depending on these two combined relations. The one law from which there is no escape, let distance and velocity change as they may during the Earth's circuit around the Sun, is that her period of revolution continues unchangeable, and therefore her mean distance also.*

We may then take the Earth's mean distance as measuring her freedom from complete solar control— complete control being understood to be such an overwhelming influence on the Earth as would force her to fall directly upon the Sun. And while the Earth's mean distance thus measures her partial freedom, the shortness of the period in which she completes her circuit measures the amount of the Sun's power over her. Her mass may be regarded as having nothing to do with either relation. Increase of mass, so far as it would be effective at all, would tend to increase the strength of the bond uniting the Earth and the Sun, and to diminish the period of the Earth's circuit. But remembering that the Earth's mass is a very minute fraction of the Sun's, it may be disregarded wholly. A body no larger and heavier than a peppercorn, if projected with the same velocity and on the same course as the Earth, would continue to travel in precisely the same path and period around the Sun.

Or we may say in preference, that the one law from which there is no escape, is the law connecting the Earth's velocity v at any time, with her distance R from the Sun at that time, and a certain fixed quantity a which is her mean distance. This law is thus expressed mathematically: where is the accelerating force of the Sun at the unit

-

R A

of distance.

We have next a law for our guidance which is of a very remarkable character, and the recognition of which will be found to throw a most important light on all the relations of the planetary scheme. It is this: Given the distance of a body from the Sun, and the velocity with which the body is travelling, then-let the course of the body be what it may (so long only as it does not bring the body into actual contact with the Sun), the period of the body's revolution is assigned.

The Earth's greatest velocity and her least correspond therefore-as truly as her mean velocity-with her period; and further, we need not trouble ourselves about the direction of her motion at any time, for if this direction were altered by the action of some external force, while yet her velocity remained unchanged, she would continue to travel in the same periodic time around the Sun, and at the same mean distance.

So that if we take the Earth's greatest velocity when she is in perihelion (18.5 miles per second), we have the velocity which is necessary in order that a body about 90,000,000 miles from the Sun may travel once in a year, or at a mean distance of 91,500,000 miles, round the Sun; and if we take her least velocity (17.9 miles per second) when she is in aphelion, we have the velocity which is necessary in order that a body about 93,000,000 miles from the Sun may travel once in a year round the Sun; while the Earth's mean velocity (18-2 miles per second) at her mean distance is the velocity necessary in order that a body at that distance may have a period of one year.

We will now take only the mean distance and the mean velocity. We see that at a distance of 91,500,000 miles a body requires a velocity of 18.2 miles per second if it is to have a mean distance of the same amount. So that clearly if it were projected square to a line from the Sun it would never change its distance; for it would have exactly the right velocity and exactly the right direction for travelling in a circle around the Sun. The Earth when at this distance, though travelling with the right velocity for the required mean distance, is not travelling square to a line from the Sun, and so does not travel in a circle. around him. But we have learned from her motion what is the just rate at which a body should be projected so as to travel in a circle round the Sun at a distance of 91,500,000 miles.

Now how much must this velocity be increased in order to enable a body at this distance of 91,500,000 miles to pass wholly from the control of the Sun? If we can determine this we shall have determined the limits of the Sun's influence at this particular distance. Over bodies moving with a velocity below that limiting velocity he is completely master; let them travel onwards as they may, increasing their distance from him more and more, there is yet a limit to this increase. Their absolute velocity will become less and less; it will become at last such, that if their direction of motion were but changed they would thenceforth continue to describe a circle around the Sun; still it will go on diminishing, until at length they reach their

extreme range of distance, after which they will be brought back through all the orders of distance they have passed through, and finally return to the place they started from, to pursue the same round for ever. But if their velocity do but equal or exceed the limit we are dealing with, they will travel onwards-with ever-diminishing velocity, it is true, but still-with ever-increasing distance, for ever.

It might be supposed that a very great increase of velocity would be required in order that the Earth should be thus (unfortunately for her inhabitants) released from the Sun's service and sent to wander freely through space. But in reality it would not even be necessary that her velocity should be doubled; an increase by one-half would be more than sufficient to free the Earth for ever from enforced periodic revolution around the centre of our planetary scheme. The exact proportion of increase necessary to effect this is represented by the proportion in which the diagonal of a square exceeds the side,* which we know to be repre

* The following simple formula conveniently expresses the relation between the mean distances a and a' of bodies which at a distance r from the Sun are travelling with the velocities v and respectively:

a' (2a—r): a (2a'—r):: v2 : v'2.

Now supposing the velocity v to be such that a circle would be described about the centre of motion, if the body travelled square to the line from that centre, then obviously a is equal to r; so that the above relation becomes

and therefore

a': 2a-r::v2 : v2;

a': r::v2: 2v2-v'2.

Now clearly if we increase until 2 is equal to 202, we make the fourth term gradually diminish until it vanishes; so that the third at last

sented numerically by the proportion in which 1414 (pretty nearly) exceeds 1000.

bears an infinitely great proportion to the fourth. But in this case the first will bear an infinitely great proportion to the second: in other words, a' will be infinitely great. Hence the period of the body will become also infinite (by Kepler's third law): or in other words the body will never return to its starting-place. Therefore, it follows that if be the velocity with which a body will describe a circle at a distance from the Sun, the greatest possible velocity which a body can have at a distance r from the Sun, so as to travel on a closed orbit around him, is v√2; and if a body is observed to travel with any greater velocity at such a distance we know with certainty that that body has entered the domain of the Sun, with a velocity imparted to it by extra-solar influences.

If in the above proportion v2 is greater than 2v2, we see that a' must have a negative value. This means that the centre of the path described by the body lies in the direction contrary to that in which the Sun lies. And knowing that the path must needs be a conic section with the Sun in a focus, it follows that the path of the body must be hyperbolic, the Sun lying at that focus which is next to the branch traversed by the body.

Of course in this case as in the former the body will never return. But there is a noteworthy distinction between the two cases. When the axis of the orbit is infinite, the body describes a parabola, and if it could be traced from the Sun as it approached from an indefinitely great distance to its nearest point and then passed away again to an infinitely great distance, the point to which it seemed to pass away would be precisely the same as that from which it seemed to come, and these coincident points would lie directly opposite the point of nearest approach. (This is obvious, because if lines be drawn from the focus of the parabola to two points of the curve equally and enormously removed from the vertex, these lines will enclose an indefinitely small angle, and will approach indefinitely near to coincidence with the axis.) On the other hand, a body approaching and then passing away on a hyperbolic orbit will seem to come from one point of the heavens and to pass away to a different point, the bisection of the celestial arc between these points lying directly opposite the point at which the body makes its nearest approach; and for a given distance of this nearest point, the arc separating the two former will be the greater as the velocity of the body when at its nearest is greater; becoming equal to two right angles when this velocity is infinitely great.