v2

If this value for

is substituted in the expression for w, formula

2

(4), then we get w expressed simply in foot-pounds.

The energy which the material point loses during the motion in the time at may consequently be represented by

dq=dQ-dw.

But this energy is only apparently lost when it seems to vanish; for, so far as the stated principle is correct, it reappears in its original magnitude merely under another form. Thus the new energy may be represented by

d2y dy

dt2ds

This equation, which may easily be put into the form

dx
dt

d2z dz
dt

shows, to begin with, that we get the whole new energy by taking the amounts of the energies which the accelerating forces in the direction of the axes will separately produce.

If we imagine the material point to be subject to any kind of resistances, and if we denote the resultant of them all by R, then this may be supposed to be resolved into two others-viz. into the resistance in the direction of the path (which I shall denote by P), and into the resistance perpendicular to the path (which I shall call P). We have then, as known,

which, substituted in the equation (5), gives

in which C is an arbitrary constant.

dt ds

(6)

Hence follows that the newly produced energy depends only on P, or the resistance in the direction of the path, whereas it is independent of P, or the resistance perpendicular to the path*.

The last result may perhaps not appear to be essentially different from what is immediately derived from formula (1), if we only look upon the material resistances in the directions of the three coordinate axes as real forces, which we might suppose to be included in the accelerating forces X, Y, and Z; but partly it is obvious that formula (1) would then represent the increment of the quantity of energy which the moveable body in fact would receive during the time t, consequently what is expressed in formula

When Xdx+Ydy + Zdz is an exact differential which we may denote by d. F(x, y, z), then formula (5) by integration simply gives

m

q=m . F (x, y, z) — — • v2+C.

2

(7)

As a particular case, I will here regard that in which the resistance P in the direction of the path is the constant, as in Coulomb's experiments with friction of metal sliding on metal; agreeably to formula (6) we then get

when q is supposed

q=P.s,

zero for s=0.

This equation shows that the newly developed energy is equal to the product of the friction and the space passed through, which is in accordance with my earlier experiments, and that the amount of this energy is independent of the velocity with which the slider is moved; and this result was likewise derived from my experiments.

The Energy in a whole System of Material Points. Let us next advert to the motion of a whole system of material points whose masses we may denote by m, m', m", &c.

After the lapse of time t, let x, y, z ; x', y', z' ; x", y", z", &c. be the coordinates of the points m, m', m", &c., the accelerating forces in direction of the axes for these points respectively X, Y, Z; X', Y', Z'; X", Y", Z", &c., and let the increments of the energies which are yielded by these points to the material resistances be respectively dq, dq', dq", &c.; then, in consequence of formula (5), we get

(3), which had then first to be subtracted from dQ in order to show the increment of the lost energy, or the energy dq appearing in a new shape: partly I have wished thereby to avoid the confounding of material resistances with real forces; for it seems to me that the material resistances are, as it were, "a lifeless thing," to which some part of real force, being the resultant of the three forces X, Y, Z in formula (1) resolved in direction of the path, is imparted during the motion of the mass m. Though it is sure that formula (6) may be regarded as a simple result of formula (1), yet I keep this formula so much the more, that the train of ideas developed above made me at first sensible of the real state of the whole.

If all these last equations are added, and if we put dq+dq'+dy" + =dq, then we get

...

in which dq, denotes the whole increment of energy which all the material parts together yield to the material resistances, and Σ denotes summation.

If there are no other resistances than those given in the system of material points in question, we have

d2x

d2y

d2z

Σm [( X − d2d') dx + ( Y — dy ) dy + (Z — 177 ) dz] =0,

diz

which shows that the system does not lose any energy on account of the internal resistances.

The energy imparted to the material resistances by the system. during the time t may, conformably to formula (8), be expressed by

9=2m)(Xdx + Ydy + Zdz) — Σ m (dx2 + dy2 + d=2) 1m

dt2

+const. (9)

If now we compare what has been set forth in equations (1) to (7) above, then we perceive that the whole internal energy which is contained in a system of material points may in all cases be represented by

dx2 + dy2 + dz2·
die

C being an arbitrary constant.

Hence results that when the energy of a body manifests itself under the shape of heat, then the contained quantity of heat may always be expressed by the vis viva contained in the material particles of the body, as we mean by vis viva half the amount of all the masses of the material particles, each multiplied by the square of its own velocity.

In a note by Ampère, "Sur la Chaleur et sur la Lumière considérés commes resultant de mouvemens vibratoires "*, the author has set forth the idea that, while all rays of light and heat advance in waves through the æther, the propagation of heat in bodies depends upon the vibrations of the atoms and their propagation from particle into particle. Thus, looking upon heat as a motion of atoms, the author compares the quantity of heat contained in bodies with the vis viva of the atoms, and thereafter shows that the general equations for the propagation of heat in a body must also hold true for the propagation of the vis viva. As I think I have proved in the above that the internal energy of a body must necessarily be equal to the vis viva contained in the *Annales de Chimie et de Physique, vol. lviii. p. 432.

particles, it also necessarily results from this that it is by no means disagreeing with nature to apply the stated principle to the propagation of heat in bodies, as, on the contrary, it leads to truths proved by experience.

Now I shall proceed to examine how the internal quantity of energy contained in a fluid must vary when the pressure and density of the fluid vary.

Let dm be the element of the liquid mass m, whose particles, according to the above, must be supposed to be in incessant internal vibration; let the coordinates of the point of mass in question, after the time t, be x, y, z, and let Xdm, Ydm, Zdm be the moving forces on dm in the direction of the three rectangular coordinate axcs; further, let the density at this instant for the said point of mass m be p, and let p be the pressure on the unit of surface; if moreover the velocities of the element din in the directions of the three coordinate axes be denoted by

and if the increments of the velocities during the time dt are equated to u'dt, v'dt, w'dt,

then, in conformity to the above, the increment of mechanical which the element dm would have received during the time dt if it had been perfectly free will be

energy

But, as the element dm is not perfectly free, in fact it only receives an increment which may be represented by

dm(u'dx+v'dy+w'dz).

During the time-element dt this clement of mass consequently loses some part of the mechanical energy which is really produced through the accelerating forces. If the energy which dm loses during the time t is represented by q. dm, then the energy lost in the time-element dt is equal to dq. dm, and thus we get

dq. dm= [(X—u')dx + (Y —v') dy + (Z−w')dz]dm. (11) But this internal energy dq. dm, which is imparted to the material resistances by the element dm during the time dt, may be put in a simpler form; for, as is well known, we have

for if the three equations (12) are added after having been multiplied respectively by dx, dy, dz, and we observe that

dp dp dp
dy dy + dz

dx+

dx

dz=dp,

then we see that the formula may be written simply

The new increment of energy developed in the unit of mass in the time dt may consequently, for the point in question, be expressed by

By means of formula (14) we are now without difficulty able to determine the amount of internal energy produced in a unit of mass of a liquid body when it is compressed through external force; and as the internal energy produced thereby chiefly appears in the shape of energy of heat, we are able to determine the quantity of heat produced by the compression of fluids.

With regard to this, I shall here call attention to the quantity of heat developed in aëriform bodies suffering compression. Let us suppose that the gas in question, during the state of equilibrium, has in every place the same density D, and that h and gmh denote the barometric height and the pressure of air answering to this density, g being the force of gravity and m the density of the mercury. Let us further, at any instant during the compression, denote the density and pressure of the gas by and p, then we have

ρ

p=D(1+s),

(15)

in which s or the degree of condensation may be either positive or negative.

If the condensation takes place so quickly that heat is neither lost nor received during the motion, and s is only a very small magnitude, then, as is known,

where y denotes the ratio between the specific heat at constant pressure and that at constant volume. From this formula, whose correctness increases in the same degree as is decreased, follows,

dp=gmh. y.ds;

and by substituting this value for dp, together with the expression for p of formula (15), in the equation (14), we get