but it would be necessary to give a figure, and to go into details, to give the results satisfactorily.

A similar examination of the equation, giving the retardation of the earth's rotation, shows that there is not so much variety of result, for the tidal friction always tends to retard the earth.

This completes the consideration of the instantaneous effects on the earth, and the next point demanding attention is the reaction, which the bodily tides have upon the disturbing bodies.

The problem is solved by the consideration that however the three bodies may interact the resultant moment of momentum of the moonearth system remains constant, except in so far as it is affected by the sun's action on the earth. The application of this principle results in an equation giving the rate of increase of the square root of the moon's distance in terms of the heights and retardations of the several bodily tides on the earth; it appears that all the tides, except the fortnightly one, tend to make the moon's distance increase with the time, but the fortnightly tide acts in the opposite direction; its effect is, however, in general very small compared with that of the other tides. It is proved, also, that the tidal reaction on the sun, which goes to modify the earth's orbit, has quite insignificant effects, and may be neglected.

I will now show, from geometrical considerations, how some of the results previously stated come to be true. It will not, however, be possible to obtain a quantitative estimate in this way.

The three following propositions do not properly belong to an abstract, since they are not given in the paper itself; they merely partially replace the analytical method pursued therein. The results of the analysis were so wholly unexpected in their variety, that I have thought it well to show that the more important of them were conformable to common sense. These general explanations might doubtless be multiplied by some ingenuity, but it would not have been easy to discover the results, unless the way had been first shown by analysis. Prop. I. If the viscosity be small the earth's obliquity increases, the rotation is retarded, and the moon's distance and periodic time increase. The figure represents the earth as seen from above the South Pole, so that S is the Pole, and the outer circle the Equator. The earth's rotation is in the direction of the curved arrow at S. The half of the inner circle which is drawn with a full line is a semi-small-circle of S. lat., and the dotted semi-circle is a semi-small-circle in the same N. lat. Generally dotted lines indicate parts of the figure which are below the plane of the paper.

It will make the explanation somewhat simpler, if we suppose the tides to be raised by a moon and anti-moon diametrically opposite to one another. Then let M and M' be the projections of the moon and anti-moon on to the terrestrial sphere.

If the substance of the earth were a perfect fluid or perfectly elastic, the apices of the tidal spheroid would be at M and M'. If, however, there is internal friction due to any sort of viscosity, the tides will lag, and we may suppose the tidal apices to be at T and T'.

Now, suppose the tidal protuberances to be replaced by two equal heavy particles at T and T', which are instantaneously rigidly connected with the earth. Then the attraction of the moon on T is greater than on T'; and of the anti-moon on T' is greater than on T. The resultant of these forces is clearly a pair of forces acting on the earth in the direction of TM, T'M'.

The effect on the obliquity will be considered first.

These forces TM, T'M', clearly cause a couple about the axis in the equator, which lies in the same meridian as the moon and anti-moon. The direction of the couple is shown by the curved arrows at L, L'.

Now, if the effects of this couple be compounded with the existing rotation of the earth, according to the principle of the gyroscope, it will be seen that the South Pole S tends to approach M, and the North Pole to approach M'. Hence supposing the moon to move in the ecliptic, the inclination of the earth's axis to the ecliptic diminishes, or the obliquity increases.

Next, the forces TM, T'M', clearly produce a couple about the earth's polar axis, which tends to retard the diurnal rotation.

Lastly, since action and reaction are equal and opposite, and since the moon and anti-moon cause the forces TM, T'M', on the earth, therefore the earth must cause forces on those two bodies (or on their equivalent single moon) in the directions MT and M'T'. These forces are in the direction of the moon's orbitual motion, and therefore her linear velocity is augmented. Since the centrifugal force of her orbitual motion must remain constant, her distance increases, and with

the increase of distance comes an increase of periodic time round the earth.

This general explanation remains a fair representation of the state of the case so long as the different harmonic constituents of the aggregate tide-wave do not suffer very different amounts of retardations; and this is the case so long as the viscosity is not great.

Prop. II. The attraction of the moon on a lagging fortnightly tide causes the earth's obliquity to diminish, but does not affect the diurnal rotation; the reaction on the moon causes a diminution of her distance, and periodic time.

The fortnightly tide of a perfectly fluid earth is a periodic increase and diminution of the ellipticity of figure; the increment of ellipticity varies as the square of the sine of the obliquity of the equator to the ecliptic, and as the cosine of twice the moon's longitude from her node. Thus the ellipticity is greatest when the moon is in her nodes, and least when she is 90° removed from them.

In a lagging fortnightly tide the ellipticity is greatest some time after the moon has passed the nodes, and least an equal time after she has passed the point 90° removed from them.

The effects of this alteration of shape may be obtained by substituting for these variations of ellipticity two attractive or repulsive particles, one at the North Pole and the other at the South Pole of the earth. These particles must be supposed to wax and wane, so that when the real ellipticity of figure is greatest they have their maximum repulsive power, and when least they have their maximum attractive power; and their positive and negative maxima are equal.

We will now take the extreme case when the obliquity is 90°; this makes the fortnightly tide as large as possible.

Let the plane of the paper be that of the ecliptic, and let the outer semicircle be the moon's orbit, which she describes in the direction of

the arrows.

Let NS be the earth's axis, which lies by hypothesis in the ecliptic, and let LL' be the nodes of the orbit. Let N be the North Pole; that is to say, if the earth were turned about the line LL', so that N rises above the plane of the paper, the earth's rotation would be in the same direction as the moon's orbitual motion.

First consider the case where the earth is perfectly fluid, so that the tides do not lag.

Let m2, m be points in the orbit whose longitudes are 45° and 135°; and suppose that couples acting on the earth about an axis at O perpendicular to the plane of the paper are called positive when they are in the direction of the curved arrow at O. Then, when the moon is at

m1 the particles at N and S have their maximum repulsion. But at this instant the moon is equidistant from both, and there is no couple about O. As, however, the moon passes to me there is a positive couple, which vanishes when the moon is at m2, because the particles have waned to zero. From m2 to ms the couple is negative; from my to m, positive; and from m, to ms negative. Now, the couple goes through just the same changes of magnitude, as the moon passes from m1 to m2, as it does while the moon passes from m, to mã, but in the reverse order; the like may be said of the arcs mm, and mзm ̧. Hence it follows that the average effect, as the moon passes through half its course, is nil, and therefore there can be no secular change in the position of the earth's axis.

But now consider the case when the tide lags. When the moon is at m, the couple is zero, because she is equally distant from both particles. The particles have not, however; reached their maximum of repulsiveness; this they do when the moon has reached M1, and they do not cease to be repulsive until the moon has reached M2. Hence, during the description of the arc m,M2, the couple round O is positive.

Throughout the arc Mam, the couple is negative, but it vanishes when the moon is at mg, because the moon and the two particles are in a straight line. The particles reach their maximum of attractiveness when the moon is at M3, and the couple continues to be positive until the moon is at M.

Lastly, during the description of the arc Mms the couple is negative. But now there is no longer a balance between the arcs m ̧M2 and M1m, nor between M1⁄2m, and mзM. The arcs during which the couples are positive are longer and the couples are more intense than in the rest of the semi-orbit. Hence the average effect of the couples is a positive couple, that is to say, in the direction of the curved arrow round O.

It may be remarked that if the arcs m,Mi, m2M2, mзMз, m ̧M had been 45°, there would have been no negative couples at all, and the positive couples would merely have varied in intensity.

Now, a couple round O in the direction of the arrow, when combined with the earth's rotation, would, according to the principle of the gyroscope, cause the pole N to rise above the plane of the paper, that is to say, the obliquity of the ecliptic would diminish. The same thing would happen, but to a less extent, if the obliquity had been less than 90°; it would not, however, be nearly so easy to show this from general considerations.

Since the forces which act on the earth always pass through N and S, therefore there can be no moment about the axis NS, and the rotation about that axis remains unaffected. This can hardly be said to amount to strict proof that the diurnal rotation is unaffected by the fortnightly tide, because it has not been rigorously shown that the two particles at N and S are a complete equivalent to the varying ellipticity of figure.

Lastly, the reaction on the moon must obviously be in the opposite direction to that of the curved arrow at 0; therefore there is a force retarding her linear motion, the effect of which is a diminution of her distance and of her periodic time.

The fortnightly tidal effect must be far more efficient for very great viscosities than for small ones, for, unless the viscosity is very great, the substance of the spheroid has time to behave sensibly like a perfect fluid, and the tide hardly lags at all.

Prop. III. An annular satellite not parallel to the planet's equator attracts the lagging tides raised by it, so as to diminish the inclination of the planet's equator to the plane of the ring, and to diminish the planet's rotation. The effects of the joint action of sun and moon may be explained from this.

Suppose the figure to represent the planet as seen from vertically over the South Pole S; let LL' be the nodes of the ring, and LRL' the projection of half the ring on to the planetary sphere.