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equal to Q, as in the case we are considering, is nothing;

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that is, no vis viva is generated in a body by the action of two equal and opposite forces. I do not know how Mr. Rankine, and those whose opinions he shares, explain this fact consistently with their belief that it is under these conditions, and these only, that P accelerates molecular motions and therefore generates heat, and does not store up energy in Q. But, "speaking under all reserve," I suppose that they contrive to continue to attribute the character of a dynamical equation to the equation (PQ)&v=0 which it had when mv2 it retained its full form (P-Q) dv= and imagine that P

2

does drive the body m along the length dv against all the efforts of Q, and that it is in doing this that it generates heat. But if this is the view of any one, I think that person will not see the truth in this matter very clearly until he has discarded it. If P could by any possibility do any one thing which Q could not prevent, then the motion P would communicate to m would be accelerated, not uniform, as it is assumed to be. Those, if there are any, who deliberately maintain such an opinion as this, have not observed that (P-Q)dv is no longer the equation of vis viva at all, but has become the statical equation of virtual velocities, and that in that equation the motion of m through dv is not the work of either of the two forces P or Q, but any arbitrary possible motion derived from an external cause. In the dynamical equation the motion is due to the forces in action. In the statical equation it is independent of and unaffected by those forces; and the supposed case of work done is a case of statics only and not of dynamics. I rely upon this, therefore, for the proof of one half of my assertion, viz. that no acceleration, and therefore no heat, is generated by the action of forces which are in equilibrium, and subject to the equation of virtual velocities, or, in other words, of forces in that condition in which they are Isaid to do work.

Mr. Rankine has adduced a mathematical demonstration, derived from the theory of the collision of elastic bodies, which he relies on as a positive argument on the other side. What I have just now given is a direct and positive proof that the force above the piston cannot accelerate, so long as it is equilibrated by the resistance below. Mr. Rankine's argument is, in form, equally direct and positive to prove that it does. I shall simply point out a very important mistake which vitiates the whole of his proof; and unless he can maintain that his reasoning is sound, the proposition he attempts to support must be taken as disproved. His mistake is this. He represents the piston, in fact, as imparting

vis viva to the particles without losing any of its own-contrary to the principle which he himself appeals to, that of the conservation of force. He finds +(u+v) for the velocity of the particle relatively to the piston after impact; and to get the absolute velocity of the particle, he adds this to u, the velocity of the piston before the impact, forgetting that that velocity has been altered by the impact. The consequence of this oversight is that the velocity of the centre of gravity of the two bodies, and also the sum of their vis viva, are both greater after impact than before it-results impossible according to the true laws of impact. I claim, therefore, at least for the present, to say that Mr. Rankine has failed to prove that his proposition is true, and that I have tendered a proof, as yet uncontroverted, that it is false.

I will now pursue the remainder of the main argument, and examine the contrast between my own opinions and those represented by Mr. Rankine, as to cases where the force meets with no resistance, and does, therefore, no work. When Q=0, the general equation of vis viva takes the form P&v= and it is

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2;

obvious that in this case all the force is employed in producing acceleration, and none of it in "storing up energy." All the force, to use the ordinary language of mechanics, is employed dynamically, or is doing dynamical work, and none is producing statical work, which is pressure. If a gun-barrel is placed horizontally and no friction or atmospheric resistance acts to resist the motion of the ball through its length, the vis viva it has at the muzzle measures the expenditure of the internal energy of the exploded charge, which has become externalized, and is now no longer internal energy in the gas, but actual energy of motion ⚫ in the bullet. All this, therefore, is energy lost to the gas. But, say the thermodynamists, this gas has met with no resistance, done therefore no work, and can have lost no heat. Have I misrepresented them? Or how do they explain such a monstrous conclusion?

Lastly, I will take the equation in its general form, where forces are employed in both the ways that we have now considered separately, viz. both statically and dynamically, in producing both pressure and motion. The use of this equation, properly treated, will be to enable us to distinguish these forces into two groups, which, as Mr. Rankine tells us, is the business of thermodynamics, but which I contend has not as yet been properly done.

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The equation (P-Q)&v= may be written

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And this, again, if we suppose Q'=Q, may be resolved into two separate equations, a statical one,

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Whenever, therefore, P is not equal to (here supposed greater than) Q, we may resolve it into two parts, Q'=Q, which is therefore the reaction of the lower surface of the piston equal and opposite to the pressure of the gas below, and P-Q', the remaining part of P, which acts upon the particles of the gas as if they were perfectly free. The effect of P-Q'upon the gas is therefore wholly dynamical; it does not alter the pressure, and its entire effect is the generation of motion. Thermodynamists teach (as far as I may venture to speak of teachings which I can give no mental assent to, and may therefore unintentionally misrepresent in consequence of misunderstanding them) that the whole gain of heat is determined by the first, or statical one, of our two equations. The force Q (part of P) presses down the piston through Sv against the resistance Q, and the work done is Qov, and it is done by Q', a part only of P; and the remainder, P-Q, being an unresisted force, has no effect either on the work done or on the gain of heat. I think this is a true representation of what they would say, because it appears to me to be what they must say if the case is reversed, and we consider, instead of a condensation by piston, the case of expansion, as the discharge of a bullet from a gun-barrel. If P is considered to represent the explosive force of the charge of a gun, and Q the resistance arising from friction, the weight of the bullet, and the pressure downwards of the external air, then Q is a fixed quantity, and the whole energy Pov expended can do no more work than Qov; that is to say, that the largest charge that can be put into the gun can do, technically, no more work, and can therefore expend no more heat, than that small charge Q' which is just able to counterbalance Q, so that the ball may move uniformly along the barrel and fall to the ground when it reaches the muzzle. In this case I apprehend thermodynamists must relegate the excess of gas-force, P-Q', over the resistance, into that group which we have already discussed, of forces which do no work and exhaust no heat.

The truth is, I have never seen anywhere any recognition among writers on this subject of the case of unequal forces acting in this way in condensing or in expanding the gas; and although the opinion is wholly untenable, I believe that the general conception is, at all events when the load on the piston descends

and condenses, that the reaction of the gas is always equal to the pressure upon the piston, whatever it may be. That this opinion, however, is wrong is evident from this, that if the pressure of the gas were always equal to the external pressure put upon it by the piston, the gas would never yield and be condensed by the descent of the piston, but would continue to sustain it. It appears to me to be certain that the internal pressure of a gas is never any thing else than that given by Mariotte's law as depending upon the density and the temperature only, and that any additional external force applied to its surface acts dynamically, and not statically, upon it.

LIII. On the Equation of Laplace's Coefficients. By R. MOON, M.A., Honorary Fellow of Queen's College, Cambridge*.

THE

HE equation of Laplace's coefficients has attracted the attention of three distinguished English mathematicians, all of whom within a comparatively brief space have passed from the scene the late Judge Hargreave, Mr. Boole, and Professor Donkin.

Having adopted Laplace's method of transformation and reduction, Mr. Hargreave gave, in the Philosophical Transactions for 1841, the first solution of the problem in finite terms.

One peculiarity of this method is that the equation actually integrated by it is not that which was originally proposed, but a derivative from the latter, the problem solved being in fact vastly more general than that proposed for solution-a circumstance of which the solution obtained by it affords ample proof+.

Proceeding by the method of separation of symbols of operation from those of quantity, Mr. Boole, by each of three independent methods, arrived at an expression for the integral; and one of great elegance, obtained on the same principle, was given by Professor Donkin in the Philosophical Transactions for 1857. In a paper published in the Philosophical Magazine for July last I showed that the equation

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d2z dz dz
+T +P +Q +Uz,
dy2 da dy

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where R, S, T, P, Q, U are functions of x and y only, in the cases which are not amenable to Monge's method will always have an integral in which z is represented by a series (finite or infinite according to circumstances) of the form

z=Ax(u)+A, Sdux(u) +A2 §§ du2x(u) + &c.,

* Communicated by the Author,

For instance, the expression thus obtained for the fiftieth coefficient involves upwards of fifty arbitrary functions.

or by a pair of such series; where x is arbitrary, u is determined by the equation

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The equation of Laplace's coefficients may readily be integrated by this general method, some of the peculiarities of which are well illustrated by its application to that equation, of which it probably offers the simplest solution which is obtainable. Retaining the original spherical coordinates, the equation may be written

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and putting

and 4 for x and y respectively in the preceding formulæ, we shall find

* In the paper above referred to, I have stated "that no constants are to be introduced," in effecting the integrations here indicated. This is true so long as in (1) we have U=0; but where this does not hold, the omission of the constants would greatly curtail the generality of the result, as a glance at the mode in which each of the quantities A1, A2, &c. is derived from its immediate predecessor will at once show.

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