5. Designating by the weight of the projectile, c the calibre or diameter of the bore, g the acceleration of the force of gravity, we have:

Substituting these values in expression (5) and putting for brevity :

6. In order to obtain a formula which shall be numerically applicable, it remains to fix the value of y. It can be obtained from the results of experiments. In fact, the empirical formulæ of the commission of Gavre established the fact that, all other elements being. constant, the initial velocity is:

1st. Proportional to the power of the charge (Memorial de l'artillerie de la Marine, Vol. IV, pp. 25 and 45.

2d. Inversely proportional to the power of the capacity S' in which the powder charge is placed (Mem. de l'art. de la Mar., Vol. II, p. 37). The velocity is then approximately proportional to

Now according to formula (6), the velocity is proportional to

should be, for values of and J, comprised between the limits of practice, sensibly proportional to, and this is also verified numerically.

which we can in practical applications replace by the following:

λ

I

This last equation is more simple and sufficiently exact. We shall adopt it in our last researches. (Formula (8) does not take into account, as in formula (7), the variations of ; but these variations are so small that in practice their influence on the velocity may be neglected.) 7. Characteristics of a powder.-Formula (8) includes two quan: tities , whose values depend upon the nature of the powder and upon the form and dimensions of the grains. These two elements completely determine the effect produced by a powder under given conditions. We can, then, call them characteristics. We will designate them hereafter by the letters a, 3, putting:

(fa)1 and

Τ

8. Numerical determination of the constants A and B.-Supposing the characteristics of a given powder to be known, it suffices to measure the velocities given by this powder under two different conditions of firing, in order to find the numerical values of A and B from formula (8). This amounts to replacing successively the various elements in the formula by the particular values relating to the two firings, giving to v the corresponding values determined by experiment, and we have two equations showing the relation between the two unknown quantities A and B.

9. In the absence of precise data on the absolute values of a and §, we may take them equal to 1 for a well-defined powder, and refer the characteristics of other powders to those of the powder adopted as a standard. From this point of view we take, for example, as a standard a sample of Wetteren powder (grains of 13 to 16 millimeters), which under the normal conditions of proof firing has given in the 24 and 14 centimeter guns initial velocities respectively of 4371 and 4384 decimeters.

The numerical values of the various elements of firing are shown in the following table:

Putting successively the particular values from this table in formula

fa

(8) and replacing and by 1, we obtain two equations from

Τ

which we find the values

(10)

T

Consequently for any powder whatever of characteristics a and ẞ the formula for velocities is the following:

10. In place of taking a and equal to unity for the standard powder, we may calculate the numerical values of these quantities according to certain particular values off, t, a, and . These last quantities are it is true imperfectly known, but no error will result in the calculation of velocities, since we only change the units to which the characteristics of a powder are referred.

This method of working has the advantage of taking into account in the valuation of characteristics the physical signification of the elements on which they depend. It permits also, in certain cases, the establishment à priori and independently of all experiment, the values of the characteristics due to the physical properties of the powder. We shall give hereafter some examples.

In order to determine the constants A and B under this hypothesis, we adopt for the standard powder the values:

(12)

f=431000, T=0.730, a=3,

[The standard powder is composed as follows: saltpetre 75 parts, charcoal 12.5 parts, and sulphur 12.5 parts. Its specific constants are

the following:

Number of grains to the kilogram,

Density in mercury,

N= 350

8=1.75

Velocity of combustion under normal pressure (decimeters) w=0.10 The grain is very irregular, but the mean form is supposed to be a sphere of radius r, and given by the formula

r

The duration of combustion is deduced from the equation: 7= W

The value ƒ=431000 is that adopted for cannon powder (No. 61, Part III). The values a=3, I are for spherical grains (No. 39, Part III)].

By equation (9) the characteristics of this powder are deduced,

(13)

1331, Po=1.370,

and the new constants A and B are found by dividing equation (10) by a., P., we have thus:

(14)

A=1.569, B=0.00795,

consequently the formula for velocities as a function of all the elements of firing becomes

This formula constitutes the definite result we had in view. We give in succeeding chapters numerous applications of this formula.

The signification of the letters on which it depends has been given in Articles 2, 5, and 7. It is essential to recollect that the units adopted are the decimeter, the kilogram, and the second. (In the original formula (5) the constants P and Q are absolute numbers, independent of the choice of units. In formula (8) however the values of the constants A and B defined by the relations of No. 5 depend upon the units cited above.)

11. Formula of pressures.—Supposing the velocity of combustion proportional to the square root of the pressure under which it takes place, the maximum of the mean interior pressure is given by the formula:

in which A denotes a numerical factor independent of the choice of units, the signification of the other letters being the same as for formula (15). (See Part III, Nos. 63 and 64. In Part III, Art. 64, the value of A deduced from experiment is given as 0.703, admitting for the Wetteren powder 70.6. This number corresponds to a velocity of combustion of 11.7 millimeters per second. Some results recently obtained, however, lead us to admit that this velocity is in reality. nearly 10 millimeters. This value which has served to establish the constants of formula (15) will be adopted in the calculations of pressures.)

Replacing, as in No. 5, m and w by their values as functions of p and c, designating by K the product of A and all the constant factors fa of the formula, and observing finally that is the square of the characteristic a defined in No. 7, it becomes

α

T

12. This formula, already very simple, may be still further simplified by remarking that for the usual values of and 4, the function

له

is very nearly proportional to 1. Adopting this interpolation, which is quite sufficient for practice, the approximate formula for pressures is

and under this form we shall employ it in the researches to follow.

13. Numerical determination of K-Supposing the characteristic a of a powder to be known, it is sufficient to measure the maximum pressure for that powder given under a single condition of firing, in order to find from formula (18) the numerical value of the constant K.

For example, this determination can be made for the standard powder referred to in No. 9. This powder fired under the conditions defined in the table of the article before cited, has given pressures which, expressed in kilograms per square centimeter, are equal to

(These numbers are the means of the readings of three crusher gauges placed in the powder chamber.)

Consequently with our units the values of P in these two cases are respectively equal to 229000 and 114000. With these numbers two