All the columns except the last are copied from Dr. Sondhauss's paper. It will be found that the observations are better represented by (VII.) than by (A); but it must be remembered that (VII.) contains an arbitrary constant, c, which acts nearly as a constant multiplier, although, if I understand Dr. Sondhauss aright, its value was not determined from this series of experiments. However this may be, it is certain that nearly all the values of n calculated from (A) are too great. The fact is that (A) is scarcely applicable to the experiments at all. In only five cases is the ratio of the diameter of the neck to the dimension of the vessel even tolerably small. These are 1, 4, 11, 12, 14; but in 1, on account of the extremely small diameter of the neck and its considerable length, the influence of friction is probably sensible; and its effect would be to lower the pitch. The body in 4 is cylindrical, and perhaps too long in proportion to the quarter wave-length. In 11, 12, and 14 the agreement is sufficiently good. I consider accordingly that there is no evidence in the Table unfavourable to formula (A), supposed to be stated with the proper restrictions. In my own experiments, made by the method of resonance, I found a very good agreement between the directly observed and the calculated pitch, the average error being under a quarter of a semitone. Even with formula (VII.) as the basis of calculation there would be a fair agreement, certainly better than is the case with Dr. Sondhauss's own experiments. The difference between (VII.) and (A) is, as I have already remarked, comparatively small, and could only be certainly distinguished under favourable circumstances. Not finding the necessary data in Dr. Sondhauss's paper, I venture to quote some experiments from the paper on Resonance. There are seven observations in which the necks were sufficiently long to bring out the difference between the formulæ, being more than four times the diameter. It will be seen that the alteration is in every case for the worse if the formula (VII.) is substituted for (A).

These experiments seem to decide the question; but it would be interesting to see if Dr. Sondhauss obtained a similar result

by the method of blowing. The difference, amounting in (VIII.) and (B) to half a semitone, is far greater than any error to be feared in the measurement of pitch or of the dimensions of the vessel, and ought therefore to give a sufficient handle to decide between the formulæ, if proper attention is given to the choice of a suitable resonator. In the foregoing remarks I have naturally dwelt most on my differences with Dr. Sondhauss; but I should be sorry to have it supposed that I write in a hostile spirit, or do not recognize the claims of one to whom the science of acoustics is so largely indebted.

Terling Place, Witham,
August 12, 1870.

Postscript, August 19.

I have since calculated the results of the experiments of Wertheim on pipes open at both ends, and find that in this case

Phil. Mag. S. 4. Vol. 40. No. 266. Sept. 1870.

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I

XXVI. On the Principles of Thermodynamics.

By the Rev. J. M. HEATH*.

HAVE for some time past been challenging the attention of scientific men to the consideration of a very important question-whether some of the most elementary principles in dynamics have not been overlooked by those who originally framed the language of thermodynamics; and whether their successors have not indolently adopted that language, and the notions which it was formed to embody, without using sufficient care to purify it of those errors which the prepossessions and imperfect knowledge of the first investigators had made almost unavoidable in them.

Mr. Rankine has replied to me, in the August Number of this Journal, to the effect that the principles I contend for are (as far as he understands me) right; and I think he must be taken as admitting, by his silence, that they were in fact ignored by the earliest original investigators in the science. But he tells me that the more modern writers, at least the best among them, had long ago perceived all that I have been pointing out, and have so altered their creed and their language as to be free from the reproach which I appear to make against them.

In support of this statement, Mr. Rankine has given us two propositions embodying the creed which is considered orthodox at the present moment. Mr. Rankine's name is so high an authority upon this point, that there can be no question that we have from him an authentic statement of what the present doctrine is, which it is said is in strict harmony with all that I have called for. If, therefore, I myself can make a similar statement of my own opinions, we shall at once be enabled to judge both whether the two doctrines are essentially one and the same, as is alleged, or, if different, which of them has the best claim to be accepted as representing the truth of the matter.

I understand Mr. Rankine's first proposition to be, that if the elasticity of a body results from the mutual attraction or repulsion of its particles acting upon one another, any forcible compression of such a body would result in "stored-up energy," but would be wholly incapable of producing (molecular motion or) heat. And I think he imagines that this is the doctrine which I have only just now come to perceive for myself and am bringing forward as somewhat of a novelty; whereas it has long ago been perceived by the best writers of the present time, and they are one and all careful in all cases to separate the amount of energy expended in this "storing-up of energy" by "overcoming the resistance of repulsive forces" from the total amount, before they estimate the remainder, which alone can produce heat.

* Communicated by the Author.

Mr. Rankine's second proposition is, that if the body's elasticity is the result of the mere motion of particles having inertia but not acting upon each other by forces of attraction and repulsion, then in the compression of such a body by external force all the force employed imparts motion to the particles of the gas, and that the addition so made to the previous vis viva of those particles is the same as the vis viva which the same force acting through the same space would have generated supposing the particles had been previously at rest,-or, in other words, that the compressing force will descend through the same space whether there be any resistance to its motion or none.

These two propositions (if indeed I have rightly understood them) appear to me to involve a conclusion completely subversive of the whole doctrine of thermodynamics, viz. that the production of heat in different bodies by compression depends upon their molecular constitution—and, furthermore, that upon no hypothesis as to the constitution of a body will the production of heat be in accordance with the true mechanical laws of the production of vis viva. If the body consists of particles which mutually repel each other, Mr. Rankine thinks no heat can ever be generated by impact or compression in such a body, although it is certain that vis viva can be generated among its particles. If the body consists of only moving and impinging particles without repulsive forces, Mr. Rankine thinks that the new heat generated by condensation in such a body will be equal to what a merely mechanical solution of the question teaches us would be the total vis viva-i. e. the sum of what there was at first, added to the addition made by the work done by the force.

I believe my own analysis of what is objectionably called the "overcoming of resistance by a force" to be true in mechanics, and to be free from each of the objections above stated. If the area of the piston by which a volume of gas is compressed is unity, and P is the external load put upon it, and p is the internal pressure of the gas opposing its descent, then P may, at every point of the descent, be considered as consisting of two parts, one =p which we will call p', and the remainder, which therefore will be P-p'.

p and p' constantly increase as the piston descends and the volume of the gas diminishes; and P-p' consequently is diminishing during the whole descent. Let P-p=0, or P=p', when has become v'. The whole amount of deduction, therefore, which has to be made from the action of P through the p, dv. The question is, What has been the employment of this force? My answer is, that it is the sum of all the pressures which, at every point of the descent, have held the elasti

v-v is

SP

space

city of the gas below in statical equilibrium. It has generated no motion whatever, although it is a force exercised by a moving body; but it has neutralized the resistance; so that they might both of them be at once entirely struck out of the equations of solution, and the problem so converted into a more simple one of the condensation of a gas possessing no elasticity by the action of the force (P-p1)dv.

S

The corrections which this solution supplies to the two propositions of Mr. Rankine respectively are, that, in the first, it admits the generation of heat by the force | (P-p1)dv, which Mr. Rankine denies; and in the second it denies the generation of heat by the force pdv, which Mr. Rankine asserts. I believe my solution to be right in both instances; but it is upon the latter point that I anticipate there will be the greatest difficulty of agreement between us, as the appeal is tacitly to a proposition which I think Mr. Rankine will at first sight consider inadmissible.

In speaking of a gas whose elasticity results from the motion of its particles, Mr. Rankine says, "work done in diminishing the capacity of this vessel wholly takes effect in accelerating the motions of the confined particles." I believe that a gas of this nature may be condensed and its pressure increased (as is required by the law of Mariotte) without increasing the vis viva of the particles, and therefore without altering the heat.

The pressure at any given point on the surface depends upon the number and frequency of the impacts of the particles, and the vis viva of each of them. We may confine our attention to the impacts of one particle only of a given force. This particle will repeat its impact upon a given spot the more frequently, the shorter the path is which it traverses between two successive impulses; in other words, it will return the more frequently, the more the volume of the gas is contracted. If, therefore, immediately after any one appulse of the particle the piston is made to take a new position (still one of rest) immediately below its former position, or to descend through the infinitesimal space dv and there remain stationary, the particle will return to it sooner than it would do in its former position. It will strike it and will be reflected by it without any increase of its motion; in other words, the Pressure may be increased and the Volume contracted without evolution of Heat.

Liphook, August 11, 1870.