Page:EB1911 - Volume 26.djvu/994

Rh consentaneous in the equatorial belt, and of opposite meanings outside the critical latitudes. We here conceive the function always to be written § -sin'}, so that outside the critical latitudes high-tide is low-water. We may in continuing the development write the 3€-ig-Z 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 develo ment as symbols of quantities to be deduced from observation. iit will be understood, therefore, that in the followin schedules the “argument” is that part of the argument which is dirived from theory, the true complete argument being the “argument ” -wc, where K is derived from observation.

Up to this point we have sup osed the moon's longitude and the earth's position to be measured) from the “ intersection "; but in order to pass to the ordinary astronomical formulae we must measure the longitude and the earth's osition from tlre vernal equinox. Hence we determine the longitude; and right ascension of the ' inter# section " 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 36-Q1 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 series of terms each of which is a function of the elements of the moon's or sun's orbit, and a function of the terrestrial latitude Exp""'””°" of the place of observation, multiplied by the cosine gzichedul” of an angle which increases uniformly with the time. W' 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; 'I, w =obliquities of equator to lunar orbit and ecliptic; p, p, f-'longitudes of lunar and solar perigees, 25, a1, =hourly increments of p, p, ; s, h=moon's and sun's mean longitudes; a, -q=hourly increments of 5, h; t=local mean solar time reduced to angle; 'y-1, =15° per hour;)= latitude of place of observation; E, v=longitude in lunar orbit, and R.A. of the intersection; N =longitude of moon's node; i- eed inclination of lunar orbit. The “ speed " of any tide Dzfined is defined as the rate of increase of its argument, and is expressible, therefore, as a linear function of -y, 17, a, U; for we may neglect ZIV, 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 gl  ia, which multiplies every term of all the schedules. Secondly, there are general coefficients, one for each schedule, viz. cos') for the semi-diurnal terms, sin 2) for the diurnal, and éfi sin” A for the terms of long period. In each schedule the third column, headed “ coefficient, " gives the functions of I and e. 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 I =¢..»; but we pass over the explanation of the mode of computing the values. The fifth column contains ar uments, linearfunctionsof t, h, s, p, v, 2. In [A, i.] 2t+2(h-ig 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 increases 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. The schedule for the solar tides is drawn up in precisely the same manner, the only difference being that the coefficients are absolute constants. In order that the comparison of the importance of the solar tides with the lunar may be complete, the same universal coefficient L;-An;  a is retained, and the special coefficient for each term is made to involve the factor   Here -r, =§ ;-Q, m, being the I

sun's mass. With

M 1' I


 * =81~S» i='46°35=;%-To

write down any term, take the universal coefficient, the general coefficient for the class of tides, the special coefficient, and Mode of multiply by the cosine of the argument! The result, Reading taken with the positive sign, is a term in the equilibrium Schedules tide, with water covering the whole earth. The transi-Exp, amed tion 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. It must be remarked that the schedule of tides is here largely abridged, and that the reader who desires fuller information must refer to the Brit. Assoc. Report for 1883, or vol. i. of G. H. Darwin's Scientijic Papers, or to Harris's Manual of Tides.

A.—Schedule of Lunar Tides.

Universal Coefficient

"la

3 a

2 M a

ne

"T

i.-Semildiurnal Tides; Ge ral Coefficient =cos' X. A:gig speed in

D'§ ;n§ ;j"= gg coescim as gg, ;§ {*;';', § '§ ';f, ,2§ “, {f°§ if 2 § g Hour.

U

Q M2. § (r-'F;e')cos' H ~45426 -2(r-§) 28'Q84104 2° Luni-solar

(lunar K2 1}(I +i§ e')§ sinzl -o3g29 3O'082I§ 72° portion)

L*§ '§§, ,, } N l ~ Ee ws' 'll ~<»s1<»6 - zo- s- <5-1»> 2s~439m6° ii.-Diurnal Tides; General Coefficient =sin 2). gg; Speed in

iD°°'§§ l§§ "'e:E C°°'“°'@“f- § L¥§§ flwiif, iisigffiii I 5, 2§ 5 Hour.l

Linfilfi' i 0 <1 ~ gen Sin 1 C<>S'»l»I 18856 -=++=f 13-¢43°3s6° Lu '- ola

(liiriar r K1 (l+§ e2)§ sin I cos I -r8rr5 -iff I5<>4lc686° L ”°“'° ) -=<:-:Hs-»> f

§§ f ;'dc M Q ge - i sin I cos' il -o;565r   +int } 13-398660g° iii.-Lon Period Tides; General Coefficient é — Siflzx. K

-~  Speed in

»-J C

Dizcsgifve 3; CoeHicient. é§ éé; Argument. ggggf »< > Q our.

U

Change uf Of vztriablepart ' 1984.

Exit — (i»{¢§§ ¢2)§ (r-2. smfl) -z5z24l §  I:;3;e long. per annum F': 'il;htly i Mf tx - MH sin' I 1 'O1821 z(.v- §) 1~oo8o330° B.—Schedule of Solar Tides.

M 3

Universal Coefficient =§ - T 5) u.

T; ' "<5;5§ Speed in

Delflcgxggw E Coefficient. Argument. "' > L23 Hour.

i.-Semi-diurnal Tides; General Coefficient = cos'>. Prxflfil. i S2 -::'(r- § e, ')} cos* 5-an -21 137 21 3o-ooooooo° L '~ r »

liglolzfi-3) K, ;4(r+3¢, ')} sin' w '01823 21-l-zh 30'O821372° portion

L“f$§, ,C i T Qt Zn 005' tw '0¥243 21-(hen) 2a-os893r4° ii.-Diurnal Tides; General Coefficient = sin 2>. i°: T § § i' i P;f i(1 - ';e, “)1} sinw cosi »}w 4:8775 t-h-Hp: I4'QS803I4, ° n1~s ar

Lisolz; K1 z¢(I+§ ¢,2)§ - sin nu cos w ~o8407 f+h'§ '¥ I5-o410686 portion) T

iii.—Long Period Tides; General Coefficient =é-Q sln') S°;$;3“' g Ssa gh - § e,2)1} sin2 w -03643 2h 0'O821372° From the fourth columns we see that the coefficients in de# scending order of magnitude are M2, Ki (both combined); 52, O, K1 (lunar), N, P, K1 (solar) K2 (both combined), Seal f K, (lunar), Mf, Q, K. (so1ar), SSa~, ,,, § , § ', ,, The tides which we omit from the schedules are efgldes ce relatively unimportant, » but nevertheless commonly evaluated in accurate tidal work, are all lunar tides, viz. the following semi-diurnal tides: the smaller elliptic tide L, the larger and smaller evectional tides v, A, the variational t1de p. Also the, following diurnal tides, viz. the smaller elliptic tide Ml, a tide of speed y-l-o'-U called ]. Also amongst the tides of long period, the luni-solar fortnightly called MSf.

The tides depending on the fourth power of the moon's parallax The mean value of this coefficient is § (r -1-ge?) (1 -'gSil'1?'i)<I - g'sin%») = -25, and the variable part is approximately - (X +§ e2) sm 1 cos z smw cos w cos N= - -0328 eos N.