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Having these curves in mind, let us imagine the earth to leave us hanging in space at some point of its orbit, our planet pursuing its course without us, until, at the end of a year, it returns to pick us up again. During the interval of its absence we, hanging in midspace, amuse ourselves by firing off balls to perform their revolutions around the sun like little planets. The result will be that all the balls we send off with a velocity less than that of the earth, that is to say, less than eighteen and six tenths miles per second, will move around the sun in closed orbits, smaller than the orbit of the earth, no matter what direction we send them in. A very simple and curious law is that these orbits will always have the same period if the velocity is the same. All the balls sent with the velocity of the earth will be one year in making their revolution and will, therefore, come together, at the point from which they started, at the same moment. If the velocity exceeds eighteen and six tenths miles a second, the orbit will be larger than that of the earth and the period of revolution will be longer the greater the velocity. With a speed exceeding about twenty-six miles a second, the attraction of the sun could never hold in the ball, which would fly away for good in one of the branches of a hyperbola. This would happen no matter in what direction we threw the object. There is, therefore, at every distance from the sun, a certain limiting velocity which, if a comet exceeds, it will fly off from the sun never to return; while, if it falls short, it will be sure to get back at some time.

The nearer we are to the sun, the greater is this limit-