same conditions of fire, give the same initial velocity, and let us compare their pressures.

According to (90) the equality of velocities requires the condition

71. Approximate value of r.-In order that (91) and (93) shall be of any practical use, we must know the value of y at least approximately.

Its value may be found from certain experimental data. Experiments made at Gavre with a 24 centimeter gun show that the initial velocities reached with projectiles of different mass, all other elements of fire being constant, vary inversely as the oths power of the mass of the projectile.

Admitting this, it follows from (90) that we must have nearly

Equation (94) enables us to compare the velocities given by powders having the same maximum pressure; the condition of equality of pressures is expressed by the equation,

Equation (95) serves to compare the maximum pressures caused by powders giving the same velocity. Putting 7 in (92), we see that the characteristics of the powder are then connected by the relation,

(97)

10

to change con

The various formula which precede can only be considered as approximate, owing to the uncertainty in the value of 7. We have found the value y from the exponent of the variable m in (90); now, it is easy to see that a very slight variation of this exponent will 27 cause the value of 7, and particularly that of siderably. Notwithstanding this, we believe that the relations furnished by theory may, with advantage, be applied to the study of ballistics, in place of the formulæ hitherto used, which have all been empirical.

I 27

72. To give an example, we shall compare the effects of two powders of the same make and differing in form and density of grain. Take two powders, the grains of one being spheres and those of the other parallelopipeds. In the latter, let the base be a square, and the side of the base double the height. We have:

By the substitution of the parallelopiped for the sphere, we may therefore increase the velocity by about without changing the pressure, or diminish the pressure by about without changing the velocity.

In the first case, the ratio of the times of burning are, from (96),

From the ratio of the times of combustion, we can deduce the num

Similarly for the second powder, calling ', v', N', the homologous parts to , v, N, and a the height of the grain, we have from No. 45,

The material of the two powders being the same, we have v=v, ', and consequently

÷ = 2 ( 3 ) +

πΝ

We deduce from this expression, and from the two values of above, that,

Ist. To obtain the same maximum pressure we must have

73. Application to a particular case.-Wetteren powder (13 to 16 mill. grains), fired in the navy 24 cm. gun, in the ordinary conditions of épreuves de réception, gives about 437 meters velocity and 2500 kilos per square centimeter pressure.

The grains of this powder are irregular, and the mean number of grains to the kilogram is 350. Suppose that we substitute for it another powder of the same material, but whose grains are parallelopipeds with square bases and heights the side of the square.

It results from what precedes:

Ist. That, if the number of grains to the kilogram of the new powder is 350 X .44154,

the initial velocity will be

437 X 1.02446 meters about,

the pressure being the same as that of the Wetteren powder. 2d. That, if the number of grains to the kilogram is

350 X.33

the maximum pressure will be

115,

2500 X.9082270 kilos about,

the initial velocity being the same as that given by the Wetteren powder.

These results would be changed if the constants f, 8, v, differed for the two powders. The general formulæ, however, will permit our calculating the effect of these elements.

We shall return to these questions of application, which seem to have a certain importance from a practical standpoint.

CHAPTER V.

NUMERICAL DETERMINATION OF THE AUXILIARY FUNCTIONS.

74. We have seen (No. 47) that the theoretical study of the movement of the projectile in the interior of a gun should be amended by the determination of certain functions purely numerical. These functions, which may be termed auxiliaries, are those which we have designated (No. 48) by the notation yo, J', '2,... We have remarked that they consist of special transcendentals, the complete numerical determination of which would permit the theoretical solution of all the problems relating to the firing of any gun, whatever may be the variable elements of the firing.

The auxiliary functions are defined by equation (67). It does not seem possible, in the actual state of the resources of analysis, to deduce the explicit form of the unknown functions of equations where they are involved not only under the characteristics of differentiation, but also under the signs of integration.

But we can apply to these equations the methods used for the numerical integration of differential equations. The application of these methods leads to long and laborious calculations, but does not present any theoretical difficulty. We will explain it briefly.

75. Determination of the function y..-We have designated by yo that one of the auxiliary functions which forms the first term of the development of the space traversed by the projectile as a function of the time taken to travel this space. It is defined by the first of equations (67), which we reproduce below, replacing X, by its value

It should vanish also, as well as its first derivative for the value x=0. For small values of x we satisfy the equation (98) and also the initial conditions by putting:

a, b, c,... m, n, p, ... to be determined. Substituting the values (99) in equation (98) and identifying, we find for the exponents,

and the coefficients a, b, c, . . . are given by the equations

The series (99) are very convergent for values of x less than unity. We can also deduce from them the values of y, and of its derivatives for x1, but beyond x1 they become rapidly divergent, and it becomes necessary to have recourse to Taylor's Theorem.

76. Suppose the functions value of x, the variable.

dy, dy.

dx2

yo d.x

calculated for a given