# Proposition 51, Theorem 39

##### 4 minutes • 849 words

## Table of contents

## HYPOTHESIS

The resistance from the lack of lubricity in a fluid is proportional to the speed of the separation of the fluid’s parts.

### PROPOSITION 51, THEOREM 39

Assume an infinitely long solid cylinder `AFL`

in a uniform and infinite fluid.

- Rotate it with a uniform motion around its axis
`S`

.

The fluid is forced around by the cylinder.

- Every part of the fluid perseveres uniformly in its motion.
- The timespan of the parts of the fluid are as their distances from, the axis of the cylinder.

The concentric circles `BGM`

, `CHN`

, `DIO`

, `EKP`

, etc., divide the fluid into innumerable concentric cylindric solid orbs of the same thickness.

The fluid is homogeneous. And so, the impressions which the contiguous orbs make upon each other mutually will be (by the Hypothesis) as their translations from each other, and as the contiguous superficies upon which the impressions are made.

If the impression made upon any orb be greater or less on its concave than on its convex side, the stronger impression will prevail, and will either accelerate or retard the motion of the orb, according as it agrees with, or is contrary to, the motion of the same.

Therefore, that every orb may persevere uniformly in its motion, the impressions made on both sides must be equal and their directions contrary.

Therefore since the impressions are as the contiguous superficies, and as their translations from one another, the translations will be inversely as the superficies, that is, inversely u the distances of the superficies from the axis.

But the differences of the angular motions about the axis are as those translations applied to the distances, or as the translations directly and the distances inversely; that is, joining these ratios together, as the squares of the distances inversely.

Therefore if there be erected the lines `Aa, Bb, Cc, Dd, Ee`

, etc., perpendicular to the several parts of (he infinite right line SABCDEQ, and reciprocally proportional to the squares of SA, SB, SO, SD, SE, etc., and through the extremities of those perpendiculars there be supposed to pass an hyperbolic curve, the sums of the differences, that is, the whole angular motions, will be as the correspondent sums of the lines Aa, Bb, Cc, Dd, Ee, that is (if to constitute a medium uniformly fluid the number of the orbs be increased and their breadth diminished in inflnitum), as the hyperbolic areas AaQ., BAQ, CcQ, DdQ, EeQ, etc., analogous to the sums; and the times, reciprocally proportional to the angular motions, will be also reciprocally proportional to those areas.

Therefore the periodic time of any particle as D, is reciprocally as the area DdQ, that is (as appears from the known methods of quadratures of curves), directly as the distance SD. Q.E.D.

**Corollary 1**

Hence the angular motions of the particles of the fluid are reciprocally as their distances from the axis of the cylinder, and the absolute velocities are equal.

**Corollary 2 **

If a fluid be contained in a cylindric vessel of an infinite length, and contain another cylinder within, and both the cylinders revolve about one common axis, and the times of their revolutions be as their semi-diameters, and every part of the fluid perseveres in its motion, the periodic times of the several parts will be as the distances from the axis of the cylinders.

**Corollary 3 **

If there be added or taken away any common quantity of angular motion from the cylinder and fluid moving in this manner; yet because this new motion will not alter the mutual attrition of the parts of the fluid, the motion of the parts among themselves will not be changed; for the translations of the parts from one another depend upon the attrition. Any part will persevere in that motion, which, by the attrition made on both sides with contrary directions, is no more accelerated than it is retarded.

**Corollary 4 **

Therefore if there be taken away from this whole system of the cylinders and the fluid all the angular motion of the outward cylinder, we shall have the motion of the fluid in a quiescent cylinder.

**Corollary 5 **

Therefore if the fluid and outward cylinder are at rest, and the inward cylinder revolve uniformly, there will be communicated a circular motion to the fluid, which will be propagated by degrees through the whole fluid; and will go on continually increasing, till such time as the several parts of the fluid acquire the motion determined in Cor. 4.

**Corollary 6 **

The fluid tries to propagate its motion still farther. Its impulse will carry the outmost cylinder also about with it, unless the cylinder be violently detained; and accelerate its motion till the periodic times of both cylinders become equal among themselves.

But if the outward cylinder be violently detained, it will make an effort to retard the motion of the fluid; and unless the inward cylinder preserve that motion bv means of some external force impressed thereon, it will make it cease by degrees.

All these things will be found true by making the experiment in deep standing water.