Aerodynamics (Lanchester)/Appendices

by Frederick William Lanchester

APPENDIX I.

INFLUENCE OF COMPRESSIBILITY OF AIR ON THE ENERGY EXPENDED IN FLIGHT.^[1]

The influence of compressibility as affecting the expenditure of energy in flight is best computed from the velocity of wave motion—sound.

The whole theory of Chapter VIII., based on the hypothesis of constant sweep, relates, strictly speaking, as set forth, to the incompressible fluid; it will be shown that the effects of compressibility can be dealt with as a correction, or rather by a preliminary correction, to the figures involved.

Let us write $U$ for the velocity of sound, and, as before, let $V$ be the velocity of flight. Then it is evident that any disturbance will travel forward relatively to the body in flight less rapidly than it will travel backward in the opposite direction in the relation ${\frac {U-V}{U+V}},$ as in the case of "Döppler's principle." Now, regarding the fluid motion as due to a field of force (Chapter IV., § 113), we have the communication of upward momentum diminished, and the communication of downward momentum increased, in like proportion.

Thus in the ideal case of Chapter IV., if we have to deal with a compressible fluid, an expenditure of power becomes necessary in accordance with a regime

$\epsilon _{1}={\frac {U-V}{U+V}}.$

(1)

(Compare § 172 et seq.)

In the extreme case when $V$ becomes equal to Uno disturbance can precede the aerofoil in its flight, and the whole reaction will be due to the communication of downward momentum; the cyclic component in the peripteral system vanishes. In the above expression when $V=U,\epsilon _{1}=$ zero, which leads to the same conclusion.

Let us take the $\epsilon$ of Chapter VIII. to be the e$\epsilon$ proper to an incompressible fluid, and let the symbol employed above, $\epsilon _{1}$, be the corresponding value when $U$ is the velocity of sound. Then from the foregoing reasoning we have—

$\epsilon _{1}={\frac {U-V}{U+V}}\ \epsilon$

(2)

This expression is in harmony with equation (1), which relates to the special case where $\epsilon =$ unity.

Example.—Dealing with the highest result tabulated, i.e., 80 ft. sec., and taking $U=$ 1120,

$\epsilon _{1}={\frac {1120-80}{1120+80}}\ \epsilon ={\frac {13}{15}}\ \epsilon ;$

that is to say, for the velocity stated the value of $\epsilon$ employed in Chapter VIII. is too high in the relation 15/13.

But it is evident from the whole argument of Chapter VIII. that the constant $\epsilon$ is not the only one of the constants involved in the equations affected by the compressibility of air. In fact, from the reasoning employed (§§ 161, 172 et seq.) it would appear that the constants $c$ and $\kappa$ will also be affected, and we may fairly make the assumption^[2] that the constants will remain related in accordance with the equation of § 177, and that consequently we may regard the influence of compressibility as degrading the effective value of ${\mbox{n}}$ from its actual value to a less value in the proportion required by equation (2).

Thus in the case under discussion, if we have to deal with an aerofoil whose ${\mbox{n}}=$ 12, we find by Table IV. $\epsilon =$ .75. Taking 13/15ths of this, we have $\epsilon =$ .65, which the table shows corresponds to ${\mbox{n}}=$ 7. That is to say (assuming the accuracy of these "plausible values"), for a speed of light of 80 feet a second the corrected values for an aerofoil of aspect ratio ${\mbox{n}}=$ 12 can be read from the various tables by taking the equivalent aspect ratio ${\mbox{n}}=$ 7.

APPENDIX II.

A NOTE ON THE COMMUNICATION OF MOMENTUM AND ON THE VELOCITY AND MOMENTUM OF SOUND.

The "Principle of No Momentum," enunciated as a proposition in § 5 of the present work, constitutes so far as the author is aware an innovation in the treatment of problems in fluid dynamics.

The proof of this proposition, indeed the principle itself, is so perfectly simple and obvious, that it is not without some hesitation that it is put forward as new. The consideration of the following examples, involving the simple application of the principle, and leading to results which certainly are not generally recognised, would seem to leave no doubt as to the fact.

Example 1.—The Vortex Atom Theory of Kelvin gives considerable trouble in the light of the Principle of No Momentum.

If the fluid be supposed incompressible and of uniform density in its parts, and if we suppose for example a single vortex ring in motion in a rigidly bounded region,^[3] it manifestly cannot carry momentum (§ 5), and equally the momentum of a number of such rings must be zero. It is of course possible that such a ring or number of rings may raise the peripheral pressure of the region, that is, the pressure on the walls of the enclosure, but the case of an incompressible fluid and a rigid enclosure is in this respect an indeterminate problem. Thus if, still regarding the fluid as incompressible, we suppose the enclosure to possess some degree of elasticity so as to exert on the fluid a pressure sufficient to prevent cavitation, then the peripheral pressure will undergo no change in consequence of the vortices, for a change of pressure on the walls of an elastic enclosure must be accompanied by a compression or dilatation of the fluid contents. Under these conditions the greater the energy of the vortex system set up in the fluid the lower will become the pressure in the internal part of the region, so that the plus and minus momentum of the equal and opposite flow taking place across any imaginary barrier plane is accounted for by the ordinary static pressure on the confines of the region, and does not give rise to any added pressure.^[4]

If we suppose the enclosure rigid and the fluid elastic, the change of pressure due to the vortices on the boundary walls depends upon the law of elasticity, and is not a function of the magnitude or energy of the vortex system alone. The result of the above reasoning is not at all in harmony with accepted views as to the behaviour of vortices as expounded in the Vortex Atom theory.^[5][6] According to the highest authorities the individual vortices carry momentum just as if they were bodies of greater density than the fluid that contains them, instead of being, according to hypothesis, composed of the fluid itself and therefore of the same density. It is not possible in the present work to go fully into the cause of this discrepancy which the author believes to be due to the mathematical theory regarding the vortex ring, as the result of an impulse distributed evenly over the disc area, instead of as the resultant of two equal and opposite impulses, the one applied over the disc area and the other to the confines of the fluid region. This is merely thrown out as a suggestion, but whatever the explanation may be, the case of a vortex ring travelling to and fro in a rigidly bounded region filled with incompressible fluid and carrying momentum is presented for consideration to the exponents of the Vortex Atom Theory as involving a flagrant violation of the third law of motion.

Example 2.—Momentum of Sound Waves.—This is a question that has been widely discussed of recent years, and one on which different authorities are not altogether in agreement.^[7]

If we take it as essential by definition that the passage of a complete wave or train of waves results in no permanent displacement of the particles of the fluid, that is to say, that each particle of the fluid occupies after the passage of the wave train the same position as before its passage,^[8] it immediately follows that the mean density of the wave train is equal to that of the undisturbed fluid.^[9]

It is therefore evident (as in § 5) that if such a wave train be supposed to travel to and fro in a box (Fig. 160), from end to end, being repeatedly reflected, no movement of the mass centre of the fluid within the box, i.e., relatively to the box, can take place, and hence such a wave train possesses no momentum.

It follows that if the wave train have an excess of compression or rarefaction so that its mean density is greater or less than that of the undisturbed fluid (the condition of the particles returning to their initial positions being departed from), momentum will be carried positive or negative, as the case may be, exactly as represented by the excess or deficit of density in the wave train.

Thus the momentum carried by any sound wave is a measure of and is measured by the displacement of matter by that sound wave, and if the displacement is zero the momentum is zero.^[10]

The question of momentum carried by wave motion is frequently
Fig. 160. regarded from the point of view of pressure developed, that is, the pressure produced in the fluid by the communication of momentum at reflection etc. This point of view is not without interest.

Taking first the case of a gas obeying Boyle's law, i.e., ${\frac {P}{\rho }}=$ constant; the mean pressure of the whole of the space can undergo no change, for $P/\rho$ is constant for each small element throughout the region, and the integration of $\rho$ being constant (since the whole mass of fluid in the enclosure is unchanged), the integration of $P$ throughout the enclosure is also unchanged. Now it by no means follows that the mean pressure throughout the region is the same as the mean pressure on the walls of the enclosure; in fact, we know from hydrodynamic principles that in many cases of fluid motion it is not so. In the case in point, however, it is manifest that the mean pressure is the same whether the integration is taken over the surface or throughout the volume, for (Fig. 160) the pressure on the walls of the box is point for point the same as for any surface parallel to these walls passing longitudinally through the region, and the pressure on the ends is of the same mean value, for the velocity of sound can be correctly computed on this basis.^[11] It is therefore evident that for a fluid obeying Boyle's law the existence of wave motion does not give rise to any change of pressure.

Under these circumstances it follows that change of pressure will take place in a region containing an ordinary gas ($P/\rho ^{\gamma }=$ constant), the magnitude of which can be calculated from the energy of wave motion that passes into, and exists in, the thermodynamic system.^[12]

There is much confusion of thought at the present time on the question of the carrying of momentum by a wave train and the generation of pressure in a fluid region occupied by wave motion. This has probably arisen from too close attention being paid to the special case of a continuous wave train, as in the Kundt's tube.

It is much more difficult to distinguish between direct momentum transference by the wave and momentum transference by the pressure generated by the wave, in the case of the Kundt's tube, where the whole region is occupied by wave motion than in the case of a limited wave train passing to and fro.

Thus in the case of a limited wave train, if it carry momentum, that momentum can be represented by some definite value of mv, and the remainder of the system with which the wave is associated must, relatively to the common mass centre, have an equal and opposite momentum at every instant of time. But a self-contained system consisting of a simple enclosure containing fluid of uniform mean density (regarding the individual waves of the train as small) cannot suffer change of momentum without infringing the third law of motion; consequently the wave train (if of the same mean density as the quiescent fluid) cannot carry momentum. This is in effect the argument of § 5.

In the case of the continuous train, as in the Kundt's tube, we lose touch with this method of argument, for the action is continuous, and a pressure increase can only be distinguished from the true carrying of momentum by the wave train by a process of mathematical analysis that is full of pitfalls.^[13]

The case of light pressure, or the carrying of momentum by electro-magnetic radiation, is not a problem in ordinary dynamics, and is untouched by a purely dynamical argument or method of demonstration such as here employed. The reason for this fundamental distinction is that when motions of the all-pervading ether are essentially involved, such a term as a self-contained system ceases to have any signification. There is only one self-contained system known to us—the Universe.

From another point of view we know that according to modern theory the momentum of any finite quantity of matter, however small, moving with the velocity of light is infinite^[14]; consequently a finite quantity of momentum will be carried at this velocity by a quantity of matter smaller than can be expressed in finite units, or, physically speaking, communication of momentum at the velocity of light becomes independent of the displacement or transference of matter. Thus the present application of the principle of no momentum is in no way antagonistic to modern views and discovery as to the transference of momentum by light and other manifestations of electric radiation.

 

ADDENDUM A.

Assuming Boyle's law, let us examine the case of an isolated compression wave travelling to and fro in a prismatic box of unit cross section and length $=l.$ Let the mass of the fluid in this wave, in excess of the normal contents of the region it occupies, be $m.$

Now since the wave carries an excess of fluid it will carry momentum, and this momentum will be represented by the mass $m$ transported with the velocity of wave propagation, which we denote by the symbol $U.$

And the presence of this excess of fluid in the enclosure will raise the mean pressure throughout the enclosure to the same extent as if it were uniformly diffused. (This has already been demonstrated.)

The proposition is to show that the pressure increase due to the wave on the ends of the enclosure is equal to the mean pressure increase throughout the enclosure.

The proof of the proposition rests in showing that the velocity of sound can be correctly calculated by the assumption of the proposition as hypothesis.

Thus the compression wave will be in equilibrium when its rate of communication of momentum to the ends of the enclosure is equal to the added pressure.

Momentum of wave	$=m\ U.$
Momentum communicated by each reflexion	$=2\ m\ U.$
Number of reflexions per second	$={\frac {U}{2\ l}}$
∴ force due to momentum of wave	$={\frac {m\ U^{2}}{l}}$	(1)

Let $P_{1}$ be the mean pressure exerted by the additional mass $m$ distributed throughout the enclosure; then, by Boyle's law, ${\frac {P_{1}}{\rho }}=k,$ where $k$ is a constant,

and	$\rho \ {\frac {m}{l}}$
or	$P_{1}={\frac {m\ k}{l}}$	(2)
by (1) and (2)	${\frac {m\ U^{2}}{l}}={\frac {m\ k}{l}}$
or	$U^{2}=k$
∴	$U={\sqrt {k}}={\sqrt {\frac {P}{\rho }}}$

which is the well-known result; by substituting for $P$ and $\rho$ (abs. units) for air at any stated temperature the Boyle's law velocity is obtained; this is, of course, subject to Laplace's correction for the actual velocity.

The above reasoning, though here given as a disproof of Larmor's theorem as a generalisation, is in reality a valid and simple method of determining the velocity of sound. If we cast aside the mythology introduced into the subject by the light radiation specialists, and treat the question as it should be treated, as a matter of ordinary dynamics, it is evident that it is the displacement of matter (if any) that gives rise to the momentum of a wave, and it is the momentum of the wave that gives rise to the pressure at reflection, and by equating the two, as has been done in the foregoing demonstration, we have the simplest known method of obtaining the expression for the velocity of sound.

The nature of the flaw in Larmor's theorem is discussed in Addendum B of the present Appendix.

The simplicity of the present method of the determination of the velocity of sound is largely due to the form in which Boyle's law is presented. It is usual to write the isothermal law (Boyle's law), for a perfect gas $PV=$ constant; now this presumes mass constant. It would be quite as correct to write $P/m=$ constant, taking the volume to remain unchanged. It is obviously best to include both mass and volume as variables and write $P/\rho =$ constant, as has been done.

The present method has much in its favour. The argument not only covers waves of small amplitude, but waves of any amplitude and any form; we may regard a wave in a fluid obeying Boyle's law as built up of a number of superposed elements, each of which conforms to the pressure-momentum equation giving the same value of $U$ for each element alone or in superposition. Consequently waves in a fluid obeying Boyle's law have no tendency to travel faster in one part than in another part; their form is permanent and velocity uniform.

In Poynting and Thomson's "Sound," a method is given for the theoretical determination of the velocity of sound, on the assumption that the pressure changes are proportional to the volume changes, and the usual well-known expression $U={\sqrt {\frac {E}{\rho }}}$ is obtained. A foot-note is given in connection with this demonstration, as follows:—

If the pressure changes are too considerable to justify the assumption that they are proportional to the volume changes, we may regard the variation from proportionality as an external force represented by $X.$ Thus in a wave of very considerable displacement and pressure excess, $X/(P_{M}-P_{N})$ can be shown to be positive, and $U$ is greater than the value in (5). This agrees with certain experimental results given below.

Fig. 161. The suggestion here appears to be that the straight line trace in the $PV$ diagram (which is the equivalent of the Poynting and Thomson hypothesis) is essential to the rigid application of theory for waves of sensible magnitude. This is contrary to the result here obtained, and surely must be incorrect. A gas obeying Boyle's law according to these authorities would share with the real gas the mutability of wave form consequent on the adiabatic law.

According to the present author the straight line diagram is to be found in the plotting of $P$ and $\rho$ for Boyle's law, Fig. 161 a, which corresponds to the hyperbola for the $PV$ diagram; and this straight line diagram is the looked-for analogue of the isochronous pendulum.

If we plot the analogous form of the adiabatic law. $P/\rho ^{\gamma }=$ constant, Fig. 161 b, we no longer have a straight line diagram, but for small amplitude we may approximate by drawing a tangent $EF$ cutting the axis of $y$ at $F.$ We may regard the point $F$ as a new origin which will give the pressures proper to the limited portion of the curve approximated on the Boyle's law basis. From geometrical considerations we have $AO$ to $AF$ in the relation 1 is to $\gamma$, the relation of the real to the fictitious pressure of the gas; this at once gives us Laplace's correction.

In this case the assumption is obviously that the amplitude is small, for otherwise the tangent $EF$ no longer approximates sufficiently to the actual curve.

The rationale of Laplace's correction may also be studied from the direct examination of the conditions. If we suppose in an adiabatic gas that a small isolated compression wave be constrained to move with the velocity proper to the gas obeying Boyle's law, the pressure during the reflection of the wave will be in excess of the momentum the wave communicates, to the extent that an adiabatic compression pressure is greater than the Boyle's law pressure for a given change of density. For small amplitude this is in the relation of $\gamma$ to unity. Obviously the wave must travel faster to supply the momentum necessary to equalise, and since the momentum communicated per unit time varies as the square of the velocity, the velocity must be multiplied by ${\sqrt {\gamma }}.$

The question of the behaviour of an adiabatic wave of sensible amplitude is one of great complication that yet awaits a general solution. The compression regions are always endeavouring to move faster and the rarefaction regions slower than the mean velocity. From the present standpoint this is evidently due to the pressure increase becoming proportionately greater than the density increase (Fig. 161), and vice versâ, thus destroying the necessary balance between the pressure reaction and the communication of momentum by which it is maintained. The more usual and equally correct point of view is to attribute the difference of velocity of different portions of the wave to the difference of temperature of its parts.

Where we have a train of waves in a gas following the adiabatic law, it has been shown that there must be a pressure increase due to the energy that enters the thermodynamic system. Where the train is continuous, as in the Kundt's tube, no complication arises from this cause, but where we are dealing with a limited train, it is difficult to see in what manner this pressure can be confined to the region occupied by the wave train; according to thermodynamic principles it must be distributed uniformly and press equally in every direction. If this is true, the wave train as a whole will expand, and the remainder of the fluid will be compressed, so that the mean density of the wave train will become less than that of the undisturbed fluid. On this basis, employing the principle of § 5, a wave train under the conditions we are now supposing must be regarded as conveying negative momentum.

 

ADDENDUM B.

In an article on radiation in the "Encyclopaedia Britannica,"^[15][16] Larmor gives a theorem which purports to be a general proof of the transmission or communication of momentum by wave motion. Poynting^[17] has given a condensed edition of this alleged proof, which may be quoted, as follows:—

Let us suppose that a train of waves is incident normally on a perfectly reflecting surface. Then, whether the reflecting surface is at rest, or is moving to or from the source, the perfect reflection requires that the disturbance at its surface shall be annulled by the superposition of the direct and reflected trains. The two trains must therefore have equal amplitudes. Suppose now that the reflector is moving forward towards the source. By Döppler's principle the waves of the reflected train are shortened, and so contain more energy than those of the incident train. The extra energy can only be accounted for by supposing that there is a pressure against the reflector, that work has to be done in pushing it forward. . . . A similar train of reasoning gives us a pressure on the source, increasing when the source is moving forward, decreasing when it is receding.

Now it is evident that the whole of this reasoning rests on the assumption that the reflector, while impervious to the waves, is freely pervious to the medium;^[18] an assumption that may be true in the case of light, but is certainly not true in the case of sound.

Poynting evidently appreciates this difficulty, for he says:—

“It is essential, I think, to Larmor's proof that we should be able to move the reflecting surface forward without disturbing the medium except by reflecting the waves.” But further on he says:—

“But for sound waves I venture to suggest a reflector which shall freeze the air just in front of it, and so remove it, the frozen surface advancing with constant velocity $u.$ Or perhaps we may imagine an absorbing surface which shall remove the air quietly by solution or chemical combination.”

Now this is the first time that the author has heard it seriously suggested that portions of any dynamic system, essentially involved in that system, may be stolen away without affecting the sequence of events; it is, at least, evident that any such assumption totally invalidates Larmor's theorem as a generalisation, and in particular in its application to ordinary dynamic wave motion. It is very surprising to find that Poynting subsequently states that he finds Larmor's proof quite convincing.

In the address from which the above quotations have been given, Poynting cites an experiment by Prof. Wood intended to demonstrate the reality of sound pressure. In this experiment the sound waves from a strong induction- spark are focussed by a concave reflector on to a set of vanes as used on a radiometer, causing them to spin round. Now it is fair to assume that the cause of the emission of sound waves by an induction-spark is the heating of the air suddenly and locally by the spark energy, and consequently the wave will primarily be a compression wave. If steps were taken to cool the air immediately after it had been heated, doubtless a rarefaction wave of equal displacement would follow, but no such steps are taken. It is true that the air initially heated by the spark is rapidly cooled by giving up its heat to the surrounding air, but this expands the air to which the heat is passed on, so that, on the principle of the author's bottle calorimeter,^[19] no loss of volume takes place. There is possibly some minute quantity of heat lost to the conductors by which the current is supplied to the spark, but except for this the waves emitted will, on the whole, be compression waves involving a displacement of matter, and carrying the momentum appropriate to the mass displaced travelling with the velocity of sound. Ultimately the heated air is carried away by convection, but this does not affect the problem.

It is therefore evident that this experiment proves nothing, except that which we know already, i.e., a displacement of matter carries with it momentum.

It is probable that other more or less successful experiments designed to demonstrate the existence of sound pressure involve some similar fallacy. It must be borne in mind that an unsymmetrical design of sound generator may conceivably emit pressure waves containing momentum in one or more directions, and rarefaction waves in others, or perhaps the air displaced by the pressure waves emitted in one direction may be replaced by a steady flow in other directions. On the other hand, it is possible that by some highly refined method the true pressure of a continuous wave train may be detected and measured, and the theoretical result that it is due to the energy passed into the thermodynamic system may some day receive confirmation.

 

ADDENDUM C.

In the foregoing Appendix and Addenda A and B the assumption has been made that the change of mean pressure within an enclosure containing a perfect gas is directly proportional to the heat added or taken away, and mention has been made of a form of calorimeter proposed by the author depending upon this principle.

It is evident that if the principle can be proved as a general proposition as relating to the total heat it is also proved in relation to heat differences, that is heat added or subtracted.

The following proof goes beyond the problem as presented by the calorimeter, and applies generally for an enclosure in which the various portions of the gas are artificially constrained to occupy given positions by any means whatever, including, for example, the case of a wave train or other dynamic disturbance.

Let the enclosure be supposed divided into a number of small equal elements, and, examining firstly the conditions that apply to each small element to which it may be supposed that a quantity of heat $h$ is supplied and distributed uniformly, giving rise to a uniform pressure $P$ and temperature $T,$ we have:—

${\frac {P}{\rho }}=T\times {\mbox{const.}}$

but for a perfect gas

$T={\frac {m}{h}}\times {\mbox{const.}}$

where $m$ is the mass of the contents, hence

${\frac {P}{\rho }}={\frac {h}{m}}\times {\mbox{const.}}$

and since

$\rho ={\frac {m}{l^{3}}}$

$P={\frac {h}{l^{3}}}$

for the element with which we are concerned.

Now,	let	$n=$ the number of elements into which the enclosure is divided.
	„	$P_{1}\ P_{2}\ P_{3}$ etc., be the pressures developed in the different elements to which quantities of heat $h_{1}\ h_{2}\ h_{3}$ have been supplied.
	„	$H=$ the total heat.
	„	$P_{m}=$ the resulting mean pressure.

Then the value of $l^{3}$ for each element will depend upon the number of elements into which the enclosure is divided, so that $l^{3}\propto {\frac {1}{n}},$ and thus

$P_{1}=k\ n\ h_{1}$

$P_{2}=k\ n\ h_{2}$

$P_{3}=k\ n\ h_{3}\ {\mbox{etc.}}$

where $k$ is a constant.

∴	$P_{1}+P_{2}+P_{3}+{\mbox{etc.}}=$ $k\ n\ (h_{1}+h_{2}+h_{3}+{\mbox{etc.}})$
But	${\frac {P_{1}+P_{2}+P_{3}+{\mbox{etc.}}}{n}}=P_{m}$
and	$h_{1}+h_{2}+h_{3}+{\mbox{etc.}}$ is total heat added $=H$
∴	$P_{m}=k\ H$

this result continues to apply when the number of elements $n$ becomes indefinitely great, hence the proposition is proved.

 

ADDENDUM D.
A Retrospective Note.

It is perhaps of some interest to state that the investigations included in the present appendix were actually made in the early part of 1905; the portion relating to the theory of sound momentum was submitted in the form of a draft paper to Professor Poynting, then President of the Physical Society, with whom the author had some correspondence on the subject.

The author did not receive sufficient encouragement to think it worth while submitting the paper, especially in view of previous experience and of the fact that not only Poynting, but Larmor, and at that time Rayleigh, were thoroughly identified with the general doctrine of sound momentum.

Referring to a warning note raised by the author, and with regard to the suggested paper, Professor Poynting wrote on June 9th, 1905, “Yes, I am quite sure about my views. But it is quite evident that we are not going to see in the same direction. I shall probably send my proof of pressure to the Physical Society some time so as to let those interested have their choice.”

Neither paper materialised. Lord Kayleigh shortly afterwards published his article (loc. cit. ante) in the Phil. Magazine, somewhat modifying his earlier conclusions^[20] and anticipating publication by the author in respect of two of the results now stated, i.e., (1) the absence of momentum in or pressure due to a wave train under the conditions of Boyle's law; (2) the pressure of sound waves in a real gas as due to energy entering the thermodynamic system.

Before going to press the author submitted the above addendum to Prof. Poynting, and received the following reply, October 7th, 1907:—

“I stick to the postcard and have no objection to its publication.”

“My proof of pressure was practically identical with Rayleigh's and gave the result (1), and therefore I suppose (2). That is why the paper did not materialise.”

This is a truly astonishing statement in view of certain correspondence and MSS. in the author's possession. The following quotations are given as throwing some light on Prof. Poynting's actual position at the time in question.

In a letter dated June 7th, 1905, referring to a draft MS.^[21] submitted by the author, Poynting says:—

“On p. 3 the paragraph marked wants, I think, a few words inserting to make it clear.”

“Say thus:—Divide the wave train into lengths each containing unit mass. The time taken by each of these lengths to pass a given point is proportional to the length. Also the time during which, etc. ... or something of the sort. But it appears to me that this is not the whole story, but that the motion docs communicate momentum. If the velocity forward is $u\ \longrightarrow ,$ momentum $\rho \ u\times u=\rho \ u^{2}$ crosses the plane $\longrightarrow .$ If the velocity is $u\ \longleftarrow ,$ momentum $\rho \ u^{2}$ crosses the plane $\longleftarrow ,$ and both of these give an addition of momentum $\longrightarrow$ to the region on the forward side of the plane.”^[22]

For the purpose of reference p. 3 of the author's original MS. is given in the accompanying footnote, the paragraph marked by Prof. Poynting being italicised. The initial and final paragraphs are completed as on pp. 2 and 4 of the MS.

It would appear from the above transcript that Poynting unquestionably held the view on June 7th, 1905, that a Boyle's law wave train would give rise to pressure increase or momentum transference, just as he held this view at the time of his address to the Physical Society in February of that year. Furthermore, it is evident that, at the time in question, result (2) was not a consequence anticipated by Poynting, for in another communication about the same date in reply to the author he says:—

I have not thought of the sound pressure as accounted for by the kinetic theory of gas. S. Tolver Preston, I think, did so somewhere. It appears to me best in the first place to get at the idea as I have done in the paper^[23] as resulting from known observable properties. Then go to the kinetic theory if you like.

The perfectly elastic solid—if by that is meant one that obeys Hook's law rigidly—would give pressure apparently from Larmor's theorem.^[24]

It is difficult to understand how Prof. Poynting can have been led to make so extraordinary a statement as that contained in his present letter in view of the facts above given, and the author trusts that he will see his way to give publicity to some adequate explanation.

APPENDIX III.

A NOTE ON DISCONTINUOUS MOTION.

In § 101 allusion has been made to the instability of a surface of kinetic discontinuity in an inviscid fluid, and at the same time the impossibility of such a surface breaking up into finite vortex filaments is pointed out.

Helmholtz^[25] has suggested that the instability takes the form of a development of convolutions of the surface of discontinuity or surface of gyration. He says:—

"An infinitely extended plane surface uniformly covered with parallel straight [infinitesimal] vortical filaments might indeed continue stable, but where the least flexure occurs at any time the surface curls itself round in ever narrowing spiral coils, which continually involve more and more distant parts of the surface in their vortex."

It is, unfortunately, not easy to form a clear picture of the continued transition that the above implies, or even of the resulting system of flow. There would appear to be no doubt, however, that Helmholtz's view is substantially correct.

APPENDIX IV.

LEAPING OE BOUNDING FLIGHT.

Many of the smaller birds habitually fly at a considerably greater velocity than would be computed from the pressure-velocity tables (Tables IX, and X.) on the lines of § 187.

The means by which this is accomplished is instructive. The bird flies briskly for a short distance and then closes its wings, continuing its flight as a simple projectile, so that the total flight consists of alternations of active flight and projectile flight. The flight path under these conditions consists of a series of leaps, as given in Fig.1 62, in which the thick lines represent the periods of active flight and the fine lines the periods when the wings are closed.

It is evident both from the form of the flight path and from the behaviour of the bird that the whole of the sustentation takes place while the wings are spread, and that during this period the wings actually sustain both the weight of the bird and the centrifugal component due to its curvilinear flight path, and the sum of these is the effective load on the wing area in the sense of §§ 185—187.

The present note is based on visual observation. The largest bird witnessed by the author as employing the leaping mode of flight is the green woodpecker {Picus viridis); the weight of this bird averages about six to seven ounces (180 grams). Larger birds, as, for instance, the partridge, glide with wings outstretched when not in active flight. The greatest length of "leap" in proportion to the corresponding active period, noted by the author, is about 3 : 1 (Fig. 162, c). In this instance the species of bird was not identified. This proportion means that the reaction sustained by the wings when in action is approximately four times the weight of the bird, on which computation the flight velocity should be about twice that proper to the actual weight and wing area measurement.^[26]

It is difficult to assess accurately the speed of flight of a bird under any circumstances, and most of all under the conditions now under discussion. Travelling at somewhat over thirty miles

Fig. 162.

per hour on a motor vehicle, it is not an uncommon sight to see a pied wagtail or other small bird endeavouring to escape directly ahead by adopting the mode of flight under discussion. When hard pressed in this way the wagtail flies low, and its motion closely resembles the bouncing of an india-rubber ball on the surface of the road, showing that the periods of active flight become very short in comparison with the length of the "leap."

Most of the smaller birds are able, by adopting the leaping mode of flight, to attain speeds of about thirty or forty miles per hour.

The probable reason for the leaping mode of flight being confined to the smaller birds is to be found in the considerations discussed in §§ 195, 196. The influence of aerofoil weight (wing weight) is less important in the case of a small aerodrome or bird than of a large one. Consequently nature can endow a small bird relatively with an extent of wing surface not "commercially " possible in the case of a larger bird, so that the smaller bird can, in normal active flight, fly slower than a large one, but by adopting the leaping mode it can, in effect, divest itself of its superfluous surface, and can then rival the larger birds in velocity. The leaping mode is, in fact, a means of adjustment, by means of which the conditions of least resistance can be approximated under considerable variations of velocity. If one of the larger birds, with its limited relative area, were to force its velocity up to the point at which leaping flight would pay, it would require an amount of energy per second far beyond its actual horse-power capacity.^[27]

APPENDIX V.

SOARING.

Authorities are generally agreed at the present time that one at least of the varieties of soaring^[28] practised by the larger birds involves the abstraction of energy from the wind fluctuation, that is to say, the soaring bird can derive the power required for its flight from the energy of turbulence of the wind (comp. §§ 37, 131).

It is clear that a bird having no horizontal force applied to it from without (in contradistinction to a kite which is connected to the earth by a string), is unable to effect any change in the total (horizontal) momentum of the air that comes within its grasp; consequently it cannot raise or lower the mean velocity of the wind, although it may be able to cause some parts to move faster and some more slowly.

It is evident that if a bird can, by altering its angle and altitude, so manipulate the wind coming within its grasp, that the portions that are moving in excess of the mean velocity have their velocity reduced, and those that are moving at less than the mean velocity are accelerated, the total energy of the wind will be reduced, and the energy thus taken from the wind may become available for the purposes of propulsion. It is further evident that if a bird can carry the procedure suggested to the extent of reducing the whole of the air handled to a uniform velocity, that is to say, to its mean velocity, it will have taken away the whole of the energy that is available; i.e., it will have removed the whole of the turbulence energy from the air within its reach. The foregoing assumes that the energy of turbulence consists wholly of motions in the direction of the main current, but the argument may, if required, be extended to include motions in the directions of the other two co-ordinate axes of space.

Without discussion of the means whereby the bird operates to play off one portion of the wind against another, we may, from the above considerations, form an outside estimate of the available energy. Thus if we prescribe some conventional form as representing the motion of turbulence, such as a simple harmonic motion in the line of flight, or a compound harmonic or circular motion of known velocity, we can calculate the turbulence energy per unit volume, and we may convert this into a thrust force per unit area of the stratum of air handled; if, then, we know the extent of this area in the case of any particular bird, and the weight of the bird, we can determine the gliding angle $\gamma ,$ the minimum value of which is a quantity otherwise known. Conversely we may, starting from the gliding angle and other data, determine the minimum velocity of turbulence on the convention chosen that will render soaring flight possible.

A question that presents some difficulty is the estimation of the area of the stratum of air handled. At first sight this might be supposed to be the "sweep" of the aerofoil, i.e., $=\kappa \ A$ (§§ 109, 160), but the energy estimated on this basis from known fluctuation data appears to be insufficient.

The conception of the peripteral area (§ 210) suggests that, as in the case of the propeller blade, the cyclic or peripteral system may distribute the momentum over a much greater mass of fluid than that coming within the sweep of the aerofoil, and so a far larger mass of the air than that coming within the sweep area will be "handled" in the sense of the present discussion. On this basis the area of the stratum from which energy may be drawn is given by the expression, ${\frac {1+\epsilon }{1-\epsilon }}\ \kappa \ A$ (comp. § 210).

In the following example the turbulence velocity is computed necessary to provide the requisite energy to a hypothetical albatros, whose data are:—

Weight		$=14\ {\mbox{lbs.}}$
Area		$=5\ {\mbox{square feet.}}$
${\mbox{n}}$		$=12,$ hence $\kappa =1.195$ and $\epsilon =.75.$
$\gamma$	taken	$=1/7$

The computation will be made both on the basis of sweep and that of peripteral area, and the figures will be given both for a simple harmonic motion and for circular motion, the assumption being in all cases that the whole of the available energy is utilised. As in all probability the bird can only utilise a comparatively moderate portion of the total available energy, the actual velocity of fluctuation will require to be very much greater than that stated in each case, in order that soaring should become possible.

Now resistance to flight = $W\ \gamma ,$ which from the foregoing data = 14 ÷ 7 = 2 pounds, or in absolute units = 64.4 poundals, or energy required per foot traversed = 64.4 ft. poundals.

Sweep = $\kappa \ A$ = 5 $\times$ 1.195 = 6 (approx.), and mass of air handled (on basis of sweep) per foot traversed = .078 = 6 = .47.

If $v$ be the velocity of mean square of turbulent motion, energy per foot traversed is

${\frac {.47\times v^{2}}{2}}$

∴	$.47\ \times v^{2}$		$=64.4\times 2$
	whence	$v^{2}$	$=274$
	or	$v$	$=16.5$

Thus if the motion of turbulence is equivalent to a superposed circular motion, that is to say, if it consist of two component horizontal simple harmonic motions at right angles, and if the bird is able to abstract the total energy of both components, then V will be the maximum velocity of either component, or the uniform velocity of the equivalent circular motion; hence under the supposed conditions the maximum velocity of turbulence = 16.5 feet per second.

If the turbulence contain only one harmonic component, or if, which amounts to the same thing, the bird is only able to take advantage of the harmonic component in the line of flight, the available energy for a given maximum velocity will be only half that on the basis of circular motion; hence, in order that the necessary energy should exist in the wind, the maximum velocity must be multiplied by ${\sqrt {2}},$ or, on simple harmonic basis, the maximum velocity (plus or minus) of fluctuation becomes 23.4 feet per second.

The above estimates are on the basis of sweep. On the basis of peripteral area we have mass of air handled per foot traversed—

$={\frac {1+\epsilon }{1-\epsilon }}\ \rho \ \kappa \ A={\frac {1.75}{.25}}\times .47=3.3\ {\mbox{lbs.}}$

∴$\quad {\frac {3.3\times v^{2}}{2}}=64.4$

$v^{2}=39,\ \ {\mbox{or}},\ \ v=6.25$

∴ on the basis of circular motion the maximum velocity of turbulence = 6.25 feet per second.

Or, on the simple harmonic basis, $6.25\times {\sqrt {2}}=8.8,$ feet per second.

In the foregoing investigation the question of the means whereby the energy is trapped, or the possible percentage of the total that is available, is left untouched. The whole subject belongs essentially to the later portions of the work, Aerodonetics, where the matter will be treated more fully; the present publication is only made as an illustration of the employment of the peripteral theory expounded in the present work.

APPENDIX VI.

AN ELECTRO-MAGNETIC ANALOGY.

The Eulerian theory of the inviscid fluid gives results that, it has already been remarked, bear but little resemblance to the behaviour of any actual liquid or gas. It is the more remarkable that these self-same results possess much that is in common with electrical phenomena. Thus the hydrodynamic plottings are true representations of the electrical and magnetic fields, and the theorem of energy and other Eulerian propositions in general apply.

The present analogy (for it is so far no more than an analogy) is one that has frequently attracted attention, and it is not without interest to follow the matter into the by-ways of hydrodynamic theory dealt with in the present work.

If we take the magnetic flux as the analogue of the flow (ψ function), then the electric current becomes a cyclic motion around the conductor. This point of analogy is emphasised by the need for a doubly or multiply connected region in both cases, in the case of the electric current for the completion of the circuit, and in the case of hydrodynamic theory in order that cyclic motion should become possible.^[29]

If the conductor be situated in a magnetic field, it will experience a force at right angles to the direction of the field just as has been shown to exist in the case of the peripteral system, so that again we find the analogy holds. Thus, let us suppose a straight conductor in a uniform rectilinear magnetic field, the conductor and the lines of force being at right angles, and let the conductor be part of a completed circuit of zero resistance, carrying a current of some stated strength; then the conductor will experience a force at right angles to the direction of magnetic flux = F. Now let us apply a force F₁ equal and opposite to F, acting from without on the conductor, so that the latter will be held stationary; we may regard this force as the analogue of the weight of an aerodone supported in an Eulerian fluid, the electric current representing the cyclic component of the peripteroid motion, and the magnetic flux the superposed translation, in accordance with the régime of §§ 80 and 122.

If we suppose now a resistance to be inserted in the electrical circuit, the current, and therefore the force F, will tend to fall off, but the applied force F₁ continues, so that the conductor is set in motion in the magnetic field and is maintained in motion, the energy expended by the applied force F₁ being accounted for as energy lost in the electrical circuit; this is in fact the principle of the generation of an electric current by means of a dynamo.

The motion of the conductor under the influence of the force F₁ corresponds in our analogy to the descent of an aerodone in its gliding path, the gliding angle being represented by the velocity of the conductor divided by the velocity of the magnetic flux.

It is difficult to carry the present analogy much further without some stretching of the imagination or distortion of fact; even thus far there are many difficulties. For example, there is nothing in the hydrodynamic analogue of the electro-magnetic system depicted to give the conductor a sense of direction in the magnetic flux; its only knowledge of its motion through the supposed hydrodynamic stream is its relative motion, and as such it is difficult to see in what manner a conductor consisting of symmetrically disposed components (molecules) can distinguish between the real and apparent directions of the magnetic stream, in other words, how it can distinguish between an impressed transverse motion and a transverse component of the magnetic field. The analogy between the Eulerian fluid and the luminiferous ether is strong, but at present is not strong enough to bear any great weight.

In spite of difficulties, it appears probable to the author that in the near future some use may be made of existing electrical theory as an auxiliary means of investigating the aerodynamics of flight. Thus, in the general dynamics of the periptery, and in connection with the relations of the strength of the cyclic motion and the magnitude of the load reaction, it may be that mathematical solutions exist, ready to hand, in the analogous electrical theory, such as appropriately interpreted may some day be found to be of service.

APPENDIX VII.

FLUID RESISTANCE STUDIED BY THE AID OF AN IMPROVED KIND OF HYPOTHETICAL MEDIUM.

In § 131 a suggestion is made that leads to a new method of treatment of problems in fluid resistance.

Let us imagine a modification of the medium of Newton in which the particles, instead of being at rest, are in a state of agitation, and in the first instance let us suppose that all the particles, moving in directions at random, have the same velocity.

Taking first the case of a normal plane travelling at a velocity greater than that of the particles, we have the resistance proportional to the energy per unit volume (§ 131), the energy being reckoned only in respect of motion in the direction of flight, of either plus or minus sign. This energy is made up of two parts, the corpuscular energy of the medium, of which one third only counts as being in the direction of the axis of flight, and the energy of translation.

Now the corpuscular energy is constant in respect of the velocity of flight, and the energy of translation varies as the square of this quantity, consequently the law of resistance for this modified Newtonian medium will be. P = k V² + n, where k and n are constants.

If the velocity of the plane, instead of being greater than that of the particles, be less, the medium will exert a pressure on the back of the plane as well as on the face, and the resistance will be due to the pressure difference.

Taking the velocity as very low, then, the pressure being due to the bombardment of the particles, it may be easily demonstrated that the pressure difference and therefore the resistance must vary directly as the velocity. This may be regarded as the equivalent of "Stokes stage" in the case of a real fluid.

If the particles of the medium have different velocities the same general principles apply, only if the method is to be interpreted quantitatively the problem becomes a trifle more complex as involving the integration of a series of some kind.

In the case of a normal plane such as we have so far considered, the components of the motion of the particles transverse to the direction of flight have no influence. In the case of a solid body or curved lamina this is not the case, the lateral bombardment cannot be without effect on the total resistance.

Without examining the problem analytically, it appears obvious to the author that if (as is the case in a real gas) the energy of the particles is equally distributed in the three "degrees of freedom," that is in the directions of the three co-ordinate axes, the resistance at high velocities will not, in respect of the corpuscular energy, depend upon the form of the surface in presentation, but will depend upon the cross sectional area only; and any relief that can be obtained by rounding off or pointing the surface in presentation will take effect only in respect of the portion of the resistance that varies as V². That is to say, in the expression, P = k V² + n, giving easy entrance lines will diminish the constant k, but will have no influence on the value of the constant n.

The modified Newtonian medium of our present hypothesis resembles in many ways the perfect gas of kinetic theory, but differs in one very important respect. The molecules of a perfect gas are not only in a state of motion, but are undergoing frequent encounters one with another. Whether these encounters are due to gross impact or to some kind of action at a distance is immaterial from the point of view of the present discussion. The particles or corpuscles of the hypothetical medium have no magnitude, consequently they do not encounter one another, and therefore the medium has no continuity.

It is probable that the difference in the behaviour of air or any other gas, and the medium, will be least at very high and very low velocities; at intermediate velocities the present mode of treatment is unlikely to be of any utility. It seems possible that the j)resent theory may find some application in relation to the flight of high velocity projectiles.

APPENDIX VIII.

PROPULSION BY SAILS.

It is scarcely necessary to point out that the peripteral theory set forth in the present work is capable of wider application than to the problems concerned in aerial flight.

The sailing boat, for example, offers a very promising field for the application of the peripteral principles of flight, and furnishes strong confirmation of the present theory. We may look upon the sailing boat, and especially the racing craft with its fin or deep keel, as an aerofoil combination in which the under-water and above-water reactions balance one another.

Laying on one side for subsequent consideration the part of the problem that relates to the heeling of the vessel and its stability, we may treat the matter in the first instance as if the under-water and above-water forces lie in one horizontal plane. Under these conditions the problem resolves itself into an aerofoil combination in which the aerofoil acting in the air (the sail spread) and that acting under water (the keel, fin, or dagger plate) mutually supply each other's reaction.

The result of this supposition is evidently that the minimum angle at which the boat can shape its course relatively to the wind is the sum of the under and above-water gliding angles.

If the boat had no body (hull), and the conditions of our supposition be complied with, this reasoning shows that the minimum angle of the course relatively to the apparent direction of the wind would be the sum of^[30] the γ for water and the γ for air, which is probably a degree or so less than 20 degrees, or rather less than two "points."

In practice, the two reactions (under and above water) not being in one plane, there is a resultant torque which has to be taken by the moment of heel due to the stability of the vessel. This results in a necessity for added surface and resistance due to the motion of the hull, both above and below water, especially the latter; the actual course is in consequence at a greater angle^[31].

It seems to the author that by taking the present view many points hitherto but partially understood appear in a new light. For example, the bulging or filling of sails beyond the line of relative wind direction, a phenomenon well known to yachtsmen and other sailors, is the strict analogue of the arched section with dipping front edge of the aerofoil so amply demonstrated in the foregoing pages.

Further, the "dagger plate," the well-known expedient of the designer of light- draught racing craft, evidently "scores" over the ordinary centre-board by reason of its greater aspect ratio.

---

1. The method here given is founded on a suggestion made by the author in his paper to the Birmingham Natural History and Philosophical Society in 1894. Owing to repeated rearrangements and revisions, it was accidentally omitted from the MS, of the present work.
2. The method is evidently no more than an approximation.
3. The mixed nature of the conception of vortex atoms in a non-atomic enclosure is possibly responsible for the difficulty pointed out. The enclosure, to carry a vortex atom theory to its logical conclusion, should itself consist of an entanglement of vortex rings or filaments.
4. The author has heard it argued that every stream of fluid passing any imaginary barrier plane carries momentum across that plane, and therefore must result in added pressure between the fluid and the enclosure. Such an argument is evidently unsound; on the fluid tension hypothesis (§ 82) we may regard these internal motions of the fluid as giving rise to tension across the barrier plane, and this tension is equal and opposite to the momentum per unit time transmitted by every current and counter current set up in the fluid, and on the principles discussed in §§ 81, 82, and 83; this applies not only for the whole region, but individually for every small element of the fluid cut by the imaginary plane. Interpreting in the usual way, we see that it is the ordinary hydrostatic pressure on the walls of the enclosure that supplies the necessary force to balance the momentum transferred per second, and that a diminution of pressure in the vicinity of the barrier arises automatically, precisely equivalent to the momentum transference taking place.
5. Nature, xxiv., p. 47, also "Motion of Vortex Rings," J. J. Thomson.
6. See: Joseph Larmor, On the Average Pressure Due to Impulse of Vortex-Rings On a Solid, Nature, Vol. xxiv. 1881, p. 47 (Wikisource contributor note)
7. Compare Poynting, Presidential Address. Physical Society. February 10th, 1905, with Eayleigh. Phil. Mag., vol, x., pp. 364, 374. September, 1905.
8. If this condition is infringed, the motion is obviously not pure wave motion, but comprises a superposed translation.
9. This is evident, for if A B C be three equidistant points on the line of propagation, the fluid in the regions A B and B C will be identically the same when the wave train has passed from the region A B into the region B C.
10. There is some want of harmony between this result and the conclusions of many eminent authorities, see Larmor. Encycl. Brit., xxxii., p. 121 b; Rayleigh. Phil. Mag., vol. iii., p. 338, 1902; and Poynting, l.c. ante. Rayleigh has amended his conclusions somewhat in a subsequent communication, l.c. ante.
11. See Addendum A.
12. If heat be added to a quantity of a perfect gas contained within an enclosure, the consequent rise of pressure is due to the quantity of heat added and is independent of its distribution. "When wave motion exists in such a gas, heat is abstracted where the gas is rarefied and added where the gas is compressed, but more heat is added than subtracted; the difference represents the work done, according to well-known thermodynamic principles. We can therefore look upon the adiabatic wave as a Boyle's law wave in which heat has been added to one part and abstracted from another part, but in sum an addition of heat has been made to the contents of the enclosure, and the mean pressure increase can be calculated therefrom.
    The fact that the distribution of added heat within a vessel does not affect the pressure increase has been taken advantage of by the author (1894) in the construction of an air calorimeter, a small quantity of gas whose calorific value is to be determined being burnt in a large vessel and the rise of pressure noted (see Addendum C). For mechanical reasons the appliance was not a success.
    The result that the pressure due to an adiabatic wave can be deduced from the energy entering into the thermodynamic system appears to have been reached independently by Lord Rayleigh.
13. Poynting. Pres. Add.. Phys. Soc, 1905, p. 397.
14. J. J. Thomson, "Electricity and Matter," Ch. II., p. 44.
15. Vol. xxxii., p. 121 (b).
16. See Vol. xxxii., p. 121 (b). (Wikisource contributor note)
17. Pres. Address. Phys. Soc, l.c, ante.
18. This fact is mentioned by Rayleigh, Phil. Mag., vol. iii., 1902, p. 338.
19. Addendum C.
20. Compare Phil. Mag., vol. iii., 1902, pp. 341, 342 (Eq. 14).
21. The portion of the author's draft paper referred to is as follows:—
    “Then, let A B be any length in the direction of propagation, at any instant, occupied by a train of waves. Let B C be the place occupied by the same train when it has advanced by the amount A B (p. 3).

    A		B		C
    ┗━━━━━━━━━┻━━━━━━━━━┛

    Then by (4) A B = B C, and by (3) the particle at B, when waves occupy A B, is identical with particle at B when waves occupy B C; therefore the train contains the same mass of fluid as that of an equal volume of undisturbed fluid.
    Considering now a supposititious wave in a medium obeying Boyle's law,—
    The volume occupied by any small unit of mass is by Boyle's law inversely as the pressure. Therefore the linear distance in the direction of propagation occupied by any small unit of mass is inversely as its pressure.
    But the time during which pressure acts across the imaginary plane is by (4) proportional to this linear distance. Therefore the time during which any pressure acts across the imaginary plane is inversely as that pressure, or $p\ t=$ constant for any small unit of mass. But $p\ t$ is the momentum communicated across the imaginary plane by pressure per unit area, and we have shown the total units of mass in any wave is the same as in undisturbed air. Consequently in a plane wave in a fluid obeying Boyle's law the momentum communicated by the pressure of the wave is precisely that communicated by the undisturbed fluid.
    And since the sum of the translation of mass $(m\ l)$ by the wave is zero, the sum of the communication of momentum by motion $(m\ v)$ is also zero.
    That is to say, the plane wave in an elastic fluid obeying Boyle's law carries no momentum.
    If the adiabatic wave is examined by the foregoing method an excess of mean pressure is found to exist, and without doubt, if the source of sound emits a continued succession of waves, momentum accompanies such waves as an ever-spreading field of excess mean pressure, but it is not clear that if the source ceases to emit, this pressure region will be confined to and move with the advancing waves; it appears more probable to the author that the air contained within the wave sphere shares in the excess pressure.”

22. This argument appears to involve a fallacy similar to that mentioned in footnote, p. 399. The note in question is the answer to an argument actually used by Poynting in conversation with the author.
23. Presidential Address. Phys. Soc., l.c. ante.
24. At this time Poynting evidently has no misgivings as to the soundness of Larmor's theorem, and therefore must still have supposed that a Boyle's law wave-train carries momentum, apart from the evidence already given.
25. "Sensations of Tone," Appendix VII., B II.
26. It has already been pointed out (§ 187) that the problem is in all likelihood modified by the conditions of active flight, so that the tabulated figures, which relate to the gliding mode, may require to be multiplied by some unknown coefficient. In all probability the velocity of least resistance for a given bird in active flight is somewhere about 20 per cent. greater than for the gliding or soaring mode.
27. Ceteris paribus, the horse-power of any animal or machine varies as the square of its linear dimension, whereas the weight varies as the cube. Thus the power per unit weight is greater for a small bird than a large one. (See "The Horse-power of the Petrol Engine in its Relation to Bore. Stroke, and Weight," "Proc. Inst. Automobile Engineers," April, 1907.)

    Incidentally it may be remarked that it is probably for this reason that the soaring mode of flight, in which energy is captured from the wind, is principally employed by the larger birds, many of which are otherwise incapable of prolonged flight.

28. Other methods of soaring are practised by many of the larger birds. In some cases soaring is accomplished by merely gliding on an up-current whose velocity is equal to or in excess of the late of fall for gliding in still air; the up-current is sometimes due to the wind ascending the slope of a mountain or cliff, or may be due to the direct ascent of hot air from, for example, a sun-baked coast region. Another form of soaring depends upon the proximity of masses of air having different velocities, as the live stream and "dead-water" region in the wake of an obstacle; the bird circles round and round, playing off the one mass of air against the other.
29. The making or breaking of an electrical circuit alters simultaneously the connectivity of the regions both internal and external to the conductors; it is the latter that is the essential according to modern views, although it is the connectivity internal to the conductor that is usually present in the mind when reference is made to the completion of the circuit.
30. An erratum published in Volume 2 has been applied: "P. 431, line 3 from foot should read, 'angle of the course relatively to the apparent direction of the wind would be the sum of.'" The original text was: "angle of the course relatively to the wind would be the sum of" (Wikisource contributor note)
31. An erratum published in Volume 2 has been applied: "P. 432, line 7 from top, delete words,—'about twice that stated even in the most carefully designed craft.'" (Wikisource contributor note)