and thickness, the subtractive term corresponding to spherical or cubical grains may be diminished by one-fourth of its value. If its absolute value is 100 meters, for example, we would gain 25 meters of velocity without increase of maximum pressure. The gain increases for decreasing values of x; if we take x=}, the subtractive term is diminished by one-third; and, supposing the same absolute value of the velocity, the increase is about 33 meters.

2d. Grains of the form of a parallelopiped.-Calling x and y the ratio of the least side to the two others, we have (No. 42),

(85)

a=1+x+y, ?=

x+y+xy
I + x + y'

and the ratio of the subtractive term to that for the spherical or cubical grain is

If the grain has a square base, we have y=x; and the above ratio becomes

This expression diminishes as x varies from 1 to o, but the change is somewhat slow. When two dimensions of the grain are double the third, x, and the value of the ratio is 1. The velocity therefore increases by 1 of the subtractive term corresponding to the spherical grain. Supposing this term 100 meters, it is about 7; it becomes 16 for x=}.

The advantage of flattened grains over those which are spherical or cubical seems then real; but the advantage appears to be less than in the case of grains which are pierced.

61. Theoretical force of some powders.—The numerical application of the preceding theories requires that the velocity of combustion and force of the powder shall be known.

The velocity of combustion must be observed under the normal atmospheric pressure, 760m. It has been determined for a great many powders by Piobert, and the figures he has given may still give a sufficient approximation in many cases. In No. 45, some experimental results, which will give at least an approximate value of this element in the ordinary conditions of practice, are given.

The processes of fabrication having, however, been greatly altered since the time of Piobert, it has seemed advisable to undertake new experiments with the new powders.

The force of the powder is the element whose direct determination offers the greatest difficulties; there is no experiment which gives its value with certainty for any powder.

In the want of more precise data, we present the theoretical values which we derived with M. Roux from the experimental determinations made at the Dépôt Centrale des Poudres et Salpêtres. These values were deduced, by (3) of the first chapter, from the temperatures of combustion and the volume of the permanent gas.

The units taken are the kilogram and decimeter. Consequently, according to the definition of the force of a powder, the figures in the table express in kilograms the pressure per square decimeter of the permanent gases of a kilogram of powder, occupying, at the temperature of combustion, the volume of one liter.

The figures of this table confirm what was said in No. 59 concerning the equality of the force of powders which have been differently made.

62. If according to No. 37, we take the velocity of combustion of powder to be proportional to the square root of the pressure, we put ain the preceding formulæ. We thus obtain new and very simple expressions which, as we shall see later, appear to represent accurately facts and the results of experiment.

63. Formula for pressures, when a=.-Replacing a by, and z by its value in (19), equation (77), for the pressure, becomes,

the time of burning of a grain under the pressure p.,

the weight of the powder,

4 the density of loading,

the density of the powder,

a a numerical coefficient depending upon the form of grain,

A an absolute number whose value is independent of the elements of fire, and of the units chosen.

64. Determination of A by experiment.-To determine A theoretically, it would be necessary to integrate the first of (66), and deduce a table of values of the function %, of which A is the maximum. This determination, which, however, will be effected later, is not necessary. The essential form of the function which gives the maximum pressure being fixed, we have only, to determine A, to measure the pressure in known conditions of fire-a single experiment is enough. The following is an example of this determination by means of a navy gun of 24 centimeters; Wetteren powder was used (13 to 16 millimeters). The values of the various quantities on which the pressure depended were as follows (kilogram, decimeter and second being the units taken):

In these conditions, the mean of three manomètres à écrasement in the powder chamber gave 2300 kilos per square cm. We have, therefore, P230000; from which we have the value

A=0.703.

It would be of interest to determine this coefficient by means of a more accurate instrument. This might be done by the use of the instrument lately invented by MM. Déprez and Sébert, and Captain Ricq.

65. Pressures produced by the same powder in similar guns.—In passing from one similar gun to another, m and a vary as the cube of

*This value is the one which, in table in No. 62, belongs to cannon powder, whose elements are in the same proportion as in the Wetteren powder. †The grain being irregular, it was assumed that the value of a for a sphere could be taken.

This value is obtained by taking a velocity of combustion of 11.7 mm. per second, and calculating the mean radius of grain by (46), with the values N=350, d= 1.8.

§g= 98.09 d.

The calibre is 2.42 d. c.

the calibre, and as its square; the density of loading 4 remains the same. If then the powder is the same for the two arms, 7 does not change; and we see, from (88), that the maximum pressure varies proportionally to the calibre. Consequently, we may lay down the following law:

In similar guns, charged with the same powder, the maximum pressure is proportional to the calibre.

This law may often be made of use. For example: experiment has shown that the maximum pressure developed by Wetteren powder (13 to 16 mill. grains) in a 24 cm. gun is about 2300 kilos per square cm.; from which we conclude that the pressure of the same powder in a 14 cm. gun will be 2300 X 1340 kilos per square cm. about.

14 24

66. Formula for velocities in the hypothesis, a=.-Putting a = } in (81), we find;

To apply this formula, the functions, and, must be known, or (67) must be integrated. But, without this, we may derive from (89) an approximate formula for initial velocities which may be useful.

67. Approximate formula for initial velocity.-The empirical formulæ by which initial velocities have been determined have generally been monomial. The form is much the simplest for calculation by logarithms, and it is not incompatible with the form which theory indicates, since any function may always, within certain limits, be considered as nearly proportional to a properly chosen power of the variable.

Adopting this method, we see that the quantity,

λ mz may be considered, as an approximation, as proportional

to a positive power y' of the first of these variables, and to a negative power of the second. We may then, calling B a numerical factor, replace it by an expression of the form

where and may be determined by experiment. We shall return to this point in a special work.

Supposing that 7 and are known (90), enables us to evaluate numerically the influence of the various elements of fire upon the effects obtained. It exhibits, in particular, the proper influence of the characteristics of the powder used. This last, it appears, could hardly be obtained by any purely empirical method.

We shall now deduce from (90) some consequences which seem worthy of remark.

68. Initial velocity attained by the same powder in similar guns.— In similar guns, u and z vary with the calibre, as the square of the calibre, and m and as its cube. Also, a, τ and λ remain the same; we therefore conclude that the initial velocity varies as the

69. Initial velocities attained by different powders giving the same maximum pressure.-Consider two powders having the characteristics, f, t,a,i, f,t,

f', r', a', x'.

In the same conditions of fire these powders would give the same maximum pressure if their characteristics satisfy the relation (82),

This being fulfilled, formula (90) shows that the corresponding initial velocities are connected by the relation,

We may thus compare various powders, and ascertain what influence the force of the powder and form of the grain exercise.

70. Maxima pressures produced by different powders giving the same initial velocity.-Consider, secondly, two powders which, in the