Corollaries

COROLLARY 1

Hence:

• the angular motions of the parts of the fluid around the axis of the globe are reciprocally as the squares of the distances from the globe’s centre.
• the absolute velocities are reciprocally as the same squares applied to the distances from the axis.

COROLLARY 2

Assume that a globe revolves with a uniform motion around an axis in a similar and infinite quiescent fluid with a uniform motion.

It will communicate a whirling motion to the fluid like that of a vortex.

• That motion will by degrees be propagated onward infinitely.

This motion will be increased continually in every part of the fluid until the periodical times of the several parts become as the squares of the distances from the centre of the globe.

COROLLARY 3

The inward parts of the vortex are continually pressing on and driving forward the external parts because of their greater speed.

Those exterior parts communicate the same quantity of motion to those parts that are still father beyond them.

This preserves the quantity of their motion continually unchanged.

The motion is perpetually transferred from the centre to the circumference of the vortex until it is swallowed up and lost in the boundless extent of that circumference.

This will cause the matter between any two spherical surfaces concentrical to the vortex to never be accelerated.

This is because that matter will be always transferring the motion it receives from the matter nearer the centre to that matter which lies nearer the circumference.

COROLLARY 4

Therefore, in order to continue a vortex in the same state of motion, some active principle is required from which the globe may receive continually the same quantity of motion which it is always communicating to the matter of the vortex.

Without such a principle it will undoubtedly come to pass that the globe and the inward parts of the vortex, being always propagating their motion to the outward parts, and not receiving any new motion, will gradually move slower and slower, and at last be carried round no longer.

COR. 5

If another globe should be swimming in the same vortex at a certain distance from its centre, and in the mean time by some fore e revolve constantly about an axis of a given inclination, the motion of this globe will drive the fluid round after the manner of a vortex; and at first this new and small vortex will revolve with its globe about the centre of the other; and in the mean time its motion will creep on farther and farther, and by degrees be propagated in infinitum, after the manner of the first vortex.

For the same reason that the globe of the new vortex was carried about before by the motion of the other vortex, the globe of this other will be carried about by the motion of this new vortex, so that the two globes will revolve about some intermediate point, and by reason of that circular motion mutually fly from each other, unless some force restrains them. Afterward, if the constantly impressed forces, by which the globes persevere in their motions, should cease, and every thing be left to act according to the laws of mechanics, the motion of the globes will languish by degrees (for the reason assigned in Cor. 3 and 4), and the vortices at last will quite stand still.

COR. 6

If several globes in given places should constantly revolve with determined velocities about axes given in position, there would arise from them as many vortices going on in infinitum.

For upon the same account that any one globe propagates its motion in infinitum, each globe apart will propagate its own motion in infinitum also; so that every part of the infinite fluid will be agitated with a motion resulting from the actions of all the globes.

Therefore the vortices will not be confined by any certain limits, but by degrees run mutually into each other; and by the mutual actions of the vortices on each other, the globes will be perpetually moved from their places, as was shewn in the last Corollary; neither can they possibly keep any certain position among themselves, unless some force restrains them.

But if those forces, which are constantly impressed upon the globes to continue these motions, should cease, the matter (for the reason assigned in Cor. 3 and 4) will gradually stop, and cease to move in vortices.

COR. 7

If a similar fluid be inclosed in a spherical vessel, and, by the uniform rotation of a globe in its centre, is driven round in a. vortex; and the globe and vessel revolve the same way about the same axis, and their periodical times be as the squares of the semi-diameters;

The parts of the fluid will not go on in their motions without acceleration or retardation, till their periodical times are as the squares of their distances from the centre of the vortex. No constitution of a vortex can be permanent but this.

COR. 8

If the vessel, the inclosed fluid, and the globe, retain this motion, and revolve besides with a common angular motion about any given axis, because the mutual attrition of the parts of the fluid is not changed by this motion, the motions of the parts among each other will not be changed; for the translations of the parts among themselves depend upon this attrition.

Any part will persevere in that motion in which its attrition on one side retards it just as much as its attrition on the other side accelerates it.

COR. 9

Therefore if the vessel be quiescent, and the motion of the globe be given, the motion of the fluid will be given. For conceive a plane to pass through the axis of the globe, and to revolve with a contrary motion.

Suppose the sum of the time of this revolution and of the revolution of the globe to be to the time of the revolution of the globe as the square of the semi-diameter of the vessel, to the square of the semi-diameter of the globe; and the periodic times of the parts of the fluid in respect of this plane will be as the squares of their distances from the centre of the globe.

COR. 10

Therefore if the vessel move about the same axis with the globe, or with a given velocity about a different one, the motion of the fluid will be given.

For if from the whole system we take away the angular motion of the vessel, all the motions will remain the same among themselves as before, by Cor. 8, and those motions will be given by Cor. 9.

COR. 11

If the vessel and the fluid are quiescent, and the globe revolves with an uniform motion, that motion will be propagated by degrees through the whole fluid to the vessel, and the vessel will be carried round by it, unless violently detained;

The fluid and the vessel will be continually accelerated till their periodic times become equal to the periodic times of the globe. If the vessel be either withheld by some force, or revolve with any constant and uniform motion, the medium will come by little and little to the state of motion defined in Cor. 8, 9, 10, nor will it ever persevere in any other state.

But if then the forces, by which the globe and vessel revolve with certain motions, should cease, and the whole system be left to act according to the mechanical laws, the vessel and globe, by means of the intervening fluid, will act upon each other, and will continue to propagate their motions through the fluid to each other, till their periodic times become equal among themselves, and the whole system revolves together like one solid body.