Encyclopædia Britannica, Ninth Edition/Tides/Chapter 4

IV. The Harmonic Analysis.

§ 22. Methods of applying Theory to Practice.

Two methods of treating observations.The comparison between tidal observations and tidal theories, and the formation of tables predicting the tidal oscillations of the sea, have been carried out in two different ways, which may be called the "synthetic" and the "analytic."

Synthetic method.The semi-diurnal rise and fall of tide with the weekly alternation of spring and neap would naturally suggest to the investigator to make his formula conform to the apparent simplicity of the phenomenon. He would seek to represent the height of water by either one or two periodic functions with a variable amplitude; such a representation is the aim of the synthetic method. That method has been followed by all the great investigators of the past, Newton, Bernoulli, Maclaurin, Laplace, Lubbock, Whewell, Airy. Since at European ports the two tides which follow one another on any one day are nearly equal, or, in other words, there is scarcely a sensible diurnal tide, these investigators bestowed comparatively little attention to the diurnal tides. If these are neglected, the synthetic method is simple, for a single function suffices to repre sent the tide. In non-European ports, however, the diurnal tide is sometimes so large as to mask the semi-diurnal, and to make only a single instead of a double high water in twenty-four hours. To represent this diurnal tide in the synthetic method we are compelled to introduce at least one more function. There should also be a third function representing the tides of long period; but until the last few years these tides have scarcely been considered, and there fore we shall have little to say of them in explaining the synthetic method. The expression for the tide-generating forces due to either sun or moon consists of three terms, involving the declinations and hour-angles of the planet. One of these terms for each goes through its period approximately twice a day, a second once a day, and the third varies slowly ( 7). The mathematical basis of the synthetic method consists of a synthesis of the mathematical formula?. The semi-diurnal term for the moon is fused with that for the sun, and the same process is carried out for the diurnal and slowly varying terms. A mass of tidal observation at a place where the diurnal tide is small, even if, as in all the older observations, it consists merely of heights and times of high and low water, soon shows that the fusion of two simple harmonic or periodic functions is insufficient to represent the state of tide; and the height and time of high water are found to need corrections for the variations of declination, of motion in right ascension, and of the parallaxes of both bodies. But when continuous tide-gauges were set up far more extended data than those of the older observations became accessible to the investigator, and more and more corrections were found to be ex pedient to adapt the formulas to the facts. A systematic method of utilizing all the data became also a desideratum. This state of matters led Sir W. Thomson to suggest the analytic method. 1 It Analytic is true that the dynamical foundations of that method have always method, lain below the surface of the synthetic method, and have constantly been appealed to for the theoretical determination of corrections; nevertheless, we must regard the explicit adoption of the analytic method as a great advance. In this method we conceive the tidal forces or potential due to each disturbing body to be developed into a series of terms each consisting of a constant (determined by the elements of the planet's orbit and the obliquity of the ecliptic) multiplied by a simple harmonic function of the time. Thus in place of the terms of the synthetic method for the three classes of tides we have an indefinitely long series of terms for each of the three classes. The loss of simplicity in the expression for the forces is far more than counterbalanced by the gain of facility for the discussion of the oscillations of the water. This facility arises from the great dynamical principle of forced oscillations, which we have explained in the historical sketch. Applying this principle, we see that each individual term of the harmonic development of the tidegenerating forces corresponds to an oscillation of the sea of the same period, but the amplitude and phase of that oscillation must depend on a network of causes of almost inextricable complication. The analytic method, then, represents the tide at any port by a series of simple harmonic terms whose period is determined from theoretical considerations, but whose amplitude and phase are found from observation. Fortunately the series representing the tidal forces converges with sufficient rapidity to permit us to consider only a moderate number of harmonic terms in the series.

Now it seems likely that the corrections which have been applied in the use of the synthetic method might have been clothed in a more satisfactory and succinct mathematical form had investigators first carried out the harmonic development. In this article we shall therefore invert history and come back on the synthetic method from the analytic, and shall show how the formulae of correction stated in harmonic language may be made comparable with them in synthetic language. One explanation is expedient before pro- Fusion of ceeding with the harmonic development. There are certain terms terms in the tide-generating forces of the moon, depending on the longi- affected tude of the moon's nodes, which complete their revolution in 18 6 by moyears. Now it has been found practically convenient, in the appli- tion of cation of the harmonic method, to follow the synthetic plan to the moon's extent of classifying together terms whose speed differs only in node, consequence of the movement of the moon's node, and at the same time to conceive that there is a small variability in the intensity of the generating forces.

§ 23. Development of Equilibrium Theory of Tides in Terms of the Elements of the Orbits.

Within the limits at our disposal we cannot do more than in- Equilidicate the processes to be followed in this development.

We have already seen in (3) that the expression for the moon's theory; tide-generating potential is elements

v _ 3??i, o _ i or orbits v 2r^ COS * Z ~ *" iutroand in (10) that ducet cos z - =

• -2M 8 where Mj.M^lLj are the direction cosines of the moon referred to axes fixed in the earth. We require to find the functions'll a M 2, ^(Mj 2 - 1I 2 2 ), &c., of the moon's direction cosines. 2 Let A, B, C (fig. 2) be the axes fixed in the earth, C being the north pole and AB the equator; let X, Y, Z be a second set of axes, XY being the plane of the moon's orbit; M the projection of the moon in her orbit; I=ZC, the obliquity of the lunar orbit to the equator; x = AX = BCY; Z=MX, the

^{[1] [2]} Moon's longitude and obliquity of orbit introduced.moon's longitude in her orbit measured from X, the intersection of the equator with the lunar orbit, hereafter called the "intersection." Then

M! = cos I cos x + sin I sin x cos 7^| M 2 = -cos Zsinx + sin IcosxcosI V (66). M,= sin I sin/

Writing for brevity ^7= cos 7, <?=sin^7. we find that

Mj 2 - M 2 2 =p* cos 2(x - Z) + 2pV cos 2x + q* cos 2(x + Z) -2MjM 2 = the same with sines in place of cosines .(67), MjMo = - M,*= Moon's distance and ec centricity intro duced. = the same with sines in place of cosines - +? 4) + 2pycos2Z

These are the required functions of MI, M 2, M 3 .

Moon's distance and eccentricity introduced.Now let c be the moon's mean distance, e the eccentricity of the moon's orbit, and let

3 m

Then, putting

2C 3 .(69). we have -^f X 2 + Y 2 -22? (71).

Corresponding to the definition of a simple tide given in § 1, the expression for each term of the tide-generating potential should consist of a solid spherical harmonic, multiplied by a simple time-harmonic. In (71) p-^rj, p^-if), &c., are solid spherical harmonics, and in order to complete the expression for V it is necessary to develop the five functions of X, Y, Z in a series of simple time-harmonics. But (71) may be simplified in such a way that the five functions are reduced to three. The axes fixed in the earth may be taken, as in 7, to have their extremities as follows: the axis C the north pole, the axis B 90 E. of A on the equator, and the axis A on the equator in the meridian of the place of observation. Thus, if X be the latitude of that place, we have

= cosX, 17=0, f=sinX.

Tide-generating potential.Then, writing a for the earth's radius at the place of observation, (71) becomes

V = e . 2 [ cos 2 X(X 2 - Y 2 ) + sin 2XXZ + f(J-sin 2 X)KX 2 +Y 2 -2Z a )] (71a).

The process of developing the three functions of X, Y, Z consists in the introduction of the formulae of elliptic motion into (66) and (70), the subsequent development of the X-Y-Z functions in a series of trigonometrical terms, and the rejection of terms which appear numerically to be negligible. The terms depending on the principal lunar inequalities evection and variation are also introduced. Finally, the three X-Y-Z functions are obtained as a series of simple time-harmonics, with the arguments of the sines and cosines linear functions of the earth's rotation, the moon's mean motion, and the longitude of the moon's perigee. The next step is to pass, according to the principle of forced oscillations, from the potential to the height of tide generated by the forces corresponding to that potential The X-Y-Z functions being simple time -harmonics, the principle of forced vibrations allows us to conclude that the forces corresponding to V in (71a) will generate oscillations in the ocean of the same periods and types as the terms in V, but of unknown amplitudes and phases. Now let I 2 - J| 2, 3EZ, i( 2 + Jf 2 - 22?) be three functions having respectively similar forms to those of

X 2 -Y 2 XZ_ 1(X 2 +Y 2 -2Z 2 ) (1-eT (T- e 2 ) 3 and 3 (1 - e 2 )

but differing from them in that the argument of each of the simple time-harmonics has some angle subtracted from it, and that the term is multiplied by a numerical factor. Height of tide at any point.Then, if g be gravity and h the height of tide at the place of observation, we must have

TO? h = y[icos 2 X(F-g 2 ) + sin2X12 + i(i- S in 2 X)|(J 2 + f 2 -2 2 )] (72).

The factor ra?/g may be more conveniently written ITTwhere Mis the earth's mass. It has been so chosen that, if the equilibrium theory of tides were fulfilled, with water covering the whole earth, the numerical factors in the -g -Z functions would be each unity and the alterations of phase would be zero. Definition of high tide of tide of long period.The terms in (P + |p - 22?) require special consideration. The function of the latitude" being - sin 2 X, it follows that, when in the northern of tide of hemisphere it is high water north of a certain critical latitude, it long is low water on the opposite side of that parallel; and the same is period, true of the southern hemisphere. It is best to adopt a uniform system for the whole earth, and to regard high tide and high water as consentaneous in the equatorial belt, and of opposite meanings outside of the critical latitudes. We here conceive the function always to be written J - sin 2 X, so that outside of the critical latitudes high tide is low water. We may in continuing the development write the 3E-H-H functions in the form appropriate to the equilibrium theory, with water covering the whole earth; for the actual case it is only then necessary to multiply by the reducing factor, and to subtract the phase alteration K. As these are unknown constants for each place, they would only occur in the development as symbols of quantities to be deduced from observation. It will be understood, therefore, that in the following schedules the " argument " is that part of the argument which is derived from theory, the true complete argument being the " argument " - K, where K is derived from observation. Up to this point we have supposed the moon's longitude and the earth's position to be measured from the intersection; but in order to pass to the ordinary astronomical formula? we must measure the longitude and the earth's position from the vernal equinox. Hence we determine the longitude and right ascension of the intersection in terms of the longitude of the moon's node and the inclination of the lunar orbit, and introduce them into our formulae for the I-g-2 functions. The expressions for the functions corresponding to solar tides may be written down by symmetry, and in this case the intersection is actually the vernal equinox. The final result of the process sketched is to obtain a scries of Explanaterms each of which is a function of the elements of the moon's or tion of sun's orbit, and a function of the terrestrial latitude of the place of schedules observation, multiplied by the cosine of an angle which increases below. uniformly with the time. We shall now write down the result in the form of a schedule; but we must first state the notation employed: e, e, = eccentricities of lunar and solar orbits; 7, w= obliquities of equator to lunar orbit and ecliptic; p,p t longitudes of lunar and solar perigees; ^,rs t = hourly increments of p, p t; s, h = moon's and sun's mean longitudes; <r, 17= hourly increments of 5, li; t = local mean solar time reduced to angle; y 77 = 15 per hour; X = latitude of place of observation;, v = longitude in lunar orbit, and R.A. of the intersection; N= longitude of moon's node; i inclination of lunar orbit. The speed of any tide is defined as the Speed rate of increase of its argument, and is expressible, therefore, as defined. a linear function of 7, ij, <r, CT; for we may neglect CT, as being very small. The following schedules, then, give h the height of tide. The arrangement is as follows. First, there is a universal coefficient ~-r.l - ) a, which multiplies every term of all the schedules. Secondly, there are general coefficients, one for each schedule, viz., cos 2 X for the semi-diurnal terms, sin 2X for the diurnal, and - sin 2 X for the terms of long period. In each schedule the third column, headed "coefficient," gives the functions of 7and e(and in two cases also of p). In the fourth column is given the mean semi-range of the corresponding term in numbers, which is approximately the value of the coefficient in the first column when 7= w; but we pass over the explanation of the mode of computing the values. The fifth column contains arguments, linear functions of t, h, s, p, v, . In [A, L] 2t + 2(h-i>) and in [A, ii.] t + (h-v) are common to all the arguments. The arguments are grouped in a manner convenient for subsequent computation. Lastly, the sixth is a column of speeds, being the hourly increase of the arguments in the preceding column, estimated in degrees per hour. It has been found practically convenient to denote each of these partial tides by an initial letter, arbitrarily chosen. In the first column we give a descriptive name for the tide, and in the second the arbitrarily chosen initial. In some cases no initial has been chosen, and here we indicate the tide by the analytical expression for its speed, or hourly increase of argument. The schedule for the solar tides is drawn up in precisely the same manner, the only difference being that the coefficients are absolute constants. The eccentricity of the solar orbit is so small that the elliptic tides may be omitted, except the larger elliptic semi diurnal tide. In order that the comparison of the importance of the solar tides with the lunar may be complete, the same universal coefficient term is made sun's mass. With - = 81-5, T -i = -46035 = *,. m T 2-17226 To write down any term, take the universal coefficient, the Mode of general coefficient for the class of tides, the special coefficient, and reading multiply by the cosine of the argument. The result, taken with schedules the positive sign, is a term in the equilibrium tide, with water explained, covering the whole earth. The transition to the actual case by the introduction of a factor and a delay of phase (to be derived from observation) has been already explained. The sum of all the terms is the complete expression for the height of tide h. oTf ~ ) a is retained, and the special coefficient for each zM c / r m

de to involve the factor - . Here T t =% 3, m t being the Schedule of Lunar Tides. [A, i.] Universal Coefficients - T>(- ) Semi-diurnal Tides; General Coefficient = cos 2 . Coefficient. "of Argument Speed in Descrip tive Name ~

| || Degrees per m.s. S

• " > 5 Hour. O

Principal K 1-4!2)cO S 4i/ 4542o -2(st } 28—9841042 lunar. Luni-solar (lunar K—> K i+i e2)|sin2/ 03929 30 OS21372 portion). Larger N i ^ cos 4 JI 08796 -2(s-)(s-p) 28 -4397296 elliptic. Smaller elliptic. 1 L I {1-12,5 cos(2p-2^)l i 01257 J where . ^ 29-5284788 Elliptic, 2 I cot2J/-6cos2(j)-^) second JN j A^ pos*ir o L173 -2(s-f>2(s-p] 27 -8953548 order. Larger evectional.2 V ft* -COS^^7 01234: 01.706 -2(!-f)+(j-j o)+2ft -23 28 5125830 Smaller evectional.

i 13" COS^^f 00176S 00330 -2(s-)-(-p) -2A+ 2S+T 29 -4556254 Variational. 4

i r >*? 00736^ 01094 -2(s-f)+ tt27 -9682084 o, [A, ii.] Diurnal Tides; General Coefficient = sin 2X. Q, Initial. -I Argument t+(h-v). Speed in y. Descrip tive Nairn-. Coefficient. III Degrees per m.s. ari Hour. Q to Lunar di -2(s-|)-(s-p) -(s-^)+Q-iT where tanQ = Jtan(p-0 -2(s-^)+(s-r^) +2/i-2s+37r we urnal. O (l-Je2) s j n / co tyl 18S56 13 9430351 7+20". oo (l-J2)Jsin/sii 12JJ 00812 16-139101> Luni-solar (lunar por KI (l+?e 2 )|sin/cos/ 18115 15-041068( tion). Larger elliptic. Q ^.JsinJcos 2 I 03651 13 398660f me cor ser Smaller elliptic. 9, e.JsinJcos 2 J.Tx 005226 01649 14 492052: mo the y+a-rs. J fe.JsinjfcosJ 01485 15 -585443: no Elliptic, IIP second 7-40-+2CT Ye 2 -isinfcos!JI 00487 12 -854286: uc fll, order. till Evectional. 7-3<r-CT-r277 }?me. J sin/cos 2 } 7 005127 00708 13-471514. eqi Al 3>n [A, iii.] Long Period Tides; General Coefficient rsin 2 X. as su Descrip tive Name. Initial. Coefficient. J 2. ^ Argument. Speed in De grees per m.s. an in of > o 3our. th( O be Change of mean level. (1 + ie2)Kl-?sin 2 /) Of variable 252248 part is N, the long, of node 19 34 per annum to Monthly. Mm 3e.J(l-?sin27) 04136 s 5443747 Th Evectional monthly. <r-217+CJ

M-Kl-fsiua/) 005809 ( -(s-p) 00755 +2s-2h 4715211 ne< to Luni-solar fort MSf 3m2j(i-|sin2r) 004229 2(3-70 1 0158958 fur nightly. 10 00621 an Fort _J? nightly. Mf ( l-fe2)isin 2 / 07827 2(*-<) 1 0980330 Ot th( Ter3 T-CT Je.Jsin 2 / 01516 (-, o)+2(s-s] 1 6424077 mensual. is vai an [B.] Schedule of Solar Tides. Solar Tides; Universal Coefficient =— 2 mc Descriptive Name. Initial. Coefficient. Value of Coefficient. Argu ment. Speed in Degrees per m.s. Hour. [L] Semi-diurnal Tides; General Coefficient = cos 2 X. Principal solar. 82 T i(l _ |e,2) cos* J w 21137 2 30 -0000000 Luni - solar T I (solar por K 2 01823 2t+2/i 30 0821372 tion). Larger el liptic. T T, 01243 2t-(h-p,) 29-9589314 pi.] Diurnal Tides; General Coefficient = sin 2X. Solar diur nal. P ^" (1 - i, 2 )i sin w cos 2 Jw 08775 t-K+fr 14 9589314 Luni-solar (solar por KI - (1 -r-f*, 2 )! sin w cos w 08407 t+h-lir 15 0410686 tion). [iii.] Long Period Tides; General Coefficient =^-f sin 2 X. Semi-an nual. Ssa (1 |e,2)j sin 2 o> 03643 Zh 0821372 1 Fused with 2y-ff+a. 2 m is the ratio of the moon's mean motion to the sun's. 3 In these three entries the lower number gives the value when the co efficients of the evection and variation have their full values as derived from lunar theory. 4 Indicated by 2MS as a compound tide (see below, 24). 5 A fusion of -y -trier, of which the latter is the tide named. 8 The upper number is the mean value of the coefficient of the tide y-ff-rs; the lower applies to the tide MI, compounded from the tides y-ff- CT and 7 The lower number gives the value when the coefficients in the evection have their full value as derived from lunar theory. 8 The mean value of this coefficient is J(l+Se2)(i_3 s i n 2i)(i_^ s in2w)= 25, and the variable part is approximately -(1+ije-) sini cosi sinw cosw cosW= -0328 cos AT. 9 The lower of these two numbers gives the value when the coefficients in the evection and variation have their full values as derived from lunar theory. W Indicated by MSf as a compound tide. From the fourth columns we see that the coefficients in de- Scale of scending order of magnitude are M 2, Kj (both combined), S 2, import0, KI (lunar), N, P, Kj (solar), K 2 (both combined), K 2 (lunar), Mf, ance of Mm, K 2 (solar), Ssa, v, M 1} J, L, T, 2N, /*, 00, 3<7 - nr, tides. 7 - So- - CT + 2i), 7 - 4<r + 2cr, a - It] + CT, 2(ff - 17), X. The tides depending on the fourth power of the moon's parallax arise from the potential Vs-j-p 3 ( cos 3 z - cos 2). They give rise to a small diurnal tide M 1; and to a small ter-diurnal tide M 3; but we shall not give the analytical development.

§ 24. Meteorological Tides, Over-Tides, and Compound Tides.

Meteorological tides.All tides whose period is an exact multiple or submultiple of a mean solar day, or of a tropical year, are affected by meteorological conditions. Thus all the tides of the principal solar astronomical series S, with speeds 7-17, 2(y-7/), 8(7 -if), &c., are subject to more or less meteorological perturbation. An annual inequality in the diurnal meteorological tide Sj will also give rise to a tide^- - 2tj, and this will be fused with and indistinguishable from the astro P; it will also give rise to a tide with speed 7, which will be indistinguishable from the astronomical part of Kj. Similarly the astronomical tide K 2 may be perturbed by a semi-annual in equality in the semi-diurnal astronomical tide of speed 2(7-77). Although the diurnal elliptic tide Sj 01-7-77 and the semi-annual and annual tides of speeds 277 and 77 are all probably quite insensible as arising from astronomical causes, yet they have been found of sufficient importance to be considered. The annual and semi annual tides are of enormous importance in some rivers, representing in fact the yearly flooding in the rainy season. In the reduction of these tides the arguments of the S series are t, 2t, 3t, &c., and of the annual, semi-annual, ter-annual tides h, 2h, 3h. As far as can be foreseen, the magnitudes of these tides are constant from year year.

Over-tides.We have in § 21 considered the dynamical theory of over-tides. The only tides of this kind in which it has hitherto been thought necessary to represent the change of form in shallow water belong to the principal lunar and principal solar series. Thus, besides the fundamental astronomical tides M 2 and So, the over-tides M 4, Mg, M 8, and S 4, S 6 have been deduced by harmonic analysis. The height of the fundamental tide M 2 varies from year to year, according to the variation in the obliquity of the lunar orbit, and this variability is represented by the coefficient cos 4 /. It is probable that the variability of M 4, M 6, M 8 will be represented by the square, cube, and fourth power of that coefficient, and theory ( 21) indicates that we should make the argument of the over-tide a multiple of the argument of the fundamental, with a constant subtracted.

Compound tidesCompound tides have been also considered dynamically in § 21. By combining the speeds of the important tides, it will be found that there is in many cases a compound tide which has itself a speed identical with that of an astronomical or meteorological tide. We thus find that the tides 0, K 1; Mm, P, M^ Mf, Q, M 1( L are liable to perturbation in shallow water. If either or both the component tides are of lunar origin, the height of the compound tide will change from year to year, and will probably vary proportionally to the product of the coefficients of the component tides. For the purpose of properly reducing the numerical value of the compound tides, we require not merely the speed, but also the argument. The following schedule gives the adopted initials, argument, and speed of the principal compound tides. The coefficients are the products of those of the two tides to be compounded. [C. ] Schedule of Compound Tides. Initials. Arguments com bined. Speed. Speed in Degrees perm.s. Hour. MO+KX Ay - 2er 41-0251728 MK MS Mj-O 4-y - Iff - 2rj 58 9841042 MSf S 2 "-M 2 2<r-2ij r-0158958 2MK M 2 +O 3y-4ff 42-9271398 8-1+ K Sy-Z-rj 45 -0410086 MX M 2 +N 47-5<r+CT 57 -4238338 S 3 +0 87 - 2<r - 2?7 43 -9430356 S 2 -0 7+2(T- 2f] 16-0569644 2SM S 4 -M 2 27+20- 47j 3r-0158958 Mo-f Sj 67 2<r 47j 88-9841042 2MS M 4 o 2 27-4(T+2i; 27 -9682084 M4-rS 2 6-y - 4<r - 2?; 87 96S2084

§ 25. On the Form of Presentation of Results of Tidal Observations.

Immediate resultof harmonic analysis.Supposing n to be the speed of any tide in degrees per mean solar diate re- hour, and t to be mean solar time elapsing since O h of the first day suit of of (say) a year of continuous observation, then the immediate result harmonic of harmonic analysis is to obtain A and B, two heights (estimated analysis, in feet and tenths) such that the height of this tide at the time t is given by A cos nt + B sin nt. If we put R = V(-^ 2 + B 2 ) and tan f = B/A, then the tide is represented by

R cos (nt - 3 V ).

In this form R is the semi-range of the tide in British feet, and f is an angle such that f/?i is the time elapsing after O h of the first day until it is high water of this particular tide. It is obvious that f may have any value from to 360, and that the results of the analysis of successive years of observation will not be com parable with one another when presented in this form.

Final But let us suppose that the results of the analysis are presented form; in a number of terms of the form

tidal fH cos ( V+ u - K),

where V is a linear function of the moon's and sun's mean longistaiits. tudes, the mean longitude of the moon's and sun's perigees, and the local mean solar time at the place of observation, reduced to angle at 15 per hour. V increases uniformly with the time, and its rate of increase per mean solar hour is the n of th,e first method, and is called the speed of the tide. It is supposed that u stands for a certain function of the longitude of the node of the lunar orbit at an epoch half a year later than O h of the first day. Strictly speaking, u should be taken as the same function of the longitude of the moon's node, varying as the node moves; but, as the varia tion is but small in the course of a year, u may be treated as a constant and put equal to an average value for the year, which average value is taken as the true value of u at exactly mid year. Together V+ u constitute that function which has been tabulated as the " argument " in the schedules of 23. Since V+ u are to gether the whole argument according to the equilibrium theory of tides, with sea covering the whole earth, it follows that K/n is the lagging of the tide which arises from kinetic action, friction of the water, imperfect elasticity of the earth, and the distribution of land. It is supposed that H is the mean value in British feet of the semi-range of the particular tide in question; f is a numerical factor of augmentation or diminution, due to the variability of the obliquity of the lunar orbit. The value of f is the ratio of the " coefficient " in the third column of the preceding schedules to the mean value of the same term. For example, for all the solar tides f is unity, and for the principal lunar tide M 2 it is equal to cos 4 /-f-cos 4 |w cos 4 i; for the mean value of this term has a coefficient cos 4 <a cos 4 i. It is obvious, then, that, if the tidal observations are consistent from year to year, H and K should come out the same from each year's reductions. It is only when the results are presented in such a form as this that it will be possible to judge whether the harmonic analysis is yielding satisfactory results. This mode of giving the tidal results is also essential for the use of a tide-predicting machine (see 38).

We must now show how to determine H and K from R and f. It is clear that H = R/f, and the determination of f from the schedules depends on the evaluation of the mean value of each of the terms in the schedules, into which we shall not enter. If F be the value of V at O h of the first day, then clearly

so that K = f + V Q -f u. Thus the rule for the determination of K is: Add to the value of f the value of the argument at Oh of the first day.

Tidal constants.The results of harmonic analysis are usually tabulated by giving H, K under the initial letter of each tide; the results are thus comparable from year to year.^[3] For the purpose of using the tide-predicting machine the process of determining H and K from R and f has simply to be reversed, with the difference that the instant of time to which to refer the argument is O h of the first day of the new year, and we must take note of the different value of u and f for the new year. Tables^[4] have been computed for f and u for all longitudes of the moon's node and for each kind of tide, and the mean longitudes of moon, sun, and lunar perigee may be ex tracted from any ephemeris. Thus when the mean semi-range H and retardation K of any tide are known its height may be com puted for any instant. The sum of the heights for all the principal tides of course gives the actual height of water.

§ 26. Numerical Harmonic Analysis for Tides of Short Period.

Treatment of tide curves.The tide-gauge (described below, § 36) furnishes us with a continuous graphical record of the height of the water above some known datum mark for every instant of time. The first operation performed on the tidal record is the measurement in feet and decimals of the height of water above the datum at every mean solar hour. The period chosen for analysis is about one year and the first measurement corresponds to noon.

If T be the period of any one of the diurnal tides, or the double period of any one of the semi-diurnal tides, it approximates more or less nearly to 24 m. s. hours, and, if we divide it into twentyfour equal parts, we may speak of each as a T-hour. We shall for brevity refer to mean solar time as S-time. Suppose, now, that we have two clocks, each marked with 360, or 24 hours, and that the hand of the first, or S-clock, goes round once in 24 S-hours, and that of the second, or T- clock, goes round once in twenty-four T-hours, and suppose that the two clocks are started at or O h at noon of the initial day. For the sake of distinctness, let us imagine that a T-hour is longer than an S-hour, so that the T-clock goes slower than the S-clock. The measurements of the tide curve give us the height of water exactly at each S-hour; and it is re quired from these data to determine the height of water at each T-hour. For this end we are, in fact, instructed to count T-timc, but are only allowed to do so by reference to S-time, and, moreover, the time is always to be specified as an integral number of hours. Commencing with O* 1 of the first day, we begin counting 0, 1, 2, &c., as the T-hand comes up to its hour-marks. But, as the S-hand gains on the T-hand, there will come a time when, the T-hand being exactly at the p hour-mark, the S-hand is nearly as far as p + . When, however, the T-hand has advanced to the p + 1 hourmark, the S-hand will be a little beyond p + l +, that is to say, a little less than half an hour before p + 2. Counting, then, in T-time by reference to S-time, we jump from p to p + 2. The counting will go on continuously for a number of hours nearly equal to 2p, and then another number will be dropped, and so on throughout the whole year. If it had been the T-haud which went faster than the S-hand, it is obvious that one number would be repeated at two successive hours instead of one being dropped. We may describe each such process as a "change."

Method of analysis.Now, if we have a sheet marked for entry of heights of water according to T-hours from results measured at S-hours, we must enter the S-measurements continuously up top, and we then come to a change; dropping one of the S-series, we go on again continu ously until another change, when another is dropped; and so on. Since a change occurs at the time when a T-hour falls almost exactly half-way between two S-hours, it will be more accurate at a change to insert the two S-entries which fall on each side of the truth. If this be done the whole of the S- series of measure ments is entered on the T-sheet. Similarly, if it be the T-hand which goes faster than the S-hand, we may leave a gap in the T-series instead of duplicating an entry. For the analysis of the T-tide there is therefore prepared a sheet arranged in rows and columns; each row corresponds to one T-day, and the columns are marked O^h, l^h, . . . 23^h; the O's may be called T-noons. A dot is put in each space for entry, and where there is a change two dots are put if there is to be a double entry, and a bar if there is to be no entry.^[5] The numbers entered in each column are summed; the results are then divided, each by the proper divisor for its column, and thus the mean value for that column is obtained. In this way 24 numbers are found which give the mean height of water at each of the 24 special hours. It is obvious that if this process were con tinued over a very long time we should in the end extract the tide under analysis from amongst all the others; but, as the process only extends over about a year, the elimination of the others is not quite complete. The elimination of the effects of the other tides may be improved by choosing the period for analysis not exactly equal to one year.

Let us now return to our general notation, and consider the 24 mean values, each pertaining to the 24 T-hours. We suppose that all the tides except the T-tide are adequately eliminated, and, in fact, a computation of the necessary corrections for the absence of complete elimination, which is given in the Tidal Report to the British Association in 1872, shows that this is the case. It is Necessary for augmenting factors,obvious that any one of the 24 values does not give the true height of the T-tide at that T-hour, but gives the average height of the water, as due to the T-tide, estimated over half a T-hour before and half a T-hour after that hour. A consideration of this point shows that certain augmenting factors, differing slightly from unity, must be applied. In the reduction of the S-series of tides, the numbers treated are the actual heights of the water exactly at the S -hours, and therefore no augmenting factor is requisite.

We must now explain how the harmonic analysis, which the use of these factors presupposes, is carried out.

If t denotes T-time expressed in T-hours, and re is 15, we express the height h, as given by the averaging process above explained, by the formula

h^Ao-f A 1 cos nt + B! sin nt + A 2 cos 2nt + B 2 sin 2nt + . . .,

where t is 0, 1, 2, . . . 23. Then, if S denotes summation of the series of 24 terms found by attributing to t its 24 values, it is obvious that

A 2 = T 1 5 Zhcos 2nt; 63 = ^ Zhsin 2nt; &c., &c.

Since n is 15° and t is an integer, it follows that all the cosines and sines involved in these series are equal to one of the following, viz., 0, ±sin 15°, ±sin 30°, ±sin 45°, ±sin 60°, ±sin 75°, ±1. It is found convenient to denote these sines by 0, ±S₁, ±S₂, ±S₃, ±S₄, ±S₅, ±1. The multiplication of the 24 h's by the various S's and the subsequent additions may be arranged in a very neat tabular form, like that given in a Report to the British Association in 1883. The A s and B s having been thus deduced, we have R= /( A 2 -l- B 2 ). R must then be multiplied by the augmenting factor. We thus have the augmented R. Next the angle whose tangent is B/A gives The addition to f of the appropriate V + u gives K, and the multiplication of R by the appropriate 1/f gives H. The reduction is then complete. An actual numerical example of harmonic analysis is given in the Admiralty Scientific Manual (1886) In the article "Tides"; but the process there employed is slightly different from the above, because the series of observations is sup posed to be a short one.

§ 27. Harmonic Analysis for Tides of Long Period.

Tides of long period.For the purpose of determining the tides of long period we have to eliminate the oscillations of water-level arising from the tides of short period. As the quickest of these tides has a period of many days, the height of mean water at one instant for each day gives sufficient data. Thus there will in a year's observations be 365 heights to be submitted to harmonic analysis. To find the daily mean for any day we take the arithmetic mean of 24 consecutive hourly values, beginning with the height at noon. This height will then apply to the middle instant of the period from O h to 23 h, that is to say, to'll h 30 m at night. The formation of a daily mean does not obliterate the tidal oscillations of short period, be cause none of the tides, except those of the principal solar series, have commensurable periods in mean solar time. A small correc tion, or "clearance of the daily mean," has therefore to be applied lor all the important tides of short period, except for the solar tides. Passing by this clearance, we next take the 365 daily means, and iiud their mean value. This gives the mean height of water for the year. We next subtract the mean height from each of the 365 values, and find 365 quantities Sh, giving the daily height of water above the mean height. These quantities are to be the subject of the harmonic analysis, and the tides chosen for evaluation are those which have been denoted above as Mm, Mf, MSf, Sa, Ssa.

Let Sh A cos (<r - CT) t + B sin (ff - cr) t + C cos 2fft + D sin 2<rt + C cos 2(<r - if)t + D sin 2(<r - i))t } (73), + E cos t)t + F sin it + G cos 2jit + H sin 2^

where t is time measured from the first'll h 30 m . If we multiply the 365 5h's by 365 values of cos (0- - ts}t and effect the summation, the coefficients of B,C,D, &c., are very small, and that of A is nearly 182. Similarly, multiplying by sin (ff- is)t, cos 2<rt, &c., we obtain 10 equations for A,B,C, &c., in each of which one coefficient is nearly 182^ and the rest small. These equations are easily solved by successive approximation. In this way A,B,C, &c., are found, and afterwards the clearance to which we have alluded is applied. Finally the cleared A,B,C, &c., are treated exactly as were the components of the tides of short period. Special forms and tables have been prepared for facilitating these operations.

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1. 1 Airy, and after him Chazallon, appear to have been amongst the first to use a kind of harmonic analysis for reducing tidal observations; but, as Airy did not emancipate himself from the use of hour-angles, declinations, &c., his work can hardly be considered as an example of the analytic method; see his " Tides and Waves," and Hatt's Phenomene des Marees, Paris, 1885.
2. 2 For further details of the analysis of this section, see the Report "On Harmonic Analysis, &c.," for 1883 to the British Association (Southport).
3. See, for example, a collection of results by Baird and Darwin, Proc. Roy. Soc., No. 239, 1885.
4. Report on Harmonic Analysis to Brit. Assoc., 1883, and more extended table in Baird's Manual of Tidal Observation, London, 1887.
5. 3 A sample page is given in the Report to the Brit. Assoc., 1883.