arises, whether it is not possible to concentrate the rays of heat artificially by means of mirrors or burning-glasses, so as to produce a higher temperature than that of the radiating bodies, and thus to effect the passage of the heat into a hotter body. The author has, therefore, thought it necessary to treat this question in a special paper, the contents of which are given in Chapter XII. Matters are still more complicated in cases when heat is transformed into work, and vice versâ, whether this be by effects such as those of friction, resistance of the air, and electrical resistances, or whether by the fact that one or more bodies suffer such changes of condition, as are connected partly with positive and partly with negative work, both internal and external. For by such changes heat, to use the common expression, becomes latent or free, as the case may be; and this heat the variable bodies may draw from or impart to other bodies of different temperatures.

If for all such cases, however complicated the processes may be, it is maintained that without some other permanent change, which may be looked upon as a compensation, heat can never pass from a colder to a hotter body, it would seem that this principle ought not to be treated as one altogether self-evident, but rather as a newly-propounded fundamental principle, on whose acceptance or non-acceptance the validity of the proof depends.

§ 8. Zeuner's later Treatment of the Subject.

The mode of expression employed by Zeuner was criticized by the author on the grounds stated in the last section, in a paper published in 1863. In the second edition of his book, published in 1866, Zeuner has therefore struck out another way of proving the second main principle. Assuming the condition of the body to be determined by the pressure p and volume v, he forms for the quantity of heat dQ, taken in by the body during an indefinitely small variation, the differential equation

dQ=A (Xdp + Ydv)..............

..(2),

where X and Y are functions of p and v, and A is the heatequivalent of work. This equation, as is well known, cannot

be integrated so long as p and v are independent variables. He then proceeds (p. 41):

But let S be a new function of p and v, the form of which may be taken for the present to be known as little as that of X and Y, but to which we will give a signification, which will appear immediately from what follows. Multiplying and dividing the right-hand side of the equation by S, we have

dQ = AS [3 dp + 3 dv]

S

.(3).

We may now choose S, so that the expression in brackets is a perfect differential; in other words, so that

1

S may be the integrating factor, or S the integrating divisor, of the expression within brackets of equation (2)."

From this it follows that in the following equation derived from (3),

Y

do = 4 [ 23 dp + 3 dv ]
Q A

S

S

..(4),

the whole right-hand side is a perfect differential, and therefore for a cyclical process we must have

In this way Zeuner arrives at an equation similar to equation (7) of Chapter IV., viz.

= 0.

The resemblance, however, is merely external. The essence of this latter equation consists in this, that T is a function of temperature only, and further a function which is independent of the nature of the body, and is therefore the same for all bodies. Zeuner's quantity S, on the contrary, is a function of both the variables, p and v, on which the bodies' condition depends; and further, since the functions X and Y, in equation (2), are different for different bodies, it must be true of S also that it may be different for different bodies. So long as this holds with regard to S, equation (5) has done nothing for the proof of the second main principle;

since it is self-evident that there must in general be an

integrating factor, which may be denoted by

1

and by S' which the expression within brackets in equation (2) may be converted into a complete differential. Accordingly, in Zeuner's proof, as he himself concludes, everything depends on the fact that S is a function of temperature only, and a function which is the same for all bodies, so that it may be taken as the true measure of the temperature.

For this purpose he supposes a body to undergo different variations, which are such that the body takes in heat whilst S has one constant value, and gives out heat whilst S has another constant value; and which together make up a cyclical process, shewing a gain or loss of heat. This procedure he compares with the lifting or dropping of a weight from one level to another, and with the corresponding mechanical work; and he proceeds (p. 68): "A further comparison leads to the interesting result that we may consider the function S as a length or a height, and the expression as a weight;

Ο

AS

in what follows therefore I shall call the above value the Weight of the Heat." Since a name has here been introduced for a magnitude containing S, in which name there is nothing which relates to the body under consideration, it appears that an assumption has here been tacitly made, viz. that S is independent of the nature of the body, which is by no means borne out by the earlier definition.

Zeuner then carries still further the comparison between the processes relating to gravity and those relating to heat, and transfers to the case of heat some of the principles which hold

Q for gravity; in so doing he treats S as a height, and as a

AS

weight, just as before. Then, having finally observed that the principles thus obtained are true if we take S to mean the temperature itself, he proceeds (p. 74): "We are therefore justified in taking as the basis of our further researches the hypothesis that S is the true measure of temperature."

It appears from this that the only real foundation of the reasonings, which in his second edition Zeuner puts forward as the basis of the second main principle, is the analogy

between the performance of work by gravity and by heat; and moreover that the point which has to be proved is in part tacitly assumed, in part expressly laid down as a mere hypothesis.

§ 9. Rankine's Treatment of the Subject.

We may now turn to those authors who have considered that the fundamental principle is not sufficiently trustworthy, or even that it is incorrect.

Here we must first examine somewhat more closely the mode of treatment which, as already mentioned, Rankine considered must be substituted for that of the author.

Rankine, like the author, divides the heat which must be imparted to the body, in order to raise its temperature, into two distinct parts. One of these serves to increase the heat actually existing in the body, and the other is absorbed in work. For the latter, which comprises the heat absorbed in the internal and in the external work, Rankine uses an expression, which in his first section he derives from the hypothesis that matter consists of vortices. Into this method of reasoning we need not enter further, since the circumstance that it rests on a particular hypothesis as to the nature of molecules and their mode of motion, makes it sufficiently clear that it must lead to the consideration of complicated questions, and thus leaves much room for doubt as to its trustworthiness. In the author's treatises he has based the development of his equations, not on any special views as to the molecular constitution of bodies, but only on fixed and universal principles; and thus, even if the above fact were the only one which could be alleged against Rankine's proof, the author would still expect his own mode of treating the subject to be finally established as the most correct. But yet more uncertain is Rankine's mode of determining the second part of the heat to be imparted, viz. that which serves to increase the heat actually existing in the body.

Rankine expresses the increase of the heat within the body, when its temperature t changes by dt, simply by the product Kdt, whether the volume of the body changes at the same time or not. This quantity K, which he calls the real specific heat, he treats in his proof as a quantity independent of the specific volume. Any sufficient ground for this

cases, Thomson developes a series of general equations, which are independent of the body's condition of aggregation, and only then passes on to more special applications.

On one point this second paper still falls short of the author's. For here also Thomson holds fast by the law of Mariotte and Gay-Lussac in the case of saturated vapour, and hesitates to accept an hypothesis with respect to permanent gases, which the author had made use of in his investigation (see Chapter II., § 2, of this work). On this he remarks*: "I cannot see that any hypothesis, such as that adopted by Clausius fundamentally in his investigations on this subject, and leading, as he shews, to determinations of the densities of saturated steam at different temperatures, which indicate enormous deviations from the gaseous laws of variation with temperature and pressure, is more probable, or is probably nearer the truth, than that the density of saturated steam does follow these laws, as it is usually assumed to do. In the present state of science it would perhaps be wrong to say that either hypothesis is more probable than the other." Some years later, after he had proved by his joint experiments with Joule that this hypothesis is correct within the limits assigned by the author, he used the same method as the author to determine the density of saturated vapourt.

Rankine and Thomson, so far as the author knows, have always recognized most frankly the position here assigned to the first labours of themselves and the author on the mechanical theory of heat. Thomson remarks in his paper‡: "The whole theory of the moving force of heat rests on the two following principles, which are respectively due to Joule and to Carnot and Clausius." Similarly he introduces the second main principle as follows: "Prop. II. (Carnot and Clausius)." He then proceeds to give a proof discovered by himself, and goes on§: "It is with no wish to claim priority that I make these statements, as the merit of first establishing the proposition on correct principles is entirely due to Clausius, who published his demonstration of it in the month

*Edin. Trans., Vol. xx. p. 277; Phil. Mag., Vol. iv. p. 111.

+ Phil. Trans., 1854, p. 321.

Edin. Trans., Vol. xx. p. 264; Phil. Mag., Vol. iv. p. 11.

§ Edin. Trans., Vol. xx. p. 266; Phil. Mag., Vol. IV. pp. 14, 242.