We can deduce from the theoretical formulæ the following result: Among the systems of values of and which give the same pressure there is one for which the velocity is a maximum.

T

To demonstrate it, consider formulæ (31) and (18) for velocities and pressures; putting for brevity:

in which the variables 4 and are the principal quantities to consider. Eliminating 4 between (44) and (45) we have

whence taking account of (39) the value of (=):

For a given value of P this value of v depends only on the variable and it passes through a maximum for the particular value

The corresponding values of the density of loading and of the velocity deduced from relations (45) and (46) are:

(50)

5

43. Replacing T, H, and h in (47), (48) and (49) by their values (33), (42) and (43), we have

To=5B2 (pu)}

с

4=

3

h

4

3

or better, taking account of the numerical values of A, B, K (we have from (14) of Art. 10, A= 1.569, B=0.00795, and from formula (22) of Art. 14 K=0.0153),

44. In the greater number of cases, notably when the value of P is considerable, formula (54) gives for the density of loading values which cannot be attained in practice. The greatest value possible of v is obtained by taking 4 equal to its limiting value.

If we suppose this value to be unity, the corresponding value of 7 is found from equation (45), which, making 4= 1, and replacing h by its value from (43), gives

Under the same hypothesis, formula (8) gives for the velocity:

We may remark that in this expression the quantity between brackets is equal, according to equation (51), to 46; taking this into account and replacing the coefficients K and A in (57) and (56) by their numerical values, we have the following system:

which should be substituted for formulæ (53), (54) and (55) when the elements of firing are such that the value of 4, (54) is greater than unity.

45. In attributing to P the limiting value which it cannot pass without compromising the safety of the piece, the above formulæ serve to calculate the greatest velocity that can be realized in a given piece, with a given form of grain, and given weights of the projectile and charge.

To this end, we calculate first the value of 4, from (54).

If this value is less than 1 we adopt it for the density of loading and calculate the duration of combustion by formula (53). If, however, the deduced value of 4 is greater than 1, we take the density of loading equal to 1, and calculate the duration of combustion by formula (58).

The velocity sought is given by formula (55) in the first case and by formula (60) in the second case.

46. We give some applications o centimeter Krupp gun.

these calculations to the 30

1st. Suppose first the limit of P is fixed at 2500 kilograms per square centimeter, then with the units adopted P= 250000.

The values of 4, for the three forms of grain already considered, and of charges of,, and the weight of the projectile, are summarized in the following table:

Designation of the form
of the grain.

Cube,

Parallelopipedon,

Pierced cylinder,

The following table gives the corresponding values of 7, 4, and the

velocity v.

W50 Kilos.

W60 Kilos.

75 Kilos.

Designation of the form of grain. Cube,

1.025 1.000 426.0 1.123 1.000 449.7 1.255 1.000 477.5 Parallelopipedon, 0.683 1.000 437.6 0.749 1.000 460.4 0.837 1.000 487.4 Pierced cylinder, 0.490 0.980 485.5 0.490 0.895 505.8 0.490 0.800 531.9

2d. Suppose in the second place the limit of P reduced to 1500 kilograms per square centimeter, which corresponds very nearly to the strength of bronze pieces.

P=

Making P 150000, formula (54) gives the values of 4, less than unity; which are adopted as the densities of loading, and v and are calculated from formulæ (55) and (53).

We thus obtain the following table:

Designation of form of grain.

Cube,

1.469 0.882 386.1 1.469 0.805 402.3 1.469 0.720 423.0 Parallelopipedon, 0.918 0.827 392.4 0.918 0.755 408.8 0.918 0.675 429.8 Pierced cylinder, 0.490 0.588 427.3 0.490 0.537 445.1 0.490 0.480 468.1

47. From the preceding calculations we discover some interesting results.

ist. We observe that, in guns of great resistance there is no advantage, with the forms of grain used, in taking the density of loading notably less than the gravimetric density of the charge. We can reduce the density of loading and retain the same velocity by a correlative diminution of the duration of combustion, but this augments the pressure.

If we regulate the duration of combustion in a manner to retain the pressure we diminish the velocity.

In guns of little strength, there is on the other hand always an advantage in adopting densities of loading less than the gravimetric density of the charge.

2d. In bronze guns, with powder grains of the form of a cube, sphere or parallelopipedon (generally used), the velocity cannot pass beyond 400 to 410 meters with a charge of the weight of the projectile. This limit may, however, be exceeded with the same charge using pierced cylindrical or prismatic grains. The calculation indicates finally that the limit of velocities is from 420 to 430 meters, firing ordinary powder with a charge the weight of the projectile. These various results may be generalized and extended to different calibres by the application of the principle of similitude.

48. Theory attributes a remarkable superiority, from a ballistic point of view, to pierced cylindrical or prismatic grains.

This theoretical result should, however, be accepted with some It is exclusively due to the particular law followed in the combustion of a grain, and in practice the law would not hold good in the case of the rupture or disintegration of the grains. Again, the method adopted in the packing of the grains in the cartridges influences their ignition.

For these reasons we shall only consider the figures obtained by prismatic powders as the superior limits of velocities which may be obtained.

PART V.

ADDITIONAL PRACTICAL FORMULE.

1. The formulæ which give the velocity and pressure in guns have already been given. To this end, a mixed method has been used; the general form of the equations having been established by theoretical considerations, and the numerical values of certain coefficients determined by experiment.

In particular, to obtain one of the coefficients, it was assumed, conformably to the results of experiments made by the committee at Gavre, that, if all the other elements are constant, the initial velocity is (1) proportional to the power of the weight of the charge, and (2)

inversely proportional to the chamber.

power of the capacity of the powder

2. Now, it is known that, in certain cases, it is better to take the velocity proportional to the power of the weight of the charge.* We shall examine, in this note, the modifications which the formulæ will undergo in this hypothesis.

The form of this new discussion differs in one point from that previously adopted. It shows that, in the ordinary conditions of fire, the two empirical laws, according to which the initial velocity depends upon the charge and the volume of the powder chamber, are not distinct.

If one of them is admitted, the other results from it; this reduces the number of data which must be determined by experiment.

3. Formula for initial velocities.—Calling

the weight of the charge of the powder,

p the weight of the projectile,

u the length the shot moves in the gun, the calibre or diameter of the bore,

4 the density of loading,

f the force of the powder,

*Memorial de l'artillerie de la marine, tome I, p. 643; tome II, p. 40;

tome V, p. 401.