Principia Mathematica (Motte Translation, 1848) — part 27 of 45

In the Scholium subjoined to the sixth Section, we shewed, by experi- ments of pendulums, that the resistances of equal and equally swift globes moving in air, water, and quicksilver, are as the densities of the fluids. We here prove the same more accurately by experiments of bodies falling in air and water. For pendulums at each oscillation excite a motion in the fluid always contrary to the motion of the pendulum in its return ; and the resistance arising from this motion, as also the resistance of the thread by which the pendulum is suspended, makes the whole resistance of a pen- dulum greater than the resistance deduced from the experiments of falling bodies. For by the experiments of pendulums described in that Scholium, a globe of the same density as water in describing the length of its semi- diameter in air would lose the 3 3^4 2 P^^^ of its motion. But by the theory delivered in this seventh Section, and confirmed by experiments of falling bodies, the same globe in describing the same length would lose only a part of its motion equal to 4 jV-g, supposing the density of water to be to the density of air as 860 to 1. Therefore the resistances were found greater by the experiments of pendulums (for the reasons just mentioned) than by the experiments of falling globes ; and that in the ratio of about 4 to 3. But yet since the resistances of pendulums oscillating in air, wa- ter, and quicksilver, are alike increased by like causes, the proportion of the resistances in these mediums will be rightly enough exhibited by the

Sec. VIL] of natural philosophy. 355

experiments of pendulumSj as well as by the experiments of falling bodies.

And from all this it may be concluded, that the resistances of bodies, moving

in any iiuids whatsoever, though of the most extreme fluidity, are, cceteris

paribus, as the densities of the fluids.

These things being thus established, we may now determine what part

of its motion any globe projected in any fluid whatsoever would nearly lose

in a given time. Let D be the diameter of the globe, and Y its velocity

at the beginning of its motion, and T the time in which a globe with the

velocity T can describe in vacuo a space that is, to the space |D as the

density of the globe to the density of the fluid ; and the globe projected

tY in that fluid will, in any other time t lose the part ^ , the part

TV

rp remaining ; and will describe a space, which will be to that de-

scribed in the same time in vacuo with the uniform velocity Y, as the

T + ^ logarithm of the number — y^ — multiplied by the number 2,302585093 is

to the number 7^, by Cor. 7, Prop. XXXY. In slow motions the resist- ance may be a little less, because the figure of a globe is more adapted to motion than the figure of a cylinder described with the same diameter. In swift motions the resistance may be a little greater, because the elasticity and compression of the fluid do not increase in the duplicate ratio of the velocity. But these little niceties I take no notice of.

And though air, water, quicksilver, and the like fluids, by the division of their parts in infinitum, should be subtilized, and become mediums in- finitely fluid, nevertheless, the resistance they would make to projected globes would be the same. For the resistance considered in the preceding Propositions arises from the inactivity of the matter ; and the inactivity of matter is essential to bodies, and always proportional to the quantity of matter. By the division of the parts of the fluid the resistance arising from the tenacity and friction of the parts may be indeed diminished ; but the quantity of matter will not be at all diminished by this division ; and if the quantity of matter be the same, its force of inactivity will be the same; and therefore the resistance here spoken of will be the same, as being always proportional to that force. To diminish this resistance, the quan- tity of matter in the spaces through which the bodies move must be dimin- ished ; and therefore the celestial spaces, through which the globes of the planets and comets are perpetually passing towards all parts, with the utmost freedom, and without the least sensible diminution of their motion, must be utterly void of any corporeal fluid, excepting, perhaps, some ex- tremely rare vapours and the rays of light.

356 THE MATHEMATICAL PRINCIPLES [BoOK II.

Projectiles excite a motion in fluids as they pass througli them, and this motion arises from the excess of the pressure of the fluid at the fore parts of the projectile above the pressure of the same at the hinder parts : and cannot he less in mediums infinitely fluid than it is in air, water, and quick- silver, in proportion to the density of matter in each. Now this excess of pressure does, in proportion to its quantity, not only excite a motion in the fluid, but also acts upon the projectile so as to retard its motion ; and there- fore the resistance in every fluid is as the motion excited by the projectile in the fluid ; and cannot be less in the most subtile aether in proportion to the density of that aether, than it is in air, water, and quicksilver, in pro- portion to the densities of those fluids.

SECTION YIII.

Of motion 'propagated through fluids,

PROPOSITION XLL THEOREM XXXII.

A pressure is not propagated through a fluid in rectilinear directions unless where the particles of the fluid lie in a right line.

If the particles a, b, c, d, e, lie in a right line, the pres- sure may be indeed directly propagated from a to e ; but then the particle e will urge the obliquely posited parti- cles / and g obliquely, and those particles / 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 I and 7n, and so on i?i infinitum. There- fore the pressure, as soon as it is propagated to particles that lie out of right lines, begins to deflect towards one hand and the other, and will be propagated obliquely in infinitum ; and after it has begun to be propagat- ed 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 as it lights on particles that do not lie exactly in a right line. Q,.E.D.

Cor. If any part of a pressure, propagated through a fluid from a given point, be intercepted by any obstacle, the remaining part, which is not in- tercepted, will deflect into the spaces behind the obstacle. This may be demonstrated also after the following manner. Let a pressure be propagat- ed from the point A towards any part, and, if it be possible, in rectilinear

Sec. YIIL]

OF NATURAL PHILOSOPHY.

357

directions ; and the obstacle NBCK being perforated in BC, let all the pressure be intercepted hut the coniform part APQ. pass- ing through the circular hole EC. 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 /§'iA on the superficies/g", 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 re-action of the second frustum fghi, as much urged and pressed on the superficies /§", 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, Prop. XIX) cannot preserve its figure, unless it be compressed with the same force on all sides. Therefore with the same force with w^hich it is press^ed 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 it- self, unless there be an ambient fluid opposing that endeavour. Therefore, by the eifort 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 su- perficies/g- towards PQ. Q.E.D.

PROPOSITION XLII. THEOREM XXXIII.

All motion propagated through a fluid diverges from a rectilinear pro- gress into the unmoved spaces. Case 1, Let a motion be propagated from the point A through the hole BC, and, if it be possible, let it proceed in the conic space BCQ^P according to right lines diverging from the point A. And let us first sup- pose this motion to be that of waves in the surface of standing water ; and let de,fg, hi, kl, &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

358 THE MATHEMATICAL PRINCIPLES [BoOK 11.

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, I, (fcc, d,f, h, k, (fcc, 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 tow?irds 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,. And 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 exper- iment in still water.

Case 2. Let us suppose that de, fg, hi, kl, nin, represent pulses suc- cessively propagated from the point A through an elastic medium. Con- ceive the pulses to be propagated by successive condensations and rarefactions of the medium, so that the densest part of every pulse may occupy a spherical superficies described about the centre A, and that equal intervals intervene between the successive pulses. Let the lines de, fg, hi, kl, (fcc, represent the densest parts of the pulses, propagated through the hole BC ; and because the medium is denser there than in the spaces on either side towards KL and 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 KIj, 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 through a mountain interposed ; and, if they come into a chamber through 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

Sec. VIIL] of natural philosophy. 359

from A through 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 reqede towards all the parts of the quiescent medium ; as Avell to the parts on each hand, as KL and NO, as to those right before, as PQ,; and by this means all the motion, as soon as it has passed through the hole BC, will begin to dilate itself, and from thence, as from its principle and centre, will be propagated directl'y every way. Q.E.D.

PROPOSITION XLIII. THEOREM XXXIV.

Every tremulous body in an elastic medium propagates the ??iotion of the pidses on every side right foriva7'd; but in a non-elastic medium excites a circular motion.

Case. 1. The parts of the tremulous body, alternately going and return- ing, 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 re- turning suffer those compressed parts to recede again, and expand them- selves. 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 infin- itum. x\nd 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 re- turn. And therefore they will not be all going and all returning at the. same instant (for in that case they would always preserve determined dis- tances from each other, and there could be no alternate condensation and rarefaction) ; but since, in the places where they are condensed, they ap- proach 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 though the parts of the tremulous body go and return in some certain and deter-, minate direction, yet the pulses propagated from thence through the medium will dilate themselves towards the sides, by the foregoing Proposition ; and

360 THE MATHEMATICAL PRINCIPLES [BoOK II.

will be propagated on all sides from tliat tremulous body, as from a com- mon centre^ in superficies nearly spherical and concentrical. An example of this we have in waves excited by shaking a finger in water, which proceed not only forward and backward 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 wliich the tremulous body would otherwise leave 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 wfinitum, 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 though 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 of the body where it is pressed will always come round in a circle to the parts that yield to it. Q.E.D.

Cor. It is 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 through 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 XLIY. THEOREM XXXV.

Jf loater ascend and descend alternately in the erected legs KL, MN, of a canal or pipe ; a?id a pendulum be constructed whose length between the point of suspension and the centre of oscillation is equal to half the length of the loater in the canal ; I say, that the water loill ascend and descend in the same ti?nes in which the pendulum oscillates.

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

Sec. YIIL]

OF NATURAL PHILOSOPHY.

361

body, VP the thread, Y the point of suspension, RPQS the cycloid which

V" KM

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 re- tarded 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 EABP, 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 Cor. Prop. LI) 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.

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

Cor. 2. If the length of the whole water in the canal be of 6 J feet of French measure, the water will descend in one second of time, and will as- cend in another second, and so on by turns iti i?ijinitum ; for a pendulum of 3y^g- such feet in length will oscillate in one second of time.

Cor. 3. But if the length of the water be increased or diminished, the time of the reciprocation will be increased or diminished in the subdupli- cate ratio of the length.

PROPOSITION XLV. THEOREM XXXVI.

The velocity of waves is in the suhduplicate 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 sus- pension and the centre of oscillation is equal to the breadth of the waves ;

362 THE MATHEMATICAL PRINCIPLES [BoOK II.

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 1 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 ABCDEP represent the surface of stagnant water ascending and descend- ing in successive waves ; and let A, C, E, (fee, be the tops of the waves ; and let B, D, F, (fee, be th^ 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, E, &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 eleva- ted water, that alternate ascent and descent will be analogous to the recip- rocal motion of the water in the canal, and observe the same laws as to the times of its ascent and descent; and therefore (by Prop. XLIY) 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, 0, E, will become the lowest in the time of one oscillation, and in the time of another oscillation will ascend again. Therefore between the pas- sage 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.EL

CoR. 1. Therefore waves, whose breadth is equal to 3j\ French feet, will advance through a space equal to their breadth in one second of time ; and therefore in one minute will go over a space of 183|- feet ^ and in an hour a space of 11000 feet, nearly.

Cor. 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 as- cend 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 XLYII. THEOREM XXXYH.

If pulses are propagated through a fluid, the several particles of the fluid, going and returning with the shortest reciprocal motion, are al- loays accelerated or retarded according to the laio of the oscillating pendulum. l^et AB, BC, CD, (fcc, represent equal distances of successive pulses ;

ABC the line of direction of the motion of the successive pulses propagated

Sec. YIIL]

OF NATURAL PHILOSOPHY.

363

n y. ?

F4-

]M-

from A to B ; E, F, G three physical points of the quiescent medium sit- uate in the right line AC at equal distances from each other ; Ee^ F/, Gg- equal spaces of extreme shortness^ through which those points go and return with a reciprocal motion in each vi- bration ; £, </), y, any intermediate places of the same points ; EF, FG physical lineolae, or linear parts of the medium lying between those pointS; and successively transferred into the places £0, 0y, 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 circum- ference of this circle and its parts, in such sort, that, when any time PH or PHSA is completed, if there be let fall to PS the perpendicular HL or /iZ, and there ^

be taken Ee equal to PL or P/, the physi- cal point E may be found in e. A point, as E, moving acccording to this law with a reciprocal motion, in its going from E throuo:he to e, and returnino^ ao^ain throuo;h

O / CD O O

e to E, will perform its several vibrations with the same de- grees of acceleration and retardation with those of an oscil- lating 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, th^, that a medium hath such a motion excited in it from any cause whatsoever, and consider what will follow from thence.

In the circumference PHSA let there be taken the equal arcs, HI, IK, or Ai, ik^ 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 perpendicu- lars 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 transferred from B to C ; if PH or PHSA be the time elapsed since the beginning of the mo- tion 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 PHS^^ the time elapsed since the beginning of the motion of the point G; and therefore Ee, F^, Gy, will be respectively equal to PL, PM, PN, while the points are going, and to PI, Fiji, F71, when the points are returning. Therefore ey or EG

• Gy — Es will, when the points are going, be equal to EG — LN,

lllll

364 THE MATHEMATICAL PRINCIPLES [BoOK 11.

and in their return equal to EG + In. But ey is the breadth or ex- pansion of the part EG of the medium in the place ey ; 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 + In or EG + LN to EG. Therefore since LN is to KH as IM to the radius OP, and KH to EG as the circumference PHSAP to BC ; that iS; if we put V for the radius of a circle whose circumference is equal to BC the interval of the pulses, as OP to V ; and, ex cequo, LN to EG as IM to Y ; the expansion of the part EG, or of the physical point F in the place £y, to the mean ex- pansion of the same part in its first place EG, will be as V — IM to Y in going, and as Y -- im ioN in its return. Hence the elastic force of the point F in the place ey to its mean elastic force in the place EG is as

:^. T^r; to r- in its 2:oino; and as ^rp— — : — to ^ in its return. And by

the same reasoning the elastic forces of the physical points E and G in going

are as ^. — ™- and :^ zr—- to ^ ; and the difference of the forces to the

mean elastic force of the medium as

YY-Y X HL-Yx KN + HL X KN

to ~: : that is, as ^-^^ to ^, or as HL — KN to Y ; if we suppose

(by reason of the very short extent of the vibrations) HL and KN to be indefinitely less than the quantity Y. Therefore since the quantity Y is given, the difference of the forces is as HL — KN ; that is (because HL — KN is proportional to HK, and OM to 01 or OP ; and because HK and OP are given) as OM ; that is, if F/ be bisected in Q, as ^0. And for the same reason the difference of the elastic forces of the physical points £ and y, in the return of the physical lineola £}', is as 9.6. But that dif- ference (that is, the excess of the elastic force of the point e above the elastic force of the point y) is the very force by which the intervening phy- sical lineola ey of the medium is accelerated in going, and retarded in re- turning ; and therefore the accelerative force of the physical lineola ey is as its distance from ^, the middle place of the vibration. Therefore (by Prop. XXXYIII, Book 1) the time is rightly expounded by the arc PI ; and the linear part of the medium ey is moved according to the law above- mentioned, 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.

Cor. 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 ey as soon as it returns to its first place is at rest ; neither will it move again, unless it

Sec. VIIL] of natural philosophy. 365

receives a new motion either from the impulse of the tremulous body, or of the 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 XL VIII. 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 me- dium 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 not exceedingly intense, the error will not be sensible ; and therefore this pro- portion may be considered 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, through spaces proportional to their contractions and dilatations, with ve- locities that are as those spaces ; and therefore the pulses, which in the time of one going and returning advance forward a space equal to their breadth, and are always succeeding into the places of the pulses that im- mediately 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 de- scribe 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 through 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 of the medium were augmented, then, because the n.iotive force is increased

366 THE MATHEMATICAL PRINCIPLES [BoOK IL

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 com- pounded 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 XL

The density and elastic force of a medium being given, to find the ve- locity of the pulses.

Suppose the medium to be pressed 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 propa- gated. Suppose a pendulum to be constructed whose 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 through a space equal to the circumference of a circle described with the radius A.