 250 279 298 326 355 14 28 58 86 104 134 162 192 221 249 279 297 326 355 Constituent OO Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hours . 29 30 29 31 30 29 28 29 29 29 28 28 29 29 30 28 29 29 30 28 29 28 29 29 58 60 57 60 58 59 58 58 59 58 58 57 58 57 59 57 58 57 58 57 58 58 57 58 86 88 86 89 87 88 86 88 88 87 87 87 88 87 88 85 87 86 88 85 87 86 87 87 104 107 103 107 104 106 104 105 105 105 105 104 106 104 107 104 106 104 106 104 106 104 105 105 134 136 133 137 132 135 132 135 134 134 133 134 135 133 135 132 135 133 135 132 135 133 135 134 163 164 162 166 162 166 161 165 163 163 162 162 163 162 164 162 164 161 165 161 164 161 163 163 192 193 192 194 191 194 190 194 191 193 191 193 193 192 192 191 193 190 193 189 193 190 193 191 221 221 220 223 220 223 219 223 220 223 219 222 221 221 222 220 222 220 224 218 222 218 222 220 250 250 250 251 249 251 249 251 249 251 248 251 250 250 250 250 251 249 252 248 252 248 251 249 280 279 280 281 278 280 277 280 278 280 277 280 278 280 278 279 279 278 281 277 280 277 281 278 298 297 297 298 297 298 297 298 297 298 296 299 296 297 296 297 297 296 298 295 298 295 298 295 326 326 327 327 326 326 325 327 326 326 325 327 326 328 325 327 326 326 327 323 326 324 327 325 355 355 355 355 355 355 355 355 355 355 355 355 355 356 354 356 354 355 356 354 356 354 356 354 369 1 369 2 _ 369 3 370 4 369 5 369 6 368 7 369 8 369 9 369 10 368 11 12 369 369 13 370 14 369 15-.. 369 16 369 17 369 18 370 19 368 20 __-- 369 21 368 22 370 23 369 292 U. S. COAST AND GEODETIC SURVEY Table 32. — Divisors for primary stencil sums — Continued Constituent P Series Hour 29 58 87 105 134 163 192 221 250 279 297 326 35 87 105 105 105 106 105 105 105 106 105 105 105 106 105 105 106 105 106 104 105 104 104 104 104 105 135 134 134 135 134 134 134 135 134 134 I 134 135 134 134 135 133 134 133 134 133 134 133 133 134 164 163 163 164 164 163 163 164 163 164 163 164 163 162 163 162 163 162 163 162 163 162 162 163 193 192 192 193 193 192 192 193 192 193 192 192 191 192 192 191 192 192 192 191 192 191 191 192 222 221 222 222 222 221 221 222 221 221 220 221 220 221 221 220 221 221 221 220 221 221 220 221 251 250 251 251 251 250 251 251 249 250 249 250 249 250 250 249 250 250 250 249 250 250 249 250 280 279 280 280 280 279 279 279 278 279 279 279 278 279 280 278 279 279 279 278 279 279 278 279 297 298 298 297 296 297 297 296 297 297 297 296 297 297 297 296 297 297 296 298 327 297 327 327 326 325 326 326 325 326 326 326 325 326 327 325 326 326 327 325 326 326 325 327 356 354 355 356 355 354 355 356 354 355 355 356 354 355 356 354 355 355 356 354 355 355 354 356 Constituent Q Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour 29 29 29 28 30 29 30 30 28 29 28 29 29 30 29 30 29 29 28 29 29 30 29 27 58 59 56 59 58 59 58 58 59 58 58 57 59 58 59 58 59 57 58 57 57 58 56 88 86 88 86 89 87 88 87 87 88 86 88 86 89 87 88 86 87 85 87 86 87 87 85 106 104 106 103 107 105 107 104 105 106 104 106 104 107 104 107 104 106 103 105 104 105 105 103 136 133 135 132 136 133 136 133 135 135 134 134 133 136 133 136 133 135 131 134 132 134 134 133 164 162 165 161 166 162 165 162 164 163 163 164 163 165 161 164 161 164 160 164 161 164 162 162 194 191 193 190 195 191 195 191 194 192 192 192 191 192 191 194 190 193 188 193 190 193 191 192 222 221 222 220 225 219 224 219 223 221 221 220 220 221 220 222 219 223 218 223 218 222 220 221 250 250 251 249 253 249 254 248 251 248 250 249 250 250 250 251 248 251 247 252 247 252 249 251 280 280 280 278 282 277 281 277 280 278 280 277 279 278 279 280 278 280 277 281 276 281 277 280 297 298 298 297 299 296 300 295 298 296 298 295 297 296 297 297 296 298 295 299 294 299 295 298 326 327 326 326 328 325 328 324 326 325 327 324 327 325 327 326 325 326 324 328 324 328 325 327 355 357 354 354 356 254 356 354 356 355 355 353 356 354 356 355 355 355 354 356 353 356 354 357 368 1 370 2 368 3 369 4 370 5 . 369 6.. 371 7 369 8 369 9. 369 10 370 11 368 12 370 13 369 14.. 371 15. 368 16 368 17 368 18 ... 367 19 370 20 367 21 370 22 368 23 . 370 HARMONIC ANALYSIS AND PREDICTION OF TIDES 293 Table 32. — Divisors for primary stencil sums — Continued Constituent 2Q 29 58 87 105 134 163 192 221 250 279 297 326 355 83 101 102 89 75 75 83 102 101 89 75 75 83 101 101 87 74 74 81 101 101 87 74 74 113 116 117 104 90 90 113 117 116 104 90 90 113 117 117 103 90 90 113 117 117 103 90 90 142 141 142 129 115 136 142 142 141 129 115 136 142 142 142 128 115 136 142 142 141 127 114 135 167 166 167 154 159 167 167 167 166 154 159 167 167 167 167 153 159 165 166 166 166 152 158 166 192 191 192 196 192 192 192 192 191 197 191 192 192 191 191 194 191 190 191 191 191 194 191 191 217 216 233 230 217 217 217 217 232 230 215 216 216 216 231 229 216 215 216 216 231 229 216 216 242 255 269 255 242 242 241 254 267 255 240 241 241 254 268 254 241 240 241 254 268 254 241 241 279 293 293 279 266 266 277 293 292 280 265 277 293 293 279 266 265 278 292 293 279 266 266 308 309 295 282 282 309 309 308 296 281 283 307 294 281 280 307 308 294 281 281 334 333 334 320 307 332 334 334 333 320 305 331 332 333 333 319 306 330 333 332 333 319 306 331 359 358 359 345 355 358 358 358 357 345 353 358 357 358 358 344 354 357 358 357 358 344 354 358 Constituent R Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour 30 29 29 29 29 29 28 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 59 59 58 58 58 58 57 58 58 58 58 58 58 58 58 58 58 57 58 58 58 58 58 58 88 88 88 87 87 86 86 87 87 87 87 87 87 87 87 87 87 86 87 87 87 87 87 87 106 106 106 105 105 104 104 105 105 105 105 105 105 105 105 105 105 104 105 105 105 105 105 105 135 135 135 135 134 133 133 134 134 134 134 134 134 134 134 134 133 133 134 134 134 134 134 134 164 164 164 164 163 162 162 163 163 163 163 163 163 163 163 163 162 162 163 163 163 163 163 163 193 193 193 193 192 192 191 192 192 192 192 192 192 192 192 191 191 191 192 192 192 192 192 192 222 222 222 221 221 221 221 221 221 221 221 221 221 221 221 220 220 220 221 221 221 221 221 221 251 251 251 250 250 250 250 251 250 250 250 250 250 250 249 249 249 249 250 250 250 250 250 250 280 280 279 279 279 279 279 280 280 279 279 279 279 279 278 278 278 278 279 279 279 279 279 279 298 298 297 297 297 297 297 298 298 298 297 297 297 296 296 296 296 296 297 297 297 297 297 297 327 326 326 326 326 326 326 327 327 327 327 326 326 325 325 325 325 325 326 326 326 326 326 326 356 355 355 355 355 355 355 356 356 356 356 356 354 354 354 354 354 354 355 355 355 355 355 355 370 1. 369 2__ 369 3-. . 369 4 369 5. 369 6 369 7 370 8 370 9 370 10 370 11 370 12 368 13 14 368 368 15 368 16 17 368 368 18 19.. 369 369 20 21.. 369 369 22 23 369 369 294 U. S. COAST AND GEODETIC SURVEY Table 32. — Divisors for primary stencil sums — Continued Constituent T Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour 29 29 29 29 29 30 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 29 28 58 58 58 58 58 59 58 58 58 58 58 58 58 58 59 58 58 58 58 58 58 58 57 57 88 87 87 87 87 88 87 87 87 87 87 87 87 87 88 87 87 87 87 87 87 86 86 86 106 105 105 105 105 106 105 105 105 105 105 105 105 105 106 105 105 105 105 105 105 104 104 104 135 134 134 134 134 135 134 134 134 135 134 134 134 134 135 134 134 134 134 134 133 133 133 133 164 163 163 163 163 164 163 163 163 164 163 163 163 163 164 163 163 163 163 163 162 162 162 162 193 192 192 192 193 193 192 192 192 193 192 192 192 192 193 192 192 192 191 192 191 191 191 191 222 221 221 221 222 222 221 221 221 222 221 221 221 221 223 221 221 220 220 221 220 220 220 220 251 250 250 250 251 251 250 250 250 251 250 250 250 250 252 250 249 249 249 250 249 249 249 250 280 279 279 279 280 280 279 279 279 281 279 279 279 279 281 278 278 278 278 279 278 278 278 279 298 297 297 297 298 298 297 297 297 299 297 297 297 297 298 296 296 296 297 297 296 296 296 297 327 326 326 326 328 327 326 326 326 328 326 326 326 325 327 325 325 325 326 326 325 325 325 326 356 355 355 355 357 356 355 355 355 357 355 355 354 355 356 354 354 354 355 355 354 354 354 355 370 1 369 2 369 3 369 4 371 5 6 370 369 7 369 8 369 9 _____ 371 10 369 11 369 12. 368 13 369 14 370 15 368 16 368 17 368 18 _. 369 19 369 20 368 gl 368 22 368 23 369 Constituent X Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour 29 29 29 31 31 29 29 29 29 30 31 28 28 28 28 29 30 28 28 28 28 29 30 28 58 57 57 59 59 59 58 57 57 58 60 59 57 57 57 58 59 60 57 57 57 58 59 58 89 87 86 88 88 88 88 88 86 87 88 87 87 87 85 86 87 88 87 86 85 86 87 87 107 106 104 105 105 105 105 106 104 104 106 105 105 105 105 104 105 106 105 105 104 104 105 105 135 134 134 136 134 134 134 135 134 135 135 134 134 134 134 134 133 134 133 133 133 134 133 133 164 162 162 165 164 162 162 163 162 163 166 162 162 162 163 163 164 164 162 162 162 164 164 163 194 191 191 194 193 193 193 192 191 192 195 193 191 190 191 191 192 194 191 190 190 192 192 192 223 221 219 222 221 221 221 223 219 220 223 221 221 221 220 220 221 223 221 221 219 221 221 221 252 250 250 252 250 250 250 252 250 250 252 249 249 249 251 249 249 251 249 249 249 250 249 249 280 278 278 282 279 278 278 280 278 279 282 278 278 278 280 279 280 280 278 278 278 281 279 277 298 296 296 300 298 297 296 298 296 297 301 297 296 296 298 296 297 299 295 295 295 298 297 296 330 325 324 328 326 326 326 326 324 325 329 326 326 324 326 325 326 329 326 324 324 327 326 326 358 355 354 357 355 355 355 358 354 354 358 355 355 354 355 353 354 357 354 354 353 355 354 354 372 1 369 2 . 369 3 371 4 369 5.. 369 6 369 7 371 8 368 9 367 10 .. 371 11 . _ -_ 368 12 368 13 368 14 . .. 371 15 367 16 368 17 371 18 368 19 368 20 368 21 .. 370 22. 368 23 368 HARMONIC ANALYSIS AND PREDICTION OF TDDES 295 Table 32. — Divisors for primary stencil sums — Continued Constituent m Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour _ 29 30 30 27 30 30 27 29 30 29 28 30 29 28 29 30 29 28 29 30 27 30 30 28 59 61 57 57 60 57 56 59 59 57 58 59 57 57 59 59 57 57 59 57 57 60 58 56 89 88 88 87 87 86 86 88 87 87 88 86 86 87 88 87 87 86 86 87 87 88 87 85 105 107 105 104 106 106 103 106 107 104 105 105 105 104 106 107 104 103 106 106 104 106 105 101 135 135 134 134 135 133 133 136 134 134 134 134 133 134 136 134 133 133 135 134 134 136 132 131 163 164 164 163 162 163 163 163 164 163 163 162 163 164 163 164 162 162 163 164 163 164 162 161 192 194 193 190 192 193 190 193 194 191 191 192 193 191 193 193 191 190 193 193 191 193 192 190 223 223 221 219 222 220 220 223 222 219 221 222 220 221 222 222 219 220 222 221 221 222 220 219 252 252 250 249 249 250 250 251 250 249 251 219 250 251 250 250 249 249 250 251 250 250 249 249 280 282 280 276 279 280 278 280 279 279 278 279 280 279 278 280 278 277 280 281 277 279 279 278 299 298 296 296 298 297 297 299 296 296 298 297 297 297 298 297 296 297 298 298 296 297 295 295 326 327 326 325 326 327 326 326 326 326 326 326 327 326 325 327 326 325 327 327 325 326 325 325 356 356 355 353 356 356 354 355 356 354 355 356 356 353 355 357 354 354 356 356 353 356 356 352 369 1 369 2 369 3 4 368 369 5. 370 6 369 7 8 369 369 9 369 10 369 11. 12 13 14 369 369 369 368 15 16 . .. _ - 370 369 17 18 19 368 369 370 20 369 21 22 23 369 369 369 Constituent 29 58 87 105 134 163 192 221 250 279 297 326 355 31 59 86 103 135 165 193 221 249 283 300 327 355 28 56 83 103 134 161 189 216 250 278 295 322 351 28 56 89 107 135 162 190 224 252 280 297 324 358 28 60 89 106 134 161 195 222 250 278 295 329 357 33 62 90 107 135 167 196 223 251 280 302 329 357 30 57 85 102 135 164 192 219 249 281 299 326 354 28 55 83 104 134 161 189 217 250 277 295 322 352 28 55 90 107 135 162 191 224 252 279 296 325 358 28 61 89 106 134 161 195 222 250 277 295 329 357 34 62 90 107 134 169 197 224 252 282 302 330 358 29 56 84 101 134 162 190 217 248 279 296 324 351 28 55 85 107 134 162 190 219 251 278 295 323 353 28 56 89 106 133 161 190 223 251 278 295 326 356 29 62 90 107 134 164 195 223 251 278 298 330 357 32 60 88 105 134 167 194 222 250 281 300 328 355 27 55 83 101 133 161 188 216 247 278 295 323 350 27 55 86 107 134 162 189 221 250 278 295 323 355 27 58 88 106 133 161 192 223 250 278 295 327 356 30 62 89 107 134 165 195 223 250 278 299 330 357 31 59 86 103 134 166 193 221 248 282 299 327 354 27 55 82 101 133 161 188 216 249 278 295 322 350 27 55 87 106 134 162 189 222 250 278 295 322 356 27 59 88 105 133 160 193 223 250 278 295 328 357 31 62 89 106 134 165 195 223 250 279 300 328 356 296 U. S. COAST AND GEODETIC SURVEY Table 32. — Divisors for primary stencil sums — Continued Constituent p Series Hour 29 58 87 105 134 163 192 221 250 279 297 326 355 87 87 107 105 105 106 106 106 106 104 105 105 104 105 104 106 104 104 106 104 104 106 104 105 105 104 135 134 134 135 135 134 136 134 135 133 133 134 133 135 134 132 135 133 133 135 132 134 135 133 164 164 162 165 163 163 165 163 165 162 162 163 162 163 163 162 164 161 162 164 161 163 164 162 193 193 191 194 192 192 193 192 194 192 191 193 190 192 193 190 193 191 191 193 190 191 193 191 222 222 220 224 221 220 223 221 222 222 220 223 219 221 221 219 222 220 220 222 219 220 222 219 251 251 249 252 250 250 252 249 251 251 249 252 249 250 250 248 250 250 248 251 249 249 251 248 279 280 278 281 279 278 282 278 280 280 278 280 279 280 280 276 279 279 277 280 277 278 281 277 297 299 295 299 297 296 299 296 296 298 297 297 299 295 298 295 297 295 296 298 295 326 328 324 328 326 325 328 325 326 327 325 327 326 326 328 324 327 327 325 326 325 324 328 323 355 356 353 357 355 354 357 354 355 357 353 356 355 355 357 354 355 356 354 356 354 353 357 352 Constituent MK Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour 1 2 30 29 29 30 29 29 30 30 30 28 29 29 28 29 29 29 28 29 29 28 29 29 28 29 59 58 58 58 58 58 58 58 59 57 58 59 58 58 59 59 57 59 58 57 57 58 57 57 88 88 88 86 87 87 86 87 88 86 86 88 86 86 88 87 86 87 88 87 86 88 87 87 105 106 105 104 106 105 105 105 106 105 104 106 105 105 106 106 105 104 105 105 103 105 105 104 135 135 134 133 134 134 134 133 135 134 133 134 134 133 134 135 133 134 134 135 133 134 135 134 164 163 164 163 163 164 164 163 164 164 163 162 163 162 163 163 162 163 162 163 162 162 163 163 192 192 192 191 192 192 193 192 193 192 192 192 192 192 193 192 192 193 191 193 191 191 191 192 222 222 221 221 222 220 221 221 221 220 221 221 220 221 222 221 220 222 220 221 221 221 221 221 251 251 249 250 251 250 250 251 251 249 251 250 250 250 250 250 249 250 249 250 249 250 250 249 279 280 278 278 280 279 279 279 280 278 279 279 279 278 280 280 278 279 279 279 278 280 280 278 297 298 297 296 298 298 297 297 299 296 297 298 297 297 297 297 296 296 296 297 296 297 297 297 325 327 326 325 326 327 326 326 327 325 326 326 326 326 326 327 326 325 326 327 325 326 327 325 355 356 355 355 356 356 356 355 355 354 355 354 355 355 355 355 355 354 354 356 354 355 355 355 368 369 369 3 . 368 4 369 S_. 369 6. 369 7 8-. 369 369 9_ 369 10 11 12 13 14 15 . 369 369 369 369 370 369 16-- 369 17 369 18 19, - 368 370 20 21 22 23- . 369 369 369 370 HARMONIC ANALYSIS AND PREDICTION OF TIDES 297 Table 32. — Divisors for primary stencil sums — Continued Constituent aMK Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour 0.. 30 29 29 29 30 59 59 59 58 59 87 87 87 86 87 106 106 106 105 105 134 134 134 133 134 164 164 163 163 164 193 193 192 193 192 221 221 221 222 221 251 250 250 251 250 280 280 279 280 279 298 298 297 298 297 326 327 325 326 326 355 356 355 355 355 369 1. 370 2 368 3.. 369 4 369 5 29 29 30 59 58 59 88 86 87 106 104 105 135 134 135 164 163 164 192 191 192 222 221 221 251 250 251 279 278 280 298 297 299 327 325 327 357 354 356 370 6 368 7 370 8 30 29 58 57 87 87 105 104 134 134 163 164 192 192 221 222 250 251 279 279 296 297 326 326 355 356 368 9 369 10 30 58 88 105 135 164 192 221 251 279 297 326 355 369 11 . 30 29 59 57 88 87 106 105 135 134 164 162 193 192 222 221 251 249 280 278 297 296 327 326 356 356 370 12 369 13 29 57 87 104 134 162 191 221 249 278 296 325 354 368 14 29 29 28 57 58 57 86 87 87 104 105 105 133 134 135 162 162 163 191 192 192 220 221 222 249 249 250 279 279 279 297 296 297 326 326 326 355 355 354 369 15 369 16 369 17 28 57 86 104 133 162 191 220 249 278 296 325 353 368 18 29 59 88 106 135 163 193 222 250 280 298 328 356 371 19 28 58 87 105 134 162 192 221 249 278 296 325 354 368 20 28 57 87 104 133 162 191 220 250 279 297 326 355 369 21 29 58 87 105 133 163 192 220 250 279 297 326 354 369 22 28 58 87 106 134 163 193 221 250 279 297 326 355 370 23 28 57 87 104 133 162 191 219 249 278 296 325 354 368 Constituent MN 29 58 87 105 134 163 192 221 250 279 297 326 355 28 56 85 104 134 163 193 223 253 283 301 328 356 30 60 90 109 139 166 194 222 250 277 296 325 355 28 56 84 101 129 158 188 218 248 278 297 326 355 30 60 91 109 139 168 198 226 254 281 299 326 355 30 59 87 104 132 159 187 217 248 277 296 325 355 28 57 88 106 136 165 195 225 256 283 301 328 356 30 61 90 109 136 164 192 220 247 277 295 325 355 28 56 83 101 130 160 190 221 250 280 298 327 355 30 61 90 109 138 168 196 224 251 279 296 325 355 29 57 84 102 129 157 187 218 247 277 295 325 355 30 60 89 108 137 167 198 228 255 283 300 328 356 31 61 89 107 134 162 190 218 247 277 295 325 356 28 55 84 102 132 162 193 222 252 282 299 327 355 30 59 89 107 137 165 193 220 248 276 294 324 355 28 55 83 100 128 159 189 218 248 278 296 327 355 30 59 89 107 137 168 198 225 253 281 298 326 356 30 58 86 103 131 159 187 216 246 276 294 325 355 28 56 86 104 135 165 195 224 254 282 299 327 355 29 59 89 107 135 163 190 218 246 276 295 325 354 27 55 83 100 131 161 190 220 250 280 299 328 355 29 59 89 108 138 168 195 223 251 279 296 325 354 28 56 84 101 129 157 186 216 246 276 295 325 354 28 58 88 107 137 167 196 226 254 282 299 327 354 29 59 88 105 133 161 188 216 246 276 295 325 354 298 U. S. COAST AND GEODETIC SURVEY Table 32. — Divisors for primary stencil sums — Continued Constituent MS Series 29 58 87 105 134 163 192 221 250 279 297 326 355 30 59 88 106 135 164 192 222 250 279 297 326 354 30 58 88 106 134 164 192 221 250 279 296 326 355 29 58 87 105 134 163 192 222 250 280 298 327 355 30 58 88 107 135 165 194 223 252 281 298 328 356 29 58 87 105 134 163 191 221 249 279 296 325 354 30 58 88 105 134 164 192 221 250 280 297 327 256 29 58 88 105 135 164 194 223 252 281 299 328 356 30 58 87 105 134 162 192 220 250 278 296 326 354 29 58 87 104 134 162 191 220 249 278 296 326 355 29 57 87 104 133 162 192 220 250 279 297 326 355 30 59 88 106 136 164 193 222 251 279 298 327 355 30 58 88 105 134 163 192 220 250 278 296 326 354 28 58 87 104 134 162 192 221 250 279 298 326 356 28 57 86 105 134 163 193 221 251 279 297 325 355 29 59 87 105 135 163 192 221 250 278 297 325 354 29 58 87 105 134 163 192 220 250 278 296 325 355 28 58 86 104 134 163 193 222 251 280 299 327 356 29 58 87 106 135 163 193 221 251 279 297 326 355 28 58 86 104 133 162 190 220 248 278 296 324 354 28 57 86 104 132 162 191 220 249 279 297 326 356 29 59 87 106 134 1B4 192 222 250 279 298 326 355 29 58 86 105 133 162 191 220 249 278 296 325 ' 354 28 58 86 104 133 162 190 220 248 278 296 325 355 28 57 86 105 133 163 192 221 250 280 297 326 356 Hour Constituent 2SM Series 29 58 87 105 134 163 192 221 250 279 297 326 355 369 Hour 28 30 28 30 28 30 28 30 29 29 30 28 30 28 30 27 30 57 60 56 60 55 60 56 60 57 59 59 57 59 57 60 55 60 56 60 57 59 57 58 58 87 88 86 89 84 90 84 90 85 89 86 87 87 87 88 85 89 84 90 85 89 85 88 86 106 106 105 107 103 107 103 109 103 108 104 106 104 105 105 103 106 102 108 102 107 102 106 103 136 133 135 135 133 135 133 137 133 137 133 136 132 135 133 133 134 132 135 132 135 132 135 132 164 163 165 163 163 163 163 164 163 165 163 164 161 164 162 163 162 162 163 162 163 162 163 162 192 193 193 193 191 193 192 193 193 193 193 192 191 191 192 191 192 191 192 192 190 192 191 192 220 223 220 223 219 223 220 223 221 222 223 220 221 219 222 219 222 219 222 220 220 221 220 222 250 252 249 253 247 253 248 253 249 252 251 249 249 249 251 247 252 247 252 248 250 249 250 250 280 280 279 281 276 282 277 283 277 282 279 279 277 279 279 277 281 276 282 276 280 276 280 278 299 297 297 298 294 299 295 300 295 300 295 298 295 298 297 296 298 295 300 295 298 295 299 295 329 325 327 326 324 326 325 328 325 328 325 327 323 328 325 326 326 325 328 325 326 325 327 325 356 355 356 355 353 355 355 356 355 356 355 355 353 356 355 355 355 3(55 355 355 354 355 355 355 369 1 2. 370 370 3 370 4 367 5 370 6 370 7.. 370 8 370 9 370 10 370 11 369 12 13 14 367 369 369 15 . 368 16 369 17—. 28 30 29 29 29 28 30 368 18 369 19 __. 369 20-. 367 21 369 22 .. 368 23 369 HARMONIC ANALYSIS AND PREDICTION OF TIDES 299 Table 33. — For construction of secondary stencils Con- stituent A. J S L Con- stituent B. OO 2SM K and P R and T MS X J hours Differ- J hours Differ- S hours Differ- S hours Differ- L hours Differ- L hours Differ- Page ence, hours ence, hours ence, hours ence, hours ence, hours ence, hours + _ db ± _ 1 0-23 3 0-23 0-23 0-23 0-23 0-23 2 10- 3 9 0-23 1 0-23 1 0-23 0-23 0-23 1 3 16- 4 15 0-23 2 0-23 1 0-23 1 17-21 0-23 1 4 23- 5 21 0-23 3 0-23 2 0-23 1 0-23 1 0-23 1 5. 5-6 3 0-23 4 0-23 2 0-23 1 0-23 1 0-23 2 6 0-23 10 0-23 5 0- 1 2 0-23 1 0-23 1 0-23 2 7 19-12 16 0-23 6 0-23 3 0-15 1 0-23 1 0-23 3 8 1-12 22 1-11 6 0-23 3 0-23 2 0-23 2 0-23 3 9 8-13 4 0-23 7 0-23 4 0-23 2 0-23 2 0-23 3 10 -- 14 10 0-23 8 0-23 4 0-23 2 0-23 2 0-23 4 11 0-23 17 0-23 9 0-23 5 0-23 2 0-23 2 0-23 4 12 3-20 23 0-23 10 0-23 5 0-23 3 0-23 2 0-23 5 13 10-21 5 0-23 11 0-23 6 0-23 3 0-23 3 0-23 5 14 16-22 11 0-23 12 0-23 6 0-23 3 0-23 3 12-20 5 15 23 17 0-23 13 0-23 7 0-23 3 0-23 3 0-23 6 16 6- 4 6- 3 13 0-23 7 0-23 4 0-23 3 0-23 6 17 12- 5 6 0-23 14 0-23 8 0-23 4 0-23 3 0-23 7 18 19- 6 12 0-23 15 0-23 8 0-23 4 0-23 4 0-23 7 19 1- 7 18 0-23 16 0- 8 8 0-23 4 0-23 4 0-23 8 20 8 0-23 17 0-23 9 0-23 4 0-23 4 0-23 8 21 0-23 7 0-23 18 0-23 9 0-23 5 0-23 4 0-23 8 22 21-14 13 0-23 19 0-23 10 0-23 5 0-23 4 0-23 9 23 4-14 19 4- 9 19 0-23 10 0-23 5 0-23 5 0-23 9 24 10-15 1 0-23 20 0-23 11 0-23 5 0-23 5 0-23 10 25 0-23 8 0-23 21 0-23 11 0-23 6 0-23 5 0-23 10 26 23-21 14 0-23 22 0-23 12 0-23 6 0-23 5 0-23 10 27 6-22 20 0-23 23 0-23 12 0-23 6 0-23 5 0-23 11 28 12-23 2 0-23 0-23 13 0-23 6 0-23 6 0-23 11 29 19- 8 0-23 1 0-23 13 0-23 7 0-23 6 0-23 12 30 1 14 0-23 2 0-23 14 0-23 7 0-23 6 0-23 12 31 8- 6 21 8- 1 2 0-23 14 0-23 7 0-23 6 0-23 12 32 15- 7 3 0-23 3 0-15 14 0-23 7 0-23 6 0-23 13 33 21- 8 9 0-23 4 0-23 15 0-23 7 0-23 7 0-23 13 34 4- 9 15 0-23 5 0-23 15 0-23 8 0-23 7 0-23 14 35 0-23 22 0-23 6 0-23 16 0-23 8 0-23 7 0-23 14 36 17-15 4 0-23 7 0-23 16 0-23 8 0-23 7 2-21 14 37 23-15 •10 0-23 8 0-23 17 0-23 8 0-23 7 0-23 15 38 6-16 16 6- 8 8 0-23 17 0-23 9 0-23 8 0-23 15 39 12-17 22 0-23 9 0-23 18 0-23 9 0-23 8 0-23 16 40 0-23 5 0-23 10 0-23 18 0-23 9 0-23 8 0-23 16 41 2-23 11 0-23 11 0-23 19 0-23 9 0-23 8 13-15 16 42 8- 17 0-23 12 0-23 19 0-23 10 0-23 8 0-23 17 43 15- 1 23 0-23 13 0-23 20 0-23 10 0-23 9 0-23 17 44 21- 2 5 0-23 14 0-23 20 0-23 10 0-23 9 0-23 18 45 0-23 12 0-23 15 0-23 20 0-23 10 0-23 9 0-23 18 46 .. 10- 8 18 10-23 15 0-23 21 0-23 10 0-23 9 0-23 19 47 17- 9 0-23 16 0-23 21 0-23 11 0-23 9 0-23 19 48 23-10 6 0-23 17 0-23 22 0-23 11 0-23 10 0-23 19 49 6-11 12 0-23 18 0-23 22 0-23 11 0-23 10 0-23 20 50 0-23 19 0-23 19 0-23 23 0-23 11 0-23 10 0-23 20 51 19-16 1 0-23 20 0-23 23 0-23 12 0-23 10 0-23 21 52 2-17 7 0-23 21 0-23 0-23 12 8-16 10 0-23 21 (53)..... 7-14 12 0-23 21 0-23 0-23 12 .0-23 11 0-23 21 300 U. S. COAST AND GEODETIC SURVEY Table 33. — For construction of secondary stencils — Continued Constit- uent A L M N O Constit- uent B MK MN 2MK V i" 2N L hours Differ- M Differ- N Differ- N Differ- O hours Differ- O hours Differ- Page enc3, hours hours ence, hours hours ence, hours hours ence, hours ence, hours ence, hours _ + + + + 1 23-10 0-23 1 20-7 0-23 0-23 0-23 2 20- 8 1 0-23 2 11-23 1 0-23 1 0-23 1 0-23 3 17- 5 2 0-23 4 2-14 2 0-23 1 0-23 1 0-23 4 15- 3 3 0-23 5 17- 6 3 0-23 1 0-23 2 0-23 5 12- 1 4 0-23 7 9-21 4 0-23 2 0-23 2 0-23 6 9-22 5 0-23 8 0-13 5 0-23 2 7- 8 2 0-23 7 7-20 6 0-23 10 15- 4 6 0-23 3 0-23 3 0-23 8 4-17 7 5- 11 6-19 7 0-23 3 0-23 3 0-23 9 2-15 8 0-23 13 22-11 8 0-23 3 0-23 4 0-23 10 23-12 9 18- 8 14 13- 2 9 0-23 4 0-23 4 0-23 11 20-10 10 0-23 16 4-18 10 0-23 4 0-23 5 0-23 12 18- 7 11 7-15 17 20- 9 11 0-23 5 0-23 5 0-23 13 15- 5 12 0-23 19 11- 1 12 0-23 5 0-23 6 0-23 14 12- 2 13 19-22 20 2-16 13 2-10 5 0-23 6 0-23 15 10- 14 0-23 22 17- 2 14 0-23 6 0-23 7 0-23 16 7-21 15 0-23 9-23 15 0-23 6 0-23 7 0-23 17 4-19 16 0-23 1 0-15 16 0-23 7 0-23 8 0-23 18 2-17 17 0-23 3 15- 6 17 0-23 7 0-23 8 0-23 19 23-14 18 0-23 4 6-21 18 0-23 8 21- 5 8 0-23 20 21-12 19 0-23 6 22-13 19 0-23 8 0-23 9 0-23 21 18- 9 20 0-23 7 13- 4 20 0-23 8 0-23 9 0-23 22 15- 7 21 0-23 9 4-20 21 0-23 9 0-23 10 0-23 23 13- 5 22 0-23 10 19-11 22 0-23 9 0-23 10 0-23 24 10- 2 23 0-23 12 11- 3 23 0-23 10 0-23 11 0-23 25 7-0 0-23 13 2-18 0-23 10 0-23 11 0-23 26 5-21 1 0-23 15 17-10 1 0-23 10 0-23 12 0-23 27 2-19 2 0-23 16 8- 1 2 0-23 11 0-23 12 0-23 28 23-16 3 0-23 18 0-16 3 0-23 11 0-23 13 0-23 29 21-14 4 6-0 19 15- 8 4 0-23 1:2 0-23 13 0-23 30 18-11 5 0-23 21 6-23 5 0-23 12 0-23 14 0-23 2 31 15- 9 6 19- 7 22 22-15 6 0-23 12 0-23 14 0-23 2 32 13-6 7 0-23 13- 6 7 0-23 13 11- 4 14 0-23 2 33 10- 4 8 7-l<5 1 4-22 8 0-23 13 0-23 15 0-23 2 34 8- 1 9 0-23 3 19-13 9 0-23 14 0-23 15 0-23 2 35 5-23 10 20-22 4 11- 5 10 0-23 14 0-23 16 0-23 2 36 2-21 11 0-23 6 2-20 11 2-20 14 0-23 16 0-23 2 37 0-18 12 0-23 8 17-12 12 0-23 15 0-23 ' 17 0-23 2 38 21-16 13 0-23 9 8- 3 13 0-23 15 0-23 17 0-23 2 39 18-13 14 0-23 11 0-18 14 0-23 16 0-23 18 0-23 2 40 16-11 15 0-23 12 15-10 15 0-23 16 0-23 18 0-23 2 41 13- 8 16 0-23 14 6- 1 16 6-9 16 0-23 19 0-23 2 42 10- 6 17 0-23 15 21-17 17 0-23 17 0-23 19 0-23 2 43 8- 4 18 0-23 17 13- 8 18 0-23 17 0-23 20 0-23 2 44 5- 1 19 0-23 18 4-0 19 0-23 18 0-23 20 0-23 2 45 3-23 20 0-23 20 19-15 20 0-23 18 1-23 20 0-23 2 46 0-20 21 0-23 21 10- 6 21 0-23 19 0-23 21 0-23 2 47 21-18 22 0-23 23 2-22 22 0-23 19 0-23 21 0-23 2 48 19-15 23 18-16 17-13 23 0-23 19 0-23 22 0-23 2 49 16-13 0-23 2 8- 5 0-23 20 0-23 22 0-23 3 50 13-10 1 6-23 3 0-20 1 0-23 20 0-23 23 0-23 3 51 11- 8 2 0-23 5 15-12 2 0-23 21 0-23 23 0-23 3 52 8- 5 3 19- 7 6 6- 3 3 0-23 21 0-23 0-23 3 (53)..... 0-23 4 0-23 8 0-23 4 0-23 21 0-23 0-23 3 HARMONIC ANALYSIS AND PREDICTION OF TIDES Table 33. — For construction of secondary stencils — Continued 301 Constituent A O Constituent B P Q 2Q Page O hours Differ- ence, hours hours Differ- ence, hours O hours Differ- ence, hours O hours Differ- ence, hours O hours Differ- ence, hours 1 . 18- 1 0-23 18-15 7-19 19-22 0-23 19-12 7-16 19 8- 5 20- 9 8-13 0-23 8- 2 20- 6 9 21-19 9-23 21- 3 0-23 21-16 10-20 22- 0-23 22-13 10-17 22- 6 10- 6 23-10 11-14 0-23 11- 3 23- 7 11 0-20 12- 0- 4 0-23 0-17 12-21 1 13-10 1-14 13-16 0-23 13- 7 2-11 14-15 2- 14- 4 2- 8 0-23 4-16 2 8 13 18 23 5 10 15 20 2 7 12 18 23 4 9 15 20 1 12 17 22 4 9 14 19 6 11 17 22 3 8 14 19 6 11 16 21 3 8 13 19 5 10 16 21 2 8 12 0-23 6- 3 18-12 7-22 19- 8 7-18 19- 3 7-13 19-23 8 0-23 8- 5 20-15 8- 1 20-10 9-20 21- 6 9-15 21- 1 9-11 0-23 10- 8 22-17 10- 3 22-13 10-22 22- 8 10-18 23- 3 11-13 23 0-23 23-20 11- 5 0-15 12- 1 0-10 12-20 0- 6 12-15 1 0-23 1-22 13- 8 1-17 13- 3 2-13 14-22 2- 8 14-18 2- 3 0-23 4-23 3 9 15 21 3 9 15 21 3 9 16 22 4 10 16 22 4 10 16 22 5 11 17 23 5 11 17 23 5 11 17 6 12 18 6 12 18 6 13 19 1 7 13 19 1 7 3 19 2 18-22 6- 8 18- 6 7-16 19- 1 7-11 19-21 7-19 19- 5 8-15 20- 8-10 20 8-18 20- 4 9-14 21-23 9 21- 7 9-17 21- 3 10-12 22 10-21 22- 6 10-16 22- 2 10-11 23-10 11-19 23- 5 11-15 23- 11-23 0- 9 12-18 0- 4 12-14 0-12 12-22 1- 7 13-17 1- 3 13 1-11 13-21 2- 6 14-16 2 14- 2-10 14-20 4 5 17 6 18 6 18 6 19 7 19 7 19 7 20 8 20 8 20 9 21 9 21 9 22 10 22 10 22 11 23 11 23 11 12 12 13 1 13 1 13 1 14 2 14 2 14 3 15 3 13 23-11 9-20 7-17 17- 5 2-14 12- 22-10 20- 6 6-18 16- 3 1-13 11-23 21- 8 19- 7 5-17 15- 2 0-12 10-22 8-20 18- 6 4-16 13- 1 23-11 22- 9 7-19 17- 5 3-14 12- 11-22 20- 8 6-18 16- 4 1-13 0-10 10-21 19- 7 5-17 15- 2 13-23 23-11 8-20 18- 6 4-16 14- 1 12- 22- 9 7-19 17- 5 3-15 1-13 11-23 21- 8 5-16 6 18 7 19 19 20 8 20 8 20 8 21 9 21 9 21 10 22 10 22 10 23 11 23 11 23 12 12 12 1 13 1 13 1 14 2 14 2 14 2 15 3 15 3 15 4 16 4 11 12-17 21- 5 7 2 19 3 4 6 15-18 1- 6 11-18 20 8 20 5 . 6 7 8 8 9 10 11 4- 7 14-19 0- 7 9-19 21 9 12 21 13 9 14 15 16 18-19 3- 8 13-20 23- 8 10 22 17 10 18 22 19 20 7- 8 17-20 2- 9 12-21 23 11 22 23 11 24 20-21 6- 9 15-21 1- 9 12 27 . 12 29 30 9-10 19-22 5-10 14-22 1 31 13 32 10 13 34 . 22-23 8-11 18-23 3-11 14 36 2 37 14 38 o 39 40 41 21- 7-12 17- 2-12 15 42 ... 3 43 15 44 3 45 46 10-12 20- 1 6-13 16- 1 4 47 48 16 4 49 . 16 50 51 0- 1 9-13 17- 3 17 52 . 5 15 246037—41- -20 302 U. S. COAST AND GEODETIC SURVEY Table 34. — For summation of long-period constituents Assignment of daily sums for constituent Mf Constituent division Days of series 1 28 55 82* 110 137 164* 192 219 246 274 301 328 356 2 29 56 84 111 138 166 193 220 248* 275 302 330* 357 3 30 57* 85 112 139 167 194 221 249 276 303 331 358 4 31 59 86 113 141* 168 195 223* 250 277 304* 332 359 5 32 60 87 114 142 169 196 224 251 278 306 333 360 6 34* 61 88 115* 143 170 197* 225 252 279 307 334 361 7 35 62 89 117 144 171 199 226 253 281* 308 335 363* 8* 36 63 90* 118 145 172 200 227 254 282 309 336 364 10 37 64 92 119 146 174* 201 228 256* 283 310 337* 365 11 38 65 93 120 147 175 202 229 257 284 311 339 366 12 39 67* 94 121 149* 176 203 230* 258 285 312* 340 367 13 40 68 95 122 150 177 204 232 259 286 314 341 368 14 42* 69 96 123* 151 178 205* 233 260 287 315 342 369 15 43 70 97 125 152 179 207 234 261 289* 316 343 16* 44 71 98 126 153 180 208 235 262 290 317 344 .... 18 45 72 100* 127 154 182* 209 236 263* 291 318 345* 19 46 73 101 128 155 183 210 237 265 292 319 347 20 47 75* 102 129 156* 184 211 238* 266 293 320 348 21 48 76 103 130 158 185 212 240 267 294 322* 349 22 49* 77 104 131* 159 186 213 241 268 295 323 350 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 23 51 78 105 133 160 187 215* 242 269 297* 324 351 24 52 79 106 134 161 188 216 243 270 298 325 352 26* 53 80 108* 135 162 189* 217 244 271* 299 326 353 27 54 81 109 136 163 191 218 245 273 300 327 355* Assignment of daily sums for constituent MSf Constituent division Days of series 1 2 3 4 5 6 7 8 9 10 11. 12 13 14 15 16 17 18 19 20 21 22 23 119 148 178 207 237 266 296 325 355 120 149* 179 208* 238 268* 297 327* 356 121 151 180 210 239 269 298 328 357 122 152 181* 211 240* 270 300* 329 359* 124* 153 183 212 242 271 301 330 360 125 154 184 213* 243 272* 302 332* 361 126 156* 185 215 244 274 303 333 362 127 157 186 216 245 275 304* 334 364* 128* 158 188* 217 247* 276 306 335 365 130 159 189 218 248 277 307 336* 366 131 160* 190 220* 249 279* 308 338 367 132 162 191 221 250 280 309 339 368* 133* 163 192* 222 252* 281 311* 340 -.... 135 164 194 223 253 282 312 341 136 165* 195 224* 254 284* 313 343* .... 137 167 196 226 255 285 314 344 .... 138 168 197* 227 256* 286 316* 345 .... 140* 169 199 228 258 287 317 346 .... 141 170 200 229 259 288* 318 348* .... 142 172* 201 231* 260 290 319 349 .... 25 55 84 114 143 173 202 232 261 291 320* 350 26 56 85* 115 144* 174 204* 233 263* 292 322 351 .... 28* 57 87 116 146 175 205 234 264 293 323 352* .... 29 58 88 117* 147 176* 206 236* 265 295* 324 354 .... 1 30 60* 89 2 31 61 90 3 33* 62 92* 4 34 63 93 5* 35 65* 94 7 36 66 95 8 37* 67 96* 9 39 68 98 10 40 69* 99 12* 41 71 100 13 42 72 101* 14 44* 73 103 15 45 74 104 17* 46 76* 105 18 47 77 106 19 49* 78 108* 20 50 79 109 21* 51 80* 110 23 52 82 111 24 53* 83 112* HARMONIC ANALYSIS AND PREDICTION OF TIDES 303 Table 34. — For summation of long-period constituents — Continued Assignment of daily sums for constituent Mm Constituent division Days of series . 1 28 2 29 3 30 4 32* 5 33 6 34 7 35 9* 36 10 37 11 38 12 40* 13 41 14 42 15* 43 17 44 18 45 19 46* 20 48 21 19 22 50 23* 51 25 52 26 53 27 54* 56 57 58 59 60 61 63* 64 65 66 67 68 71* 72 73 74 75 76 77* 79 80 81 82 83 84 85* 87 88 89 90 91 92 94* 95 96 97 98 99 100 102* 103 104 105 106 107 108* 110 111 138 112 139* 113 141 114 142 115 143 116* 144 118 145 119 146 120 147* 121 149 122 150 123 151 125* 152 126 153 127 154 128 156* 129 157 130 158 131 159 133* 160 134 161 135 162 136 164* 137 165 166 167 168 169 170* 172 173 174 175 176 177 178* 180 181 182 183 184 185 187* 188 189 190 191 192 193 195* 196 197 198 199 200 201* 203 204 205 206 207 208 209* 211 212 213 214 215 216 218* 219 220 221 222 223 224 226* 227 228 229 230 231 232* 234 235 236 237 238 239 240* 242 243 244 245 246 247 249* 250 251 252 253 254 255 257* 258 259 260 261 262 263* 265 266 267 268 269 270 271* 273 274 275 276 277 278 280* 281 282 283 284 285 286 288* 289 290 291 292 293 294* 296 297 298 299 300 301 302* 304 305 306 307 308 309 311* 312 313 314 315 316 317 319* 320 321 322 323 324 325* 327 328 329 330 331 332 333* 335 336 337 338 339 340 342* 343 344 345 346 347 348 350* 351 352 353 354 355 356* 358 359 1 360 2 361 3 362 4 363 5 364* 6 366 7 367 8 368 9 369 10 11 12 13 14 15 17 18 19 20 . 22 23 Assignment of daily sums for constituent Sa Constituent division Days of series Constituent division Days of series f 1- 8 \ 359-369 9- 23 24- 38 39- 53 54- 69 70- 84 85- 99 100-114 115-129 130-145 146-160 161-175 12 176-190 .... . - 13. 1 14 206-221 2 15 222-236 3 16 237-251 4 17 252-266 5 18 267-282 6 19 283-297 7 20 298-312 8 21 313-327 9 .. 22 . . 328-342 10 23 343-358 11 304 U. S. COAST AND GEODETIC SURVEY Table 35.— Products I : < Vo\ n 444 '"(4) Constituent Time meridian in hours =S-f- 15 1.000 2.000 3.000 6.000 Products, in degrees 6.500 M 2 S 2 N 2 Kl M4------ Oi M 6 (MK)i. S4 (MN) 4 - vi S 6 H2 (2N) 2 — (OO),.. X2 Si Mi Ji Mm Ssa Sa MSf.._. Mf pi Qi Ta R 2 (2Q)i — Pi (2SM) 2 . M 3 L 2 (2MK)j K 2 M 8 (MS)*.. 28.98 30.00 28.44 57.97 60.00 58.88 86.95 90.00 85.32 115.94 120. 00 113.76 144. 92 150.00 142. 20 159.41 165. 00 156.42 173.90 180.00 170. 64 15.04 57.97 13.94 . 30. 08 115.94 27.89 45.12 173. 90 41.83 60.16 231. 87 55.77 75.21 289. 84 69.72 82.73 318. 83 76.69 90.25 347. 81 83.66 86.95 44.03 60.00 57.42 173.90 88.05 120. 00 114. 85 260. 86 132. 08 180. 00 172. 27 347. 81 176. 10 240. 00 229. 70 74.76 220. 13 300. 00 287. 12 118. 24 242. 14 330. 00 315. 83 161.71 264. 15 0.00 344. 54 28.51 90.00 27.97 27.90 57.03 180. 00 55.94 55.79 85.54 270. 00 83.90 83.69 114. 05 0.00 111.87 111. 58 142. 56 90.00 139. 84 139. 48 156. 82 135. 00 153. 83 153. 42 171. 08 180. 00 167. 81 167. 37 16.14 29.46 15.00 14.50 32.28 58.91 30.00 28.99 48.42 88.37 45.00 43.49 64.56 117. 82 60.00 57.99 80.70 147. 28 75.00 • 72.48 88.77 162. 01 82.50 79.73 96.83 176. 73 90.00 86.98 15.59 0.54 0.08 31.17 1.09 0.16 46.76 1.63 0.25 62.34 2.18 0.33 77.93 2.72 0.41 85.72 2.99 0.45 93.51 3.27 0.49 0.04 1.02 1.10 13.47 0.08 2.03 2.20 26.94 0.12 3.05 3.29 40.41 0.16 4.06 4.39 53.89 0.21 5.08 5.49 67.36 0.23 5.59 6.04 74.09 0.25 6.10 6.59 80.83 13.40 29.96 30.04 12.85 26.80 59.92 60.08 25.71 40.20 89.88 90.12 38.56 53.59 119.84 120. 16 51.42 66.99 149. 79 150. 21 64.27 73.69 164. 77 165. 23 70.70 80.39 179. 75 180. 25 77.13 14.96 31.02 43.48 29.53 29.92 62.03 86.95 59.06 44.88 93.05 130. 43 88.59 59.84 124. 06 173.90 118. 11 74.80 155. 08 217. 38 147. 64 82.27 170. 59 239. 12 162. 41 89.75 186. 10 260. 86 177. 17 42.93 30.08 115.94 58.98 85.85 60.16 231. 87 117.97 128. 78 90.25 347. 81 176. 95 171.71 120. 33 103. 75 235. 94 214. 64 150. 41 219. 68 294.92 236. 10 165. 45 277. 65 324. 41 257. 56 180. 49 335. 62 353. 90 188. 40 195. 00 184. 86 97.77 16.79 90.63 205. 19 286. 16 30.00 13.25 185. 33 225. 00 181. 79 181. 32 104. 90 191. 46 97.50 94.23 101. 31 3.54 0.53 0.27 6.60 7.14 87.56 87.09 194. 73 195. 27 83.55 97.23 201. 60 282. 60 191. 94 279. 03 195. 53 33.59 23.40 HARMONIC ANALYSIS AND PREDICTION OF TIDES 305 Table 35.— Prod uc *(•© for Form 444 — Continued Constituent M 2 S 2 N 2 Ki M 4 Oi M 6 (MK) 3 - S4 (MN)4- V2 S 6 (12 (2N) 2 — (00),.. x 2 Si Mi Ji Mm Ssa Sa MSf— . Mf pi Qi T 2 Rs (2Q)i — Pi (2SM) 2 . M 3 L 2 (2MK) 3 K 2 Ms (MS)4- Time meridian in hours = S-M5 7.000 8.000 9.000 10.000 10.500 11.000 11.500 Products in degrees 202. 89 210.00 105. 29 45.78 97.60 248. 67 308. 18 60.00 41.97 199. 59 270. 00 195. 78 195. 27 112.97 206. 19 105. 00 101.48 109. 10 3.81 0.57 0.29 7.11 7.69 94.30 93.79 209. 71 210. 29 104. 71 217. 11 304. 33 206. 70 300. 49 210. 57 91.55 52.89 231.87 240. 00 227. 52 260. 86 270. 00 255. 96 289. 84 300. 00 284. 40 304. 33 315.00 298. 62 318.83 330. 00 312.84 333. 32 345.00 327. 06 120.33 103.75 111.54 135. 37 161.71 125.49 150.41 219.68 139.43 157. 93 248. 67 146. 40 165 45 277. 65 153. 37 172.97 306. 63 160. 34 335. 62 352. 20 120.00 99.39 62.57 36.23 180. 00 156.81 149. 52 80.25 240. 00 214. 24 193. 00 102 26 270. 00 242.95 236.48 124. 28 300. 00 271. 66 279. 95 146. 29 330. 00 300. 37 228. 10 0.00 223. 75 223. 16 256. 61 90.00 251. 71 251. 06 285. 13 180. 00 279. 68 278. 95 299. 38 225. 00 293. 67 292. 90 313. 64 270. 00 307. 65 306. 85 327. 89 315.00 321. 63 320. 80 129. 11 235. 65 120.00 115.97 145. 25 265. 10 135. 00 130. 47 161.39 294. 56 150. 00 144. 97 169. 46 309. 28 157. 50 152.22 177. 53 324.01 165. 00 159. 46 185. 60 338. 74 172. 50 166. 71 124. 68 4.35 0.66 140. 27 4.90 0.74 155. 85 5.44 0.82 163. 65 5.72 0.86 171.44 5.99 0.90 179. 23 6.26 0.94 0.33 8.13 8.78 107. 77 0.37 9.14 9.88 121. 24 0.41 10.16 10.98 134. 72 0.43 10.67 11.53 141. 45 0.45 11.17 12.08 148. 19 0.47 11.68 12.63 154.92 107. 19 239. 67 240. 33 102. 83 120. 59 269. 63 270. 37 115. 69 133.99 299. 59 300. 41 128. 54 140. 69 314. 57 315. 43 134.97 147. 39 329. 55 330. 45 141.40 154. 08 344. 53 345. 47 147. 82 119.67 248. 13 347.81 236. 23 134. 63 279. 14 31.29 265. 76 149. 59 310. 16 74.76 295. 28 157.07 325. 67 96.50 310. 05 164. 55 341. 17 118.24 324. 81 172. 03 356. 68 139. 98 339. 58 343.42 240. 66 207. 49 111.87 26.34 270. 74 323. 43 170. 86 69.27 300.82 79.36 229. 84 90.73 315.86 137. 33 259. 33 112.20 330. 90 195. 30 288. 83 133. 66 345.94 253. 27 318.32 12.000 347. 81 0.00 341.28 180. 49 335. 62 167. 32 323. 43 168. 30 0.00 329. 09 342. 15 0.00 335. 62 334. 74 193. 67 353. 47 180. 00 173.96 187. 03 6.53 0.99 0.49 12.19 13.18 161.66 160. 78 359. 51 0.49 154. 25 179. 51 12.19 161.71 354.34 155. 13 0.99 311.24 347. 81 306 U. S. COAST AND GEODETIC STJRVEY Table 36. — Angle differences for Form 445 Constituent Jan. 1 Feb. 0^, to 1,0* Feb . 1, 0^ to Dec, 31, 24* Jan 1, b , to Dec. 31, 24* Common year Leap year Common year Leap year M 2 o +324. 2 +279 +31 +288 +294 +253 +355 +243 +333 +288 +234 +127 +315 +342 +76 +45 +61 +31 +36 +97 +303 +249 +329 +31 +204 +329 +36 +306 +9 +258 +61 +217 +324 o -35.8 -81 -329 -72 -66 -107 -5 -117 -27 -72 -126 -233 -45 -18 -284 -315 -299 -329 -324 -263 -57 -111 - 31 -329 -156 -31 -324 -54 -351 -102 -299 -143 -36 o +136. 6 +93 +329 +274 +167 +49 +106 +230 +317 +274 +49 +131 +316 +248 +12 +44 +299 +329 +223 +162 +348 +123 +31 +329 +80 +31 +223 +25 +18,0 +304 +299 +186 +137 -223.4 -267 -31 -86 -193 -311 -254 -130 -43 -86 -311 -229 -44 -112 -348 -316 -61 -31 -137 -198 -12 -237 -329 -31 -280 -329 -137 -335 -180 -56 -61 -174 -223 +112. 2 +56 +330 +225 +142 +336 +82 +168 +281 +225 +359 +159 +303 +236 +27 +57 +300 +330 +248 +188 +311 +85 +30 +330 +28 +30 +248 +348 +169 +254 +300 +89 +112 o -247.8 -304 -30 -135 -218 -24 -278 -192 -79 -135 -1 -201 -57 -124 -333 -303 -60 -30 -112 -172 -49 -275 -330 -30 -332 -330 -112 -12 -191 -106 -60 -271 -248 +100. 8 +12 +202 +101 +302 +101 +113 +290 +202 +283 +258 +271 +230 +88 +89 +259 +259 +291 +12 +284 +259 +331 +189 +202 +43 +101 o -259. 2 -348 -158 -259 -58 -259 -247 -70 -158 -77 -102 -89 -130 -272 -271 -101 -101 -69 -348 -76 -101 -29 -171 -158 -317 -259 o +76.4 +335 +1 +153 +76 +229 +77 +51 +254 +153 +233 +286 +258 +218 +103 +102 1 +1 +284 +285 +254 +334 +259 +1 +232 +359 +284 +294 +178 +152 +1 +306 +76 o -283. 6 S 2 N 2 -25 Ki -359 M 4 -207 Oi -284 Me -131 (MK)j -283 S4 (MN)i -309 -106 S 6 -207 (2N) 2 -127 (OO)i -74 x 2 -102 Si Mi -142 Ji . . -257 Mm . -258 Ssa -359 Sa -359 MSf -76 Mf -75 -106 Qi -26 T 2 . -1 R 2 -359 (2Q)i -128 Pi -1 (2SM) 2 -76 Ms -66 L 2 -182 (2MK) 3 -208 K 2 -359 M 8 -54 (MS) 4 - -284 HARMONIC ANALYSIS AND PREDICTION OF TIDES 307 Table 37. — U. S. Coast and Geodetic Survey tide -predicting machine No. 2 GENERAL GEARS AND CONNECTING SHAFTS Gears and Shafts Face or diameter Number of teeth Pitch Period of rotation Remarks Inches 0.56 0.56 0.50 0.50 0.50 0.41 0.41 0.50 0.38 0.38 0.38 0.38 0.38 0.38 0.27 0.27 0.38 0.17 0.17 0.15 0.17 0.17 0.15 0.17 0.17 0.15 0.17 0.17 0.15 0.17 0.17 0.15 0.31 0.25 0.25 0.25 0.25 0.41 0.41 0.44 0.38 0.38 0.38 0.38 0.28 0.28 0.50 0.28 0.28 0.38 40 120 120 24 24 24 72 72 24 24 75 75 30 30 75 75 30 30 75 75 30 30 60 120 48 48 84 84 48 48 180 60 48 48 240 60 48 48 60 120 48 48 1 366 110 110 110 30 30 30 75 75 30 30 75 75 30 30 Dial hours 4 4 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 24 24 24 24 24 12 4 4 1 1 12 24 24 24 24X366 24X366 12 Hand crank shaft for operating machine. Spur gear on shaft 1. Spur-stud gear. Spur gear on shaft 2. Short horizontal shaft. Bevel gear on shaft 2. Bevel gear on shaft 3. Diagonal shaft connecting with middle section. Bevel gear on shaft 3. Bevel gear on shaft 4. Short vertical shaft through desk top. Bevel goar 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 8. 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, 0.56 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 at lower end of feeding roller. Bevel gear on shaft 3. Bevel gear on shaft 13. Main vertical shaft of middle section. Spur gear on shaft 13. Spur stud gear. Spur stud gear on shaft 14. Front vertical shaft of rear section. Bevel gear on shaft 14. Bevel gear on shaft 15. Main connecting horizontal shaft of rear sec- tion. Bevel gear on shaft 15. Bevel gear on shaft 16. Rear vertical shaft of rear section. 308 U. S. COAST AND GEODETIC SURVEY Table 38. — U. S. Coast and Geodetic Survey tide -predicting machine No. 2 CONSTITUENT GEARS AND MAXIMUM AMPLITUDE SETTINGS Constit- uents Teeth in gear wheels Theoretical speed per hour Vertical shafts Intermediate shafts Crank shafts I II III IV o 15. 5854433 107 90 52 119 15. 0410686 61 73 51 85 30. 0821372 122 80 96 146 29. 5284788 104 61 56 97 14.4920521 103 85 59 148 28. 9841042 103 74 59 85 43.4761563 86 62 70 67 57. 9682084 118 74 103 85 86. 9523126 140 62 86 67 115.9364168 118 37 103 85 28. 4397296 65 46 53 79 27. 8953548 68 58 46 58 13.9430356 92 89 58 129 16. 1391016 134 131 71 135 14. 9589314 91 73 50 125 13. 3986609 84 88 51 109 12.8542862 127 114 50 130 30. 0410686 85 50 43 73 15. 0000000 63 75 50 84 30. 0000000 70 70 70 70 60. 0000000 75 45 60 50 90.0000000 90 48 80 50 29. 9589314 81 50 45 73 29. 4556254 131 65 57 117 27.9682084 125 82 74 121 28. 5125830 89 69 70 95 13.4715144 69 70 41 90 44. 0251728 120 81 105 106 42. 9271398 81 52 79 86 57. 4238338 135 42 53 89 58.9841042 118 61 62 61 31. 0158958 69 47 50 71 1.0980330 84 45 1 51 1. 0158958 149 80 1 55 0. 5443747 93 41 1 125 51 f 149 \ 125 1 60 0. 0410686 120 0. 0821372 51 149 1 125 Gear speed per dial Error per hour o 15. 5854342 -0.08 15. 0410959 + .24 30. 0821918 + .48 29. 5284773 - .01 14. 4920509 - .01 28. 9841017 -.02 43. 4761675 +.10 57. 9682035 -.04 86. 9523351 +.20 115. 9364070 +.09 28. 4397358 +.05 27. 8953627 +.07 13.9430363 +.01 16. 1391009 -.01 14.9589041 -.24 13. 3986656 +.04 12. 8542510 -.31 30. 0410959 +.24 15. 0000000 .00 30. 0000000 .00 60.0000000 .00 90. 0000000 .00 29. 9589041 -.24 29. 4556213 -.04 27. 9681516 -.50 28. 5125858 +.02 13. 4714286 -.75 44. 0251572 -.14 42. 9271020 -.33 57. 4237560 -.68 58. 9841440 -.35 31. 0158825 -.12 1. 0980392 +.05 1.0159091 +.12 0. 5443902 +.14 0. 0410738 +.05 0. 0821477 +.09 Maximum amplitude settings of cranks J,____ Ki___ K 2 __, U-- *Mi_. M 2 ._ Mj.. M 4 __ M 6 _. M 8 -_ N 2 — 2N... 0,___ 00.. Pl___ Qi— 2Q— R 2 — Si___ Sa.._ 84— Be— Ti.._ X 2 — - /JL2-— VI PI— - MK_ 2MK MN. MS- 2SM, Mf_. MSf. Mm. Units 1.4 11.0 3.9 2.4 1.0 20.0 1.4 4.0 1.0 0.4 6.0 1.0 9.0 0.8 4.8 3.0 0.6 0.4 2.0 9.8 1.0 0.4 1.0 0.4 1.2 2.0 0.8 1.9 1.4 0.7 2.0 1.4 4.0 2.0 3.0 8.0 3.0 'Designed for one-half of speed of M2. HARMONIC ANALYSIS AND PREDICTION OF TIDES 309 Table 39.— Synodic periods of constituents DIURNAL CONSTITUENTS Ji Ki Mi Oi 00 Pi Qi 2Q Si Ki Days. 27. 555 13. 777 9.133 27. 093 23. 942 6.859 5.492 25.622 7.096 Days. Days. Days. Days. Days. Days. Days. Days. Mi 27. 555 13.661 13. 661 182. 621 9.133 6.859 365. 243 9.557 0, 27. 555 9.133 32. 451 13. 661 9.133 29. 803 14. 632 00 6.830 14. 765 27. 555 13. 777 14. 192 31.812 Pi 12. 710 5.474 4.566 13. 168 5.623 Qi -- 9.614 7.127 365. 243 10. 085 2Q 27. 555 9.367 205. 892 Si 6.991 24. 302 pi 9.814 SEMIDIURNAL CONSTITUENTS K 2 L 2 M 2 N 2 2N R 2 S 2 T 2 x 2 M2 VI L 2 Days. 27. 093 13. 661 9.133 6.859 365. 225 182. 621 121.748 23. 942 7.096 9.557 16. 064 Days. Days. Days. Days. Days. Days. Days. Days. Days. Days. M 2 .-- 27. 555 13. 777 9.185 29. 263 31.812 34. 847 205. 892 9.614 14. 765 10. 085 N 2 27. 555 13.777 14. 192 14. 765 15. 387 31.812 14. 765 31.812 7.383 2N 27. 555 9.367 9.614 9.874 14. 765 31.812 205. 892 5.823 Pv 2 6.991 7.127 7.269 9.614 205. 892 24. 302 4.807 s 2 365. 259 182. 630 25. 622 7.236 9.814 15. 387 T 2 365. 259 27. 555 7.383 10. 085 14. 765 A2 29. 803 7.535 10.371 14. 192 M2 V2 2SM... 10.085 15.906 9.614 27.555 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 1 day] Day of month. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. 1 2 3. 4 5 6 7. 8 9 10 11 12 13 14 15 16 17 IS 19 20 21 22 23 24 25 26 27 28 29 30 31 21 52 22 53 23 54 24 55 25 56 26 57 27 58 28 59 29 30 31 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 169 170 171 111 141 172 112 142 173 113 143 174 114 144 175 115 145 176 116 146 177 117 147 178 118 148 179 90 119 149 120 150 151 LSI 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 237 268 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 289 290 291 292 298 299 300 301 302 303 304 305 306 307 308 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 294 325 295 326 296 327 297 328 329 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 310 U. S. COAST AND GEODETIC SURVEY Table 41. — Values of h in formula h=(l + r 2 + 2r cos x)i X r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 X 10 20 1.000 1.000 1.000 1.100 1.099 1.095 1.200 1.197 1.190 1.300 1.296 1.286 1.400 1.396 1.383 1.500 1.495 1.480 1.600 1.594 1.577 1.700 1.694 1.675 1.800 1.793 1.773 1.900 1.893 1.871 2.000 1.992 1.970 o 360 350 340 30 40 50 1.000 1.000 1.000 1.088 1.079 1.067 1.177 1.160 1.139 1.269 1.245 1.215 1.361 1.331 1.294 1.455 1.420 1.376 1.549 1. 510 1.460 1.644 1.601 1. 546 1.739 1.693 1.634 1.835 1.786 1.723 1.932 1.879 1.813 330 320 310 60. 70 80 1.000 1.000 1.000 1.054 1.038 1.022 1.114 1.085 1.053 1.179 1.138 1.093 1.249 1.197 1.140 1.323 1.262 1.193 1.400 1.331 1.252 1.480 1.403 1.316 1.562 1.479 1.385 1.646 1. 557 1.457 1.732 1.638 1.532 300 290 280 90 100 110 1.000 1.000 1.000 1.005 0.988 0.970 1.020 0.985 0.950 1.044 0.993 0.941 1.077 1.010 0.941 1.118 1.037 0.953 1.166 1.073 0.974 1.221 1.117 1.006 1.281 1.167 1.045 1.345 1.224 1.093 1.414 1.286 1.147 270 260 250 120 130 140 1.000 1.000 1.000 0.954 0.939 0.926 0.917 0.885 0.856 0.889 0.839 0.794 0.872 0.804 0.740 0. 866 0.779 0.696 0.872 0.767 0.664 0.889 0.768 0.646 0.917 0.782 0.644 0.954 0.808 0.657 1.000 0.845 0.684 240 230 220 150 160 170 1.000 1.000 1.000 0.915 0.907 0.902 0. 833 0.815 0.804 0.755 0.725 0.706 0.684 0.639 0.610 0.620 0.557 0.515 0.566 0.482 0.422 0.527 0.418 0.334 0.504 0.369 0.254 0.501 0.344 0.193 0.518 0.347 0.174 210 200 190 180 1.000 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 180 Table 42. — Values of k in formula k=tan~ 1 r sin x 1 + r cos x [When x is between ISO and 360° , tabular values are negative] X r 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 o o o o o o o o o o o 10 20 0.00 0.00 0.00 0.00 0.90 1.78 0.00 1.66 3.30 0.00 2.30 4.58 0.00 2.85 5.68 0.00 3.33 6.63 0.00 3.75 7.47 0.00 4.12 8.22 0.00 4.45 8.88 0.00 4.73 9.47 0.00 5.00 10.00 360 350 340 30 40 50 0.00 0.00 0.00 2.63 3.42 4.12 4.87 6.36 7.73 6.78 8.92 10.90 8.45 11.13 13.70 9.90 13.08 16.17 11.17 14.80 18.35 12.30 16.32 20.30 13.30 17.68 22.03 14.20 18.90 23.60 15.00 20.00 25.00 330 320 310 60 70 80 0.00 0.00 0.00 4.72 5.20 5.53 8.95 9.98 10.78 12.73 14.33 15.68 16.10 18.30 20.22 19.10 21.87 24.37 21.78 25.07 28.15 24.18 27.95 31. 58 26.33 30. 55 34.67 28.27 32.88 37.47 30.00 35.00 40.00 300 290 280 90 100 110 0.00 0.00 0.00 5.71 5.72 5.55 11.31 11.53 11.41 16.70 17.32 17.43 21.80 22.95 23.53 26.57 28.33 29.55 30.96 33.42 35.35 34.99 38.12 40.85 38.66 42.45 45.98 41.99 46.42 50.70 45.00 50.00 55.00 270 260 250 120 130 140 0.00 0.00 0.00 5.22 4.68 3.98 10.89 9.97 8.63 17.00 15.90 14.05 23.42 22.42 20.33 30.00 29. 45 27.52 36.58 36.80 35.52 43.00 44.27 44.13 49.10 51.60 53.02 54.78 58.57 61.77 60.00 65.00 70.00 240 230 220 150 160 170 0.00 0.00 0.00 3.13 2.17 1.10 6.90 4.81 2.48 11. 45 8.13 4.23 17.02 12.37 6.53 23.80 17.88 9.70 31.98 25.20 14.28 41.63 34.98 21.37 52.48 47.82 33.22 63.88 63.38 53.98 75.00 80.00 85.00 210 200 190 180 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 180 EXPLANATION OF SYMBOLS USED IN THIS BOOK Although the following list is fairly comprehensive, some of the symbols given may at times be used in the text to represent other quantities not listed below, but such application will be made clear by the context. A (1) General symbol for a tidal constituent or its amplitude. It is some- times written with a subscript to indicate the species of the constituent (par. 52). (2) General symbol with an identifying subscript for a constituent term in the development of the lunar tide-producing force (par. 66) . (3) The particular tidal constituent being cleared by the elimination process (par. 245). (4) Azimuth of tide-producing body reckoned from the south through the west (par. 80) . (5) Azimuth of horizontal component of force in any given direction (par. 85). a (1) Speed or rate of change in argument of constituent A. (2) Mean radius of earth. B (1) Tidal constituent following constituent A in a series. (2) General symbol with an identifying subscript for a constituent term in the development of the solar tide-producing force (par. 117). (3) General symbol for disturbing constituents in elimination process (par. 245). b Speed or rate of change in argument of constituent B. C (1) Mean constituent coefficient (par. 74). (2) General symbol for coefficients of cosine terms in Fourier series (par. 187). c Reciprocal of mean value of 1/d. Ci Reciprocal of mean value of 1/di. D Declination of moon or sun. d Distance from center of earth to center of moon. d\ Distance from center of earth to center of sun. E (1) Mass of earth. (2) Argument of tidal constituent (same as V-\- u) . e Eccentricity of moon's orbit. e\ Eccentricity of earth's orbit. F Reduction factor, reciprocal of node factor / (par. 78) . F a Horizontal component of tide-producing force in azimuth A. When numerals are annexed, the first digit (3 or 4) signifies the power of the parallax of the moon or sun involved in the development and the second digit (0, 1, 2, or 3) indicates the species of the terms included in the group. Thus F aZ0 represents that part of the horizontal component in azimuth A that comprises the long-period terms depending upon the cube of the parallax. F a South horizontal component of tide-producing force. (See F & for explana- tion of annexed numerals.) F v Vertical component of tide-producing force. (See F & for explanation of annexed numerals.) F w West horizontal component of tide-producing force. (See F & for explana- tion of annexed numerals.) / Node factor (par. 77) . G (1) Greenwich epoch or phase lag of a tidal constituent (par. 226). (2) Gear ratio of predicting machine (par. 396) . g (1) Mean acceleration of gravity on earth's surface. (2) Modified epoch of tidal constituent, same as k' (par. 225) . H Mean amplitude of a tidal constituent (par. 143) . 311 312 IT. S. COAST AND GEODETIC SURVEY H Mean water level above datum used for tabulation. h (1) Mean longitude of sun; also rate of change in same. (2) Height of tide at any time. h 3 Height of equilibrium tide involving cube of moon's parallax. A second digit in the subscript limits the height to that due to terms of the single species indicated by this digit (pars. 97 and 101). h 4 Height of equilibrium tide involving 4th power of moon's parallax. A sec- ond digit in the subscript has the same significance as in the case of h 3 . I Obliquity of lunar orbit with respect to earth's equator. i Inclination of lunar orbit to the ecliptic. Ji Tidal constituent. j Longitude of moon in its orbit reckoned from lunar intersection. Ki, K 2 Tidal constituents. KJ 2 , KP b KQi Tidal constituents. k Difference between mean and true longitude of moon (par. 59). L Longitude of place; positive for west longitude, negative for east longitude. L 2 , LPi Tidal constituents. M Mass of moon. Mi, M a , M 3 , M 4 , M 6 , M 8 Tidal constituents. Mf Tidal constituent. MK 3 , 2MK 3 , MK, Tidal constituents. Mm Tidal constituent. MN 4 , 2MN 6 , MNS 2 Tidal constituents. MPj Tidal constituent. 2MS 2 , MS 4 , 2MS 6 , 3MS 8 , 2(MS) 8 Tidal constituents. MSf Tidal constituent. MSN 6 , 2MSN 8 Tidal constituents. m Ratio of mean motion of sun to that of moon (par. 62) . N Longitude of moon's node; also rate of change in same. N 2 , 2N 2 , NJi Tidal constituents. Oi, OOi Tidal constituents. P Mean longitude of lunar perigee reckoned from lunar intersection (par. 122) . Pi Tidal constituent. p (1) Mean longitude of lunar perigee; also rate of change in same. (2) Numeral indicating species of constituent, frequently written as the subscript of the constituent symbol. In special case used with long- period constituents to show number of periods in month or year. Pi Mean longitude of solar perigee; also rate of change in same. Q Term in argument of constituent Mi (par. 123). Q a Factor in amplitude of constituent Mi (par. 122). Q n Term in argument of constituent Mi (par. 122). Qi, 2Qi Tidal constituents. R (1) Amplitude of constituent pertaining to a particular time (par. 143). (2) Term in argument of constituent L 2 (par. 129). i? a Factor in amplitude of constituent L 2 (par. 129). R 2 , RPi Tidal constituents. r Distance of any point from center of earth. S (1) Mass of sun. (2) Longitude of time meridian; positive for west longitude, negative for east longitude. (3) General symbol for coefficients of sine terms in Fourier series (par. 187). (4) Working scale factor of predicting machine. S' Solar factor UJU (par. 118). Si, S 2 , S3, S 4 , S 6 , S 8 Tidal constituents. Sa Tidal constituent. SK 3 Tidal constituent. 2SM 6 Tidal constituent. HARMONIC ANALYSIS AND PREDICTION OF TIDES 313 SOi, S0 3 Tidal constituents. Ssa Tidal constituent. s Mean longitude of moon; also rate of change in same. s' True longitude of moon in orbit referred to equinox (par. 59). T (1) Number of Julian centuries reckoned from Greenwich mean noon, December 31, 1899. (2) Hour angle of mean sun. (3) Time expressed in degrees of constituent reckoned from phase zero of Greenwich argument (par. 439). T 2 Tidal constituent. t (1) Hour angle of tide-producing body. (2) Time reckoned from beginning of tidal series. U Basic factor (M/E) (a/c)K Ui Factor (S/E) (a/c,)K u Part of constituent argument depending upon variations in obliquity of lunar orbit (par. 71). V (1) Principal portion of constitute argument (par. 71). (2) Velocity of current (par. 330). (V+u) Constituent argument at any time. (V -\-u) Constituent argument at beginning of a tidal series. V g Potential due to gravity at earth's surface (par. 96). Vz Tide-producing potential involving cube of moon's parallax (par. 94) . F 4 Tide-producing potential involving 4th power of moon's parallax (par. 94.) X Longitude of observer reckoned in celestial equator from lunar inter- section. Y Latitude of observer. When combined with a subscript consisting of a letter and numerals, it represents the latitude factor to be used with the tidal force component similarly marked (par. 79). z Geocentric zenith distance of tide-producing body. a (Alpha) General symbol for the initial phase of tidal constituent A. 13 (Beta) Initial phase of constituent B. y (Gamma) Initial phase of constituent C. 8 (Delta) Initial phase of constituent D. e (Epsilon) Initial phase of constituent E. f (Zeta) The explement of the initial phase of a constituent (par. 221). 0! (Theta) Tidal constituent, same as XOi. k (Kappa) Local phase lag or epoch of tidal constituent (par. 144). k' Modified epoch of tidal constituent (par. 225) . X 2 (Lambda) Tidal constituent. ix (Mu) Attraction of gravitation between unit masses at unit distance. fj. 2 Tidal constituent, same as 2MS 2 . v (Nu) Right ascension of lunar intersection (par. 24). v' Term in argument of lunisolar constituent Ki (par. 133). 2v" Term in argument of lunisolar constituent K 2 (par. 135). v 2 Tidal constituent. £ (Xi) Longitude in moon's orbit of lunar intersection (par. 24) . 7T (Pi) An angle of 3.14159 radians or 180°. ttj Tidal constituent, same as TKi. Pi (Rho) Tidal constituent, same as vK\. a 1 (Sigma) Tidal constituent, same as *>Ji. r (Tau) Length of series in mean solar hours (par. 248). Moon's ascending node. INDEX A Page Adams, J. C 1 Airy, George B 1 Amplitude of constituent 2, 49 Analysis of high and low waters 100 Analysis of monthly sea level. _ 98, 114 Analysis of observations 49 Analysis of tidal currents 118 Anomalistic month, year 4 Approximation, degree of 8 Argument. (See Equilibrium argu- ment.) Astres fictifs 23 Astronomical data 3, 153, 162 Astronomical day 3 Astronomical periods 163 Astronomical tide 30 Augmenting factors 71, 91, 157, 228 B Basic factor 24 C Calendars 4 Civil day 3 Coefficients 24 Component. (See Constituent tides.) Component of force, horizontal 26 Component of force, vertical 15 Compound tides 47, 167 Constituent day 3 Constituent hour 4 Constituent tides 2, 16, 87 Formulas 21,35,39 Tables 153,164,167 Currents, analysis 118 Currents, prediction 147 D Darwin, G. H 1 Datum for prediction 124, 144 Day, several kinds 3 Day of year, table 309 Declinational factor 17 Degree of approximation 8 Development of tide-producing force 10 Diurnal constituents 16 E Eccentricity of orbit 4 Eclipse year 4 Elimination 84, 116, 158, 236 Elliptic factor 24 Epoch of constituent 49, 75 Equations of moon's motion 19 Page Equilibrium argument 22, 50, 75, 108, 124, 157, 204 Equilibrium theory 28 Equilibrium tide 28, 38 Equinox 6 Eudoxas 1 E vection 4 Explanation of tables 153 Explanation of tidal movement 2 Extreme equilibrium tide 33 Extreme tide-producing force 13 Factor F. (See Reduction factor.) Factor f. (See Node factor.) Ferrel, William 1, 127 Forms for analysis of tides 104 Forms for predicting machine 143 Fourier series 62 Fourth power of moon's parallax-- 34 Fundamental astronomical data. _ 153, 162 Fundamental formulas 10 G General coefficient 24 General explanation, tidal move- ment 2 Gravitational tide 30 Greatest equilibrium tide 33 Greatest tide-producing force 13 Greenwich argument 76 Greenwich epoch 77 Gregorian calendar 4 H Harmonic analysis 3, 49, 112 Harmonic constants 3, 49, 143 Harmonic prediction 3, 123 Harris, Rollin A 1 High and low water analysis 100 Historical statement 1 Horizontal component, tide-pro- ducing force 26, 37 Hour, several kinds 4 Hourly heights 104 Hydraulic current. 148 H y drographic datum 144 I Inclination of moon's orbit 6, 155, 173 Inference of constants 78, 114 315 316 INDEX Julian calendar. Page 4 K Ki and K 2 tides 44 Kelvin, Lord 1, 126 Laplace 1 Latitude 6 Latitude factors 17, 24, 154, 168 Length of series 51 Lesser lunar constituents 35 Lesser solar constituents 40 Lesser tide-producing force 34, 40 Longitude 6 Longitude, lunar and solar ele- ments 162,170 Long-period constituents 16, 87, 302 L 2 -tide 43, 156, 177, 192 Lunar constituents 21, 35 Lunar day 3 Lunar hour 4 Lunar intersection 6 Lunar node 6, 8 Lunisolar tides 44 M Mean constituent coefficient 24 Mean longitude 7 Meteorological tides 46 Month, several kinds 4 Monthly sea-level analysis 98, 114 Moon's motion, equations 19 Moon's node 6, 8 Moon's parallax, 4th power 34 Mi-tide 41, 156, 179, 192 N Node, lunar 6, 8 Node factor 25 Compound tides 47 Constituent Ki 45 Constituent K 2 46 Constituent L 2 44 Constituent Mj___ 43 Lesser tide-producing force 36 Predictions 124 Table 199 Nodical month 4 O Obliquity factor 24 Obliquity of ecliptic 6 Obliquity of moon's orbit 6 Observational data 50 Over tides 47 Period of consi ituent 3 Periods, astronomical 163 Phase lag 49, 75 Phase of constituent 2 Poor, Charles Lane 1 Potential 30 I Page Predicting machine. (See Tide- predicting machine.) Prediction of tidal currents 147 Prediction of tides 123 Principal lunar constituents 21 Principal solar constituents 39 R Record of observations 50 Reduction factor 25, 111, 156, 186 S Secondary stencils 57, 159, 299 Semidiurnal constituents 16 Settings for tide-predicting ma- chine 145, 306 Shallow-water constituents 46, 167 Shidy, L. P 53 Sidereal day 3 Sidereal hour, month, year 4 Solar day 3 Solar f actor 40 Solar hour 4 Solar tides 39 South component, tide-producing force 26, 37 Species of constituent 16 Speed of constituent 3, 23 Stationary wave 2 Stencil sums 107 Stencils 53, 106, 158, 268 Summarized formulas: Equilibrium tide 33 Lesser tide-producing force 36 Principal tide-producing force _ 33 Summation for analysis 52 Surface of equilibrium 30, 32 Symbols used in book 311 Synodical month 4 Synodic periods of constituents 161, 309 T Tables 162 Explanation 153 Terdiurnal constituents 34 Thomson, Sir William 1, 126 Tidal currents 118, 147 Tidal movement 2 Tide-predicting machine 126 Adjustments 139 Automatic stopping device 135 Base 127 Constituent cranks 130 Constituent dials 131 Constituent pulleys 132 Constituent sliding frames 131 Datum of heights 141 Day dial 128 Dial hour 128 Dimensions 127 Doubling gears 132 Forms used 143 Gear speeds 129 Gearing 128, 160, 307 Graph scale 137 INDEX 317 Tide-predicting machine — Con. Page Height formula 126 Height predictions 134 Height scale 134,141 Height side 128 High and low water marking device 139 Hour marking device 139 Marigram gears 137, 141 Marigram scale 137 Nonre versing ratchet 136 Operation of machine 142 Paper 136,142 Pens 138,142 Plane of reference 141 Positive and negative direc- tions 131 Predicting 142 Releasable gears 130 Scale, amplitude settings. 132, 140 Scale, height dial 134, 141 Scale, marigram 137 Scale, table 138 Scale, working 135 Setting machine 140 Stopping device 135 Summation chains 133 Tide-predicting machine — Con. p age Summation wheels 133 Tide curve 136 Time dials 128 Time formula 126, 132 Time prediction 135 Time side 128 Verification of settings 142 Tide-producing force 10 Tide-producing potential 31 Tropical month, year 4 V Variation inequality 4 Vernal equinox 6 Vertical component, tide-produc- ing force 15, 34 W West component, tide-producing force 26, 37 Y Year, several kinds 4 Young, Thomas 1 o 24G037— 41- -21