which is equivalent to N1 = ∞. This equation determines the admissible values of ƒ (=o/2w). The constants in

(11) are then given by

It is easily seen that when ẞ is infinitesimal the roots of (26) are given by

One arithmetically remarkable point remains to be noticed. It might appear at first sight that when a value of ƒ has been found from (26) the coefficients B3, B5, B,, ... could be found in succession from (15) and (16), or by means of the equivalent formula (18). But this would require us to start with exactly the right value of f and to observe absolute accuracy in the subsequent stages of the work. The above argument shews, in fact, that any other value, differing by however little, if adopted as a starting point for the calculation will inevitably lead at length to values of N, which approximate to the limit 1*.

217. It is shewn in the Appendix to this Chapter that the tidegenerating potential, when expanded in simple-harmonic functions of the time, consists of terms of three distinct types.

The first type is such that the equilibrium tide-height would be given by

} = H' (} — cos2 8). cos (σt + €)†.

...

.(29)

The corresponding forced waves are called by Laplace the 'Oscillations of the First Species'; they include the lunar fortnightly and the solar semi-annual tides, and generally all the tides of long period. Their characteristic is symmetry about the polar axis, and they form accordingly the most important case of forced oscillations of the present type.

If we substitute from (29) in (7), and assume for

In strictness, here denotes the geocentric latitude, but the difference between this and the geographical latitude may be neglected consistently with the assumptions introduced in Art. 214.

expressions of the forms (11) and (12), we have, in place of (14), (15),

=

whilst (16) and its consequences hold for all the higher coefficients. It may be noticed that (31) may be included under the general formula (16), provided we write B_1 2H'. It appears by the same argument as before that the only admissible solution for an ocean covering the globe is the one that makes N = 0, and that accordingly N, must have the value given by the continued fraction in (24), where ƒ is now prescribed by the frequency of the disturbing forces.

In particular, this formula determines the value of N1. Now

in other words, this is the only value of A which is consistent with a zero limit of N,, and therefore with a finite velocity at the poles. Any other value of A, if adopted as a starting point for the calculation of B1, B3, B5, ... in succession, by means of (30), (31), and (16), would lead ultimately to values of N, approximating to the limit 1. Moreover, since absolute accuracy in the initial choice of A and in the subsequent computations would be essential to avoid this, the only practical method of calculating the coefficients is to use the formulae

where the values of N1, N2, N3, ... are to be computed from the continued fraction (24). It is evident à posteriori that the solution thus obtained will satisfy all the conditions of the problem, and that the series (12) will converge with great rapidity. The most convenient plan of conducting the calculation is to assume a roughly approximate value, suggested by (19), for one of the ratios N, of sufficiently high order, and thence to compute

N1-1, N-2, ... N2, N1

in succession by means of the formula (23). The values of the constants A, B1, B3, ..., in (12), are then given by (32) and (33). For the tidal elevation we find

In the case of the lunar fortnightly tide, f is the ratio of a sidereal day to a lunar month, and is therefore equal to about, or more precisely 0365. This makes f2=00133. It is evident that a fairly accurate representation of this tide, and à fortiori of the solar semi-annual tide, and of the remaining tides of long period, will be obtained by putting f = 0; this materially shortens the calculations.

The results will involve the value of 8,

=

4wagh. For 8 = 40, which

corresponds to a depth of 7260 feet, we find in this way

=

¿/H' — ·1515 — 1·0000μ2 + 1·5153μ- 1·2120μ+6063μ3·2076μ10 +0516μ12-0097μ140018μ160002μ18, ......(35) *

H'

Since the polar and equatorial values of the equilibrium tide are and H', respectively, these results shew that for the depths in question the long-period tides are, on the whole, direct, though the nodal circles will, of course, be shifted more or less from the positions assigned by the equilibrium theory. It appears, moreover, that, for depths comparable with the actual depth of the sea, the tide has less than half the equilibrium value. It is easily seen from the form of equation (7) that with increasing depth, and consequent diminution of ẞ, the tide-height will approximate more and more closely to the equilibrium value. This tendency is illustrated by the above numerical results.

* The coefficients in (35) and (36) differ only slightly from the numerical values obtained by Darwin for the case f=0365.

It is to be remarked that the kinetic theory of the long-period tides was passed over by Laplace, under the impression that practically, owing to the operation of dissipative forces, they would have the values given by the equilibrium theory. He proved, indeed, that the tendency of frictional forces must be in this direction, but it has been pointed out by Darwin* that in the case of the fortnightly tide, at all events, it is doubtful whether the effect would be nearly so great as Laplace supposed. We shall return to this point later.

218. When the disturbance is no longer restricted to be symmetrical about the polar axis, we must recur to the general equations (1) and (2) of Art. 214. We retain, however, the assumptions as to the law of depth and the nature of the boundaries introduced in Art. 215.

If we assume that N, u, v, Č all vary as ei(ot+s+e), where s is integral, the equations referred to give

It appears that in all cases of simple-harmonic oscillation the fluid particles describe ellipses having their principal axes along the meridians and parallels of latitude, respectively.

Substituting from (3) in (2) we obtain the differential equation in ' :

219. The cases = 1 includes, as forced oscillations, Laplace's 'Oscillations of the Second Species,' where the disturbing potential is a tesseral harmonic of the second order; viz.

=

H" sin 0 cos 0. cos (at ++ €),

. . . . . . . . . (1) where o differs not very greatly from w. This includes the lunar and solar diurnal tides.

In the case of a disturbing body whose proper motion could be neglected, we should have σ = w, exactly, and therefore f = . In the case of the moon, the orbital motion is so rapid that the actual period of the principal lunar diurnal tide is very appreciably longer than a sidereal day*; but the supposition that f = simplifies the formulae so materially that we adopt it in the following investigation. We find that it enables us to calculate the forced oscillations when the depth follows the law

h

=

(1 − q cos2 0) ho,

where q is any given constant.

(2)

Taking an exponential factor ei(w++e), and therefore putting s = · 1, ƒ = }, in Art. 218 (3), and assuming

Substituting in the equation of continuity (Art. 218 (2)), we get

which is consistent with the law of depth (2), provided

(3)

.(4)

(5)

One remarkable consequence of this formula is that in the case of uniform depth (q= 0) there is no diurnal tide, so far as the rise and fall of the surface is concerned. This result was first established (in a different manner) by Laplace, who attached great importance to it as shewing that his kinetic theory was able to account for the relatively small values of the diurnal tide

*It is to be remarked, however, that there is an important term in the harmonic development of for which σ =w exactly, provided we neglect the changes in the plane of the disturbing body's orbit. This period is the same for the sun as for the moon, and the two partial tides thus produced combine into what is called the 'luni-solar' diurnal tide.

†Taken with very slight alteration from Airy, "Tides and Waves," Arts. 95..., and Darwin, Encyc. Brit. 9th ed., t. xxiii. p. 359.