the

represent the masses of the earth and the moon, r the equatorial radius of the former, D the distance between the centres of both bodies, A the difference of terrestrial longitude between the crest of the tide-wave and that place where the moon is vertical, difference of longitude between the crest of the wave and any point in the channel; and let y be the variable height of the tide at this point; while the breadth of the channel is denoted by b, and is supposed to be small in comparison to the earth's dimensions. Then brdo will express the magnitude of the infinitesimal portion of the water which rises above the proper geometrical boundary of the terrestrial spheroid, and its attractive force on the moon will be

in which g measures the attraction exerted at the distance k by a volume of water assumed as the unit of matter and having a spherical form. Of the force represented by formula (5), the component acting horizontally and tending to change the direction of terrestrial gravity on the moon will be

k2bgyr2 sin (A-4) dp

(D2—2Dr cos (A$) +r2 cos2 (A—†) )**

Making this expression equal to df, transforming it into a series of which it is necessary to retain only the first two terms, and putting C for k2gb, there results

df=

Cyr sin (A-4)do, 3Cyr sin 2(A-)do

D3

+

(7)

The value of the force f which occasions the secular change in the lunar movement may be obtained by integrating the last equation; but for this purpose y must be expressed in terms of ; and in ascordance with the theory of the tides and the laws of periodicity in their movements, y may be assumed equal to h cos 24+h' sin 24; h and h' being two constant quantities, the latter small in comparison to the first, and depending on the effect of friction. Equation (7) becomes, on the substitution of this value of y,

Chr2 sin (A+)do

2D8

3Chr3 sin 2Ado

4D4

Chr2 sin (34—A)dp

2D3

3Chr3sin(44-2A)do

df=

+

4D4

Now, on integration within the limits of $=0 and $=2π, the terms containing this variable angle disappear; and since A is constant, there results

If the waters of our globe, instead of being confined to the vicinity of the equator, were made to occupy a number of regular channels ranging with the parallels of latitude, and if br denote the breadth of one of these channels, O the polar distance of its middle part, while the notation already given is retained for the remaining items, the tidal swelling of the fluid confined under the given parallel will have its tangential action on the moon expressed by

3πk2gbr3 sin3 O
2D4

This may be easily found by using in the foregoing investigation r sin O, the radius of the parallel of latitude, instead of r the radius at the equator. Now instead of the complicated case which our terraqueous world presents, I shall, like most writers on the tides, take an equivalent one in which the entire globe is supposed to be covered with water having a depth equal in all places, or varying regularly with the latitude according to some obvious law. We may suppose this hypothetical ocean divided into watery zones or canals by partitions parallel to the equator; as the number of these divisions become infinite, the breadth of each will be represented by rdO, and its tangential force on the moon by dF. From formula (10) there is thus obtained

dF =

3πk2gr sin3 Odo

2D4

(h sin 2A+h' cos 2A). (11)

The angle A may without much error be regarded as constant for all latitudes; but it is proper to consider the greatest height of the tide-wave as depending on the distance from the equator; and supposing it proportional to the cosine of the latitude, we must substitute for h and h' in the last equation h sin O and h' sin O. Making this substitution and integrating within the limits of O=0 and O=90°, we obtain

Some idea of the small effect of the tides on lunar motion may be derived from a numerical estimate of the value of F in the last equation, supposing the angle A to be 45°, and the greatest height of the tides equal to three feet. For this purpose it will be most convenient to take for the unit of attracting matter a

sphere equal to the earth in size and having the density of water; k would then be equal to r; and according to the most recent estimates of the earth's density g would indicate a velocity of five 2.4 feet a second generated in a second of time. Now is nearly

ᎠᏎ

equal to 12960000, and the product 2kh sin 2A will represent a quantity of matter which, with the unit of measure I have assumed, will be 3000000 The velocity due to the force F in a second of time will be expressed by the following fraction of a foot per second:

27648000000000•

This insignificant force acting on the moon for three millions of years would change her velocity a little more than 1 per cent.; and through the indirect influence on her orbit an increase of about 3 per cent. would be then occasioned in her period of revolution around our planet. If in this estimate I have assigned too low a value for the height of the equatorial tides, there is an ample compensation for the error by giving to the angle A the value necessary for producing a maximum effect.

The relation already exhibited between the change in lunar and terrestrial motion may be also deduced by investigating the loss in the earth's rotation from the reciprocal attraction of the moon on the protuberant tidal waters in a channel either coincident with the equator or parallel to it. To arrive at an approximate estimate of this loss, in the first case the tangential force proceeding from lunar attraction must, with the notation already 3k2gmr sin 2 (A-6) used, be expressed by ; and the momentary

2D3

decrease in the momentum of terrestrial matter from the action of this force on the small portion of the fluid represented by bryde will be

3k gbyr m sin 2(A-)do dt

(13)

On making the substitutions already employed for y and k2gh, the formula becomes

Reducing and integrating with reference to do, taking the limits of 0 and 27, there results

3πChmr2 sin 2Adt 3πCh'mr2 cos 2Adt

2D3

+

2D3

(15)

This is the loss of terrestrial momentum in the instant of time dt; and the loss must fall on the water alone if its movements were wholly unimpeded by friction. It thus appears that the waste of motive force is sixty times as great to the earth as to the moon, as may be seen by comparing the last formula with equation (9), after multiplying the latter by m. The same conformity to the law for the preservation of areas may be shown for zones of water parallel to the equator in every latitude.

While the enlargement of the moon's orbit through tidal influence converts her apparent gain into an actual loss of velocity, a corresponding result of indirect action would be exhibited in our liquid domain if no friction retarded its movements. Were the terrestrial waters confined to regular channels ranging with the equator or the parallels of latitude, the constant loss of motion would serve to increase the gravity and the pressure of the fluid. But if an ocean of uniform depth covered the entire earth, and if its bottom were perfectly smooth, its waters, though losing some velocity by tidal movements, would have their velocity of rotation increased by retiring towards the polar regions as the centrifugal force declined. In the aqueous envelope of the earth there would accordingly be a gain of momentum, while a loss occurred to the moon on a corresponding scale and from the same cause. But the result is much modified by friction, which makes the oceanic waters partake of the velocity of our planet, and occasions a consumption of motion proportional to the calorific energy of the tides.

The effects of the impediments to the great movement of our seas may be readily understood from what is known to attend the collision of imperfectly elastic bodies. If a large meteorite moving from west to east directly over the equator, and having a circular orbit coincident with the verge of our atmosphere, were to have its planetary career arrested by striking a very high mountain, the collision would occasion no loss of momentum; for whatever the body parted with must be gained by the earth; but the sum of the living forces which the earth and the meteorite possess, and which are measured by the masses multiplied by the square of the velocities, would be diminished in proportion to the amount of heat developed as the meteorite struck the mountain or incorporated with our planet in any other way. There is a similar destruction of living force and a corresponding development of heat from the rolling of the vast bodies of water over the asperities in the bed of the ocean; and motion is ever annihilated in giving birth to calorific energy. Yet nothwithstanding the effects of friction, much of the velocity which the moon gives the liquid domain is retained for some time and exhibited in the production of oceanic currents; but as the force

of these currents is called into requisition for the works of nature or of art, and the water is made to partake of the velocity of the bed on which it rests, the store of force in our planet must be wasted and the length of our day augmented.

A more definite relation between the destruction of force in this manner and the consequent change in planetary motion may be shown by investigating the extent to which a satellite revolving close to its primary has its orbit altered by tidal action arising from the eccentricity of the ellipse which it describes, supposing the rotation adjusted for keeping the same side always turned to the central body. To this problem other solutions may be given besides that which I presented in the Philosophical Magazine for December 1851. To seek for evidence of the correlation of forces by physical inquiries of cases hitherto untried or relating to phenomena presented in distant systems is as legitimate as the course pursued by Newton and his followers, who applied all the vast resources of mathematics, not for calculating the course of projectiles near the earth's surface, but for determining the orbits which solar attraction would give bodies moving with immense velocities through the distant realms of space.

Cincinnati, January 21, 1869.

[To be continued.]

XXX. The Story of an Equation in Differences of the Second Order. By J. J. SYLVESTER*.

MY

Y recent researches into the order of the various systems of equations which serve to determine the forms of reducible cyclodes have led me to notice an equation in the second order of differences which I imagine is new, and possesses a peculiarly interesting complete integral.

If we call

and (i, j, k,... &c. being given) determine a, b, c,... &c. so as to make (fx)+(f'x)2 a complete square, and if we suppose the indices i, j, k,... to consist of A integers of one value, μ integers of a second value, v of a third, and so on, the number of solutions of the problem will in general depend not on i, j, k, but on the derived integers λ, u, v,...; and we may denote the maximum value of this number by the type [, u, v, . . .]†. . * Communicated by the Author.

+ Ex. gr. if

ƒx = (x2 — a2)' (x2 — b2ƒ3 (x2 — c2)1(x2 — d2)',

the type is [1, 1, 1, 1], of which the maximum value is 9; but if the sum Phil. Mag. S. 4. Vol. 37. No. 248. Mar. 1869.

Q