^<^ /^-/^ 
 
 Compliments of 
 
 ELI W. BLAKE. 
 
ORIGINAL SOLUTIONS 
 
 SEVERAL PROBLEMS 
 
 AERODYNAMICS. 
 
 By ELI W. BLAKE. 
 
 NEW HAVEN: 
 TUTTLE, MOREHOUSE & TAYLOR, PRINTERS. 
 
 1882. 
 
.0 
 
 ^\^ 
 
PREFACE. 
 
 The employment of the laws and principles of dynamics in dis- 
 cussing the movements of the parts of an aerial medium caused 
 by the action of a local force impressed upon them, or by their 
 own inherent elastic force causing them to flow towards a total 
 or partial vacuum, has heretofore been thought by mathemati- 
 cians to be attended with great difficulties ; so great indeed that 
 few attempts were made to surmount them, and none that were 
 attended with full and complete success. Hence many problems 
 in aerodynamics, that were of great interest to science and the 
 arts, remained unsolved. • Some years since I became deeply 
 interested in an attempt to solve one of these problems, and 
 after much thought succeeded in devising a method whereby a 
 satisfactory solution was accomplished. Pert^eiving that the 
 same method, with suitable modifications, was applicable to the 
 solution of other problems, I gave my thoughts to others from 
 time to time as I found leisure amid the active pursuits of life ; 
 and the results of these investigations, as they were respectively 
 reached, were published in the then current numbers of the 
 American Journal of Science. 
 
 Believing that the methods employed in those investigations 
 might be employed with success in solving other important 
 problems, and desiring to bring this field of research to the 
 attention of physicists, 1 recently prepared another paper per- 
 taining to this branch of physics and offered it for publication 
 in the Journal of Science ; but the editors, finding in it positions 
 and conclusions which conflicted with their pre-conceived 
 opinions ; and not being disposed to publish that which they 
 Avere not ready to endorse, declined the article. 
 
 Under these circumstances it was decided to print the paper, 
 prefixing to it the articles that had been published in the 
 Journal of Science pertaining to aerodynamics, arranged in the 
 order of their respective dates ; and in this collective, but some- 
 what disjointed form, to present them as a contribution toward 
 a more full development of this interesting and important 
 branch of Physics. 
 
ARTICLE I. 
 
 A Theoretical Determination of the Law which governs 
 the Flow of Elastic Fluids through Orifices* 
 
 The subject announced at the head of this article, is not only 
 interesting considered simply as a subject of scientific inquiry, 
 but it is also a matter of practical importance in its relations to 
 several branches of mechanism. Among these may be instanced, 
 as perhaps first in importance, the bearing of the subject upon 
 the construction of the steam engine. The size of the pipes and 
 valves which conduct the steam to and from the working cylin- 
 der, should be properly adjusted to the size and velocity of the 
 piston. In general, the larger these pipes and valves the better, 
 so far as respects the power of the engine. But there are incon- 
 veniences attendant on making them large; and in order to 
 make a due compromise between the inconvenience that may be 
 incurred on the one hand and the amount of power that may be 
 sacrificed on the other, it becomes necessary to understand cor- 
 rectly the law which governs the flow of elastic fluids through 
 orifices. Treatises on the dynamics of fluids have not omitted to 
 give rules for the determination of such questions; but it will be 
 seen in the course of this article that those rules are very defec- 
 tive. 
 
 If the velocity with which a fluid flows through an orifice from 
 one vessel into another be represented by Y, the density under 
 which it passes the orifice by D, and the area of the orifice by S, 
 then the product VDS is the measure of the quantity of fluid 
 discharged in a given time. It is an established law in the 
 dynamics of fluids, that the velocity of the flow is directly as the 
 
 * Published A. D. 1848, in the American Jotirnal of Science, second series, vol. 
 V, page 78. 
 
 2 
 
square root of the pressure and inversely as the square root of 
 the density. If, then, the eflScient pressure which produces this 
 flow be represented by P, the general law expressed by symbols 
 will be, 
 
 Dsyp 
 
 The above expression is in accordance with the received theory, 
 and properly understood it is correct and applicable to all fluids, 
 elastic as well as inelastic. But it must be observed that D in 
 this expression must in all cases represent the density utider 
 which the fluid passes the orifice. 
 
 In all the treatises on the dynamics of fluids that I have exam- 
 ined, the quantity D in the foregoing expression represents the 
 density of the fluid in the discharging vessel: it being assumed 
 that the fluid passes the orifice without change of density. This 
 assumption is correct so far as respects inelastic fluids, but as 
 respects elastic fluids it is far otherwise. A particle cannot even 
 begin to approach the orifice without a change of density. Sur- 
 rounded by other particles, it will not begin to move until the 
 pressure before it becomes less than the pressure behind it. If 
 the pressure before it is less than the pressure behind it, then the 
 density there is less also, and consequently the density of the 
 particle itself is diminished, for that must be intermediate be- 
 tween the density before and behind it; and as it cannot begin 
 to move without a change of density, so for the same reasons its 
 motion cannot be accelerated without a further change of density. 
 Thus for every increment of its velocity in its approach to the 
 orifice, there must be a corresponding decrement of its density. 
 Hence it is evident that the fluid passes the orifice under a den- 
 sity less than the density in the discharging vessel. 
 
 Again, there is an error in the received theory, in considering 
 the efticient pressure which causes the discharge (represented by 
 P in the above expression), as equal in all cases to the difference 
 of pressure in the two vessels. The true amount which is to be 
 deducted from the pressure in the discharging vessel, in order to 
 find the eflicient pressure that produces the discharge, is the 
 elastic force that is due to the density which the fluid has in its 
 passage through the orifice; for it is obvious that that alone 
 reacts against the pressure in the discharging vessel. From this 
 consideration also we may arrive at the same conclusion as was 
 deduced in the last paragraph, viz: that the fluid must pass the 
 
orifice with a diminished density; for otherwise the elastic force 
 of the fluid in the orifice would he a perfect counterpoise to the 
 pressure, and there could be no flow. 
 
 From what precedes it will be apparent that in the application 
 of the general expression, 
 
 VDS 
 
 -v/D 
 
 to the case of elastic fluids, the density of the fluid in the orifice, 
 represented by D, is an unknown quantity, having a value some- 
 where intermediate between cipher and the density of the fluid 
 in the discharging vessel; also that the efficient pressure which 
 produces the discharge, represented by P, is an unknown quantity 
 whose value is dependent on that of D. We will now proceed to 
 elicit a general rule for the determination of the value of D and 
 P in every case that can occur. 
 
 In the annexed figure let z/ be the dis- 
 charging vessel containing fluid whose 
 density is Zf, and let d be the receiving 
 vessel, which for the present we will 
 consider a vacuum. Let the smallest 
 place in the passage leading from one to 
 the other be the orifice, and let its area 
 be S, and let D represent the unknown density with which the 
 fluid passes the orifice. Since the pressure is as the density, the 
 density may be employed to express the pressure. Then it fol- 
 lows from the preceding observations that the efficient pressure 
 which produces the discharge is z/ — D. Since the velocity will 
 be directly as the square root of the pressure and inversely as 
 the square root of the density, we have, 
 
 x/(z.r-D) 
 
 V 
 
 VO 
 
 .Multiplying this expression by D and reducing, we have, 
 
 VD X -v/(^D-D^)- 
 
 Now if we conceive several sections to be made across the 
 passage at different points on each side of the orifice, and if the 
 areas of these sections are respectively S', S", etc., the velocities 
 of the fluid in them V, V ', etc. and the densities D', D ", etc. 
 V'D'S', V"D"S", etc. are the measures of the quantities of fluid 
 that pass through these sections respectively in a given time. 
 
8 
 
 l>ut when the current is established, the same quantity flows 
 through each in a given time. Therefore VD'S'^Y'D'S' 
 =VDS. Now VDS being a constant quantity, if each of the 
 factors vary, VD will be a maximum when S is a minimum. 
 But S is a minimum at the orifice; and therefore VD is a maxi- 
 mum at the orifice. But we have before found YD x v'C^D — D^); 
 and therefore when VD is a maximum \/(^D— D^) must likewise 
 
 be a maximum. Now, when /^^(z^D— D^) is a maximum D = — . 
 
 Hence, when the discharge is into a vacuum, the density of the 
 fluid at the orifice is equal to half the density in the discharging 
 vessel. 
 
 For convenience in the illustration, we have made the passage 
 from one vessel to the other in the figure, divergent each way 
 from the orifice; but our reasoning would obviously be equally 
 applicable if the orifice opened directly from one vessel into the 
 other, without the intervention of the divergent tubes. 
 
 Let us now inquire, what will be the value of D when the 
 receiving vessel contains fluid of any density less than that in 
 the discharging vessel. 
 
 Let the density in the receiving vessel be d. Theji J is a limit 
 beyond which the fluid cannot expand, either before or after it 
 passes the orifice; so that D can never be less that d. As in the 
 preceding case /s/{AT>—J}^) was the maximum for all the values 
 that can be assigned from cipher to z/, so in this case, and for 
 the same reasons, \/(zJD— D^) must be the maximum for all the 
 values of D that can be assigned between d and Z'. But we 
 
 found in the other case that the maximum occurs when D = — . 
 
 2 
 
 If, then, this value of D is assignable between d and J, the 
 
 maximum must in this case also occur when D = — . But this 
 
 2 
 
 value of D will always be assignable between d and J, if d be 
 
 not greater than — . Therefore if d be any quantity not greater 
 
 A A •. . 
 
 than — , D will be equal to — . In other words, if the density in 
 
 the receiving vessel be not greater than half the density in the 
 discharging vessel, the density in the orifice will be equal to half 
 the density in the discharging vessel. 
 
 Again, from the nature of maxima and minima, it is obvious 
 
9 
 
 that \^[/]l>—D^) will be a maximum when, of all the values that 
 are assignable to D, that value is assigned which differs least 
 
 from — . Hence, if d exceed — , so that D must have a value 
 
 greater than — , then \/(z?D— D^) wall be a maximum when D has 
 
 the smallest value that is assignable to it. Now the smallest 
 value that is assignable to it in this case is D = d. Hence, if the 
 density in the receiving vessel exceed half the density in the 
 discharging vessel, the density under which the fluid passes the 
 orifice is equal to the density in the receiving vessel. 
 
 Thus we have found for the value of D, D = — if c? is not 
 
 greater than — ; if otherwise, D = d. And for the value of P 
 
 (since P = z/-D) we have P = — if d is not greater than — ; if 
 otherwise, P = z/ — f7. 
 
 In applying the general dynamic law VDS x — to the 
 
 case of elastic fluids, the values of D and P should therefore be 
 assigned in accordance with this rule. 
 
 We have already remarked that treatises on the dynamics of 
 fluids, in applying the above general expression to elastic fluids, 
 put D as equal to J, and P as equal to A— d in all cases. This, 
 as will appear from the above rule, makes D too large in all 
 cases; and P also too large whenever d is less than half z/. In 
 .the case of a discharge into a vacuum, it makes each of these 
 quantities double what it should be. In constructing a formula 
 to express the velocity of the flow into a vacuum, these errors 
 balance each other; so that in that particular case the result is 
 the same as if the values of these quantities were assigned in 
 
 2 
 accordance with our rule; for by our rule V x -- = 1, and by 
 
 ^ 2 
 
 the old rule V x ~^-^ = 1. 
 \/z/ 
 
 2 . 
 Again, since — -^ is a constant quantity however z/ may vary, 
 
 ^2" 
 
10 
 
 it follows from our rule that the velocity of the flow into a 
 vacuum is a constant quantity, being the same for every density 
 in the discharging vessel. The same also results from the old 
 theory. 
 
 But in constructing a formula to express the quantity that will 
 flow into a vacuum in a given time, the results of the two rules 
 will differ widely. For since the rule gives the velocity of the 
 flow correctly, and at the same time puts its density at double 
 what it should be, it follows that the old rule makes the quantity 
 discharged in a given time double what it should be. 
 
 Hence it appears that in a steam engine, the valves and pipes 
 which convey the steam from the working cylinder to the con- 
 denser, must be of double the size that, would be assigned to 
 them by the old rule in order to discharge the contents of the 
 working cylinder in a given time, without increased reaction 
 upon the piston. 
 
 It appears from the rule as stated above, for finding the values 
 of D and P, that the quantity c?, which expresses the density of 
 the fluid in the receiving vessel, will not enter at all into the 
 formula which expresses the velocity of the flow, nor into that 
 which expresses the quantity discharged in a given time, pro- 
 vided it be equal to, or less than, half the density in the discharg- 
 ing vessel. Hence it follows that the fluid in the receiving vessel, 
 if its density does not exceed half the density in the discharging 
 vessel, will have no effect whatever upon the flow. Consequently 
 air or steam will rush into a vacuum no faster than into a vessel 
 containing flifid of half its density. On the contrary, both the 
 velocity of the flow and the quantity discharged in a given time 
 will be the same in both cases; and so, also, if the density in the 
 receiving vessel is any quantity less than half the density in the 
 discharging vessel, the flow will be the same in velocity, quantity, 
 and density, as into a vacuum. Accordingly, a vessel containing 
 steam of a density due to a pressure of ten or any other number 
 of atmospheres, will empty itself no faster into a vacuum than 
 into the open air, until in the progress of the discharge, the den- 
 sity is reduced below that due to the pressure of two atmospheres. 
 It will be readily seen that these conclusions have an important 
 bearing upon the construction of the steam engine. 
 
ARTICLE II. 
 
 A Determination of the general law of the Propagation 
 of Pulses in Elastic Media* 
 
 Sir Isaac Newton in his Principia, Book II, prop. 47, 48, 49, 
 has determined the velocity of a pulse, propagated in the atmos- 
 phere, whose intensity differs from cipher only by a very small 
 quantity; or rather, as we shall see in the course of this article, 
 whose intensity is cipher. This velocity he shows to be that 
 which a body would acquire by falling over half the atmospherical 
 subtangent (or half the height of a homogeneous atmosphere); 
 and this he assumes would be the velocity of sound, provided the 
 atmosphere were perfectly pure and perfectly elastic. Lagrange 
 and others have since investigated the same case by different 
 processes, but with the same result, and have concurred with 
 Newton in regarding that result as showing the true theoretical 
 velocity of sound. Both Newton and Lagrange in their respect- 
 ive solutions of the case assume, in effect, that the velocity of 
 the pulse is irrespective of its intensity. Newton at one point 
 (prop. 48, case 1), appears to recognize the fact that an increased 
 intensity would make a difference, but thinks that vmless the 
 pulse were " exceedingly intense," the error would not be sensible. 
 But Lagrange says (Mecanique Analytique, Part II, sec. 1 2, art. 
 14), "the velocity of the pulse is constant and independent of 
 the primitive movement, which is confirmed by experience, as all 
 sounds strong or weak appear to be propagated with the same 
 velocity." 
 
 It is proposed in this article to show that the velocities of 
 pulses vary with their intensities, and to determine, in general 
 
 * Published A. D. 184P, in the American Journal of Science, second series, vol. 
 V, p. 372. 
 
12 
 
 form, the relation which subsists between the velocity and inten- 
 sity of pulses. This we shall do by solving the general problem 
 of the velocity of pulses by a new process, which comprehends 
 the intensity of the pulse as an essential element. Having done 
 this, we shall see the relation in whicii the case solved by Newton 
 stands to the general law; and if we mistake not, it will then be 
 apparent that the velocity found in that case is not the velocity 
 of sound, but a lintit below which its velocity cannot fall. 
 
 A pulse, considered as propagated through a line of particles, 
 and considered with reference to its physical condition at any 
 instant of time, consists of a series of contiguous particles in 
 that line, greater or less in number, which are more dense than 
 the particles before and behind them on the line, and which are 
 in motion with some velocity, while the particles before and 
 behind are at rest and in their natural state of density. This 
 series of particles, as it advances, encounters successively the 
 stationary particles, compressing them to the same density, put- 
 ting them in motion, and thus adds them to the series. In the 
 mean time, an equal number of the posterior particles of the 
 series expand, resuming their natural density, and come to rest. 
 It is obvious that if the propelling force due to the reaction of 
 the particles expanding from the }iosterior extremity of the series, 
 is equal to the retarding force of the particles encountered by 
 the anterior extremity (as must always be the case if the elasticity 
 is perfect, and if the action is confined to the particles in the 
 line), then the pulse will continue to advance indefinitely, and 
 with a uniform velocity; a velocity however which, as we shall 
 see, is not independent of the degree of condensation to which 
 the particles are brought. 
 
 Fig. 1. 
 C D 
 
 A JL I Tl 
 
 Let C be a point where the density is a maximum in a pulse 
 which is moving toward B. It is not material to our present 
 purpose to inquire whether the place of maximum density is a 
 mere point, or whether it extends over some finite space on the 
 line AB. In either case, somewhere in advance of C we shall 
 find particles in every stage of density from the natural to the 
 maximum state; and these will be arranged in the order of their 
 density, the more dense being toward C Each of these })articles 
 will be accelerated so long as the density of the jjarticle behind 
 
13 
 
 it is greater than that of the particle before it, and no longer- 
 Consequently, each will have its maximum velocity when it 
 reaches its maximum density. Therefore, if C is a point where 
 the density is a maximum, it is likewise a point where the velocity, 
 which the pulse gives successively to all the particles, is a 
 
 maximum. 
 
 Fig. 2. 
 
 Let mn be the space which a particle in its natural state occu- 
 pies on the line AB, fig. 1 and let sn be that which it occupies in 
 its most condensed state. Let D be a point in the line AB, in 
 advance of C, where the particles are at rest, not having yet felt 
 the influence of the approaching pulse; and let the two imaginary 
 points C and D be conceived to move with the same velocity as 
 the pulse. Then each travels over the same space and through 
 the same number of particles in a given time. Consequently 
 while C moves over a space equal to that occupied by a particle 
 in its natural state, it only moves through one particle in its most 
 condensed state; that is to say, while C moves over the space mn, 
 it moves, relatively to the particle through which it passes, only 
 over sn. Consequently the particle itself moves in the same time 
 over a space equal to ms. Hence when mn represents the velocity 
 of the pulse, ms represents the final velocity which the pulse 
 gives to every particle through which it passes. 
 
 Let H be the atmospherical subtangent; or the length of the 
 column of particles of the natural density whose weight is equal 
 to the elastic force of a particle in its natural state, and let Yi-\-h 
 be the length of a similar column whose weight is equal to the 
 elastic force of a particle at the maximum density. Since the 
 space occupied by a particle is inversely as the compressing force, 
 we have, 
 
 mn : soi : : Yl-\-h : H or nis-^sn :sn :: H-\-h : H ; 
 
 sn X h 
 
 consequently ms : sn : : A : H. Whence H= 
 
 ins 
 
 The force which accelerates all the particles in advance of C, 
 which have felt the influence of the pulse but have not yet 
 reached their maximum velocity, is the difference of the elastic 
 forces which correspond to the natural and the maximum densi- 
 ties. This is a constant force, and must evidently be that force 
 which is competent to give the velocity ms to all the particles in 
 3 
 
14 
 
 any space in the time in which the pulse runs over that space. 
 Let us suppose the pulse runs over the space h. Then the pulse 
 runs over h in the time in which the difference of those elastic 
 forces will give to all the particles in h the velocity ms. But the 
 difference of the elastic forces is equal to the weight of all the 
 particles in h. Therefore the pulse runs over A in the time 
 in which those particles would in falling by their own weight 
 acquire the velocity ms. 
 
 The time in which a falling body acquires the velocity ms is to 
 the time in which it would acquire the velocity of the pulse or 
 mn, as ms to mn, and the spaces over which the pulse would run 
 in these times are as the times and therefore as ms to mii. There- 
 fore putting S for the space which the pulse would run over 
 while a falling body would acquire the velocity of the pulse, we 
 have ms :mti :: h :S. Whence 
 
 g, mnXh ms-\-snXh snX^h , . 
 
 o=- =z =: \-n 
 
 ms ms ms 
 
 But we have before found H = ^^^^- Substituting H for its 
 
 7ns 
 
 value in the preceding equation, we have S = H-fA. If then 
 
 we put \\-{-h for tlie velocity of the pulse, the space through 
 
 which a body must fall to acquire that velocity will be — — 
 
 In this expression h is to be regarded as the intensity of the 
 pulse: it being the length of that column of particles which must 
 be superadded to the height of a homogeneous atmosphere, in 
 order to produce in the air that degree of increased condensation 
 which the pulse effects in the particles through which it passes. 
 
 If in the expression last found we make A = 0, the expression 
 
 TT 
 
 becomes — . This is the result arrived at by Newton, and which, 
 
 as we have already remarked, was regarded by him and is now 
 generally received as the theoretical formula for the space through 
 which a body must fall to acquire the velocity of sound. But it 
 is evident from our demonstration that the velocity due to that 
 space, instead of being the velocity of any assignable pulse is 
 simply a limit below which no pulse can be propagated in an 
 elastic fluid whose subtangent is H.* 
 
 * Newton's demonstration of this problem has been regarded by several dis- . 
 tmguished maihematieians as obscure and ineonchi.sive. It commences witli the 
 hypothesis that a particle put in motion b}' a pulse is accelerated and retarded 
 
15 
 
 A pulse which produces the sensation of sound must produce 
 real motion in the particles through which it passes. In such a 
 pulse h must have some finite magnitude. Nor can that mag- 
 nitude be by any means the smallest that is competent to pro- 
 duce motion in the air; for if such were the fact, then the 
 slightest impulse given to the air by a vibratory movement, 
 even waving the hand in it, should produce the sensation of 
 sound. The intensity of a pulse which is competent to produce 
 that sensation, will of course vary with the sensibility of the 
 ear which is to receive it; and consequently the nature of the 
 case does not allow us to assign any definite magnitude to the 
 minimum intensity of sonorous pulses; but we know by experi- 
 ence that the velocity of the particles in which a pulse originates 
 must be great in order to produce the sensation of sound in the 
 most delicate ear. We also know that in the case of the heavier 
 sounds, as the report of cannon, the condensation of the particles 
 in which the pulse originates is very intense. 
 
 This view of the subject may throw some light upon the 
 discrepancy between the theoretical velocity of sound, as deter- 
 mined by Newton and others, and its real velocity as found by 
 experiment. The velocity according to that theory should be 
 about 944 feet in a second; varying slightly from this according 
 to the state of the barometer, thermometer, and hygrometer. 
 
 The velocities 
 found by experi- \ 
 ment, by 
 
 Roberts, 1300 
 
 Boyle, 1200 
 
 Mersenne, .... 1474 
 
 Flamsteed and Halley, 1 142 
 
 Florentine Academy, 1148 
 
 French Academy, .. 1172 
 
 according to the law of the oscillating pendulum. Gabriel Cramer (see Glasgow- 
 edition of Xewton's Principia, Book II, prop. 48, notes), objects to the result 
 arrived at by Newton, that it flows from his hypothesis and not from the nature 
 of things. To prove this he deduces the same result upon the hypothesis that 
 the panicle is accelerated and retarded by a constant force. The fact that 
 Cramer's hypothesis answered just as well for the solution of the problem as 
 Newton's, seems not a little to have puzzled the editor of the edition of the 
 Principia referred to, who devotes several pages of notes to the vindication of 
 Newton's result from its supposed bearing. But the enigma is solved when we 
 consider that both Newton and Cramer regard the space through which the 
 particle vibrates as an infinitesimal quantity. In such case, evidently it can make 
 no difEerence what is assumed as the law of acceleration ; this being a point at 
 whieh all laws coalesce. 
 
16 
 
 Newton adopts the smallest of these experimental velocities, 
 viz: 1142 feet per second, as the true practical velocity of sound; 
 and to this he reconciles his theory; in part, by a hypothesis that 
 the air consists, to a certain extent, of solid particles, through 
 which the pulse is transmitted instantaneously; and in part by 
 another hypothesis, that the pulse does not give motion to the 
 foreign matter which the air contains, and so is transmitted so 
 much the faster through the true air as there is less of it in a 
 given space. This explanation of the matter has not been sat- 
 isfactory to those who have followed Newton in investigating 
 this subject. They have justly thought that such causes might 
 retard, but could not accelerate the pulse. Various other hy- 
 potheses have been successively proposed and rejected, and a 
 vast amount of labor has been exijended in the effort to reconcile 
 theory and practice in this case. Of these hypotheses, we Avill 
 mention only that which Laplace is said to have regarded as the 
 true one, viz: the increased elasticity in the air produced by the 
 heat evolved by condensation. A little reflection will serve to 
 show that this cause also may retard, but cannot accelerate the 
 pulse. In order that the force of the pulse may be maintained 
 without loss, the propelling force derived from the reexpan- 
 sion of the particles must be not less than their retarding force 
 in being compressed; and in order to this, whatever heat is 
 evolved during their compression, must be reabsorbed in their 
 reexpansion. If, then, any time is required either for the evolu- 
 tion of the heat, or for its absorption, so that the specific heat 
 of the particle does not instantaneously conform to the change 
 of density; or if any portion of the heat evolved is radiated and 
 lost, so as not to be present to be reabsorbed, then the evolution 
 of heat must retard the pulse. Otherwise it cannot affect the 
 velocity of its propagation. 
 
 In the view of the subject we have taken, it will cease to be 
 a matter of surprise that the velocity of sound should be found 
 to be greater than that assigned by Newton's theory; as, also, 
 that the experimental velocities should be found to differ greatly 
 among themselves, however carefully the experiments may have 
 been tried. 
 
 When the velocity of a pulse is given, we may find its intensity 
 
 V2 
 
 by the formula hz= H. We may find the maximum density, 
 
 Oil 
 
 V« 
 
 the natural density being 1, by the formula, max. dens. = ^^fj« 
 
17 
 
 We may find the space occupied by a particle at the maximum 
 density, its natural extent being 1, by the formula, extent at 
 
 32H 
 
 max. dens. = ^v"„— . 
 
 V ^ 
 
 When the velocity is 1142 feet per second, if Ave put the sub- 
 tangent H = 27818 feet, we have, 
 
 Intensity of the pulse, h = 12937 feet. 
 Maximum density, 1*465. 
 Extent of particle, 0*682. 
 
 If we take for the given velocity, that Avhich a body Avould 
 acquire by falling through the subtangent, or that with Avhich 
 air would rush into a vacuum, we shall find h = H, and the par- 
 ticles will be compressed into half their natural size, arid the 
 density Avill be double the natural density. 
 
 A condensing force equivalent to the pressure of a column 
 12937 feet high, and which compresses the particles into about 
 -^-^ of their natural size, is a pulse "exceedingly intense" as 
 compared with that which Newton supposes; but if our solution 
 of the problem is correct, it is physically impossible that a pulse 
 of less intensity should propagate itself in our atmosphere with 
 a velocity of 1142 feet in a second. 
 
>^ OP TFTR 
 
 ;UNIVBIlSIT7l 
 
 ARTICLE III. 
 
 On the Mode of Expansion of Elastic Fluids as controlled 
 by Dynamic Laws* 
 
 That under the controlling influence of the known laws of 
 motion, elastic fluids must expand according to some definite and 
 invariable law, is an obvious truth and one which has often been 
 recognized by mathematicians. But the determination of that 
 law is a problem which hitherto, it is believed, has not been 
 solved. There are many interesting points in mechanics and 
 physics, in relation to which the present state of knowledge is 
 imperfect, which depend for their correct and complete develop- 
 ment, in part at least, on a solution of this problem. It is there- 
 fore a point of some interest to science. It is our purpose in this 
 article to solve this problem; and we shall do so by employing 
 a method similar in part to that employed in solving the problem 
 of the propagation of pulses in elastic media, in the American 
 Journal of Science, second series, vol. v, page 372. 
 
 Before entering upon the investigation we will here state one 
 curious and remarkable fact which the investigation discloses. 
 We revert to it hert because a fact so much at variance with pre- 
 conceived notions may be interesting to those readers who will 
 not care to follow out the mathematical details of this article. 
 
 When a fluid passes by free expansion from one state of density 
 to another, we should naturally suppose that it must pass through 
 all the intermediate states of density that can be assigned between 
 the two. Such appears to have been the notion of every writer 
 who has made reference to this point; and at first view it would 
 seem absurd to suppose that the fact could be otherwise. But 
 
 * Published A. D. 1850, in the Amerioan Journal of Science, second series, vol. 
 ix, p. 334. 
 
20 
 
 such is not tlie way in which elastic fluids expand. On the con- 
 trary the parts of the fluid successively and inMantcweoiisly 
 change their density, to the extent of one-half (when free to 
 expand to that extent) without passing into the intermediate 
 slates. As vapor is thrown off from the surface of water in a 
 tenuous state ab initio, and without having first passed into those 
 states of density which are intermediate between the density of 
 the w ater and that of the vapor, so a column of rarefied fluid is 
 thrown of from the front of a denser colunin; each infinitesimal 
 element of the highest order of the denser column, being success- 
 ively and instantaneously transformed to the more rare state. 
 And as the change of density is instantaneous, so likewise the 
 entire velocity due to that change is imparted instantaneously to 
 each element successively. 
 
 But to proceed with the investigation; suppose a straight tube 
 of uniform calibre extending indefinitely in both directions from 
 a given point. Suppose the tube on one side of this point to be 
 filled with a column of fluid of the density D, indefinitely expan- 
 sible, and always maintaining the same ratio between its density 
 and elastic force when it expands; and suppose the other portion 
 of the tube a perfect vacuum. It is required to determine the 
 law according to which the fluid expands into the vacuum; so 
 that we may be able to assign the precise state, of the fluid, in 
 respect to density and velocity, at each and every point of the 
 tube after the lapse of any given time from the commencement 
 of the expansion. 
 
 Since the elastic force is always as the density, D may repre- 
 sent both the density and the elastic force. The force D acts 
 during the first instant in every part of the column, and in every 
 direction; and therefore during that instant every part of the 
 column is kept in equilibrio except the first element. Conse- 
 quently in the first instant expansion takes place in the first 
 element only, and as the w^hole force D acts during that instant, 
 the parts of this element must receive such velocities that the 
 sura of their momenta shall be equal to that due to the action of 
 the constant force D during that time. It is obvious that the 
 termination of the first instant coincides with the commencement 
 of motion in the second element; also that motion will not com- 
 mence in the second element until the density in front of it has 
 been to some extent reduced. Let the ratio in which it is reduced 
 
 before motion begins in the second element be represented l)y . 
 
21 
 
 Then the density of the posterior part of the expanded element 
 at the end of the first instant is — Now for reasons which will 
 
 X 
 
 soon be apparent, all the other parts of the expanded element, 
 whatever may be their present state of density, may be con- 
 sidered as having passed first into the density — But at the 
 
 same time that the grade — began to form in front of the 
 column, that grade itself must have begun to expand again in 
 the same ratio, forming another grade — • And at the same 
 
 time that the grade — ^ began to form, that likewise must have 
 
 begun to expand in the same ratio forming a grade — , ? and so 
 
 on ad infinitum. The grades therefore will correspond to the 
 terms of an infinite series, in decreasing geometrical progression. 
 All of them originate simtdtaneouslii in the first element; and 
 yet every grade respectively may be considered as having passed 
 into and out of all the grades which precede it ; inasmuch as each 
 in its origin is a constituent part of that which precedes it. The 
 fluid which passes into any one of these grades in the first instant 
 does not all of it pass into the next in the same time; for equal 
 quantities by measure expand in equal ratios in equal times; and 
 since a given quantity by measure in any one grade becomes a 
 larger quantity by measure when expanded into the next grade, 
 a portion will have been left at the end of the first instant in each 
 grade which has not expanded into the next. Hence at the end 
 of the first instant the first element of the column will have been 
 distributed into portions or grades, having their respective densi- 
 ties corresponding to the terms of the infinite series 
 
 D D D I) 
 
 x^' ^' a;" ic" ^^^* ^ ' 
 
 If we extend this series backward one term, we obtain the 
 series 
 
 ^ D D D D 
 
 ^' x' ^' ^' x'' ^^''- ^^^ 
 
 tK' wj *Kf ftt/ 
 
 Since equal, quantities by measure pass out of each of thesg 
 states in a given time, if s be the space occupied by the original 
 4 
 
22 
 
 element, and if we multiply each of the terms of the series (B) 
 by s, then the terms of the resulting series 
 
 ^ D« Ds Ds D« 
 
 Ds, — , —5-, —5-, — J-, etc. (C) 
 
 X X X X 
 
 will severally express the quantities of fluid that expand from 
 each grade respectively into the next. Now since fluids expand- 
 ing in equal ratios acquire equal velocities, equal velocities are 
 acquired in each of these expansions. If then we find that 
 velocity and by it multiply the sum of the series (C), the product 
 will be the sum of the momenta generated in, or imparted to, the 
 parts of the first element in the time in which the point of expan- 
 sion recedes through s. 
 
 If the quantity Ds be expanded from the density J) to the den- 
 sity — , the space it will occupy will be increased in the inverse 
 
 ratio of these densities; and, therefore, — :D : :s: sx. Hence s and 
 
 X 
 
 sx are respectively the spaces occupied by the element before and 
 after the first expansion. Now the velocity which the mass D« 
 receives in this expansion, is obviously that which would carry it 
 over the difference between these spaces in the time in which the 
 expansion takes place; that is, the velocity imparted in the first 
 expansion is sx — s:=8.x—l; and the same velocity is imparted in 
 every other expansion. If, then, we multiply the sum of the 
 series (C) by s.x—\, the product will be equal to the sum of all 
 the momenta generated in the parts of the element. This pro- 
 duct is Ds^x. Therefore Ds^x is the entire amount of momentum 
 which the force D is competent to generate in the time in which 
 the point of expansion recedes through s. 
 
 We will now proceed to find another expression for the mo- 
 mentum which the force D is competent to generate in the same 
 time, in order that by comparing it with that just found, we may 
 ascertain the value of x. 
 
 Let H be the height of a column of fluid of the density D, 
 whose weight is equal to the elastic force D; and let II — A be 
 the height of another column of the same density whose weight 
 
 is equal to the elastic force — . Then — :D: :H:H— A. Let mw 
 
 ^ XX 
 
 be the space occupied by the first element at the density D, 
 
 ?u n s . 
 
 I I I 
 
23 
 
 « 
 
 and ms that which it occupies when expanded to the density—. 
 Since the spaces occupied by the element in these states are 
 inversely as the densities, mn : ))ts ::--:]):: H— A : H, and there- 
 
 fore, ijis—.^n -.uis: : II— A: 11: whence we obtain II = . In 
 
 the time in which the point of expansion recedes through inn, 
 the element Ds receives a velocity which will carry it over sn in 
 the same time. If, then, 7n)i represent the velocity of the point 
 of expansion, sn will represent the velocity imparted to the fluid 
 by the first expansion. Consequently, the retrogressive velocity 
 of the point of expansion must be such that in the time in which 
 
 it passes over any space, the force D may give to all the fluid 
 
 in that space the velocity sn. Hence the point of expansion will 
 
 run over h in the time in which the force D will give to all 
 
 the fluid in h the velocity sn. But the force D is equal to 
 
 the loeight of all the fluid in A. Therefore the point of expansion 
 runs over A in the time in which the mass A would in falling by 
 its own gravity acquire the velocity sn. The time in which a 
 falling body acquires the velocity S7i is to that in which it would 
 acquire the velocity of the point of expansion, or inn as sri to ?//«/ 
 and the spaces over which the point of expansion would run in 
 these times are in the same ratio. Therefore, putting S for the 
 space which the point of expansion w^ould run over while a falling 
 body would acquire the velocity of the point of expansion, we 
 have s)i : mn : : A : S, or, sn : ms — sn : : A : S; whence we obtain 
 
 hXr"S , r, ^ 1 1 +• 4.- ^ Ti f'iXrns ,„. 
 
 b= A. Hut we have before lound H= . Ihere- 
 
 sn sn 
 
 fore S=II— A; that is, the point of expansion will run over H— A 
 
 in the time in which a falling body will acquire the same velocity. 
 
 Consequently the velocity of the point of expansion is that which 
 
 XT I 
 
 a body will acquire by falling through . 
 
 If the force D act on the mass H during the time that mass 
 would fall through H, it would give that mass a velocity which 
 would carry it over 2H in the same time, because the force D is 
 equal to the weight of the mass. The mean velocity of a body 
 falling through H is that w hich will be acquired by falling through 
 
24 
 
 — . If, then, the point of expansion moved with the velocity 
 
 H . ' 
 
 acquired by a body in falling through — , in the time of passing 
 
 over H, the force D would be competent to give to all the fluid 
 in H a velocity which would carry it over 2H in the same time; 
 and, consequently, in the time of passing over s it would give to 
 the mass l)s a velocity which would in the same time carry it 
 over 25. But the point of expansion, as before shown, moves 
 
 with the velocity acquired by falling through — - — . Xow the 
 
 , . , H . ^ ^ H-A /H /H-A 
 
 velocity due to — - is to that due to — — as 4/ -j^^ \/ — x — ; and 
 
 the times in which the point of expansion would move over s 
 
 T H 
 with these velocities, are inversely as these velocities, or as — 
 
 /i 
 
 1 
 
 to — ^-. But D:—: :H:H— A, and therefore these times are 
 2 X 
 
 as — to — , or as 2 to a/2x. The velocitv which the force 
 4 2x ^ 
 
 D can impart in these times is as the times respectively. And 
 
 since it has been shown that in the former of these times the 
 
 velocity 25 will be imparted by the force D, we have 2',^2x;i 
 
 2s\/2x 
 2s', —:=sW2x. That is, the velocity which the force D is 
 
 competent to impart to the mass Ds in the time in which the 
 point of expansion recedes through s, is 5y^2.»'. Consequently 
 the momentum which the force D can imjjart in the same time is 
 Ds* a/2.1'. But we have before found this momentum to be Ds^x. 
 Therefore \)s^x—\)s^ ^2x, whence x=y/2x and a'=2. 
 
 Having thus found the absolute value of a*, if we substitute 
 this value for x in the series (A) we shall have, for the densities 
 of the several parts or grades into which the first element will 
 have been distributed at the end of the first instant, the respect- 
 ive terms of the following series, viz: 
 
 D D D D D 
 
 2'T' 8' 16' 64' 
 
 We found the velocity of the point of expansion to be that 
 
25 
 
 which a body will acquire hy falling through — — ; the value of 
 
 H-A H 
 
 h being dependent on the value of x. But when a;=:2, — — =— . 
 
 Therefore the absolute velocity of the point of expansion is that 
 
 TT 
 
 which a body will acquire by falling through — , or one-fourth of 
 
 the subtangent of the fluid. 
 
 Since the extent of the element is doubled by the first expan- 
 sion, the velocity of the first grade will be equal to the velocity 
 of the point of expansion, or that due to one-fourth of the sub- 
 tangent; and an equal additional velocity is imparted in each 
 succeeding expansion. If, then, v represent the velocity due to 
 one-fourth the subtangent of the fluid, the absolute, velocities of 
 the several grades respectively will be expressed by the respect- 
 ive terms of the series v, 2v, 3y, 4u, 5v, etc. 
 
 Since one element =s by measure passes from each grade into 
 the next, and becomes =2s in the next, the length of each grade 
 at the end of the first instant =2s—s=iS. That is, the length of 
 each grade is equal to that of the original element; and the place 
 of the first grade is that which was occupied by the original 
 element, the other grades succeeding it in continuous order. 
 
 Having now ascertained the state of things at the end of the 
 first instant, let us inquire what takes place in the second instant. 
 
 It is obvious that during the second instant tlie front of the 
 second element of the column, and also the front of each grade 
 respectively is a point of expansion from which one element =.s 
 by measure passes into the next grade. Thus in the second 
 instant each grade receives an addition of 2s to its rear and loses 
 Is from its front. The same takes place in every succeeding 
 instant. Since the increment of the length of the grades for each 
 instant is s, the velocity of the increase is v. The length of the 
 grades is therefore always equal to the space through which the 
 point of expansion has receded in the column. Thus while the 
 length of the grades increases with the uniform velocity v, their 
 number, velocity and density remain unchanged. Consequently 
 no other gradations of density can exist in front of a column 
 expanding into a vacuum, but those which correspond to the 
 
 terms of the infinite geometrical series — , — , — , — , etc.: and no 
 
 2' 4' 8' 16 ' 
 
 other gradations of velocity but those which correspond to the 
 
 terms of the infinite arithmetical series v, 2v, 8y, 4w, etc. 
 
26 
 
 The point of expansion in the colunm recedes with the velocity 
 v; and since the length of the first grade is always equal to the 
 space through which that point of expansion has moved, it fol- 
 lows that the point of expansion from the first to the second 
 grade is stationary. And since the second grade increases in 
 length with the velocity v, the third point of expansion moves 
 forward with the velocity v; and since all the other grades 
 increase in length with the same velocity v, the velocities of the 
 several points of expansion will be expressed by the following 
 series — v, 0, v, 2v, Sv, 4w, etc. 
 
 In order to give a synopsis of the results to which we have 
 come, let AB be a column of fluid of the density T), expanding 
 into a vacuum toward C. Let the velocity due to a height equal 
 to one-fourth of the subtangent of the fluid be v. Suppose 
 expansion to have commenced at B, and the point of expansion 
 to have receded to any distance ?i. Set off from B an infinite 
 number of spaces B«, ah, be, cd, etc., each equal to, B;?. Then 
 the points n, B, a, h, c, d, etc., are the plac^ of the points of 
 expansion, and the boundaries of the several grades, or parts 
 having different degrees of density and velocity, into which the 
 original mass Bn has been distributed. 
 
 A n B « & c (Z etc. 
 
 C^ 
 
 Between these points respectively the densities are 
 
 D D D D D 
 
 2 ■ 4 ■ 8 * 16 * 64 ■ ^ ^' 
 The velocities are 
 
 y . 2u. 3w. 4y . 5y . etc. 
 
 These points move toward C with the velocities 
 
 — V, 0, V, 2u, 3v, 4y, etc., 
 
 and relatively to each other, and to the fluid, with the velocities 
 
 V, V, V, V, W, V. 
 
 As corollaries from the preceding investigations we may state 
 the following propositions: 
 
 1. No other gradations of density can exist in front of a col- 
 umn of fluid which is expanding toward a vacuum except tliose 
 which are found by successive divisions of the original density 
 by 2. 
 
27 
 
 2. The change of density in the fluid in passing from one of 
 these grades to the next is not gradual but instantaneous; so 
 that .the grades are constantly separated from each other by a 
 mere imaginary plane. 
 
 3. No other velocities can exist among the parts of a fluid 
 which is expanding toward a vacuum but such as are multiples 
 of the velocities which a body will acquire by falling through 
 one-fourth of the subtangent of the fluid. 
 
 4. The velocity imparted to the particles of an expanding fluid 
 is not the result of a continual and gradual acceleration, but of 
 successive instantaneous increments equal to that which a body 
 will acquire by falling through one-fourth of the subtangent of 
 the fluid. 
 
 It now remains to consider the mode of expansion when the 
 fluid is not free to expand indefinitely, but has its expansion 
 arrested at some given density d. 
 
 It is obvious that if d correspond in value to any of the terms 
 of the series, the manner of expansion up to that point will be 
 the same as if the expansion were continued indefinitely. There 
 will therefore be in the expanding fluid, in such case, so many 
 grades corresponding to the terms of the series, as there are of 
 complete terms intervening between D and d. But let us inquire 
 what takes place when d does not correspond to any terms of the 
 series. First, suppose d to be greater than the first term. Then 
 from what has been before shown, the velocity of the point of 
 expansion i* that which a body will acquire by falling thi-ough 
 
 TT L 
 
 when H - h is the height of a column whose weight is 
 
 equal to the elastic force of the expanded fluid; also that the 
 
 LT 
 
 velocity of the point of expansion is that due to the height — 
 
 when the expansion is from D to — . These velocities are as 
 
 4/ — to ^ — - — and since H : II — A : : D : f/, those velocities 
 
 are as 4/ — to a/ —. The velocity due to — is v. Hence we 
 
 have \/ -y '• \/ — ' '■ V : va/ -^ =z velocity of the point of expan- 
 sion in this case. 
 
 Let us next find the velocity of the fluid. The times of run- 
 
28 
 
 niiig over s by the point of expansion, with the velocities 4/ — 
 
 and |/ - are inversely as these velocities ; and the velocities 
 
 imparted to the mass D.s- in these times are as the products of 
 the times by the respective forces. When the velocity of the 
 
 point of expansion was 1/-7- the force was — and the velocity of 
 
 the fluid was v. The force in the present case is D — f?. Hence 
 D 
 
 ~2 D-d B-d , . 
 
 we have — -^ : — — - : : y : vy'2 . —jf^ = velocity of the fluid in 
 
 VT Vj ^ 
 
 this case. 
 
 Secondly, suppose the value of d to fall between any two con- 
 secutive terms of the series. It is obvious that we have now 
 only to substitute in the expression last found that term of the 
 series which is next greater than d for D, and it will then express 
 the acceleration due to expansion from the last complete term 
 into the fractional grade. 
 
 To find the retrogressive velocity of the point of expansion, 
 relatively to the fluid, in the grade which precedes the fractional 
 grade, we must make the like substitution of the last complete 
 
 /2(l 
 term for D in the quantity u^/ ^ found above. The retrogres- 
 sive velocity of the point of expansion in the grade which pre- 
 cedes the fractional grade is greater than in the other grades, and 
 of course that grade will be shorter than the others in the same 
 ratio. This is the only modification which a fractional grade 
 produces in those that precede it. In all other respects the mode 
 of expansion, up to the fractional grade, corresponds to the view 
 presented in the foregoing synopsis. 
 
 We are now prepared to construct a formula for the final 
 velocity of a fluid which expands from any density D to any 
 other density d. 
 
 Let V be the final velocity; v the velocity due to a height 
 
 equal to one-fourth the subtangent of the fluid; ji the number of 
 
 . D D D D ^. ^ . 
 
 complete terms or the series — , — , — , — , etc., which intervene 
 
 between D and d. Then u» is obviously the velocity of the 
 grade which precedes the fractional grade, if there be a fractional 
 
29 
 
 grade. When the first grade is fractional we found its velocity 
 
 to be V a/2. — TT^j'y and we also found that to suit this expres- 
 
 sion to the case of a fractional grade occurring elsewhere in the 
 range of the series, we are to substitute for D that term of the 
 series which is next greater than d. Now the value of that term 
 
 will be — . Making the substitution accordingly, the expression 
 
 for the additional velocity due to expansion into the fractional 
 
 grade becomes, after reducing wy'2 . - y y- ^J adding this 
 
 quantity to vn we obtain the final velocity of the fluid, resulting 
 from its expansion from any density D to any other density d. 
 Hence the formula is 
 
 , /^ D — 2"(7 
 
 When there is no complete term of the series between D and d, 
 71 = and the above formula becomes 
 
 When there is no fractional grade, that is, when d is equal to 
 some term of the series, that part of the formula beyond n equals 
 0, and then the above formula becomes V=: vn. 
 
 From the general principles here developed it is obvious that, 
 as in expansion, so likewise in condensation, the transition of 
 an elastic fluid from one density to another is not by gradations 
 which may be represented by a curve, but abrupt, instantaneous, 
 per saltum vel saltus. Pulses, therefore, which are propagated in 
 elastic fluids partake of the same character; that is, the condensa- 
 tion and subsequent reexpansion of the successive elements 
 through which the wave moves is instantaneous. This fact was 
 not known when the article on the propagation of pulses, referred 
 to at the commencement of this article, was written. It however 
 does not affect the validity of the reasoning in that article. 
 
ARTICLE IV. 
 
 Experimental Demonstration of the Law of the Flow of 
 Elastic Fluids which was deduced theoretically in 
 Article I.* . 
 
 Ix volume V, second series, of the American Journal of 
 Science, page 78, I proposed a new theory of the flow of elastic 
 fluids through orifices, differing essentially from that heretofore 
 received. The chief object of the present article is to give an 
 account of an experiment instituted for the purpose of testing 
 the truth of that theory. 
 
 The fundamental points of difference between the old theory 
 and the new, are as follows: 
 
 1. The old theory regards the constant force which expels the 
 fluid as being, in all cases, equal to the difference between the 
 elastic forces of the fluids in the two vessels. 
 
 The new theory regards it as equal to that difference only 
 when the less exceeds half the greater; and in all other cases as 
 equal to half the greater. . 
 
 2. The old theory considers the fluid as passing the orifice with 
 a density equal to that in the discharging vessel. 
 
 The new theory considers it as passing the orifice with a 
 density equal to that in the receiving vessel, whenever this last 
 is equal to or greater than half the density in the discharging 
 vessel; and in all other cases with half the density in the dis- 
 charging vessel. 
 
 The formula for the quantity discharged in a given time, 
 predicated upon the new theory, gives, in all cases, less than 
 
 * Published A. D. 1851, iu the American -Journal of Science, second series, 
 Vol. xii, p. 186. 
 
32 
 
 that predicated upon the old theory. In tlic case of a flow into 
 a vacuum, the difference amounts to one-half. 
 
 The scheme devised to test the relative merits of the two 
 theories, was founded upon the following considerations, viz: 
 When air rushes from the atmosphere into a receiver wholly or 
 partially exhausted, passing on its way through a small inter- 
 mediate vessel or chamber, entering that chamber and passing 
 out of it through equal orifices, it will take in that chamber a 
 density somewhere intermediate between that of the atmosphere 
 and that in the receiver. For each relation that may at any 
 moment subsist between the density of the atmosphere and that 
 in the receiver, thf density in the chamber will have a certain 
 definite and determinate value, such that the chamber may 
 receive through one orifice and discharge through the other 
 simultaneously the same quantity of air. Nom' since in order to 
 this equal simultaneous flow the two theories respectively demand 
 quite different densities in the chamber, the object of ray experi- 
 ment was to ascertain the actual densities in such a chamber 
 under various relations of the density in the receiver to the 
 density of the atmosphere, in order to compare the densities thus 
 ascertained experimentally with those demanded by each theory 
 respectively in like circumstances. 
 
 To try the experiment, I constructed the apparatus shown in 
 figure 1. A i-^ a vessel or receiver of tlie capacity of about 
 fifty gallons, so arranged that it may be exhausted l)y the 
 
83 
 
 air-pump or otlierwise. B is an elbow formed of lead pipe of 
 one inch calibre, one branch of which opens into the receiver, 
 and the end of the other branch at C is covered by a brass plate 
 or disc about ^ig^th of an inch in thickness, through which is an 
 orifice of about jig-th of an inch in diameter. Another similar 
 plate with an orifice of the same size intersects the pipe at D, 
 thus forming a chamber between the two plates. Two short 
 tubes are inserted into the lower side of the pipe; one on each 
 side of the plate D. With these short tubes two glass tubes in 
 and », each thirty-tliree inches in length, are connected by means 
 of pieces of India rubber hose. These glass tubes are open at 
 both ends and terminate at the bottom in a vase of mercury, A 
 rod {not shown in the sketch) graduated to inches and tenths is 
 placed beside the glass tubes, sustained upon a float resting upon 
 the surface of the mercury, so adjusted that zero of the gradua- 
 tion may coincide with the surface of the mercury. 
 
 If the orifice at C be closed by a stopper and the receiver 
 exhausted, the mercury will rise in the tubes; and if the density 
 of the atmosphere at the time of the experiment be expressed in 
 inches of mercury, the height of the mercury in the tubes as 
 read upon the graduated rod will be equal to the difference 
 between the density of the atmosphere and that in the receiver. 
 If we now remove the stopper from the orifice at C, the column 
 of mercury in the tube m will instantly subside to a point which 
 indicates the difference between the density of the atmosphere 
 and the density in the chamber when an equal quantity of air 
 flows through the two orifices; while at the same time the column 
 of mercury in the tube ?i will only have begun to subside very 
 slowly as the density in the receiver increases. Having noted 
 the height of the barometer at the time of the experiment, if we 
 note the simultaneous heights of these two columns of mercury, 
 and deduct them respectively from the height of the barometer, 
 we shall have the density in the chamber necessary to an equal 
 flow through the two orifices under the relation which subsists 
 at the moment of notation between the density in the receiver 
 and the density of the atmosphere. And if we note the simul- 
 taneous heights of these columns at various times during the 
 filling of the receiver, so many densities in the chamber shall we 
 find corresponding to the different relations of the other two 
 densities. 
 
 At the time of the exi^eriment the height of the barometer, or 
 
34 
 
 density of the atmosphere was thirty inches. In consequence of 
 leaks in the receiver, I was unable to exhaust it so as to raise the 
 column in the tube w higher than twenty-six inches. I noted 
 the simultaneous altitudes of the two columns at the moment 
 when the column n coincided with each successive inch-mark 
 upon the graduated rod, and thence ascertained the densities in 
 the chamber under twenty-six different relations between the 
 density in the receiver and that of the atmosphere. These 
 results I have placed in the table beyond, in w^hich the first 
 column shows the densities in the receiver at the times of nota- 
 tion, and the second the densities in the chamber corresponding 
 thereto. 
 
 In order to ascertain what these densities should have been 
 according to the old theory, I constructed a formula as follows: 
 Let /I be the height of the barometer at the time of the experi- 
 ment, D the density in the receiver, d the density in the cham- 
 ber, V the velocity through the first orifice, v the velocity 
 through the second orifice. Then according to the old theory 
 the force which drives the air through the first orifice is A — d 
 and that which drives it through the second orifice is c?— D. 
 But since an equal quantity flows through both, these forces are 
 as the velocities, that is A — d : d—T> : : V : y. 
 
 Again, according to the old theory the density with which the 
 
 air passes the first orifice is A, and that with which it passes the 
 
 second orifice is d. But since the orifices are equal and the 
 
 quantities which j^ass through them are also equal, the products 
 
 of the velocities by the densities are equal, that is AY^dv and 
 
 dv 
 V=— . Substituting this value of V in the preceding couplet 
 
 and then finding the value of d, we have the following formula 
 for determining the densities in the chamber according to the 
 old theory, viz : 
 
 d=.\/ 
 
 4 2 * 
 
 The several densities in the chamber computed by this formula 
 are placed in the fourth column of the table. 
 
 In order to ascertain what the densities in the chamber should 
 have been according to the new theory, I constructed a formula 
 as follows, preserving the same notation as above. 
 
 By the new theory the force which drives the air through the 
 
35 
 
 first orifice is A — d whenever d is not less than — . But d is 
 
 2 
 
 never less than — when an equal quantity flows tlirough both 
 
 orifices, for if it were so the chamber would, according to our 
 theory, be receiving as much as could flow into a vacuum under 
 the pressure A, and must therefore discharge into the receiver 
 as much as would flow into a vacuum under a pressure A; in 
 order to which the density in the chamber must be equal to A, 
 
 and therefore greater than --. Consequently, the force which 
 
 drives the air through the Jirst orifice is in this arrangement 
 always A — d. Again, the force expended in driving the air 
 
 through the second orifice by the new theory is — whenever D 
 
 d 
 is not greater than — . Let us first construct a formula for the 
 2 
 
 cases in which D is not greater than — . In these cases the den- 
 sities under which the air passes the orifices are respectively 
 
 d 
 
 A—d and — •. Since the forces are as the velocities, 
 
 2 ' 
 
 and since the quantities are equal, dY^=—-, and V=— . Substi- 
 
 4 
 tuting this value of V in the couplet, we have d=^—^; a constant 
 
 5 
 
 quantity. Hence while the density in the receiver varies from 
 
 2 
 to — A, the density in the chamber is a constant quantity and 
 
 4 
 equal to — A. Let us now construct a formula for finding the 
 
 d 
 value of d when D is greater than — . In these cases the forces 
 
 are A—d and d— D and we have for the couplet A—d:d—D: :Y:v. 
 
 The densities in the orifice are d and D and we have dY=T)v 
 
 Dv 
 and Y=---. Substituting this last quantity in the couplet we find 
 
 ■V D' 
 
 J-Bl' A-D 
 
 + -T^+-2- 
 
36 
 
 as the formula for the value of d hy the uew theory when D 
 
 2 
 exceeds — J. The densities in the chamber computed by these 
 
 formulae are placed in the third column of the table. 
 
 Density of the Athosphebe during Experiment 30. 
 
 Density 
 
 In the 
 
 Receiver. 
 
 Henslty in the 
 
 chamber as found 
 
 by experiment. 
 
 Density in the 
 chainbeV due to 
 the new theory. 
 
 Density in the 
 chauibiT due to 
 the old theory. 
 
 Deviation of the 
 experiment from 
 the new theory. 
 
 
 
 
 24 
 
 18.541 
 
 ~ 
 
 I 
 
 
 
 24 
 
 18.820 
 
 
 
 2 
 
 
 
 24 
 
 19.106 
 
 
 
 3 
 
 
 
 24 
 
 19.397 
 
 
 
 4 
 
 24.58 
 
 24 
 
 19.695 
 
 .58 
 
 5 
 
 24.58 
 
 24 
 
 20. 
 
 .58 
 
 6 
 
 24.58 
 
 24 
 
 20.31 1 ■ 
 
 .58 
 
 7 
 
 24.53 
 
 24 
 
 20.628 
 
 .58 
 
 8 
 
 24.58 
 
 24 
 
 20.953 
 
 .58 
 
 9 
 
 24.58 
 
 24 
 
 21.284 
 
 .58 
 
 ]0 
 
 24.58 
 
 24 
 
 21.622 
 
 .58 
 
 11 
 
 24.60 
 
 24 
 
 21.968 
 
 .60 
 
 12 
 
 24.64 
 
 24 
 
 22.320 
 
 .64 
 
 13 
 
 24.70 
 
 24.032 
 
 22.713 
 
 .668 
 
 14 
 
 24.77 
 
 24.124 
 
 23.048 
 
 .646 
 
 15 
 
 24.89 
 
 24.270 
 
 23.423 
 
 .62 
 
 16 
 
 25.03 
 
 24.464 
 
 23.805 
 
 .566 
 
 17 
 
 25.21 
 
 24.700 
 
 24.194 
 
 .51 
 
 18 
 
 25.43 
 
 24.974 
 
 24.594 
 
 .456 
 
 19 
 
 25.69 
 
 25.280 
 
 25. 
 
 .41 
 
 20 
 
 25.96 
 
 26.615 
 
 25.414 
 
 .345 
 
 21 
 
 26.30 
 
 25.976 
 
 25.835 
 
 .324 
 
 22 
 
 26.65 
 
 26.360 
 
 26.265 
 
 .29 
 
 23 
 
 27.03 
 
 26.764 
 
 26.699 
 
 .266 
 
 24 
 
 27.43 
 
 27.186 
 
 27.149 
 
 .244 
 
 25 
 
 27.81 
 
 27.625 
 
 27.604 
 
 .185 
 
 26 
 
 28.20 
 
 28.076 
 
 28.066 
 
 .124 
 
 27 
 
 28.65 
 
 28.541 
 
 28.533 
 
 .109 
 
 28 
 
 29.08 
 
 29.017 
 
 29.016 
 
 .063 
 
 29 
 
 29.55 
 
 29.504 
 
 29.504 
 
 .046 
 
 30 
 
 30. 
 
 30 
 
 30. 
 
 
 
 The affinity of the experimental results to those derived from 
 the new theory, is obvious upon inspection of the table; and the 
 want of affinity to those derived from the old theory, is not less 
 evident. The comparative relation of the two theories to the 
 results of experiment, is more readily seen in the annexed cut (fig. 
 2), where they are respectively delineated by a curve. The upper 
 curve represents the densities or elastic forces in the chamber, 
 as found by experiment; the next curve those due to the new 
 theory, and the lower curve those due to the old theory. 
 
 Notwithstanding the near approximation of the exi)erimental 
 
be 
 
 mm 
 mm 
 
 imi 
 
 m 
 
 ■H 
 
 ■■■L 
 
 ■■■■»■■■ 
 
 ■■■■■HHB 
 
 IH 
 
\ 
 
 38 
 
 results to those due to t\e new theory, there is yet a small but 
 distinct deviation, which 'nholds throughout. This deviation in- 
 dicates either that there i^ some cause affecting the flow which 
 the theory does not take ir^io account, or that in the structure of 
 the apparatus or in trying%^e experiment, there was some failure 
 to comply with the requisil^^ conditions. 
 
 Although the apparatus/ as rude in its structure, yet care had 
 been taken to secure a cogwliance with the conditions on which 
 the experiment was base^f and in conducting the experiment I 
 was assisted by my friend. Prof. A. C. Twining, a gentleman 
 distinguished for his accuracy in such matters. The experiment, 
 moreover, was several times repeated, with no important differ- 
 ence in the results. For these reasons, in seeking the cause of 
 the deviation, my first inquiry was whether it might be attri- 
 buted to a change in the ratio of elastic force to density; the 
 theory being predicated upon the assumption that this ratio is 
 constant. It has been ascertained by experiment, that when air 
 is condensed and then suffered to lose the heat evolved by con- 
 densation, the ratio of its elastic force to its density will be 
 diminished. Hence it is certain that a part or the whole, or 
 possibly even more than the whole heat evolved by condensation 
 will be required to prevent that ratio from being diminished. 
 Still, however, it has generally been assumed by philosophers 
 (I know not on what grounds) that if air is suddenly condensed, 
 so as not to allow the heat evolved by condensation to escape, 
 the ratio of elastic force to density will be increased. This 
 assumption was made by • Laplace when he attributed to this 
 cause, in part, the velocity of sound. Let us suppose then, for 
 the present, that in sudden condensation the ratio of elastic force 
 to density is increased. It will then follow that in sudden ex- 
 pansion, the ratio of elastic force to density will be diminished. 
 But if that ratio were diminished, then the deviation in the table 
 should be in the opposite direction; that is, the experimental 
 results, instead of being greater than the theoretical, should be 
 less. The deviation, therefore, is not accounted for by this sup- 
 position; on the contrary, the experiment seems to prove that 
 the ratio is not diminished by expansion, and therefore cannot 
 be increased by condensation, as Laplace supposed. 
 
 Let us next take the contrary supposition, Viz: that the ratio 
 of elastic force to density is increased by expansion. This would 
 cause a deviation in the same direction as we find in the table. 
 
39 
 
 In order to ascertain whether the deviation in question is due to 
 this cause, we must next inquire whether a deviation arising 
 from this cause, would vary in the same manner throughout the 
 table, as does the observed deviation. Now if we go through 
 the table and assign for each observation severally, the manner 
 in which the ratio of elastic force to density must increase, in 
 order to satisfy that observation, we shall find very nearly one 
 and the same increment of the ratio demanded for all the obser- 
 vations. Hence if we attribute the deviation to this cause we 
 should be obliged to conclude that one and the same change in 
 the ratio takes place, whether the expansion be greater or less. 
 But such a conclusion is obviously inadmissible. We cannot, 
 therefore, attribute the deviation in question to a change in that 
 ratio, either by increase or diminution. 
 
 Nor can we ascribe the deviation to that which is the chief 
 cause of deviation from theory in the case of the flow of liquids, 
 viz: the contraction of the stream in passing an orifice. For if 
 that cause operated, it would affect the flow in the same ratio in 
 both orifices, and therefore would not, in this case, affect the 
 indications of the mercurial columns. Moreover, I think it can 
 be shown, from consideration a 2)^'iori, that the cause which 
 produces the contraction of the stream in liquids, could not 
 operate to affect the flow of expansible fluids. 
 
 Having satisfied myself that the deviation was not due to the 
 (Causes above named, my next inquiry was, whether a difference 
 in the sizes of the orifices (hitherto assumed to be equal) would 
 cause a deviation corresponding to that in the table. In examin- 
 ing this point, I found that the experimental results would be 
 very nearly satisfied throughout the table, by the assumption 
 that the area of the second orifice was less than that of the first, 
 in about the ratio of .933 to 1. As the two orifices had been 
 made as nearly equal as they could be by forcing the same steel 
 plug through both, I was confident that, as originally formed, 
 they could not differ to this extent. But it occurred to me that 
 some accidental circumstance might have occurred to diminish 
 the inner orifice, and I suspected that the workman, in handling 
 the brass plate after the orifice was made, had got dirt into it, 
 and had omitted to cleanse it before soldering on the outer plate. 
 To ascertain whether such was the case, I divided the tube near 
 the second orifice, and, upon examining it with a microscope, 
 discovered that there was dirt adhering around its inner periph- 
 
40 
 
 ery sufficient, I think, to cause a diminution of its area to the 
 extent above named. Unfortunately this discovery was made 
 after the arrangements for trying the experiment had been 
 removed; and I have not since found leisure to replace them and 
 try the experiment anew. But for this accidental circumstance 
 no doubt there would have been a still nearer approximation of 
 the experimental results to those derived from the formula. The 
 coincidence, however, is sufficiently near to Establish the truth 
 of the new theory, so far as respects those points of difference 
 between the two theories specified in the first part of this article. 
 
 The fifth column of the table shows the several differences 
 between the experimental results, and those due to the new 
 theory. It will be noticed that these differences increase slightly 
 between density 10 and 13 in the receiver, before they began to 
 decrease. This, I think, indicates a slight obstruction to the 
 flow through the second orifice, when the density in the receiver 
 becomes equal or nearly equal to that of the effluent stream. 
 This increment at its maximum amounts to .088, corresponding 
 to the pressure of that portion of an inch of mercury, and is, I 
 think, the measure of the obstruction or resistance due to that 
 circumstance. If this view of the subject is correct, then there 
 would have been a deviation to this extent in this part of the 
 table, even if the orifices had been equal. 
 
 It is desirable that further experiments of this kind should be 
 tried by those who have better means at command than I had to 
 do justice to the subject. To such as may be disposed to under- 
 take it, I would suggest that a perfect equality of the two orifices 
 might be secured by interchanging the discs, varying the sizes of 
 the orifices until they gave the same indications in both positions. 
 
 If, after thus securing the equality of the orifices, there should 
 still be a deviation in that part of the table where the elastic 
 force in the chamber is constant, such deviation, I think, must be 
 attributed to a change in the ratio of elastic force to density; 
 and if so, its amount would furnish the means of determining 
 the law according to which that ratio varies. 
 
 I would also suggest that a modification of this experiment 
 would furnish perhaps the best possible means of determining 
 the law according to which the ratio of elastic force to tempera- 
 ture varies, when the absolute amount of heat is constant. In 
 an arrangement for this purpose, the bulb of a thermometer 
 should be inserted into the chamber, and the outer orifice should 
 
41 
 
 be so constructed that it may be enlarged or diminished at pleas- 
 ure. With this arrangement, we may cause the air in the cham- 
 ber to assume, almost instantly, any elastic force we may choose, 
 between the elastic force of the atmosphere, and a little more 
 than twice the elastic force in the receiver; and we may keep 
 that force constant in the chamber during any time that may be 
 required to cool the thermometer down to the corresponding 
 temperature, the continual flow through the chamber in the mean- 
 time carrying off not only the heat which flows in from extrane- 
 ous sources, but also that derived from the thermometer itself. 
 We may thus ascertain the relation of elastic force to temperature 
 at as many points as we please within this range, and thereby 
 determine the law of their variation when the absolute amount 
 of heat remains constant. 
 
ARTICLE V. 
 
 The Form, Formation and Movement of Sonorous Waves* 
 
 There can be no doubt that the movements of aerial particles 
 resulting from forces impressed upon them, and the transmission 
 of force from one to another are in perfect accordance with 
 Dynamic laws. But the various attempts which have been made 
 to determine by the application of these laws the precise order 
 and extent of their respective movements have not been success- 
 ful. While by the application of Dynamic laws to other matter 
 we are able to trace with the utmost precision the paths of bodies 
 in the remotest regions of space, we have as yet no definite 
 knowledge respecting the movements of the air which surrounds 
 us; — not even of those movements on which we are constantly 
 dependent for the transmission to each other of our oral com- 
 munications. There are many problems pertaining to this 
 department of Physics the solution of which would be of great 
 interest to science, but to solve which no attempts have as yet 
 been made because of the apparent difficulties involved in the 
 investigation. Whatever of thought has been bestowed in this 
 field of inquiry has been devoted chiefly or wholly to the deter- 
 mination of the velocity of sound. But the attempts to solve 
 even this problem, initiated by Newton, have been attended with 
 so little success that now, after the lapse of two centuries, during 
 which continual efforts have been made to reconcile the widely 
 differing results of theory and experiment, scientists are not 
 agreed whether this, or that, or both, require the correction 
 which should bring them into harmony. 
 
 * Read at a meeting of the Connecticut Academy of Arts and Sciences, Decem- 
 ber 21, 1881. 
 
44 
 
 The delay to develop this branch of Physics has been due, no 
 doubt, to the peculiar and complex nature of aerial matter; its 
 tenuity, its fluidity, its compressibility and its elasticity. These 
 properties have seemed to scientists to conspire to embarrass the 
 application of Dynamic laws to the air; while its total invisi- 
 bility has compelled them to pursue their investigations in the 
 dark, by abstract thought alone, and without the aid of the 
 senses to check the aberrations of reason. 
 
 On account of these properties of aerial matter the problem 
 to find the velocity of sound has always been regarded by mathe- 
 maticians as one of great difficulty. The late Professor Peirce, 
 in his Treatise on Sound, says, " The problem to investigate the 
 general laws of the propagation of sound is one of the utmost 
 complexity, and has been resolved only under . very restricted 
 conditions." After alluding to the profound researches which 
 have been bestowed on this subject by Euler, Lagrange and 
 others, he proceeds to investigate the problem restricted to the 
 case of propagation in a tube of uniform calibre. After conr 
 structing an elaborate series of differential equations to this 
 end, he at Ifength arrives at one which he says " is altogether 
 intractable and incapable of integration." In order to proceed 
 further he finds it necessary to restrict the conditions of the 
 problem still further by confining it to the case in which the 
 elastic force of the wave exceeds that of the medium in which it 
 is propagated by an infinitesimal quantity only; that is, by con- 
 fining it to a sound of infinitesimal intensity. With the problem 
 thus restricted he goes on, and after a very elaborate process, 
 arrives at the conclusion that the velocity of sound is that which 
 a heavy body will acquire by falling through half the height of 
 a homogeneous atmosphere. This is the same conclusion as that 
 arrived at by Newton, Euler, D. Bernouilli, Lagrange, Poisson, 
 Laplace, and I know not by how many others, all of whom found 
 it necessary to restrict the problem to like conditions. 
 
 When nature is questioned respecting her laws, even by the 
 most profound mathematicians and by the most recondite pro- 
 cesses of analysis, if those processes are based upon assumptions 
 not consonant with her laws she will refuse to respond except 
 in equations " intractable and incapable of integration." It is 
 the purpose of this communication to show that those who have 
 attempted to solve this problem have made an erroneous assump- 
 tion respecting the manner in which an impinging force imparts 
 
46 
 
 motion to aeinal matter, and that upon a correction of this error 
 the difficulties which they encountered in their attempts to give 
 a general solution to the problem will vanish, and we shall be 
 able to arrive at such a solution in a direct and simple way. 
 
 The mistake to which I have referred consists in having as- 
 sumed that motion is imparted by a finite force to the infinit- 
 esimal particles of aerial matter in the same manner as to finite 
 masses of matter; that is by acceleration: or, in other words, 
 that in the former as well as in the latter case motion begins 
 with an infinitesimal velocity which is augmented to a finite 
 velocity by infinitesimal increments. Such a notion is not con- 
 sistent with Dynamic laws. These laws demand that when a 
 force of finite magnitude acts upon an infinitesimal quantity of 
 matter a finite velocity proportionate to the force should be 
 imparted in an infinitesimal time. An impinging force therefore 
 does not impart motion to aerial matter by acceleration. 
 
 But (it may be asked here), has not Newton in the 4'7th pro- 
 position of the Second Book of the Principia, demonstrated that 
 each infinitesimal particle of air that is put in motion by a sono- 
 rous wave is accelerated and retarded according to the law of the 
 vibrating pendulum? 
 
 It is true that the proposition referred to purports to demon- 
 strate this; but a careful examination will show that it fails to 
 do it. In this proposition Newton at the outset assumes in his 
 hypothesis the point to. be proved and the conclusion at which 
 he arrives results from that assumption. It is in short a notable 
 example of "reasoning in a circle." This will be evident upon 
 a close scrutiny of the argument: and it is also clearly shown by 
 Gabriel Cramer* who, to prove the inconclusive character of 
 Newton's reasohing in this proposition, shows that by a precisely 
 similar course of reasoning {et, mutatis mutandis, in totidem 
 verbis) it may be made to appear that the law of acceleration is 
 that which pertains to a uniform force, as in the case of bodies 
 falling by their own gravity. 
 
 We may conclude then that it has not been demonstrated by 
 Newton that the particles of air put in motion by a sonorous 
 wave acquire their velocity by acceleration; and therefore we 
 may accept the conclusion to which we were led by Dynamic 
 laws, that each particle as successively encountered receives its 
 full velocity instantaneously. 
 
 * See Principia, Glasgow Edition, page 273, note. 
 
 1 
 
i6 
 
 Let us now consider the modus operandi by which the force so 
 imparted is passed along from particle to particle; and to facili- 
 tate our conceptions of the process let us suppose it to take 
 place in a tube of uniform calibre. Let PL, Fig. 1 be such a 
 tube, and let P be a piston fitted to it. 
 
 Fig. ]. 
 
 If a uniform velocity v be instantaneously imparted to the 
 piston, the plate of air which is in contact with its front Avill 
 instantaneously receive the velocity of the piston and at the 
 same time be condensed by a force equal to that required to 
 overcome the inertia of the plate. Suppose the velocity r to be 
 such that this condensing force is sufficient to reduce the thick- 
 ness of the plate ^; then will this plate, thus reduced in thick- 
 ness, be added on to the front of the piston and go on with it, 
 condensing the next plate, giving it the velocity v and pushing 
 it on in front of itself, and so on. Thus it will be seen that 
 when the condensing force is that here supposed, the point where 
 the condensation takes place moves forward in the tube just ten 
 times as fast as the piston. When therefore the piston has 
 moved -^ of an inch there will be j^ of an inch of condensed air 
 in front of it moving with the same velocity as the piston. If 
 now, at this juncture, the motion of the piston be arrested, the 
 condensed air in front of it will continue to move on by its 
 momentum condensing and adding plates to its front as before, 
 and in the meantime the rear end of this condensed air having 
 advanced beyond the piston will have space to expand and • 
 resume its normal volume and come to rest: and as it respects 
 each infinitesimal plate, the restoration will be as instantaneous 
 as was its condensation. That such will be the manner of 
 restoration need not be shown here, since it will fully appear 
 from what is shown in an article on the mode of Expansion of 
 P^lastic Fluids in the American Journal of Science, second series, 
 vol. ix, page 034.* Thus we have a self-sustaining, self-propagat- 
 ing wave, the quantity condensed in front in a given time being 
 just equal to the quantity expanded in the rear in the same time; 
 the latter by its reaction furnishing the power to keep up the 
 motion and condense the plates in front. 
 
 * The third of the preceding articles. 
 
4t 
 
 A wave then, in the sense in which we shall employ the term 
 in this paper, consists of a quantity of air of a uniform density 
 greater than that of the medium in which the wave is propagated, 
 and having the absolute velocity that is due to the action of the 
 force which effected its increased condensation. 
 
 It may be well here to define a few other terms or phrases, 
 some of which we may have occasion to employ in a sense some- 
 what different from that attached to them by other writers on 
 this subject. 
 
 The breadth of a icave is the space occujiied by the condensed 
 air measured in the direction in which it moves. 
 
 The intensity of a wave is the excess of its elastic force or 
 density over that of the medium. 
 
 The velocity of a icave is the velocity of the point where the 
 particles are condensed and put in motion. 
 
 The velocity of the particles is the absolute velocity instanta- 
 neously impressed upon the particles successively by the wave. 
 
 In the use of the terra particles we are not to be understood as 
 indicating any theory respecting the constitution of the atmos- 
 phere, — we mean small parts or portions, — not molecules or 
 atoms. 
 
 In showing how a wave may be formed in a tube we have sup- 
 posed the motion of the piston to be arrested after moving ^ of 
 an inch. If it had been arrested after moving any smaller dis- 
 tance the breadth of the wave would have been smaller in the 
 same ratio, but in every other respect it would have been the 
 same; it would have had the same intensity, the same velocity, 
 and it would have imparted the same velocity to the particles; 
 these quantities being dependent only upon the velocity of the 
 piston, and not on the distance through which it moved before 
 its motion was arrested. 
 
 We have thus shown how a self -propagating wave may be 
 formed in a tube. The process of the formation of such a wave 
 in the open air is, in principle, precisely the same, and the form 
 of the wave and laws of its propagation are the same. 
 
 It is well-known that sounds of small intensity propagated 
 through tubes have the range of their audibility extended to a 
 greater distance than it could reach if propagated in the open 
 air; but this function of the tube does not involve any modifica- 
 tion of the laws of propagation on which the velocity of the 
 wave depends. The reason why a tube has this effect may be 
 
48 
 
 readily illustrated. Suppose we present the end of a tube to a 
 vibrating body or other agent that may originate a wave; then 
 a portion of the condensed air which constitutes the wave at its 
 origin will enter the tube, constituting a self-propagating Avave 
 therein; the rest of it will spread out forming a spherical shell 
 or wave whose thickness measured on the radius of the sphere is 
 the breadth of the wave; and since there can be no increase in 
 the quantity of this condensed air after the wave is formed, the 
 thickness of the shell or breadth of the wave will be inversely as 
 the square of the distance from the place of its origin. Thus we 
 see that the breadth of the wave which is i)ropagated outside of 
 the tube diminishes in a rapid ratio; but the breadth of that 
 which is propagated inside of the tube undergoes no diminution 
 except such as may be due to friction or imperfect elasticity. 
 Now it is evident that, other things being equal, the audibility 
 of a sound must be as the breadth of the wave; and that a 
 sufficient breadth of wave may be maintained in the tube after 
 the breadth of that outside shall have been so far reduced as to 
 be incapable of producing the required action upon the acoustic 
 organs. This is the only difference there is between waves pro- 
 pagated in the open air and those which are. propagated in a 
 tube, and as we have shown, it is not a difference that affects the 
 laws of their propagation. 
 
 From the law of the transmission of force from particle to 
 particle, as we have shown it to be, we will proceed to deduce 
 the law of the propagation of sonorous waves. 
 
 Prop. I. The velocity of a wave of whatever intensity is to 
 the absolute velocity which it impresses instantaneously on every 
 particle over which it passes, as the space occupied by the con- 
 densed air before condensation to its loss of space by condensa- 
 tion. 
 
 The truth of this proposition may be drawn as a corollary 
 from what was shown in describing the formation of a wave in 
 a tube, and therefore we need to give no further proof of it here. 
 
 Fig. 2. 
 
 Let he fig. 2 be a column of air of the density and elastic force 
 due to the pressure of the atmosphere, and whose length is equal 
 to the height of a homogeneous atmosphere. Let this column 
 
49 
 
 be extended to any distance a; and let a wave whose elastic 
 force exceeds the pressure of the atmosphere in the same ratio 
 as ac exceeds he pass over the column ae. 
 
 Prop. n. The velocity of a wave like that above described 
 will be to the velocity which it impresses on the particles over 
 which it passes as ac to ah. 
 
 For it is evident that such a wave will condense the particles 
 in the ratio of ac to he, and therefore the space occupied by the 
 condensed particles before condensation will be to their loss of 
 space by condensation as ac to ah, and therefore by Prop. I the 
 velocity of the wave is to that which it impresses on the j^articles 
 as ac to ah. 
 
 Prop. III. The velocity impressed upon the particles is that 
 which a falling body would acquire in the time in which the 
 wave passes over ah. 
 
 For during that time the excess of elastic force whicli is a 
 constant force and which is equal to the weight of the air in ah 
 is acting upon that air, and therefore must impart to it the same 
 velocity as it would acquire by falling during the same time; 
 and that velocity is the same as would have been acquired in the 
 same time by any other falling body. 
 
 Prop. IV. The velocity of the wave is that which a falling 
 body will acquire in the time in which the wave runs over ac. 
 
 For it was shown in Prop. Ill that in the time of running over 
 ah a falling body would acquire the velocity which the wave 
 impresses upon the particles, and since by Prop. II that velocity 
 is to the velocity of the wave as ah to ac, it follows that in the 
 time of running over ac the falling body will acquire the velocit\' 
 of the wave. 
 
 Prop. V. The velocity of a wave is that which a body will 
 acquire by falling through a height which exceeds half the 
 height of a homogeneous atmosphere in the same ratio as the 
 elastic force of the wave exceeds the elastic force of the medium 
 in which the wave is propagated. 
 
 It was showTi in the last proposition that in the time in which 
 the wave runs over ac a falling body would acquire the velocity 
 of the wave, and since the mean velocity of falling bodies is half 
 their final velocity, it follows that in the time in which the wave 
 runs over ac, the falling body in acquiring the velocity of the 
 wave, will have fallen through a space equal to half a c ; and 
 since 5 c is by construction equal to the height of a homogeneous 
 
atmosphere, and since a c exceeds h e in the ratio in which the 
 elastic force of the wave exceeds that of the air, it follows that 
 half of a c exceeds half the height of a homogeneous atmosphere 
 in the same ratio. Therefore the velocity of a sonorous wave 
 is that which a body will acquire in falling through a height 
 which exceeds half the height of a homogeneous atmosphere in 
 the same ratio as that in which the elastic force of the waAe 
 exceeds the elastic force of the medium in which the wave is 
 propagated. 
 
 Thus we have a general solution of the problem relating to the 
 propagation of sound — a solution limited by no conditions. 
 From this solution it appears that an excess of elastic f wee above 
 that of the medium is essential to the existence of a loave ; and 
 also that the greater this excess, the greater will he the velocity of 
 the wave. 
 
 If we apply this general solution to the particular case of a 
 wave whose intensity (or excess of elastic force), is so small that 
 its effect on the velocity is inappreciable, Me shall find the 
 velocity of such a wave to be that which is due to half the 
 height of a homogeneous atmosphere. This is the velocity found 
 by Newton and those who succeeded him in this investigation, 
 all of whom, in consequence of their assumption of the gradual 
 acceleration of the particles, found it necessary to confine their 
 investigations to waves of this small intensity. It seems obvious 
 that a result thus obtained can be legitimately applied only to 
 such waves ; but we find that these scientists have regarded and 
 treated it as equally applicable to all waves. They maintain 
 that the velocity which a body acquires by falling through half 
 the height of a homogeneous atmosphere is the theoretical 
 velocity of all Avaves so far as their velocity depends on the laws 
 of Dynamics. 
 
 I think it may be shown hoM' those scientists have been led to 
 this conclusion as a consequence of their acceptance of the theory 
 of gradual acceleration. It is evident that in a wave originated 
 and propagated by gradual acceleration (if such a wave can exist, 
 which I do not admit), the foremost particles can have only 
 an infinitesimal condensation, however dense the particles may be 
 furthest back. Now it appears both from the general solution 
 we have hei'e given and from the more limited one given by 
 others, that particles of so small condensation can only transmit 
 their force from one to another with the velocitv due to half the 
 
51 
 
 height of a homogeneous atmosphere ; and since a wave can 
 move no faster than its foremost particles it would thence fol- 
 low that no wave would move faster than with the velocity due 
 to half the height of a homogeneous atmosphere, and conse- 
 quently that all waves have that velocity. 
 
 We see then that if we adopt the theory of gradual accelera- 
 tion we cannot avoid the conclusion that all sounds have the 
 same velocity. But we shall see how absurd are the results to 
 which we should be driven by accepting the conclusion that all 
 Avaves have the same velocity. 
 
 According to Professor Peirce the theoretical velocity of 
 sound is 916 feet per second, and its actual or observed velocity 
 is 1090 feet per second (Peirce on Sound, page 31). The great 
 difference between these velocities was a mystery to mathema- 
 ticians from the time of Newton to that of Laplace. Laplace 
 suggested that the difference was due to the heat evolved by 
 condensation. This suggestion, coming from so high a source, 
 has been accepted by most mathematicians without examina- 
 tion as an adequate supplement to the theory of Newton, and 
 sufficient to reconcile the results of theory with those of obser- 
 vation. Others who have investigated the question have 
 doubted its sufficiency for that purpose. Professor Peirce, after 
 remarking that the heat evolved by sudden condensation may be 
 much greater than we should expect in the case of such small 
 condensations as are contemplated in the theory of sound, 
 accepts with manifest misgivings and reluctance the sufficiency 
 of Laplace's explanation, and employs it to bridge over this 
 otherwise impassable gulf in the theory of the propagation of 
 sound. 
 
 Now let us see how much the temperature of a wave must be 
 raised by evolved heat to increase its velocity from 916 to 1090 
 feet per second. According to Professor Peirce the velocity of 
 sound increases 0*96 feet per second for every degree of temper- 
 ature above 32° Fahrenheit. Then by the formula 
 
 1090 — 916 
 
 we find that 181.25 degrees is the elevation of temperature 
 required to increase the velocity of a wave from 916 to 1090 feet 
 per second. And since under the theory of gradual acceleration 
 the front of every wave must consist of particles only infinitesi- 
 
62 
 
 mally condensed, and since this part must have its velocity 
 increased as much as any other part, it would follow that an in- 
 finitesimal condensation would cause an evolution of heat that 
 would raise the temperature 181 degrees ! It would further fol- 
 low that we could not converse with a friend, even in the mildest 
 whispers, without pouring into his ears waves of a temperature 
 of 181 -1-32 = 213 degrees, or hotter than boiling water ! ! More- 
 over it would also follow that upon the slightest sudden change 
 of barometric pressure we should be immersed in air of a temper- 
 ature of 213 degrees ! ! 
 
 Such are some of the absurd conclusions which follow from the 
 doctrine of gradual acceleration supplemented by evolved heat. 
 
 The doctrine that the particles which constitute the wave 
 have all their motion impressed upon them instantaneously, leads 
 to no such absurd conclusions and presents no such hiatus to be 
 bridged over by evolved heat or otherwise; as we \\411 now 
 attempt to show. 
 
 It follows from our solution of the problem of the propagation 
 of sound that the intensity of the wave (an item which was elim- 
 nated in the old theory), is a quantity not to be disregarded in 
 determining the velocity of sonorous waves. Let us see if this 
 item, when given its proper place in the investigation, is not 
 sufficient to effect a reconciliation between the results of theory 
 and those of observation ; and that without calling in the aid of 
 evolved heat. 
 
 Prof. Peirce gives 916 feet per second as the theoretic axiA 1090 
 as the observed velocity of sound. But we should here note the 
 significant fact that 916 is the computed velocity of a wave so 
 weak that its intensity may be regarded as cipher; while 1090 
 is the observed velocity of the report of a cannon, one of the 
 most intense sounds known to us. According* to both the old 
 and the new theory, 916 is the theoretic velocity of the weaker 
 of these waves. If then we can show that under the new theory 
 1090 is also the theoretic velocity of the report of a cannon, we 
 shall have effected a full reconciliation between theory and ob- 
 servation without calling in the aid of evolved heat. In order 
 to do this the question to be solved is this, — is the intensity of 
 the wave produced by the discharge of a cannon sufficient to 
 account for the difference between 916, the velocity of a wave 
 whose intensity is regarded as cipher, and 1090, the observed 
 velocity of the report of a cannon. If we knew what is the 
 
53 
 
 intensity of the "wave so produced we could readily ascertain 
 whether it is sufficient for this purpose; but as we have no means 
 of learning this, let us inquire what the intensity of such a wave 
 must be in order that its velocity may be 1090 feet per second. 
 Referring to fig. 2 where he represents the height and density of 
 a homogeneous atmosphere and ah the intensity of the wave, the 
 question to be solved is, what must be the ratio of ah to he when 
 the velocity due to half the height he is 916 feet, and that due 
 to half the height ac is 1090 feet? By an obvious process, which 
 need not be given in detail here, it will be found that ah must in 
 such case be about ^^ of hc\ that is, the density of the wave 
 must exceed the density of the air by about 41 per cent. 
 
 That the density of the air in front of the muzzle of a cannon 
 is upon its discharge increased as much as 41 per cent., and 
 often very much more than 41 per cent., there can be no doubt. 
 It is evident that the degree of that condensation is not always 
 the same, but must depend on various considerations, such as 
 whether the cannon be charged with shot, — whether the wad be 
 so compressed as to require great force to drive it out, — the 
 length of the piece, — tlie quantity and quality of the powder, 
 &c. In the records of the various velocities observed no refer- 
 ence is made to these considerations. Prof. Peirce gives 1090 as 
 the mean of seven selected observations, none of which varied 
 materially from 1090. But there are many other observations 
 recorded giving observed velocities much higher than those which 
 he selected. These are rejected by Prof. Peirce as unreliable; for 
 what reason is not apparent. For my own part I doubt not that 
 waves are often produced by the discharge of cannon whose 
 intensity is such that under our theory their velocity of propaga- 
 tion would far exceed 1090 feet per second. If this be so, we 
 shall need no aid from evolved heat until, in the case of any 
 particular wave it can be shown that its velocity is greater than 
 under our theory can be due to its intensity. It may be asked 
 here whether I reject the notion that the velocity of sound is 
 increased by evolved heat? I answer that I do not; but I think 
 that under the old theory the effect of heat has been greatly 
 overrated. I think its effect must be in proportion to the quan- 
 tity of heat evolved, and that the quantity evolved must be in 
 proportion to the degree of condensation. I think therefore that 
 when the condensation is infinitesimal the effect of the evolved 
 heat will be inappreciable. If this be so, then, whenever a 
 
54 
 
 method shall be devised whereby we may find by experiment the 
 velocity of waves of infinitesimal condensation, w^e shall find it 
 not to differ materially from that which a body will acquire by 
 falling through half the height of a homogeneous atmosphere; 
 — the velocity found by Newton. 
 
 Those who have investigated the propagation of sound under 
 the old theory seem to have supposed that the evolved heat has 
 a thermo-dynamical effect; that it is converted into mechanical 
 power and so acts mechanically to increase the velocity of sound; 
 but it can have no such action, for the supposed force of the heat 
 is expended within and upon the very matter which is supposed 
 to be propelled by it, and therefore acts backward as much as 
 forward. A vessel sails no faster because she is on fire in her 
 hold; the earth revolves no faster upon its axis because of a fire 
 on its surface or within it; the heat of the fire under the boiler, 
 or of the steam within it does nothing to propel a steamer, until, 
 through the intervention of machinery it is made to react upon 
 matter without. Archimedes could move the world, but he must 
 have something outside of the world for his lever to react upon. 
 In the case of the heated wave there is nothing outside of the 
 wave for the heat to react upon, and therefore it can have no 
 mechanical effect to increase its velocity. The mechanical power 
 of a wave to perform the work of reproducing itself is given to 
 it by the power which originates the Avave, and it can neither 
 be increased or diminished by the heat evolved by condensation. 
 What then is the modus operandi in which the velocity of a 
 wave is increased by evolved heat? It is simply by increasing 
 the ratio of the elastic force of the wave to the density of the 
 medium in which it is propagated. The elastic force of the wave 
 is increased while the density of the medium is unchanged. The 
 result of this change of ratio is that the velocity imparted to the 
 particles by the wave is diminished, while the number of parti- 
 cles acted upon in a given time is increased in the same ratio, so 
 that the amount of mechanical work performed is neither in- 
 creased or diminished by the evolved heat. 
 
 In the book published a few years since, entitled, '' The (.'or- 
 relation of P^orces" it was stated that the eminent German scientist 
 Mayer had deduced the mechanical equivalent of heat from the 
 velocity of sound; and with a result coinciding very nearly with 
 the equivalent found in other ways. This statement was appar- 
 ently regarded by the author as being highly confirmatory of 
 
m 
 
 the doctrine of his book. This computation by Mayer, however, 
 could only be founded on the assumption that heat acts mechani- 
 cally in increasing the velocity of sound; and therefore it is 
 evident from what has been shown that whatever other ground 
 the doctrine of the correlation of forces may have to rest upon, it 
 can derive no legitimate support from this computation by Mayer, 
 A^^e have another example of a sfmilar mistake in the recent 
 aimouncement that another scientist has deduced the velocity of 
 sound from the mechanical equivalent of heat. 
 
 I had intended to present here several other interesting con- 
 clusions at which I had arrived on this subject, but I forbear, 
 lest I should extend this paper to too great length; and I pass 
 them by with the less reluctance, because any one who will be 
 likely to take the trouble to read this paper will be able himself 
 to arrive at the same results. 
 
 In conclusion I think proper to mention the fact that more 
 than thirty years since I contributed to the Journal of Science 
 an article* in which I demonstrated in a manner quite different 
 from that pursued in this article, but not less conclusive, that all 
 waves have not the same velocity, but that their velocities vary 
 with their intensity; and that the velocity of a wave was that 
 which a body would acquire by falling through a height greater 
 than half the height of a homogeneous atmosphere in the same 
 ratio as the density of the wave exceeds the density of the 
 medium in which it is propagated: — results precisely similar to 
 those deduced in this article. 
 
 That article seems to have attracted little or no attention : for 
 up to the present moment all scientific periodicals continue to 
 speak of the velocity of sound as if it were a quantity determin- 
 ate and invariable, sought for, but not yet ascertained with 
 precision and certainty; and scientists, by ingenious theoretical 
 and experimental devices still continue to search for the mythical 
 number as the Alchemists did for the Philosopher's stone. Pos- 
 sibly this article may have as little effect as the former to bring 
 those labors to an early close; but however this may be I cannot 
 doubt that sooner or later the fact will be recognized and 
 accepted, that sound waves do not all move with one and the 
 same velocity; and that the difference in their velocities is due 
 chiefly to the difference in the intensities of the forces by which 
 
 * Journal of Science, second series, vol. v, page 312. The second of the pre- 
 ceding articles. 
 
56 
 
 they are respectively originated, and only in a much smaller 
 degree to the elevation of the temperature of the wave by com- 
 pression : and it will be seen that we have no need to resort to 
 the theory of Laplace to account for the difference between the 
 velocity of the wave formed by the report of a cannon and that 
 of a wave of infinitesimal intensity. 
 
Answers to the principal objections which have been made 
 to the last of the foregoing Articles. 
 
 Objection I. 
 
 In the foi'egoing articles it is maintained that sound waves 
 which diffei' in intensity are not propagated with the same 
 velocity. To this doctrine it has been objected that when a 
 tune is played at a great distance from the hearer, the notes 
 reach his ear in regular order and in their proper sequence in 
 respect to time ; which, it is alleged, would not be the case if 
 the waves moved with different velocities. Let us inquire into 
 the force of this objection. 
 
 If we consider the minute spaces through which strings, reeds, 
 etc., vibrate in giving origin to musical sounds, and the limited 
 number of vibrations made per second, whatever may be the 
 pitch, we shall not be able to make out that the velocity of the 
 vibrating string or reed is in any case greater than one or two 
 feet per second. In the computation upon which we are enter- 
 ing, the greater we assume that velocity to be the more will the 
 result favor the objection ; and as we can afford to be very liberal 
 to the objector we will assume that the greatest velocity that 
 can pertain to any vibrating body that sends forth musical notes, 
 may be 10 feet per second. The vibrating body imparts to the 
 air its own velocity ; and, from what is shown in Article V., it 
 follows that in putting air in motion with a velocity of 10 feet 
 per second, its density will be increased only so far as to exceed 
 the normal density by about one per cent. Now suppose that in 
 the weakest waves that can transmit an audible sound there is a 
 condensation of one-tenth as much : then the densities of the 
 strongest and weakest sonorous waves that can be produced by 
 musical instruments will be to each other as 101 to lOO'l, and 
 their velocities of propagation as .y/101 to y'lOOl. Consequently, 
 if two waves, thus differing in density, be originated simulta- 
 neously at a distance of 1000 feet from the observer, while the 
 
58 
 
 denser wave runs over the 1000 feet, the less dense wave will 
 
 run over "^ — = 995*534 feet, and will then be behind 
 
 ^101 
 
 the other 1000— 995'534 = 4-466 feet in (h'stanee ; and in time 
 
 (if the velocity of the denser wave is 1000 feet per second), 
 
 -—-=^^ of a second. 
 
 995-534-J-4-466 ^"^ 
 
 Thus it appears that two waves having as great a difference of 
 intensity as can exist between any two waves produced by musi- 
 cal instruments, if originated simultaneously at the same place 
 and thence propagated 1000 feet, Avill then be separated by 4-466 
 feet in space and by ^^ of a second in time. 
 
 But the impression of a musical note on the ear is not pro- 
 duced by a single icave, but by a series or group of waves ; occu- 
 pying in their production seldom or never less than ^ of a 
 second, and therefore extending through a space of i|^=50 
 feet. Consequently the two groups of waves, of which the single 
 waves we have been considering respectively constitute the first, 
 overlap each other and are coincident for more than ^ of their 
 length. 
 
 That displacements as great as that above considered do not 
 perceptibly impair the regularity and harmony of a tune is 
 evident from the fact that in every musical performance however 
 perfect, there are constantly recurring deviations from the true 
 time of the tune to the extent of not less than ^^ of a second ; 
 and such deviations must produce displacements of like extent, 
 even though we assume that the velocities of all waves are pre- 
 cisely equal. 
 
 Again — in the case of a number of musicians performing in 
 concert, although we should assume that all waves move with 
 the same velocity and that there is not the slightest deviation 
 from the true time, yet if the difference in the distances of the 
 hearer from the respective performers should amount to 4*466 
 feet — say 4^ feet, — there would result an equal displacement of 
 the notes. 1 think it will not be claimed that the most fastidious 
 critic of musical performances could perceive a distortion of the 
 music in such a case. 
 
 Our conclusion then, is that the apparent regularity in the 
 order and chronic sequence of the notes of a tune, performed at 
 any distance from which it can be heard, does not go to show 
 that all sound waves are propagated with the same velocity. 
 
59 
 
 Objbctiqx II. 
 
 Those who have heretofore attempted to investigate the law of the 
 propagation of sound have assumed that a sonorous wave consists 
 of contiguous particles of compressed air whose increased density 
 is greatest at or near the centre of the wave, diminishing thence 
 to cipher at the front and rear, and whose particles respectively 
 have the velocities due to the respective forces by which they 
 have been compi-essed; and they have illustrated their view of the 
 nature and constitution of a wave by a diagram constructed sub- 
 stantially as follows; where AB is a line of particles over which 
 
 c d 
 
 a wave is passing from A to B; cc? is the breadth of the wave; 
 and an ordinate from the curve above to any point in cd repre- 
 sents the compressing force and velocity of the i)article at that 
 point. 
 
 In the iifth of the preceding articles a sonorous wave is defined 
 as consisting of contiguous particles of air, all equally com- 
 pressed and having the same velocity. Sucli a wave, shown by 
 a diagram constructed on the same plan as before, would show 
 the wave in the form presented on the line CD; with a hori- 
 zontal line above and precipitous termini in front and rear. 
 
 C____ I I D 
 
 According to the view of a wave as presented on AB, the qui- 
 escent particles of air as they are successively encountered by 
 the advancing wave commence their motion with an infinitesimal 
 velocity which is augmented by insensible degrees, reaching its 
 maximum at the centre of the wave; but a wave of such form as 
 is shown on CD requires the full maximum velocity to be imparted 
 to the quiescent particles instantaneously. 
 
 To this last view of a wave the objection is made that a 
 finite velocity cannot be impressed upon matter instantaneously. 
 
 In making answer to this objection it is proper to say at the 
 outset that the term instantaneously, as used in the foregoing 
 articles, does not import an absolute negation of time. It is 
 employed in its more common sense, to indicate a time shorter 
 than any assignable time. Thus understood the view objected 
 to is in perfect harmony with that universal law of dynamics, 
 
 FT X MY 
 
60 
 
 or, The product of force and time is as the product of mass and 
 velocity. 
 
 For, let V represent the absolute velocity imparted to the 
 ])articles of air encountered by the wave ; let F represent the 
 constant force required to overcome the inertia of those particles 
 in being put in motion with the velocity V ; and let M be the 
 mass of air put in motion with that velocity in any time, T. Then 
 F will be as Y, and M will be as T, and consequently if we make 
 M infinitesimal, T will be an infinitesimal of the same order. We 
 have then, a finite velocity, V, imparted by a finite force, F, to 
 an infinitesimal mass, M, in an infinitesimal time, T; thus sus- 
 taining the view which was objected to. Tn this case I regard the 
 infinitesimal mass as taking the velocity V, without first passing 
 through the smaller velocities that may be assigned between V 
 and cipher; and I know of no law of nature which precludes the as- 
 sumption that such is the fact. If, however, the objector thinks that 
 even in this case the velocity V, is acquired by gradual acceleration 
 and can conceive how this gradual process can be carried through 
 in an infinitesimal time of the highest order and within the limits 
 of a space equally minute, such a view will not conflict with the 
 views maintained in the fifth article nor call for any modification 
 of the form of the wave as presented on the line CD. 
 
 Objection III. 
 
 It has been objected to that part of the fifth article where I 
 have estimated the extent to which the tem})erature of a M'ave 
 must be raised by evolved heat, in order that its velocity may 
 be increased from 916 to 1090 feet per second, that I have 
 assumed that an elevation of the temperature of the wave one 
 degree by evolved heat will have no more etfect to increase the 
 velocity of the wave than an elevation of the temperature of the 
 medium one degree ; it being alleged by the objector that one 
 degree of heat, confined to, and acting wholly within the wave 
 itself must have a vastly greater effect to increase the velocity 
 of the wave than the same amount of heat diffused throughout 
 the medium. 
 
 This objection seems to ignore the distinction between degree 
 of heat and quantity of heat : but I forbear to assume that the 
 objector has overlooked a distinction so wide and so obvious, 
 
61 
 
 Other things being equal, the velocity of a wave is directly as 
 the square root of its elastic force and inversely as the square 
 root of the specific gravity of the medium in which it is propa- 
 gated. An elevation of temperature, whether of the wave or of 
 the medium, has no effect to increase the velocity of the wave, 
 except so far as it increases the ratio of the elastic force of the 
 wave to the specific gravity of the medium. To elevate the tem- 
 perature of the vmve increases that ratio by increasing the elastic 
 force of the wave. To elevate the temperature of the medium, 
 increases that ratio by decreasing the specific gravity of the 
 medium. Hence the question raised by the objector is whether 
 equal increments of temperature in the two cases result in equal 
 increments of that ratio. I maintain that the effects of the 
 elevation of temperature are precisely the same in both cases to 
 increase that ratio, and to increase the velocity of the wave. 
 For an elevation of the temperature of the wave one degree, by 
 increasing the elastic force, gives to the air of the wave a ten- 
 dency to expand and reduce its specific gravity in the same ratio 
 that the heat had increased its elastic force. An elevation of the 
 temperature of the medium one degree causes it to expand and 
 reduce its specific gravity in that same ratio. The effect there- 
 fore upon the velocity of the wave is the same in both cases. 
 
UNIVERSITY OF CALIFORNIA LIBRARY 
 BERKELEY 
 
 Return to desk from which borrowed. 
 This book is DUE on the last date stamped below. 
 
 JUN 4 194J 
 
 IrEC'D ld 
 
 LD 21-100m-9,'47(A57028l6)476