undetermined, and use the equations in that form. Any such hesitation would not, in the author's opinion, be justifiable, and in what follows T will always be used in the place of T. Equations VII. and VIII. will then be written in the following forms, which have already been given under Equations V. and VI. of the last chapter, viz.

CHAPTER V.

FORMATION OF THE TWO FUNDAMENTAL EQUATIONS.

§ 1. Discussion of the Variables which determine the Condition of the Body.

In the general treatment of the subject hitherto adopted we have succeeded in expressing the two main principles of the Mechanical Theory of Heat by two very simple equations numbered III. and VI. (pp. 31 and 90).

We will now throw these equations into altered forms which make them more convenient for our further calculations.

Both equations relate to an indefinitely small alteration of condition in the body, and in the latter it is further assumed that this alteration is affected in such a way as to be reversible. For the truth of the first equation this assumption is not necessary: we will however make it, and in the following calculation will assume, as hitherto, that we have only to do with reversible variations.

We suppose the condition of the body under consideration to be determined by the values of certain magnitudes, and for the present we will assume that two such magnitudes are sufficient. The cases which occur most frequently are those in which the condition of the body is determined by its temperature and volume, or by its temperature and pressure, or lastly by its volume and pressure. We will not however tie ourselves to any particular magnitudes, but will at first

assume that the condition of the body is determined by any two magnitudes which may be called x and y; and these magnitudes we shall treat as the independent variables of our calculations. In special cases we are of course always free to take one or both of these variables as representing either one or two of the above-named magnitudes, Temperature, Volume and Pressure.

If the magnitudes x and y determine the condition of the body, we can in the above equations treat the Energy U and the Entropy S as being functions of the variables. In the same way the temperature T, whenever it does not itself form one of these variables, may be considered as a function of the two variables. The magnitudes W and Q on the contrary, as remarked above, cannot be determined so simply, but must be treated in another fashion.

The differential coefficients of these magnitudes we shall denote as follows:

These differential coefficients are definite functions of x and y. For suppose the variable x is changed into x+dr while y remains constant, and that this alteration of condition in the body is such as to be reversible, then we are dealing with a completely determinate process, and the external work done in that process must therefore be also determinate, whence it follows that the quotient must

dW

dx

equally have a determinate value. The same will hold if we suppose y to change to y+dy while x remains constant. If then the differential coefficients of the external work W are determinate functions of x and y it follows from Equation III. that the differential coefficients of the quantity of heat Q taken in by the body are also determinate functions of x and y.

Let us now write for dW and dQ their expressions as functions of dx and dy, neglecting those terms which are of a

higher order than do and dy. We then have,

and we thus obtain two complete differential equations, which cannot be integrated so long as the variables x and y are independent of each other, since the magnitudes m, n and M, Ñ do not fulfil the conditions of integrability, viz.

The magnitudes W and Q thus belong to that class which was described in the mathematical introduction, of which the peculiarity is that, although their differential coefficients are determinate functions of the two independent variables, yet they themselves cannot be expressed as such functions, and can only be determined when a further relation between the variables is given, and thereby the way in which the variations took place is known.

§ 2. Elimination of the quantities U and S from the two Fundamental Equations.

Let us now return to Equation III., and substitute in it for dW and dQ expressions (3) and (4); then, collecting together the terms in dx and dy, the equation becomes,

Mda+Ndy = (d+m) dx + (dy + n) dy.

As these equations must hold for all values of dx and dy,

we must have,

Differentiating the first equation according to y, and the second according to x, we obtain,

We may apply to U the principle which holds for every function of two independent variables, viz. that if they are differentiated according to both variables, the order of differentiation is a matter of indifference. Hence we have

Subtracting one of the two above equations from the other we obtain,

We

may now treat Equation VII. in the same manner. Putting for dQ and dS their complete expressions, it be

This equation divides itself, like the last, into two, viz.

Differentiating the first of these according to y, and the second according to x, we obtain,