properties depend essentially upon the choice of the two auxiliary variables a and i; therefore, they ought to be chosen in such a way as to give the best possible graphical representation of the cycles of transformation, i. e., in such a way as to enable us to determine graphically the greatest possible number of the physical elements depending upon the transformation, by means of geometrical magnitudes depending only upon the form and position of the path described by the body in the adopted system of coördinates. system, let us examine what conditions must be fulfilled by Before defining this the functions A, u, v, in order that the auxiliary variables a and i be respectively the amplitude and the period of the vibratory motion.

4. Denoting by m the mass of one of the particles composing the substance and by u the mean velocity of the vibratory motion, the expression mu is the actual kinetic energy of the heat (the sum being extended to all the particles). Dividing this sum by the mechanical equivalent of heat E, the result is equal to the amount of heat actually contained in the body.

This amount of heat is proportional to the absolute temperature, hence:

K being a constant.

mu2= KTE

(2)

Denoting by the mean value of the force producing the vibratory motion, the formulæ :

fu = mu2 f=2x3m/=/

a

(3)

can be established without difficulty, since in all vibratory motions of small amplitude, the force producing the vibration is proportional to the displacement of the particles.

Combining equations (2) and (3) and noticing that Em = 1 and that the mean velocity u is the same for all the particles of the substance, we shall obtain :

T=

π' α KE2

(4)

which is the expression of T in terms of a and i, and is therefore identical to the third of equations (1); in other words, when the two auxiliary variables a and i denote the amplitude and the period of the vibratory motion, the function v is no longer arbitrary, and the equation to the thermodynamic surface is:

P = λ(a, i)
V = μ(a, i)

T=

π' α

KE ¿2

(5)

the functions and being still submitted to the condition that the result of eliminating a and i between equations (5) be: F(P, V, T)= 0.

When the functions and have been determined for a particular substance, equations (5) do not only represent the thermodynamic surface, but also the value of the two elements (a and ) of the vibratory motion, corresponding to any state of the substance defined by experimental data (P, V, T).

The last of equations (5) is the same for all substances, except that the value of the constant K changes from one substance to another. The determination of the functions À and will be investigated after we shall have studied the properties of the graphical representation, which properties can be found by assuming that these functions are known.

5. When a substance undergoes an elementary and reversible transformation, the amount of heat dII, absorbed by the unit of mass, is composed of two parts: the variation of the actual energy of the heat contained in the substance, and the amount of heat absorbed by the total work (external and internal). The first part is the elementary variation of the expression mu', as found above; the second is the heat absorbed by the work done by the force ƒ for a variation da of the amplitude. Hence:

1

E

This relation shows that the constant K is the quotient of the variation of the actual amount of heat contained in the substance, by the corresponding variation of temperature, so that Kis by definition the absolute specific heat of the substance. We have also, from preceding formulæ :

Such is the expression of d in terms of 7 and a; equation (4) gives by differentiation :

The formulæ contained in this paragraph have been already established in "La Thermodynamique et ses principales applications," by J. Moutier, Paris, 1885; we recall them here, as we will have to use them in some of the demonstrations.

By the aid of this equation and of the preceding one, the value of dH can also be obtained in terms of T and or of a and i:

6. The two variables, which we intend to take as coördinates. in this graphical study, depend directly upon the amplitude a and the period of the vibratory motion of heat; denoting these variables by and s, we shall define them by the equations:

By the aid of equation (7), any of the formulæ given above. and involving a and i, can be transformed into corresponding formulæ involving and s.

For instance, by solving equations (7) with respect to a and , and substituting the result in equations (5), we shall obtain the equation to the thermodynamic surface in terms of 8, as follows:

φ and

All the other equations can be transformed in the same. manner, so that it is understood that the two independent variables are now and s, and that these two quantities shall be taken as the coördinates of the point representing the physical state of the substance.

As these variables have been defined in an arbitrary manner, let us first investigate their physical nature. To reach this end, we must consider the vibratory motion of the particles as the projection of a uniform circular motion on one of its diameters; the radius of the circle is then equal to the amplitude of the vibration, and the velocity of the uniform motion is equal to the maximum velocity of the vibratory motion. It can readily be seen, that the centripetal force of the circular motion is equal to the mean value f of the force supposed to produce the vibratory motion.

The total work (external and internal) absorbed by the substance during an elementary transformation is fda as seen above. We can write identically:

Let

f

277α

=4m. Since 2лa is equal to the length of the circumference, represents geometrically the value of the centri4 m petal force referred to the unit of length, i. e., a pressure, of so many pounds per foot, supposed to be acting on the circumference of the circle.

Let 2лada = ds oг ñа2= 8. Then s is the area of the circle. According to these definitions:

Σfda = 2qds

ƒ

2πα

If now in the equation m = f be replaced by its value as given in equation (3), the result is :

The last two equations are precisely the ones by which and

& have been first defined.

is the

Since Y = 4m when m = 1, and since Ут pressure supposed to be acting on the circumference of the circle corresponding to the particle of mass m, we can define the physical nature of and s as follows: If the unit of mass of a substance be represented geometrically or symbolically by a circle, the physical state of said substance can be completely defined by the areas of the circle and by a pressure, supposed to be acting on the circumference of the circle. The two data, thus defining the state of the substance, are precisely the coördi nates and 8, which determine the position of the point representing this state. For this reason, the abscissa s shall be called the "symbolical volume" and the ordinate the "symbolical pressure" of the substance.

As the considerations developed in this paragraph are somewhat abstract, it must not be forgotten that the graphical method, which is the object of this study is quite independent of these theoretical considerations, since the two variables

and can always be regarded as two variables defined by the

π

equations: 4 = and 8 = a, whatever be their physical
nature; moreover, we shall find other reasons for regarding
as a pressure and s as a volume.

Properties of the graphical method.

7. Let M be the point representing any physical state of a substance, and and s its coördinates, then according to the previous definitions:

a =

1 and i= V

π

π

So that a and i, hence the state of the vibratory motion, are readily obtained from the actual value of the coördinates of point M.

Denoting by R the total work absorbed during a transformation, we have found that:

Whence by integration :

i.

1.

dR = Σfda

e., if AB be the curve (fig. 1) representing the path of the substance referred to the coördinates and 8, the total work ❤ (external and internal) absorbed during the transformation is equal to the area AabB limited by the path, the axis of s and the two extreme ordinates.

β

n

B

mb S

Comparing this result with the property of Clapeyron's graphical method, we see that the symbolical pressure and the symbolical volume are in the same relation with the total work, as the ordinary pressure and volume are with the external work.

8. The last of equations (8):

ps KTE

holding true for all substances, shows that the area of the rectangle Mm On formed by the coördinates and s, is equal to the actual amount of energy contained in the substance at the physical state M. Thus, if AB represents the path of the substance the area of the rectangles AaOa and BbO3 is equal to the energy of the heat contained in the substance at its initial and final states.