sible to discuss the results with exactness. It is likely that the experiments on the pressure of the gases of the powder in a closed vessel are those that were described by Captain Noble in a lecture before the Royal Society of Great Britain, or, at least, that they were determined in the same manner, that is, by crusher gauges, and the readings of these instruments being open to discussion, the results found should be accepted with some reserve. We add that the results differ completely from those which Rumford has deduced from his experi

ments.

In the meantime, formula (3) represents the facts with an exactness such that it is allowable to consider as very plausible the hypothesis according to which the products of the combustion of the powder will be, under certain circumstances, partly solid and partly gaseous. The pressure observed being due to the permanent gases alone, may be calculated by Mariotte's law, taking into account the volume occupied by the solid residue.

We show, in the following chapter, how, under this hypothesis, the equation of the movement of the projectile in the interior of a gun may be established.

5. Temperature of the products of combustion.-The value of f being determined by experiment, we deduce from it by equation (2) the value of To. We have, in fact:

According to MM. Noble and Abel, the value of v, was 280 litres for the powder experimented with. By making

f=219300 V%280

Po 103.33

we find for the absolute temperature of the gases

T=2070°

6. In order to calculate this temperature theoretically, it is necessary to know:

Ist. The heat lost by the products of the combustion of the unit of weight of powder without production of exterior work by being lowered from the temperature 7, of combustion to a determined temperature, say zero centigrade or 273° of absolute temperature.

2d. The mean specific heat of the products of combustion between these limits of temperature.

In fact, designating by the heat of combustion, and by the specific heat, we have the relation:

The heat of combustion can easily be determined by the burning of powder in a calorimeter. MM. Noble and Abel found it equal to 705 calories (French units of heat) for the experimental powder.

With regard to the specific heat, it is unknown. MM. Bunsen and Schischkoff have admitted the value c=0.185 for the products of combustion of a powder similar to our sporting powder. But this quantity corresponds to a temperature nearly the mean atmospheric temperature, and, following MM. Noble and Abel, it should vary, increase, probably, with the temperature.

Therefore, in default of more precise data, we admit the value c= 0.185 with Q705, and we find

T=4080°,

that is to say, a value nearly double that which has been deduced. from the measure of the pressures. One would be led to admit, then, that the mean specific heat has a value nearly double that which MM. Bunsen and Schischkoff have adopted, and, consequently, since the specific heat of the gases under constant volume is independent of the temperature, that the specific heat of the solid residue at the temperature of combustion is more than double what it is at ordinary tempera

tures.

This result may seem excessive. We shall see later, however, in studying the cooling of the gases by the envelope, that one can explain the difference between the theoretic temperature and that deduced from the measure of the pressures, without necessarily admitting this enormous variation of specific heats.

7. Volume of the solid residue at the temperature of combustion. The coefficient a, the value of which is determined (4), represents the volume in cubic decimeters, at To, of the solid products of combustion of a kilogram of powder. Also, after MM. Noble and Abel, the volume of these products at ordinary temperature is 0.3, and their weight is 0.57 kilogram.

It is easy to conclude from these numbers and from the value a= 0.6833:

Ist. That the mean coefficient of cubic expansion of the solid residue between the temperature 273° and 2070° is 0.000624, or roo nearly;

2d. That their specific weight at the temperature T=2070° is equal to 0.903.

This last result does not agree with one determined by MM. Bunsen and Schischkoff. These experimenters have, in fact, found, by a method which seems to be a direct experimental determination, that the specific weight of the solid residue is 1.520 at the temperature of 2808°. It seems difficult, at this stage of our knowledge, to decide which of these two numbers is the nearer the truth.

I

1600

It is, however, to be remarked, that the value found for the coefficient of expansion notably exceeds analogous coefficients which correspond, in the ordinary limits of determinations, to the greater number of solid bodies. It may then be possible that this coefficient was really too large. The same may be true of a, the coefficient which served to determine it. We shall see, in fact, that an error of this kind can result from the cooling of the products of combustion by the wall of the vessel enveloping them.

1.8

8. We will close this discussion with a remark on the comparative volumes of the powder and the solid residue produced by the combustion. The volume of a kilogram of powder of a density 1.8 is I =0.55. The volume of the solid residue resulting from its combustion is 0.68, according to MM. Noble and Abel, and 0.44 according to MM. Bunsen and Schischkoff, and the volume of the powder is somewhere between these two values. One may conclude from this, that if the solid residue of the combustion of the powder is formed under the same conditions as those of actual service, it would not be a great error to suppose that the volume of the residue, at the temperature of the flame, is equal to that of the powder itself. In the following chapter the utility of this remark will be manifest.

9. Effect of the cooling of the products of the combustion of powder by the wall of the envelope.-It has generally been supposed, in theoretical researches on the effects of powder and other explosive substances, that the cooling of the products of combustion by the wall which enveloped them could nearly always be neglected. But this effect cannot be neglected.

We have referred in our preceding work to the calorimetric experiments by which M. Saint-Robert ascertained that, in firing a gun, the

heat absorbed by the walls of the bore could be taken as nearly onefourth the heat of the combustion of the charge.

The loss of heat, relatively less in guns of large calibre, is certainly a sensible quantity, and Captain Noble, in a lecture on this subject, is disposed to attribute to this cause considerable losses of work observed in certain experiments.

M. Berthelot also estimates that, in the application of processes which we arrange to produce very high temperatures, a large proportion of the living calorific force is lost through the wall of the envelope. This eminent chemist expresses himself as follows, and adds that the facts which support the statement will perhaps not be without interest for the study of the reactions produced by the combustion of powder in the bores of guns:

"L'existence des hautes températures en principe et la possibilité de les réaliser, me paraissent devoir être distinguées avec soin.

"En principe, nos théories actuelles indiquent qu'une masse gazeuse donnée peut acquérir une force vive indéfiniment croissante, c'est-adire une température illimitée .....

"Mais, en fait, il se peut que l'intensité des radiations de toute nature augmentant avec une extrême promptitude à mesure que la température s'élève, et par suite les déperditions de la force vive qui se communique aux milieux environnants devenant de plus en plus considérables, rendent irréalisable toute température qui passerait une limite voisine de 2500 ou 3000 degrés observés dans les expériences de M. Sante-Claire Deville."

10. It seems to us probable that in all the circumstances of the combustions of the powder, the intensity of the thermic state realized is such that, in spite of the brevity of the phenomenon, the temperature can be lowered considerably in an extremely short time, and that perhaps to this rapid lowering of the temperature, varying with the mass of the powder, the surface of the envelope, and the period of combustion, may be attributed the difference, which the results present, of various authors who have sought to measure the pressure of the powder gases in a closed vessel.

We submit, while on this subject, some considerations, which without pretending to a rigorously exact determination, will serve, we believe, to give a notion of what these perturbing influences may be.

II. We consider a weight of powder burning in a closed vessel. The combustion of the powder is not instantaneous; it is done progressively, so that after a time t counted from the beginning of igni

tion, the quantity of powder burned is a function F(t) of which the form depends upon the physical qualities of the powder and upon the circumstances of combustion.

During combustion the envelope absorbs heat, and, after the time t, the temperature 7 of the products of combustion is less than To, which would have been the temperature had combustion taken place instantaneously. We will endeavor to express this difference. To this end, denote by

the total surface of the envelope,

h the velocity with which the heat is imparted for a unit of surface of the wall of the envelope,

The heat absorbed by the envelope after the time t is:

after the time t is: Sh.dt.

σ

Again, the heat lost by the weight F(t), the temperature of the products of which is lowered from T, to T, is cF (t) (T. — T), c being the mean specific heat under constant volume.

.We have then :

(5)

cF(1)(T.—T)=6

Sh.dt.

The coefficient of cooling h is a function of the temperatures of the products and of the envelope, vanishing with their difference.

If one admits that the cooling takes place by radiation from the products of combustion through a superficial layer of the envelope, and that this radiation follows the law of Dulong and Petit, we are led to put (6)

h=Hee' (aT- a11),

designating by,

T the absolute temperature of the products of combustion,

T that of the envelope;

the emissive power of the products of combustion;

s' the absorbent power of the substance of the envelope;

a is a constant always equal to 1.0077;

H is a constant, common to all bodies, and of which the value is 0.000237, if we take for units the decimeter and the second, the emissive and absorbent powers being referred to those of lampblack. The unit of heat is the calorie (the quantity of heat necessary to raise the temperature of a kilogram of water one degree centigrade).

NOTE I.-I. The numerical value adopted for the constant H of formula (6) Chapter I, has been deduced from the results of experiments by Dulong and Petit, in their researches on the laws of cooling. Consider a body at the absolute temperature 7, cooling in an enclosure to the temperature T. Calling its