自然哲学的数学原理-40

tion of the point E, then will PI or PHSi be the time elapsed since thebeginning of the motion of the point F, and PK or PHSA; the time elapsedsince the beginning of the motion of the point G; and therefore Ee, F0,Gy, will be respectively equal to PL, PM, PN, while the points are going,and to PI, Ptn, Pn, when the points are returning. Therefore ey or EG4- Gy Et will, when the points are going, be equal to EG LN364 THE MATHEMATICAL PRINCIPLES [BOOK II.and in their return equal to EG + In. But ey is the breadth or expansion of the part EG of the medium in the place ey ; and therefore theexpansion of that part in its going is to its mean expansion as EGLN to EG; and in its return, as EG -f In or EG + LN to EG.Therefore since LN is to KH as IM to the radius OP, and KH to EGas the circumference PHSAP to BC ; that is, if we put V for theradius of a circle whose circumference is equal to BC the interval of thepulses, as OP to V and, ex cequo, LN to EG as IM to V ; the expansionof the part EG, or of the physical point F in the place ey, to the mean expansion of the same part in its first place EG, will be as V IM to Vin going, and as V -f im to V in its return. Hence the elastic force of thepoint P in the place ey to its mean elastic force in the place EG is as 11. 11, v fivf* v m 1 ^s Somo>an<^ as v i^ v in lts re^urn. And by V J.1VJL V V -f Iffl Vthe same reasoning the elastic forces of the physical points E and G in goingare as . qrand ^ ==~ to T, ; and the difference of the forces to themean elastic force of the medium as T^ VV-V X HL-Vx KN + HL X KN1 HL KN 1to ~ ;that is, as : to^,or as HL KN to V ; if we suppose(by reason of the very short extent of the vibrations) HL and KN to beindefinitely less than the quantity V. Therefore since the quantity V isgiven, the difference of the forces is as HL KN ;that is (because HLKN is proportional to HK, and OM to OI or OP ; and because HKand OP are given) as OM ; that is, if F/ be bisected in ft, as ft</>. Andfor the same reason the difference of the elastic forces of the physical pointse and y, in the return of the physical lineola ey, is as ftr/>. But that difference (that is, the excess of the elastic force of the point e above theelastic force of the point y) is the very force by which the intervening physical lineola ey of the medium is accelerated in going, and retarded in returning ; and therefore the accelerative force of the physical lineola ey isas its distance from ft, the middle place of the vibration. Therefore (byProp. XXXVIII, Book 1) the time is rightly expounded by the arc PI ;and the linear part of the medium sy is moved according to the law abovementioned,that is, according to the law of a pendulum oscillating ; andthe case is the same of all the linear parts of which the whole medium iscompounded. Q,.E.D.COR. Hence it appears that the number of the pulses propagated is thesame with the number of the vibrations of the tremulous body, and is notmultiplied in their progress. For the physical lineola ey as soon as itreturns to its first place is at rest; neither will it move again, unless iiSEC. V11I.J OF NATURAL PHILOSOPHY. 36receives a new motion either from the impulse of the tremulous body, orof the pulses propagated from that body. As soon, therefore, as the pulsescease to be propagated from the tremulous body, it will return to a stateof rest, and move no more.PROPOSITION XLVIII. THEOREM XXXVIII.The velocities of pulses propagated in an elastic fluid are in a ratincompounded of the subduplicate, ratio of the elastic force directly, andthe subduplicate ratio of the density inversely ; supposing the elasticJorce of thefluid to be proportional to its condensationCASE I. If the mediums be homogeneous, and the distances of the pulsesin those mediums be equal amongst themselves, but the motion in one medium is more intense than in the other, the contractions and dilatations ofthe correspondent parts will be as those motions ;not that this proportionis perfectly accurate. However, if the contractions and dilatations are notexceedingly intense, the error will not be sensible ; and therefore this proportion may be considered as physically exact. Now the motive elasticforces are as the contractions and dilatations ; and the velocities generatedin the same time in equal parts are as the forces. Therefore equal andcorresponding parts of corresponding pulses will go and return together,through spaces proportional to their contractions and dilatations, with velocities that are as those spaces ; and therefore the pulses, which in thetime of one going and returning advance forward a space dq aal to theirbreadth, 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 onemedium than in another, let us suppose that the correspondent parts describe spaces, in going and returning, each time proportional to the breadthsof the pulses ; then will their contractions and dilatations be equal : andtherefore if the mediums are homogeneous, the motive elastic forces, whichagitate them with a reciprocal motion, will be equal also. Now the matterto be moved by these forces is as the breadth of the pulses ; and the spacethrough which they move every time they go and return is in the sameratio. And, moreover, the time of one going and returning is in a raticcompounded of the subduplicate ratio of the matter, and the o-uwuupncatcratio of the space ; and therefore is as the space. But the pulses advancea space equal to their breadths in the times of going once and returningonce; that is, they go over spaces proportional to the times, and thereforeare 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 ofthe medium were augmented, then, because the motive force is increased366 THE MATHEMATICAL PRINCIPLES [BoOK 11in the ratio of the elastic force, and the matter to be moved is increased inthe ratio of the density, the time which is necessary for producing thesame motion as before will be increased in the subduplicate ratio of thedensity, and will be diminished in the subduplicate ratio of the elasticforce. 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 thefollowing Problem.PROPOSITION XLIX. PROBLEM XLThe. density and elastic force of a medium being given, to find the, velocity of the pulses.Suppose the medium to be pressed by an incumbent weight after the mannerof our air ; and let A be the height, of a homogeneous medium, whoseweight is equal to the incumbent weight, and whose density is the samewith the density of the compressed medium in which the pulses are propagated. Suppose a pendulum to be constructed whose length between thepoint of suspension and the centre of oscillation is A : and in the time inwhich that pendulum will perform one entire oscillation composed ofits going and returning, the pulse will be propagated right onwardsthrough a space equal to the circumference of a circle described with theradius A.For, letting those things stand which were constructed in Prop. X.LV11,if any physical line, as EF, describing the space PS in each vibration, beacted on in the extremities P and S of every going and return that itmakes by an elastic force that is equal to its weight, it will perform itsseveral vibrations in the time in which the same might oscillate in a cycloid whose whole perimeter is equal to the length PS ; and that becauseequal forces will impel equal corpuscles through equal spaces in the sameor equal times. Therefore since the times of the oscillations are in thesubduplicate ratio of the lengths of the pendulums, and the length of thependulum 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 lengthis A in the subduplicate ratio of the length ^PS or PO to the length A.But the elastic force with which the physical lineola EG is urged, when itIs found in its extreme places P, S, was (in the demonstration of Prop.XLVII) to its whole elastic force as HL KN to V, that is (since thepoint K now falls upon P), as HK to V: and all that force, or which isthe same thing, the incumbent weight by which the lineola EG is compressed, is to the weight of the lineola as the altitude A of the incumbentweight to EG the length of the lineola ; and therefore, ex ctquo, the forceSEC. VII1.I OF NATURAL PHILOSOPHY. 367with which the lincola EG is urged in the places P and Sis to the weight of that lineola as HK X A to V X EG ;oras PO X A to VV; because HK was to EG as PO to V.Therefore since the times in which equal bodies are impelledthrough equal spaces are reciprocally in the subduplicateratio of the forces, the time of one vibration, produced bythe 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 X A, and therefore to the timeof the oscillation of a pendulum whose length is A in thesubduplicate ratio of VV to PO X 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 onevibration composed of the going and returning of the pendulum, the pulse willbe propagated right onward through aspace equal to its breadth BC. Therefore the time in which a pulse runs overthe space BC is to the time of one oscillation composed ofthe 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 BCis to the time in which it will run over a length equal tothat circumference in the same ratio; and therefore in thetime of such an oscillation the pulse will run over a lengthequal to that circumference. G,.E.D.COR. 1. The velocity of the pulses is equal to that whichheavy 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 aspace that will be equal to the whole altitude A ; and therefore in thetime of one oscillation composed of one going and return, will go over aspace 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 circleto its circumference.COR. 2. Therefore since that altitude A is as the elastic force of thefluid directly, and the density of the same inversely, the velocity of thepulses will be in a ratio compounded of the subduplicate ratio of the density inversely, and the subduplicate ratio of the clastic force directly.368 THE MATHEMATICAL PRINCIPLES |BoOK ILPROPOSITION L. PROBLEM XII.Tofind the distances of the pulses.Let the number of the vibrations of the body, by whose tremor the pulsesare produced; be found to any given time. By that number divide thespace which a pulse can go over in the same time, and the part found willbe the breadth of one pulse. Q.E.I.SCHOLIUM.The last Propositions respect the motions of light and sounds ;for sincelight is propagated in right lines, it is certain that it cannot consist in action alone (by Prop. XLI and XLIl). As to sounds, since they arise fromtremulous bodies, they can be nothing else but pulses of the air propagatedthrough it (by Prop. XLIII) ; and this is confirmed by the tremors whichsounds, 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 easilyexcited. But it is well known that any sounds, falling upon strings inunison with the sonorous bodies, excite tremors in those strings. This isalso confirmed from the velocity of sounds; for since the specific gravitiesof rain-water and quicksilver are to one another as about 1 to 13f, andwhen the mercury in the barometer is at the height of 30 inches of ourmeasure, the specific gravities of the air and of rain-water are to oneanother as about 1 to 870, therefore the specific gravity of air and quicksilver are to each other as 1 to 11890. Therefore when the height ofthe quicksilver is at 30 inches, a height of uniform air, whose weight wouldbe sufficient to compress our air to the density we find it to be of, must beequal to 356700 inches, or 29725 feet of our measure ; and this is thatvery height of the medium, which I have called A in the construction ofthe foregoing Proposition. A circle whose radius is 29725 feet is 186768feet in circumference. And since a pendulum 39} inches in length completes one oscillation, composed of its going and return, in two seconds oftime, as is commonly known, it follows that a pendulum 29725 feet, or356700 inches in length will perform a like oscillation in 190f seconds.Therefore in that time a sound will go right onwards 186768 feet, andtherefore in one second 979 feet.But in this computation we have made no allowance for the crassitudeof the solid particles of the air, by which the sound is propagated instantaneously. Because the weight of air is to the weight of water as 1 tc870, and because salts are almost twice as dense as water ;if the particlesof 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 ;thediameter of one particle of air will be to the interval between the centresSEC. VIIL] OF NATURAL PHILOSOPHY. 369of 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 tothe above calculation, a sound will advance forward in one second of time,。ve may add ^- 9-, or about 109 feet, io compensate for the cra-ssitude of theparticles of the air : and then a sound will go forward about 1088 feet inone second of time.Moreover, the vapours floating in the air being of another spring, and adifferent tone, will hardly, if at all, partake of the motion of the true airin which the sounds are propagated. Now if these vapours remain unmoved, that motion will be propagated the swifter through the true air alone,and that in the subduplicate ratio of the defect of the matter. So if theatmosphere consist of ten parts of true air and one part of vapours, themotion of sounds will be swifter in the subduplicate ratio of 11 to 10, orvery nearly in the entire ratio of 21 to 20, than if it were propagatedthrough eleven parts of true air : and therefore the motion of sounds abovediscovered must be increased in that ratio. By this means the sound willpass through 1 142 feet in one second of time.These things will be found true in spring and autumn, when the air israrefied by the gentle warmth of those seasons, and by that means its elastic force becomes somewhat more intense. But in winter, when the air iscondensed 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 advancein one second of time about 1142 feet of English measure, or 1070 feet ofFrench measure.The velocity of sounds being known, the intervals of the pulses are knownalso. For M. Sauveur, by some experiments that he made, found that anopen pipe about five Paris feet in length gives a sound of the same tonewith a viol-string that vibrates a hundred times in one second. Thereforethere are near 10J pulses in a space of 1070 Paris feet, which a sound runsover in asecond of time ; and therefore one pulse fills up a space of about 1 T7-Paris feet, that is, about twice the length of the pipe. From whence it isprobable 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 Prop. XLVIt appears the reason whythe sounds immediately cease with the motion of the sonorous body, andwhy they are heard no longer when we are at a great distance from thesonorous bodies than when we are very near them. And besides, from theforegoing principles, it plainly appears how it comes to pass that sounds areso mightily increased in speaking-trumpets ; for all reciprocal motion useato be increased by the generating cause at each return. And in tubes hindering the dilatation of the sounds, the motion decays more slowly, and24370 THE MATHEMATICAL PRINCIPLES [BOOK II.recurs more forcibly ; and therefore is the more increased by the new motion impressed at each return. And these are the principal phasr. )mena oisounds.SECTION IX.Of the circular motion offluids.HYPOTHESIS.The resistance arisingfrom the want of lubricity in the parts of afluid,is, casteris paribus, proportional to the velocity with which the parts ofthefluid are separated fro?n each other.PROPOSITION LI. THEOREM XXXIX.If a solid cylinder infinitely long, in an uniform and infinite fluid, revolvewith an uniform motion about an axis given in position, and the fluidbe forced round by only this impulse of the cylinder, and every partof the fluid persevere uniformly in its motion ; I say, that the periodictimes of the parts of thefluid are as their distances Jrom the axis ofthe cylinder.Let AFL be a cylinder turning uniformly about the axis S, arid let theconcentric circles BGM, CHN, DIO,EKP, &c., divide the fluid into innumerable concentric cylindric solid orbsof the same thickness. Then, becausethe fluid is homogeneous, the impressions which the contiguous orbs makeupon each other mutually will be (bythe Hypothesis) as their translationsfrom each, other, and as the contiguoussuperficies upon which the impressionsare made. If the impression made upon any orb be greater or less on itsconcave than on its convex side, the stronger impression will prevail, andwill either accelerate or retard the motion of the orb, according as it agreeswith, or is contrary to, the motion of the same. Therefore, that every orbmay persevere uniformly in its motion, the impressions made on both sidesmust be equal and their directions contrary. Therefore since the impressions are as the contiguous superficies, and as their translations from oneanother, the translations will be inversely as the superficies, that is, inverselyas the distances of the superficies from the axis. But the differences ofSEC. IX] OF NATURAL PHILOSOPHY. 371the angular motions about the axis are as those translations applied to thedistances, or as the translations ctly arid the distances inversely ; thatis, joining these ratios together, as the squares of the distances inversely.Therefore if there be erected the lines A", B&, Cc, !.)</, Ee, &c., perpendicular to the several parts of he infinite right line SABCDEQ,, and reciprocally proportional to the squares of SA, SB, SO, SO, SE, &c., andthrough the extremities of those perpendiculars there be supposed to passan hyperbolic curve, the sums of the differences, that is, the whole angularmotions, will be as the correspondent sums of the lines Ati, B6, Cc1, DC/, Ed?,that is (if to constitute a medium uniformly fluid the number of the orbsbe increased and their breadth diminished in infinitum。 as the hyperbolicareas AaQ, B6Q,, CcQ,, Dc/Q,, EeQ, &c., analogous to the sums ; and thetimes, reciprocally proportional to the angular motions, will be also reciprocally proportional to those areas. Therefore the periodic time of anyparticle as I), is reciprocally as the area Dc/Q,, that is (as appearsfrom the known methods of quadratures of curves), directly as the distance SD. Q.E.D.COR. 1. Hence the angular motions of the particles of the fluid are reciprocally as their distances from the axis of the cylinder, and the absolutevelocities are equal.COR. 2. If a fluid be contained in a cylindric vessel of an infinite length,and contain another cylinder within, and both the cylinders revolve aboutone 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 thecylinders.COR. 3. If there be added or taken away any common quantity of angular motion from the cylinder and fluid moving in this manner; yet becausethis 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 thetranslations of the parts from one another depend upon the attrition.Any part will persevere in that motion, which, by the attrition madeon both sides with contrary directions , is no more accelerated than it is retarded.COR. 4. Therefore if there be taken away from this whole system of thecylinders and the fluid all the angular motion of the outward cylinder, weshall have the motion of the fluid in a quiescent cylinder.COR. 5. Therefore if the fluid and outward cylinder are at rest, and theinward cylinder revolve uniformly, there will be communicated a circularmotion to the fluid, which will be propagated by degrees through the wholefluid; and will go on continually increasing, till such time as the severalparts of the fluid acquire the motion determined in Cor. 4.COR. 6. And because the fluid endeavours to propagate its motion stil!372 THE MATHEMATICAL PRINCIPLES [BOOK 11.farther, its impulse will carry the outmost cylinder also about with it, Tinlessthe cylinder be violently detained; and accelerate its motion till theperiodic times of both cylinders become equal among themselves. But ifthe outward cylinder be violently detained, it will make an effort to retardthe motion of the fluid ; and unless the inward cylinder preserve that motion by means of some external force impressed thereon, it will make it3ease by degrees.All these things will be found true by making the experiment in deepstanding water.PROPOSITION LIL THEOREM XL.If a solid sphere, in an uniform and infinite fluid, revolves about an axisgiven in position with an uniform motion., and thejiuid be forced roundby only this impulse of the sphere ; and every part of the fluid perseveres uniformly in its motion ; I say, that the periodic times of theparts of thefluid are as the squares of their distances from the centreof the sphere.CASE 1. Let AFL be a sphere turning uniformly about the axis S, and letthe concentric circles BGM, CHN, DIO,EKP, &cv divide the fluid into innumerable concentric orbs of the samethickness. Suppose those orbs to besolid ; and, because the fluid is homogeneous, the impressions which the contiguous orbs make one upon anotherwill be (by the supposition) as theirtranslations from one another, and thecontiguous superficies upon which theimpressions are made. If the impression upon any orb be greater or lessupon its concave than upon its convex side, the more forcible impressionwill prevail, and will either accelerate or retard the velocity of the orb, according as it is directed with a conspiring or contrary motion to that of