Substituting this in (7), and replacing C by its value from (8), we have the following equation, which gives the motion of a projectile in a gun,

(u

d2 u dt2

= '( di )2 = fy

2

dt

m

The problem is then reduced to finding the integral of this equa

du

tion; it being evident that it and the first derivative must vanish dt when to.

6. If we consider the combustion instantaneous, the weight of powder burned y becomes equal to the weight of the charge w, and the gas formed occupies immediately the whole of the powder chamber. 7. If then we call u, the reduced length of the powder chamber, that is to say, the height of an equivalent cylinder having for its base the section of the bore w, we have z=u, in (10). Consequently (12) becomes

(13)

d'u
dt2

- I

foo

(x+16) "+" (du)2="
=

in this form it is directly integrable.

dt

m

Let the velocity of the projectile, then we have

:

where C is a constant which may be determined by the condition that

This formula is the same as (58) of Part I. It may be put in the same form by means of (49) of Part I.

8. To suppose that the combustion is instantaneous, is not however permissible. If it were so, the gases being suddenly formed in a volume very nearly equal to that of the powder, would produce enormous pressures which the gun could not withstand. It is only by the use of progressive powder that the high velocities and low pressures of recent times have been attained.

We must, therefore, in (12), consider y and z as functions of t. In this case, however, the integration by ordinary methods becomes impossible.

We can obtain, however, approximate values of the integral in two extreme cases when the displacement which it represents is very great or very small compared to the reduced length of the chamber.

These two cases are, moreover, very important. The first corresponds to the ordinary cases in practice, where the chamber is generally only a rather small fraction of the volume of the bore. The examination of the second case leads to some interesting results concerning the law of tension near the origin of motion, and consequently concerning the high pressure which it is generally believed is produced at that time. The first of these cases only will be treated here.

9. Before attempting the approximate solution of (12), we may remark that it depends upon the form of the two functions F (t) and F(t), which will vary according to the manner of combustion of the charge.

We shall resume first, adding some new considerations, the notions we now have concerning these functions.

CHAPTER II.

ON THE FORM OF THE FUNCTION y=F(t), REPRESENTING THE COMBUSTION OF THE CHArge.

10. The exact determination of the function F(t) presents great difficulties. Its form depends at the same time on the physical properties of the powder, on the form and dimensions of the charge, on the position of the vent, etc.

Piobert was the first to give its form by formulas which he had obtained under the supposition that the portion of the charge not ignited retained at each instant the form which it had before the charge was ignited, and further, that the ignition was propagated spherically with a constant velocity around each of the points of the exterior surface successively reached by the flame.

We are able to clear the problem of these restrictions and to establish, following the track of Piobert, several very complete simple formulas, rigorously determining the function F(t) by the aid of two new functions y(t) and 4(t), defined as follows:

11. We designate by the weight of the charge of powder, ø(t) the fraction of this charge which is in ignition at any time t reckoned from the beginning of ignition, (t) the fraction of one of the grains (supposed equal) of the charge which is burned after a time t, counting from the instant when the exterior surface of this grain is reached by the flame.

Then let be any given epoch of the combustion and x any previous time; after the time x the weight of the powder in ignition is q=y(x). If the time x increases by dx, q receives the increase dq=wy'(x)dx.

During the time t-x which elapses between the time x and t, a portion of the element dq is burned, and this portion is equal to dq.4(t—x)=y'(x)4(t— x)dx.

The total quantity of powder burned is obtained by integrating this expression between the limits o and t, and we have

(16)

t

F(t) = f '4'(x) 4 (t− x) dx.

and 4, we

Before making hypotheses with regard to the functions shall deduce from this formula some general properties of the function F.

12. The function y=F(t) is discontinuous. In fact, designate: Ist, by the time at which all the grains are ignited;

2d, by the duration of the combustion of a grain.

For values of x greater than we have always 4(x)= 1, and consequently '(x)=o. The corresponding elements of the integral being also o, it suffices to integrate between the limits o and 0. We have then for values of greater than 0,

(17)

F(t)=σ
= ƒ¢'(x) 4 (t − x) dx.

If we suppose in the second place >, the value of (t-x) reduces to unity for those values of x such that t—x>, that is to say, for such values of x as are less than -. It results then, for values of x between o and t—, the element of the integral reduces to '(x)dx, so that we have:

t

F(1) =w['q'(x) dx+w f'q'(x) 4 (1 − x) dx,

that is to say,

(18)

F(1) = wç (t− = ) + ¤ ƒ 'y' (x) 4 (t — x) dx.

From formulas (16), (17) and (18) it is easy to conclude that the function F() is generally represented by three functions, which are substituted the one for the other during the total duration of combustion. We have in fact :

1st. If <T,

from to to t=0;

from t0 to t=r;

(19)

F(t) = wf '4'(x) 4' (t− x) dx

from 1 =: to 1=0+: ; F (t) = ƒ¢'(x) 4 (t− x) dx

2d. If >,

117

+wy (t − r).

from 1=0 to 1=0+7; F(t)=¤ ƒ¢'(x) 4 (t− x)dx

Τ

the discontinuities corresponding to the values and of the time. 13. For each of the values 0 and the first derivatives of two functions which are substituted the one for the other have equal values. This remarkable property of the function F(f) is established without difficulty by differentiating the expressions (19) and (20) with respect to and substituting the particular values of and in the results obtained.

It follows then, that if we represent by a curve the general form of the function F(1), it will be composed of three successive arcs having a common tangent at the points of discontinuity.

14. The case in which the duration of ignition bears a very small ratio to the duration of combustion of a grain.

When the charge is very small or when the interstices between grains offer an easy passage to the burning gases, it is conceivable that the necessary time for all the grains to be reached by the flame is very small, and may be neglected in comparison with the total period of the combustion of a grain.

We can then suppose that all the grains are ignited simultaneously, and it follows that the combustion of the whole charge would be represented by the same function as that of the combustion of each grain. We have then, very nearly,

15. The case in which the duration of ignition is very great with respect to the duration of the combustion of a grain.

We suppose that is very great with respect to T, we then can only consider the value of F(t) between the limits of and 0, that is to say, the second of the values (20), which for values of t notably greater than, reduces to the following:

16. General form of the function F(t).-The functions and y can in general be developed by McLaurin's Theorem, following the powers of t; it follows then, from (21) and (22), the function F(t) is of the form

(23)

F(t)=(at+bt2 + ct3 + . . .),

when the duration of ignition of the charge is very great or very small with respect to the duration of combustion of a grain.

When these two periods are comparable the combustion operates in three distinct phases; and it is easy to establish by the aid of formulas (19) and (20), that during the first phase, the function F (t) may be reduced to the form,

(24)

F(t)=(at +bt3 +...),

that is to say, it has its first term proportional to the square of the time, if each of the functions and have, following formula (23), their first term proportional to this variable.