ture is put the better; but the farther back the time of the heating, the hotter it must have been. The best for those who draw most largely on time is that which puts it farthest back, and that is the theory that the heating was enough to melt the whole. But even if it was enough to melt the whole, we must still admit some limit, such as fifty million years, one hundred million years, or two or three hundred million years ago.1 Beyond that we cannot go. The argument described (§ 19) above regarding the earth's rotation shows that the earth has not gone on as at present for a thousand million years. Dynamical theory of the sun's heat renders it almost impossible that the earth's surface has been illuminated by the sun many times ten million years. And when finally we consider underground temperature we find ourselves driven to the conclusion in every way, that the existing state of things on the earth, life on the earth, all geological history showing continuity of life, must be limited within some such period of past time as one hundred million years.

1 Thomson and Tait, Appendix D. § (r.)

APPENDIX.

ON THE

OBSERVATIONS AND CALCULATIONS REQUIRED TO FIND THE TIDAL RETARDATION OF THE EARTH'S ROTATION.1

THE first publication of any definite estimate of the possible amount of the diminution of rotatory velocity experienced by the earth through tidal friction is due, I believe, to Kant. It is founded on calculating the moment round the earth's centre of the attraction of the moon, on a regular spheroidal shell of water symmetrical about its longest axis, this being (through the influence of fluid friction) kept in a position inclined backwards at an acute angle to the line from the earth's centre to the moon. One of the simplest ways of seeing the result is this:-First, by the known conclusions as to the attractions of ellipsoids, or still more easily by the consideration of the proper "spherical harmonic "2 (or Laplace's coefficient) of the second degree, we see that an equipotential surface lying close to the bounding

1 From the Rede Lecture, Cambridge, May 23, 1866, “On the Dissipation of Energy."

2 Thomson and Tait's Natural Philosophy, § 536 (4).

F

surface of a nearly spherical homogeneous solid ellipsoid is approximately an ellipsoid with axes differing from one another by three-fifths of the amounts of the differences of the corresponding axes of the ellipsoidal boundary. Now it is known1 that a homogeneous prolate spheroid of revolution attracts points outside it approximately as if its mass were collected in a uniform bar having its ends in the foci of the equipotential spheroid. If, for example, a globe of water of 21,000,000 feet radius (this being nearly enough the earth's radius) be altered into a prolate spheroid with longest radii exceeding the shortest radii by two feet, the equipotential spheroid will have longest and shortest radii differing by of a foot. The foci of this latter will be at 7,100 feet on each side of the centre; and therefore the resultant of gravitation between the supposed spheroid of water and external bodies will be the same as if its whole mass were collected in a uniform bar of 14,200 feet length. But by a well-known proposition, a uniform line FF' (a diagram is unnecessary) attracts a point M in the line M K bisecting the angle F M F'. Let CQ be a perpendicular from C, the middle point of F'F, to this bisecting line M K. If C M be 60 × 21 × 106 (the moon's distance), and if the angle F C M be 45° we find, by elementary geometry, CQ='02 of a foot (about

2

1 Thomson and Tait's Natural Philosophy, § 501 and § 480 (e). 2 Ibid., § 480 (b) and (a).

inch). The mass of a globe of water equal in bulk to the earth is 97 x 1021 tons. And, the moon's mass being about

attraction of the moon on

of the earth's, the

a ton at the earth's

80 602 290,000

of a ton force, if, for

brevity, we call a ton force the ordinary terrestrial weight of a ton-that is to say, the amount of the earth's attraction on a ton at its surface. Hence the whole force of the moon on a globe of water equal in bulk to the earth is '97 × 1021

290,000

or 3.3 × 1015 tons force. If, then, the tidal disturbance were exactly what we have supposed, or if it were (however irregular) such as to have the same resultant effect, the retarding influence of the moon's attraction would be that of 33 × 1015 tons force acting in the plane of the equator and in a line passing the centre at of a foot distance. Or it would be the same as a simple frictional resistance (as of a friction-brake) consisting of 33 × 1015 tons force acting tangentially against the motion of a pivot or axle of about inch diameter. To estimate the retardation produced by this, we shall suppose the square of the earth's radius of gyration,

50

1 In stating large masses, if English measures are used at all, the ton is convenient, because it is 1000 kilogrammes nearly enough for many practical purposes and rough estimates. It is 1016 047 kilogrammes; so that a ton diminished by about 16 per cent. would be just 1000 kilogrammes.

instead of being, as it would be if the mass were homogeneous, to be of the square of the radius of figure, as it is made to be, by Laplace's probable law of the increasing density inwards, and by the amount of precession calculated on the supposition that the earth is quite rigid. Hence (if we take g=322 feet per second generated per second, and the earth's mass =5'3 × 1021 tons) the loss of angular velocity per second, on the other suppositions we have made, will be

The loss of angular velocity in a century would be 31 × 108 times this, or 8.5 x 10-12, which is

107 86400

the present angular vel

ocity. Thus in a century the earth would be rotating so much slower that, regarded as a timekeeper, it would lose about 116 seconds in ten million, or 3.6 seconds in a year. And the accumulation of effect of uniform retardation at that rate would throw the earth as a time-keeper behind a perfect chronometer (set to agree with it in rate and absolute indication at any time) by 180 seconds at the end of a century, 720 seconds at the end of two centuries, and so on. In the present very imperfect state of clock-making (which scarcely produces an astronomical clock two or three times. more accurate than a marine chronometer or good