retical value

nightly and

elliptic

is more convenient to regard the spherical harmonic as itself The theoslowly varying between certain limits, and thus to amalgamate of the forta number of terms together. The last term of (23) § 808 will monthly give the theoretical equilibrium values of the fortnightly decli- tides. national tide, and of the monthly elliptic tide. The full expansion of this term would involve a certain part going through its period in 19 years, in which time the lunar nodes complete a revolution. This part will, according to Sir William Thomson, be most conveniently included by conceiving the inclination of the lunar orbit to the equator to undergo a slow oscillation in that period. In practice an average value for the inclination, the average being taken over a whole year, is sufficiently exact.

(a) In what follows, the descending node of the equator on the lunar orbit will be called "the intersection." If the lunar orbit were identical with the ecliptic, the intersection would be the vernal equinox or T.

of symbols.

The following is a summary of the notation employed below:- Definition
For the moon :

=

M= mass; D= radius vector; c=mean distance; σ = mean motion; true longitude in the orbit; inclination of lunar orbit to the ecliptic: N= longitude of ascending node on the ecliptic; longitude of perigee in the orbit; e eccentricity of orbit; longitude of "the intersection" in the orbit; v = right ascension of "the intersection"; 8 = declination.

=

Observe that longitudes "in the orbit" are measured along the ecliptic as far as the lunar node, and thence along the orbit; or are measured altogether in the movable orbit from a point therein, which is at a distance behind the node equal to the distance of the node from T.

For the earth:

E = mass; a = mean radius; 7, λ = latitude and W. longitude of places on the earth's surface; = obliquity of ecliptic; I-inclination of equator to lunar orbit.

=

For both bodies together, let Ma3/Ec3. And let the time t be measured from the instant when the moon's mean longitude vanishes.

The readers of the Tidal Reports of the British Association

Long-period tides.

for 1868, 1870, 1871, 1872, 1876* may find it convenient to note that the symbols employed are frequently the Greek initials of the corresponding words: thus,-y, a, n [yâ, oelým, Aos] for the rotation and mean motions of earth, moon, and

sun.

We may now write the last term of (23) § 808, thus

It is obvious from the solution of a right-angled spherical triangle that

Whence

sin d = sin I sin (0 – §).

1-3 sin' 8 = 1-3 sin' I + sin I cos 2 (0-)(2).

By the theory of elliptic motion, we have, on neglecting the

solar perturbation of the moon, which causes the 'evection,' the

'variation' and other inequalities,

음(1-0)= 1 + e cos (0 - )

....

(3).

In proceeding to further developments, e and sin3 I will be treated as small quantities of the first order, and those of the

second order will be neglected. Thus in terms of the first order we have

h

H

· {1 + 3e cos (σt − a)} { 1 − 3 sin2 I + 1⁄2 sin2 I cos 2 (σt - §)} -1- sin' I+3e (1-3 sin' I) cos (ot-w) + sin' I cos 2 (σt-)

.....

In this expression the first term 1-3 sin I oscillates with a period of 19 years about the mean value 1-3 sin'w, the

* Also of papers presented to the British Association by Sir W. Thomson and Capt. Evans, R.N., in 1878 (reprinted in Nature, Oct. 24, 1878), and by Mr G. H. Darwin in 1882.

tides.

maximum and minimum values of I being w+i and w- -i. Long-period It represents a small permanent increase to the ellipticity of the oceanic spheroid, on which is superposed a small 19-yearly tide. This part of the expression has no further interest in the present investigation. The last term of (5) goes through a double period in nearly 27.3 m. s. days and constitutes the fortnightly declinational tide. If the approximation were carried to terms of the second order, which may very easily be done, this term would have involved a factor 1-e. The middle term goes through a single period in something over 27-3 days, the angular motion of the lunar perigee being 40° 40′ per annum. This term as it stands in (5) is complete to the second order.

Thus we may

write the expressions to the second order of small quantities, for the fortnightly and monthly elliptic tides, thus:

to

for the

longitude

(b) We must now show how to compute I and έ, and it Formula will be expedient (as will appear below) at the same time compute v.

The accompanying figure exhibits the relation of the three planes to one another.

and R. A. of the intersection.

Equator

Orbit
Ecliptic

the longitude of I in the orbit is TN-NI, and v the right ascension of I is TI.

Now from the spherical triangle TNI, we have

Long-period tides.

Also tan έ

(cot In sin N-cos N) sin N
cos N cot In sin N + sin N

Formulas for the longitude

and R.A. of the intersection.

Substituting in which from (7), and effecting some reductions in the result, and in (8), we have

These formulas are rigorously true, but since i is small, being about 5°9′, we may obtain much simpler approximate formulæ, sufficiently accurate for all practical purposes. Treating then sin i and tan i as equal to one another, and to the circular measure of 5o 9′, equations (10) become approximately,

tan § = i cot w sin No – sin 2N1-sin' w

sin w

COS @

(11).

tan v = i cosec o sin N-sin 2N

And

The second terms of these expressions are very nearly equal to one another, because cos w = 1-sin2 w approximately. v-έ is a small angle, which is to a close degree of approximation equal to i tan sin N.

Numerical calculation shows that i tan is 1°4′; hence =v - 1o 4'sin N very nearly.

same.

In the Tidal Report of the British Association for 1876 the treatment of this subject, with notation involving a symbol ), is somewhat different from the above, but the result is the The symbol denotes "the equatorial mean moon's" right ascension at the epoch when t=0; which it may be ob served is not the same epoch as that chosen here. This fictitious mean moon moves in the equator with an angular velocity equal to the moon's mean motion, and it is at the "intersection" at the instant when the moon's mean longitude is equal to the longitude "in the orbit" of the intersection. In other words, if we take a second fictitious moon moving in the plane of the lunar orbit with an angular velocity equal to the moon's mean motion, and coinciding with the actual moon at the instant when the moon's mean longitude vanishes, then the equatorial

mean moon coincides with this orbital mean moon at the inter- Long-period

section.

It is obvious then that the right ascension of the equatorial mean moon will always differ from the moon's mean longitude by vέ, and thus

) = moon's mean longitude at the epoch + 1o 4' sin N.

Therefore with the epoch of the Report of 1876 (pp. 299, 302)

ot +) - v = moon's mean longitude + 1°4′ sin N − v

= moon's mean longitude – §.

tides.

Now according to the Report (p. 305), the fortnightly tide is Formulas expressed, (by means of H as defined in (1) above), in the form

Hsin' I cos 2 (ot + ) − v).

This only differs from (6) in the term ge2, which is the correction for the eccentricity of the lunar orbit.

It is to be remarked that in the report) is the moon's mean anomaly at the epoch, and therefore' is equal to the mean longitude of the moon's perigee + 1°4′ sin N, and not simply the mean longitude of the moon's perigee, as defined in the last line of p. 302. Since the moon's mean anomaly is only involved in the arguments of the elliptic tides, which are all small, this correction in ' has no practical importance. It is however important, in regard to clear ideas of the notation and the spherical trigonometry of the subject.

In consequence of not at first apprehending properly the nature of the fictitious "equatorial mean moon," I overlooked the term 1°4′ sin N in ), and in the reductions made below have used instead of §. Since the difference between v and

is clearly of little importance in respect to the numerical values of the fortnightly tide, I have not repeated the computations with the correct value of ), or, in the present notation, with έ in place of v.

та

(c) The factor H or τa[(1+C) – sin3] involves the function E, which depends on the distribution of land and water on the earth's surface. By (21) § 808

1

for the

longitude

and R. A. of the intersection.

(3 sin l-1) cos l dl dλ

Ω

where

is the total area of ocean, and where the double integral

is taken all over the surface of the ocean.