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The Mathematical Principles of Natural Philosophy (1729)/Circular motion of fluids

HYPOTHESIS. The resistance, arising from the want of lubricity in the parts of a fluid, is, caeteris paribus, proportional to the velocity with which, the parts of the fluid are separated from each other.

Proposition LI. Theorem XXXVIII. If a solid cylinder infinitely long, in an uniform and infinite fluid, revolve with an uniform motion about an axis given in position, and the fluid be forced round by only this impulse of the cylinder, and every part of the fluid persevere uniformly in its motion; I say, that the periodic times of the parts of the fluid are as their distances from the axis of the cylinder.

Let AFL (Plate 9. Figure 2.) be a cylinder turning uniformly about the axis S, and let the concentric circles BGM, CHN, DIO, EKP, &c. divide the fluid into innumerable concentric cylindric solid orbs of the same same thickness. Then, because the fluid is homogeneous, 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 as 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 ratio's together, as the squares of the distances inversely. Therefore if there be erected the lines Aa, Bb, Cc,Dd,Ee, &c. perpendicular to the several parts of the infinite right line SABCDEQ and reciprocally proportional to the squares of SA, SB, SC, SD, SE, &c and thro' the extremities of these 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 encreased and their breadth diminished in infinitum) as the hyperbolic area's AaQ, BbQ, CcQ, DdQ, EeQ, &c. 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 semidiameters, 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 thro' the whole fluid; and will go on continually encreasing, till such time as the several parts of the fluid acquire the motion determined in corollary 4.

Corollary 6. And because the fluid endeavours 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 by 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.

Proposition LII. Theorem XL. If a solid sphere, in an uniform and infinite fluid, revolves about an axis given in position with an uniform motion, and the fluid be forced round by only this impulse of the sphere; and every part of the fluid perseveres uniformly in its motion: I say, that the periodic times of the parts of the fluid are as the squares of their distances from the centre of the sphere.

Case 1. Let AFL be a sphere turning uniformly about the axis S, and let the concentric circles BGM, CHN, DIO, EKP, &c. divide the fluid into innumerable concentric orbs of the same thickness. Suppose those orbs to be solid; and because the fluid is homogeneous, the impressions which the contiguous orbs make one upon another, will be (by the supposition) as their translations from one another, and the contiguous superficies upon which the impressions are made. If the impression upon any orb be greater or less upon its concave than upon its convex side; the more forcible impression will prevail, and will either accelerate or retard the velocity of the orb, according as it is directed rested with a conspiring or contrary motion to that of the orb. Therefore that every orb may persevere uniformly in its motion, it is necessary that the impressions made upon both sides of the orb should be equal, and have contrary directions. Therefore since the impressions are as the contiguous superficies, and as their translations from one another; the translations will be inversly as the superficies, that is, inversly as the squares of the distances of the superficies from the centre. 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 inversly; that is, by compounding those ratio's, as the cubes of the distances inversly. Therefore, if upon the several parts of the infinite right line SABCDEQ there be erected the perpendiculars Aa, Bb, Cc, Dd, Ec, &c. reciprocally proportional to the cubes of SA, SB, SC, SD, SE, &c. the sums of the differences, that is, the whole angular motions, will be as the corresponding sums of the lines Aa, Bb, Cc, Dd, Ee, &c. that is, (if to constitute an uniformly fluid medium the number of the orbs be encreased and their thickness diminished in infinitum) as the hyperbolic areas AaQ, BbQ, CcQ, DdQ, EeQ, &c. analogous to the sums; and the periodic times being reciprocally proportional to the angular motions, will be also reciprocally proportional to those areas. Therefore the periodic time of any orb DIO is recipfocally as the area DdQ, that is, (by the known methods of quadratures) directly as the square of the distance SD. Which was first to be demonstrated.

Case 2. From the centre of the sphere let there be drawn a great number of indefinite right lines, making given angles with the axis, exceeding one another by (equal differences; and, by these lines revolving about the axis, conceive the orbs to be cut into innumerable annuli: then will every annulus have four annuli contiguous to it, that is, one on its inside, one on its outside, and two on each hand. Now each of these annuli cannot be impelled equally and with contrary directions by the attrition of the interior and exterior annuli unless the motion be communicated according to the law which we demonstrated in case 1. This appears from that demonstration. And therefore any series of annuli, taken in any right line extending itself in infinitum from the globe, will move according to the law of case 1. except we should imagine it hindered by the attrition of the annuli on each side of it. But now in a motion, according to this law, no such attrition is, and therefore cannot be any obstacle to the motion's persevering according to that law. If annuli at equal distances from the centre revolve either more swiftly or more slowly near the poles than near the ecliptic; they will be accelerated if slow, and retarded if swift, by their mutual attrition; and so the periodic times will continually approach to equality, according to the law of case 1. Therefore this attrition will not at all hinder the motion from going on according to the law of case 1. and therefore that law will take place, that is, the periodic times of the several annuli will be as the squares of their distances from the centre of the globe. Which was to be demonstrated in the second place.

Case 3. Let now every annulus be divided by transverse sections into innumerable particles constituting a substance absolutely and uniformly fluid; and because these sections do not at all respect the law of circular motion, but only serve to produce a fluid substance, the law of circular motion will continue the same as before. All the very small annuli will either not at all change their asperity and force of mutual attrition upon account of these sections, or else they will change the same equally. Therefore the proportion of the causes remaining the same, the proportion of the effects will remain the same also; that is, the proportion of the motions and the periodic times. Q.E.D. But now as the circular motion, and the centrifugal force thence arising, is greater at the ecliptic than at the poles, there must be some cause operating to retain the several particles in their circles; otherwise the matter that is at the ecliptic will always recede from the centre, and come round about to the poles by the outside of the vortex, and from thence return by the axis to the ecliptic with a perpetual circulation.

Corollary 1. Hence the angular motions of the parts of the fluid about the axis of the globe, are reciprocally as the squares of the distances from the centre of the globe, and the absolute velocities are reciprocally as the same squares applied to the distances from the axis.

Corollary 2. If a globe revolve with a uniform motion about an axis of a given position in a similar and infinite quiescent fluid with an uniform motion, it will communicate a whirling motion to the fluid like that of a vortex, and that motion will by degrees be propagated onwards in infinitum; and this motion will be encreased continually in every part of the fluid, till the periodical times of the several parts become as the squares of the distances from the centre of the globe.

Corollary 3. Because the inward parts of the vortex are by reason of their greater velocity continually pressing upon and driving forwards the external parts, and by that action are perpetually communicating motion to them, and at the same time those exterior parts communicate the same quantity of motion to those that lie still beyond them, and by this action preserve the quantity of their motion continually unchanged; it is plain that the motion is perpetually transferred from the centre to the circumference of the vortex, till it is quite swallowed up and lost in the boundless extent of that circumference. The matter between any two sphærical superficies concentrical to the vortex will never be accelerated; 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.

Corollary 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 force 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. And 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. Afterwards, 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 corollary 3 and 4.) and the vortices at last will quite stand still.

Corollary 6. Is 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 corollary 3 and 4.) will gradually stop, and cease to move in vortices.

Corollary 7. If a similar fluid be inclosed in a sphærical 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 semidiameters; 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.

Corollary 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.

Corollary 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 thro' the axis of the globe, and to revolve with a contrary motion; and 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 semidiameter of the vessel to the square of the semidiameter 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.

Corollary 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 corollary 8. and those motions will be given by corollary. 9.

Corollary 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; and 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 corollary 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.

Scholium.

In all these reasonings, I suppose the fluid to consist of matter of uniform density and fluidity. I mean that the fluid is such, that a globe placed any where therein may propagate with the same motion of its own, at distances from it self continually equal, similar and equal motions in the fluid, in the same interval of time. The matter by its circular motion endeavours to recede from the axis of the vortex; and therefore presses all the matter that lies beyond. This pressure makes the attrition greater, and the separation of the parts more difficult; and by consequence diminishes the fluidity of the matter. Again, if the parts of the fluid are in any one place denser or larger than in the others, the fluidity will be less in that place, because there are fewer superficies where the parts can be separated from each other. In these cases I suppose the defect of the fluidity to be supplied by the smoothness or softness of the parts, or some other condition; otherwise the matter where it is less fluid, will cohere more, and be more sluggish, and therefore will receive the motion more slowly, and propagate it farther than agrees with the ratio above assigned. If the vessel be not sphærical, the particles will move in lines, not circular, but answering to the figure of the vessel, and the periodic times will be nearly as the squares of the mean distances from the centre. In the parts between the centre and the circumference, the motions will be slower where the spaces are wide, and swifter where narrow; but yet the particles will not tend to the circumference at all the more for their greater swiftness. For they then describe arcs of less curvity, and the conatus of receding from the centre is as much diminished by the diminution of this curvature, as it is augmented by the increase of the velocity. As they go out of narrow into wide spaces they recede a little farther from the centre, but in doing so are retarded; and when they come out of wide into narrow spaces they are again accelerated; and so each particle is retarded and accelerated by turns for ever. These things will come to pass in a rigid vessel. For the state of vortices in an infinite fluid is known by corollary 6. of this proposition.

I have endeavoured in this proposition to investigate the properties of vortices, that I might find whether the celestial phænomena can be explained by them. For the phænomenon is this, that the periodic times of the Planets revolving about Jupiter, are in the sesquiplicate ratio of their distances from Jupiter's centre; and the same rule obtains also among the Planets that revolve about the Sun. And these rules obtain also with the greatest accuracy, as far as has been yet discovered by astronomical observation. Therefore, if those Planets are carried round in vortices revolving about Jupiter and the Sun, the vortices must revolve according to that law. But here we found the periodic times of the parts of the vortex to be in the duplicate ratio of the distances from the centre of motion; and this ratio cannot be diminished and reduced to the sesquiplicate, unless either the matter of the vortex be more fluid, the farther it is from the centre, or the resistance arising from the want of lubricity in the parts of the fluid, should, as the velocity with which the part of the fluid are separated goes on increasing, be augmentted with it in a greater ratio than that in which the velocity increases. But neither of these suppositions seem reasonable. The more gross and less fluid pairs will tend to the circumference, unless they are heavy towards the centre. And tho', for the sake of demonstration, I proposed, at the beginning of this Section, an hypothesis that the resistance is proportional to the velocity, nevertheless, 'tis in truth probable that the resistance is in a less ratio than that of the velocity. Which granted, the periodic times of the parts of the vortex will be in a greater than the duplicate, ratio of the distances from its centre. If, as some think, the vortices move more swiftly near the centre, then slower to a certain limit, then again swifter near the circumference, certainly neither the sesquiplicate, nor any other certain and determinate ratio can obtain in them. Let philosophers then see how that phænomenon of the sefquiplicate ratio can be accounted for by vortices.

Proposition LIII. Theorem XLI. Bodies, carried about in a vortex and returning in the same orb, are of the same density with the vortex, and are moved according to the same law with the parts of the vortex, as to velocity and direction of motion.

For if any small part of the vortex, whose particles or physical points preserve a given situation among each other, be supposed to be congealed; this particle will move according to the same law as before, since no change is made either in its density, vis insita, or figure. And again, if a congealed or solid part of the vortex be of the same density with the rest of the vortex, and be resolved into a fluid, this will move according to the same law as before, except in so far as its particles now become fluid, may be moved among themselves. Neglect therefore the motion of the particles among themselves, as not at all concerning the progressive motion of the whole, and the motion of the whole will be the same as before. But this motion will be the same with the motion of other parts of the vortex at equal distances from the centre; because the solid, now resolved into a fluid, is become perfectly like to the other parts of the vortex. Therefore a solid, if it be of the same density with the matter of the vortex, will move with the same motion as the parts thereof, being relatively at rest in the matter that surrounds it. If it be more dense, it will endeavour more than before to recede from the centre; and therefore overcoming that force of the vortex, by which, being as it were kept in equilibrio, it was retained in its orbit, it will recede from the centre, and in its revolution describe a spiral, returning no longer into the same orbit. And by the same argument, if it be more rare it will approach to the centre. Therefore it can never continually go round in the same orbit, unless it be of the same density with the fluid. But we have shewn in that case, that it would revolve according to the same law with those parts of the fluid that are at the same or equal distances from the centre of the vortex.

Corollary 1. Therefore a solid revolving in a vortex, and continually going round in the same orbit, is relatively quiescent in the fluid that carries it.

Corollary 2. And if the vortex be of an uniform density, the same body may revolve at any distance from the centre of the vortex.

Scholium.

Hence it is manifest, that the Planets are not carried round in corporeal vortices. For according to the Copernican hypothesis, the Planets going round the Sun, revolve in ellipses, having the Sun in their common focus; and by radii drawn to the fun describe areas proportional to the times. But now the parts of a vortex can never revolve with such a motion. Let AD, BE, CF, (Plate 9. Figure 3.) represent three orbits described about the Sun S, of which let the utmost circle CF be concentric to the Sun; and let the aphelia of the two innermost be A, B; and their perihelia D, E. Therefore a body revolving in the orb CF, describing, by a radius drawn to the Sun, areas proportional to the times, will move with an uniform motion. And according to the laws of astronomy, the body revolving in the orb BE will move slower in its aphelion B, and swifter in its perihelion E; whereas, according to the laws of mechanics, the matter of the vortex ought to move more swiftly in the narrow space between A and C, than in the wide space between D and F; that is, more swiftly in the aphelion than in the perihelion. Now these two conclusions contradict each other. So at the beginning of the sign of Virgo, where the aphelion of Mars is at present, the distance between the orbits of Mars and Venus is to the distance between the same orbits at the beginning of the sign of Pisces, as about 3 to 2; and therefore the matter of the vortex between those orbits ought to be swifter at the beginning of Pisces, than at the beginning of Virgo, in the ratio of 3 to 2. For the narrower the space is, thro' which the same quantity of matter passes in the same time of one revolution, the greater will be the velocity with which it passes thro' it. Therefore if the Earth being relatively at rest in this celestial matter should be carried round by it, and revolve together with it about the Sun, the velocity of the Earth at the beginning of Pisces would be to its velocity at the beginning of Virgo in a sesquialteral ratio. Therefore the Sun's apparent diurnal motion at the beginning of Virgo, ought to be above 70 minutes; and at the beginning of Pisces less than 48 minutes. Whereas on the contrary that apparent motion of the Sun is really greater at the beginning of Pisces than at the beginning of Virgo, as experience testifies; and therefore the earth is swifter at the beginning of Virgo than at the beginning of Pisces. So that the hypothesis of vortices is utterly irreconcileable with astronomical phænomena, and rather serves to perplex than explain the heavenly motions. How these motions are performed in free spaces without vortices, may be understood by the first book,; and I shall now more fully treat of it in the following book of the System of the World.


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The Mathematical Principles of Natural Philosophy (1729)/Section 8

Proposition XLI. Theorem XXXII. A pressure is not propagated thro' a fluid in rectilinear directions, unless where the particles of the fluid lie in a right line. Plate 8. Fig. 1.

If the particles a, b, c, d, e, lie in a right line, the pressure may be indeed directly propagated from a to e; but then the particle e will urge the obliquely posited particles f and g obliquely, and those particles f and g will not sustain this pressure, unless they be supported by the particles h and k lying beyond them; but the particles that support them, are also pressed by them; and those particles cannot sustain that pressure, without being supported by, and pressing upon those particles that lie still farther, as l and m, and so on in infinitum. Therefore the pressure, as soon as it is propagated to particles that lie out of right lines, begins to deflect towards one hand and t'other, and will be propagated obliquely in infinitum; and after it has begun to be propagated obliquely, if it reaches more distant particles lying out of the right line, it will deflect again on each hand; and this it will do as often often as it lights on particles that do not lie exactly in a right line. Q.E.D.

Corollary. If any part of a pressure, propagated thro' a fluid from a given point, be intercepted by any obstacle; the remaining part, which is not intercepted, will deflect into the spaces behind the obstacle. This maybe demonstrated also after the following manner. Let a pressure be propagated from the point A (Plate 8. Figure 2.) towards any part, and, if it be possible, in rectilinear directions; and the obstacle NBCK being perforated in BC, let all the pressure be intercepted but the coniform part APQ passing thro' the circular hole BC. Let the cone APQ be divided into frustums by the transverse planes de, fg, hi. Then while the cone ABC, propagating the pressure, urges the conic frustum degf beyond it on the superficies de, and this frustum urges the next frustum fgih on the superficies fg, and that frustum urges a third frustum, and so in infinitum it is manifest (by the third law) that the first frustum defg is, by the reaction of the second frustum fgbi, as much urged and pressed on the superficies fg, as it urges and presses that second frustum. Therefore the frustum degf is compressed on both sides, that is, between the cone Ade and the frustum fhig; and therefore (by case 6. proposition 19.) cannot preserve its figure, unless it be compressed with the same force on all sides. Therefore with the same force with which it is pressed on the superficies de, fg, it will endeavour to break forth at the sides df, eg; and there (being not in the least tenacious or hard, but perfectly fluid) it will run out, expanding itself, unless there be an ambient fluid opposing that endeavour. Therefore, by the effort it makes to run out, it will press the ambient fluid, at its sides df, eg, with the same force that it does the frustum fghi; and therefore the pressure will be propagated as much from the sides df, eg into the spaces NO, KL this way and that way, as it is propagated from the superficies fg towards PQ. Q.E.D.

Proposition XLII. Theorem XXXIII. All motion propagated thro' a fluid, diverges from a rectilinear progress into the unmoved spaces. Plate 8. Figure 3.

Case 1. Let a motion be propagated from the point A thro' the hole BC, and, if it be possible, let it proceed in the conic space BCQP according to right lines diverging from the point A. And let us first suppose this motion to be that of waves in the surface of standing water; and let de, fg, hi, kj, &c. be the tops of the several waves, divided from each other by as many intermediate valleys or hollows. Then, because the water in the ridges of the waves is higher than in the unmoved parts of the fluid KL, NO, it will run down from off the tops of those ridges e, g, i, l, &c. d, f, h, k, &c. this way and that way towards KL and NO; and because the water is more depressed in the hollows of the waves than in the unmoved parts of the fluid KL, NO, it will run down into those hollows out of those unmoved parts. By the first deflux the ridges of the waves will dilate themselves this way and that way, and be propagated towards KL and NO. And because the motion of the waves from A towards PQ is carried on by a continual deflux from the ridges of the waves into the hollows next to them; and therefore cannot be swifter than in proportion to the celerity of the descent; and the descent of the water on each side towards KL and NO must be performed with the same velocity; it follows, that the dilatation of the waves on each side towards KL and NO will be propagated with the same velocity as the waves themselves go forward with, directly from A to PQ. Artel therefore the whole space this way and that way towards KL and NO will be filled by the dilated waves rfgr, shis, tklt, vmnv, &c. Q.E.D. That these things are so, any one may find by making the experiment in still water.

Case 2. Let us suppose that de, fg, hi, kj, mn, represent pulses successively propagated from the point A thro' an elastic medium. Conceive the pulses to be propagated by successive condensations and rarefactions of the medium, so that the densest part of every pulse may occupy a sphærical superficies described about the centred, and that equal intervals intervene between the successive pulses. Let the lines de, fg, hi, kj, &c. represent the densest parts of the pulses, propagated thro' the hole BC; and because the medium is denser there, than in the spaces on either side towards KL arid NO, it will dilate itself as well towards those spaces KL, NO on each hand, as towards the rare intervals between the pulses; and thence the medium becoming always more rare next the intervals, and more dense next the pulses, will partake of their motion. And because the progressive motion of the pulses arises from the perpetual relaxation of the denser parts towards the antecedent rare intervals; and since the pulses will relax themselves on each hand towards the quiescent parts of the medium KL, NO, with very near the same celerity; therefore the pulses will dilate themselves on all sides into the unmoved parts KL, NO, with almost the same celerity with which they are propagated directly from the centre A; and therefore will fill up the whole space KLON. Q.E.D. And we find the same by experience also in sounds, which are heard tho' a mountain interpose; and if they come intb a chamber thro' the window, dilate themselves into all the parts of the room, and are heard in every corner; and not as reflected from the opposite walls, but directly propagated from the window, as far as our sense can judge.

Case 3. Let us suppose lastly, that a motion of any kind is propagated from A thro' the hole BC. Then since the cause of this propagation is, that the parts of the medium that are near the centre A disturb and agitate those which lie farther from it; and since the parts which are urged are fluid, and therefore recede every way towards those spaces where they are less pressed, they will by consequence recede towards all the parts of the quiescent medium; as well to the parts on each hand, as KL and NO, as to those right before as PO: and by this means all the motion, as soon as it has passed thro' the hole BC, will begin to dilate itself, and from thence, as from its principle and centre, will be propagated directly every way. Q.E.D.

Proposition XLIII. Theorem XXXIV. Every tremulous body in an elastic medium propagates the motion of the pulses on every fide right forward; but in a non-elastic medium excites a circular motion.

Case 1, The parts of the tremulous body alternately going and returning, do in going urge and drive before them those parts of the medium that lie nearest, and by that impulse compress and condense them; and in returning suffer those compressed parts to recede again and expand themselves. Therefore the parts of the medium that lie nearest to the tremulous body, move to and fro by turns, in like manner as the parts of the tremulous body itself do; and for the same cause that the parts of this body agitate these parts of the medium, these parts being agitated by like tremors, will in their turn agitate others next to themselves, and these others agitated in like manner, will agitate those that lie beyond them, and so on in infinitum. And in the same manner as the first parts of the medium were condensed in going, and relaxed in returning, so will the other parts be condensed every time they go, and expand themselves every time they return. And therefore they will not be all going and all returning at the same instant, (for in that case they would always preserve determined distances from each other, and there could be no alternate condensation and rarefaction;) but since in the places where they are condensed, they approach to, and in the places where they are rarefied, recede from, each other; therefore some of them will be going while others are returning, and so on in infinitum. The parts so going, and in their going condensed, are pulses, by reason of the progressive motion with which they strike obstacles in their way; and therefore the successive pulses produced by a tremulous body, will be propagated in rectilinear directions; and that at nearly equal distances from each other, because of the equal intervals of time in which the body, by its several tremors, produces the several pulses. And tho' the parts of the tremulous body go and return in some certain and determinate direction, yet the pulses propagated from thence thro' the medium, will dilate themselves towards the sides, by the foregoing proposition; and will be propagated on all sides from that tremulous body, as from a common centre, in superficies nearly sphærical and concentrical. An example of this we have in waves excited by shaking a finger in water, which proceed not only, forwards and backwards agreeably to the motion of the finger, but spread themselves in the manner of concentrical circles all round the finger, and are propagated on every side. For the gravity of the water supplies the place of elastic force.

Case 2. If the medium be not elastic, then, because its parts cannot be condensed by the pressure arising from the vibrating parts of the tremulous body, the motion will be propagated in an instant towards the parts where the medium yields most easily; that is, to the parts which the tremulous body leaves for some time vacuous behind it. The case is the same with that of a body projected in any medium whatever. A medium yielding to projectiles does not recede in infinitum, but with a circular motion comes round to the spaces which the body leaves behind it. Therefore as often as a tremulous body tends to any part, the medium yielding to it comes round in a circle to the parts which the body leaves; and as often as the body returns to the first place, the medium will be driven from the place it came round to, and return to its original place. And tho' the tremulous body be not firm and hard, but every way flexible; yet if it continue of a given magnitude, since it cannot impel the medium by its tremors any where without yielding to it somewhere else; the medium receding from the parts where it is pressed, will always come round in a circle to the parts that yield to it. Q.E.D.

Corollary. 'Tis a mistake therefore to think, as some have done, that the agitation of the parts of flame conduces to the propagation of a pressure in rectilinear directions thro' an ambient medium. A pressure of that kind must be derived, not from the agitation only of the parts of flame, but from the dilatation of the whole.

Proposition XLIV. Theorem XXXV. If water ascend and descend alternately in the erected legs KL, MN of a canal or pipe; and a pendulum be constructed, whose length between the point of suspension and the centre of oscillation is equal to half the length of the water in the canal: I say, that the water will ascend and descend in the same times in which the pendulum oscillates. Plate 8. Figure 4.

I measure the length of the water along the axes of the canal and its legs, and make it equal to the sum of those axes; and take no notice of the resistance of the water, arising from its attrition by the sides of the canal. Let therefore AB, CD represent the mean height of the water in both legs; and when the water in the leg KL ascends to the height EF, the water will descend in the leg MN to the height GH. Let P be a pendulous body, VP the thread, V the point of suspension, RPQS the cycloid which the pendulum describes, P its lowest point, PQ an arc equal to the height AE. The force, with which the motion of the water is accelerated and retarded alternately, is the excess of the weight of the water in one leg above the weight in the other; and therefore, when the water in the leg KL ascends to EF, and in the other leg descends to GH, that force is double the weight of the water EABF, and therefore is to the weight of the whole water as AE or PQ to VP or PR. The force also with which the body P is accelerated or retarded in any place as Q of a cycloid, is (by corollary, proposition 51.) to its whole weight, as its distance PQ from the lowest place P to the length PR of the cycloid. Therefore the motive forces, of the water and pendulum, describing the equal spaces AE, PQ are as the weights to be moved; and therefore if the water and pendulum are quiescent at first, those forces will move them in equal times, and will cause them to go and return together with a reciprocal motion. Q.E.D.

Corollary 1. Therefore the reciprocations of the water in ascending and descending, are all performed in equal times, whether the motion be more or less intense or remiss.

Corollary 2. If the length of the whole water in the canal be of 61/9 feet of French measure, the water will descend in one second of time, and will ascend in another second, and so on by turns in infinitum; for a pendulum of 31/18 such feet in length will oscillate in one second of time.

Corollary 3. But if the length of the water be increased or diminilhed, the time of the reciprocation will be increased or diminished in the subduplicate ratio of the length.

Proposition XLV. Theorem XXXVI. The velocity of waves is in the subduplicate ratio of the breadths.

This follows from the construction of the following proposition.

Proposition XLVI. Problem X. To find the velocity of waves.

Let a pendulum be constructed, whose length between the point of suspension and the centre of oscillation is equal to the breadth of the waves; and in the time that the pendulum will perform one single oscillation, the waves will advance forward nearly a space equal to their breadth.

That which I call the breadth of the waves is the transverse measure lying between the deepest part of the hollows, or the tops of the ridges. Let ABCDEF (Plate 8. Figure 5.) represent the surface of stagnant water ascending and descending in successive waves; and let A, C, E, &c. be the tops of the waves; and let B, D, F, &c. be the intermediate hollows. Because the motion of the waves is carried on by the successive ascent and descent of the water, so that the parts thereof, as A, C, £, &c. which are highest at one time, become lowest immediately after; and because the motive force, by which the highest parts descend and the lowest ascend, is the weight of the elevated water, that alternate ascent and descent will be analogous to the reciprocal motion of the water in the canal, and observe the same laws as to the times of its ascent and descent; and therefore (by proposition 44.) if the distances between the highest places of the waves A, C, E, and the lowest B, D, F, be equal to twice the length of any pendulum, the highest parts A, C, E, will become the lowest in the time of one oscillation, and in the time of another oscillation will ascend again. Therefore between the passage of each wave, the time of two oscillations will intervene; that is, the wave will describe its breadth in the time that pendulum will oscillate twice; but a pendulum of four times that length, and which therefore is equal to the breadth of the waves, will just oscillate once in that time. Q.E.I.

Corollary 1. Therefore waves, whose breadth is equal to 31/18 French feet, will advance thro' a space equal to their breadth in one second of time; and therefore in one minute will go over a space of 1831/3 feet; and in an hour a space of 11000 feet, nearly.

Corollary 2. And the velocity of greater or less waves will be augmented or diminished in the subduplicate ratio of their breadth.

These things are true upon the supposition, that the parts of water ascend or descend in a right line; but in truth, that ascent and descent is rather performed in a circle; and therefore I propose the time defined by this proposition as only near the truth.

Proposition XLVII. Theorem XXXVII. If pulses are propagated thro' a fluid, the several particles of the fluid, going and returning with the shortest reciprocal motion, are always accelerated or retarded according to the law of the oscillating pendulum. Plate 9. Figure 1.

Let AB, BC, CD, &c. represent equal distances of successive pulses; ABC the line of direction of the motion of the successive pulses, propagated from A to B; E, F, G three physical points of the quiescent medium situate in the right line AC at equal distances from each other; Ee, Ff, Gg equal spaces of extreme shortness, thro' which those points go and return with a reciprocal motion in each vibration; ε φ γ any intermediate places of the same points; and EF, FG physical lineolae, or linear parts of the medium lying between those points, and successively transfer'd into the places εφ, φγ, and ef, fg. Let there be drawn the right line PS equal to the right line Ee. Bisect the same in O, and from the centre O, with the interval OP, describe the circle SIPi. Let the whole time of one vibration, with its proportional parts, be expounded by the whole circumference of this circle and its parts; in such sort, that when any time PH or PHSh is compleated, if there be let fall to PS the perpendicular HL or hl, and there be taken Eε equal to PL or Pl, the physical point E may be found in ε. A point as E moving according to this law with a reciprocal motion, in its going from E thro' ε to e, and returning again thro' ε to E, will perform its several vibrations with the same degrees of acceleration and retardation with those of an oscillating pendulum. We are now to prove, that the several physical points of the medium will be agitated with such a kind of motion. Let us suppose then, that a medium hath such a motion excited in it from any cause whatsoever, and consider what will follow from thence.

In the circumference PHSh let there be taken the equal arcs HI, IK, or hi, il>, having the same ratio to the whole circumference as the equal right lines EF, FG have to BC the whole interval of the pulses. Let fall the perpendiculars IM, KN or im, kn; then because the points E, F, G are successively agitated with like motions, and perform their entire vibrations composed of their going and return, while the pulse is transferr'd from B to C; if PH or PHSh be the time elapsed since the beginning of the motion of the point E, then will PI or PHSi be the time elapsed since the beginning of the motion of the point F, and PK or PHSk the time elapsed since the beginning of the motion of the point G; and therefore Eε, Fφ, Gγ will be respectively equal to PL, PM, PN, while the points are going, and to Pl, Pm, Pn, when the points are returning. Therefore εγ or EG + Gγ - Eε will, when the points are going, be equal to EG - LN, and in their return equal to EG + ln. But εγ is the breadth or expansion of the part EG of the medium in the place εγ; and therefore the expansion of that part in its going, is to its mean expansion as EG - LN to EG; and in its return as EG or EG + LN to EG. Therefore since LN is to KH as IM to the radius OP, and KH to EG as the circumference PHShP to BC; that is, if we put V for the radius of a circle whole circumference is equal to BC the interval of the pulses, as OP to V; and, ex aquo, LN to EG as IM to V; the expansion of the part EG or of the physical point F in the place εγ to the mean expansion of the same part in its first place EG, will be as V - IM to V in going, and as V + im to V in its return. Hence the elastic force of the point F in the place εγ to its mean elastic force in the place EG, is as 1	V - IM  to  1	V  in its going, and as  1	V + im  to  1	V in its return. And by the same reasoning the elastic forces of the physical points E and G in going, are as 1	V - HL and  1	V - KN  to  1	V ; and the difference of the forces to the mean elastic force of the medium, as  HL - KN	VV - V.HL - V.KN + HL.KN  to  1	V ; that is, as  HL - KN	VV  to  1	V, or as HL - KN to V; if we suppose (by reason of the very short extent of the vibrations) HL and KN to be indefinitely less than the quantity V. Therefore since the quantity V is given, the difference of the forces is as HL - KN; that is, (because HL - KN is proportional to HK, and OM to OI or OP; and because HK and OP are given) as OM; that is, if Ff be bisected in Ω, as Ω&phi';. And for the same reason the difference of the elastic forces of the physical points ε and γ in the return of the physical lineola εγ, is as a Ωφ. But that difference (that is, the excess of the elastic force of the point ε above the elastic force of the point γ) is the very force by which the intervening physical lineola εγ of the medium is accelerated in going, and retarded in returning; and therefore the accelerative force of the physical lineola εγ is as its distance from Ω the middle place of the vibration. Therefore (by proposition 38, book 1.) the time is rightly expounded by the arc PI; and the linear pare of the medium εγ is moved according to the law abovementioned, that is, according to the law of a pendulum oscillating; and the case is the same of all the linear parts of which the whole medium is compounded. Q.E.D.

Corollary. Hence it appears that the number of the pulses propagated is the same with the number of the vibrations of the tremulous body, and is not multiplied in their progress. For the physical lineola εγ as soon as it returns to its first place is at rest; neither will it move again, unless it receives a new motion, either from the impulse of the tremulous body, or of rhe pulses propagated from that body. As soon therefore as the pulses cease to be propagated from the tremulous body, it will return to a state of rest; and move no more.

Proposition XLVIII. Theorem XXXVIII. The velocities of pulses propagated in an elastic fluid, are in a ratio compounded of the subduplicate ratio of the elastic force directly, and the subduplicate ratio of the density inversely; supposing the elastic force of the fluid to be proportional to its condensation.

Case 1. If the mediums be homogeneous, and the distances of the pulses in those mediums be equal amongst themselves, but the motion in one medium is more intense than in the other: the contractions and dilatations of the correspondent parts will be as those motions. Not that this proportion is perfectly accurate. However, if the contractions and dilatations are nor exceedingly intense, the error will not be sensible; and therefore this proportion may be consider'd as physically exact. Now the motive elastic forces are as the contractions and dilatations; and the velocities generated in the same time in equal parts are as the forces. Therefore equal and corresponding parts of corresponding pulses will go and return together, thro' spaces proportional to their contractions and dilatations, with velocities that are as those spaces: and therefore the pulses, which in the time of one going and returning advance forwards a space equal to their breadth, and are always succeeding into the places of the pulses that immediately go before them, will, by reason of the equality of the distances, go forward in both mediums with equal velocity.

Case 2. If the distances of the pulses or their lengths are greater in one medium than in another; let us suppose that the correspondent parts describe spaces, in going and returning, each time proportional to the breadths of the pulses: then will their contractions and dilatations be equal. And therefore if the mediums are homogeneous, the motive elastic forces, which agitate them with a reciprocal motion, will be equal also. Now the matter to be moved by these forces is as the breadth of the pulses; and the space thro' which they move every time they go and return, is in the same ratio. And moreover, the time of one going and returning, is in a ratio compounded of the subduplicate ratio of the matter, and the subduplicate ratio of the space; and therefore is as the space. But the pulses advance a space equal to their breadths in the times of going once and returning once, that is, they go over spaces proportional to the times; and therefore are equally swift.

Case 3. And therefore in mediums of equal density and elastic force, all the pulses are equally swift. Now if the density or the elastic force os the medium were augmented, then because the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of the density; the time which is necessary for producing the same motion as before, will be increased in the subduplicate ratio of the density, and will be diminished in the subduplicate ratio of the elastic force. And therefore the velocity of the pulses will be in a ratio compounded of the subduplicate ratio of the density of the medium inversely and the subduplicate ratio of the elastic force directly. Q.E.D.

This proposition will be made more clear from the construction of the following problem.

Proposition XLIX. Problem XI. The density and elastic force of a medium being given, to find the velocity of the pulses.

Suppose the medium to be press'd by an incumbent weight after the manner of our air; and let A be the height of a homogeneous medium, whose weight is equal to the incumbent weight and whose density is the same with the density of the compressed medium in which the pulses are propagated. Suppose a pendulum to be constructed, whole length between the point of suspension and the centre of oscillation is A: and in the time in which that pendulum will perform one entire oscillation composed of its going and returning, the pulse will be propagated right onwards, thro' a space equal to the circumference of a circle described with the radius A.

For letting thole things stand which were constructed in Proposition 47. if any physical line as EF (Plate 9, Figure 1.) describing the space PS in each vibration, be acted on in the extremities P and S of every going and return that it makes by an elastic force that is equal to its weight; it will perform its several vibrations in the time in which the same might oscillate in a cycloid, whose whole perimeter is equal to the length PS: and that because equal forces will impel equal corpuscles thro' equal spaces in the same or equal times. Therefore since the times of the oscillations are in the subduplicate ratio of the lengths of the pendulums, and the length of the pendulum is equal to half the arc of the whole cycloid; the time of one vibration would be to the time of the oscillation of a pendulum, whose length is A, in the subduplicate ratio of the length ½PS or PO to the length A. But the elastic force, with which the physical lineola EG is urged, when it is found in its extreme places P, S, was (in the demonstration of proposition 47.) to its whole elastic force as HL - KN to V, that is, (since the point K now falls upon P) as HK to V: and all that force, or, which is the same thing, the incumbent weight by which the lineola EG is compress'd, is to the weight of the lineola as the altitude A of the incumbent weight to EG the length of the lineola; and therefore, ex aequo, the force with which the lineola EG is urged in the places P and S, is to the weight of that lineola as HK.A to V.EG; or as PO.A to VV; because HK was to EG as PO to V. Therefore since the times, in which equal bodies are impelled thro' equal spaces, are reciprocally in the subduplicate ratio of the forces, the time of one vibration, produced by the action of that elastic force, will be to the time of a vibration, produced by the impulse of the weight, in a subduplicate ratio of VV to PO.A, and therefore to the time of the oscillation of a pendulum whose length is A, in the subduplicate ratio of VV to PO.A, and the subduplicate ratio of PO to A conjunctly; that is, in the entire ratio of V to A. But in the time of one vibration composed of the going and returning of the pendulum, the pulse will be propagated right onwards thro' a space equal to its breadth BC. Therefore the time in which a pulse runs over the space BC, is to the time of one oscillation composed of the going and returning of the pendulum, as V to A, that is, as BC to the circumference of a circle whose radius is A. But the time in which the pulse will run over the space BC, is to the time in which it will run over a length equal to that circumference, in the same ratio; and therefore in the time of such an oscillation, the pulse will run over a length equal to that circumference. Q.E.D.

Corollary 1. The velocity of the pulses is equal to that which heavy bodies acquire by falling with an equally accelerated motion, and in their fall describing half the altitude A. For the pulse will, in the time of this fall, supposing it to move with the velocity acquired by that fall, run over a space that will be equal to the whole altitude A; and therefore in the time of one oscillation composed of one going and return, will go over a space equal to the circumference of a circle described with the radius A: for the time of the fall is to the time of oscillation, as the radius of a circle to its circumference.

Corollary 2. Therefore since that altitude A is as the elastic force of the fluid directly, and the density of the same inversely; the velocity of the pulses will be in a ratio compounded of the subduplicate ratio of the density inversely, and the subduplicate ratio of the elastic force directly.

Proposition L. Problem XII. To find the distances of the pulses.

Let the number of the vibrations of the body, by whose tremor the pulses are produced, be found to any given time. By that number divide the space which a pulse can go over in the same time, and the part found will be the breadth of one pulse. Q.E.I.

Scholium.

The last propositions respect the motions of light and sounds. For since light is propagated in right lines, it is certain that it cannot onsist in action alone, (by proposition 41 and 42.) As to sounds, since they arise from tremulous bodies, they can be nothing else but pulses of the air propagated thro' it. (by proposition 43.) And this is confirmed by the tremors, which sounds, if they be loud and deep, excite in the bodies near them, as we experience in the sound of drums. For quick and short tremors are less easily excited. But it is well known, that any sounds, falling upon strings in unison with the sonorous bodies, excite tremors in those strings. This is also confirmed from the velocity of sounds. For since the specific gravities of rain-water and quick-silver are to one another as about 1 to 132/3, and when the mercury in the barometer is at the height of 30 inches of our measure, the specific gravities of the air and of rain-water are to one another as about 1 to 870: therefore the specific gravity of air and quicksilver are to each other as 1 to 11.890. Therefore when the height of the quick-silver is at 30 inches, a height of uniform air, whose weight would be sufficient to compress our air to the density we find it to be of, must be equal to 356.700 inches or 29.725 feet of our measure. And this is that very height of the medium, which I have called A in the construction of the foregoing proposition. A circle whose radius is 29.725, feet is 186.768 feet in circumference. And since a pendulum 391/5 inches in length compleats one oscillation, composed of its going and return, in two seconds of time, as is commonly known; it follows that a pendulum 29.725 feet or 356.700 inches in length will perform a like oscillation in 1903/4 seconds. Therefore in that time a sound will go right onwards 186.768 feet, and therefore in one second 979 feet.

But in this computation we have made no allowance for the crassitude of the solid particles of the air, by which the found is propagated instantaneously. Because the weight of air is to the weight of water as 1 to 870, and because salts are almost twice as dense as water; if the particles of air are supposed to be of near the same density as those of water or salt, and the rarity of the air arises from the intervals of the particles; the diameter of one particle of air will be to the interval between the centres of the particles, as 1 to about 9 or 10, and to the interval between the particles themselves as 1 to 8 or 9. Therefore to 979 feet, which, according to the above calculation, a sound will advance forward in one second of time, we may add 979/9 or about 100 feet, to compensate for the crassitude of the particles of the air: and then a sound will go forward about 1088 feet in one second of time.

Moreover, the vapors floating in the air, being of another spring, and a different tone, will hardly, if at all, partake of the motion os the true air in which the sounds are propagated. Now if these vapors remain unmoved, that motion will be propagated the swifter thro' the true air alone, and that in the subduplicate ratio of the defect of the matter. So if the atmosphere consist of ten parts of true air and one part of vapors, the motion of sounds will be swifter in the subduplicate ratio of 11 to 10, or very nearly in the entire ratio of 21 to 20, than if it were propagated thro' eleven parts of true air: and therefore the motion of sounds above discovered must be encreased in that ratio. By this means the found will pass thro' 1142 feet in one second of time.

These things will be found true in spring and autumn, when the air is rarefied by the gentle warmth of those seasons, and by that means its elastic force becomes comes somewhat more intense. But in winter, when the air is condensed by the cold, and its elastic force is somewhat remitted, the motion of sounds will be slower in a subduplicate ratio of the density; and on the other hand, swifter in the summer.

Now by experiments it actually appears that sounds do really advance in one second of time about 1142 feet of English measure, or 1070 feet of French measure.

The velocity of sounds being known, the intervals of the pulses are known also. For M. Sauveur, by some experiments that he made, found that an open pipe about five Paris feet in length, gives a sound of the same tone with a viol-string that vibrates a hundred times in one second. Therefore there are near 100 pulses in a space of 1070 Paris feet, which a sound runs over in a second of time, and therefore one pulse fills up a space of about 107/10 Paris feet, that is, about twice the length of the pipe. From whence it is probable, that the breadths of the pulses, in all sounds made in open pipes, are equal to twice the length of the pipes.

Moreover, from the corollary of proposition 47. appears the reason, why the sounds immediately cease with the motion of the sonorous body, and why they are heard no longer when we are at a great distance from the sonorous bodies, than when we are very near them. And besides, from the foregoing principles it plainly appears how it comes to pass that sounds are so mightily encreased in speaking-trumpets. For all reciprocal motion uses to be encreased by the generating cause at each return. And in tubes hindering the dilatation of the sounds, the motion decays more slowly, and recurs more forcibly; and therefore is the more encreased by the new motion impressed at each return, And these are the principal phenomena of sounds.

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