small fraction of the first. The formula for initial velocity then becomes 32. We might be led to neglect, in the above value of £,, the terms containing 2 and those following, as has been done in substituting the approximate value (58) for the value of v in (57). But the value of €, would then be too small in guns where the ratio has a large value. 20 We shall satisfy, however, experimental facts if we use (63) and the approximation of (58). It appears that the agreement of theory and practice requires that, in place of (57) where the value of v is too small, we should use (58), which is greater than the exact square root. 33. Formula of initial velocities.-It only remains, to obtain our final formula, to substitute in (61), for A and b, their values, (45) and (47). Putting then, f is the force of the powder, the weight of the charge, m the mass of the projectile, u the space passed over by the projectile, a, λ, are defined by the relation, y=wat (1+it+...), which repre sents the weight of powder burned after the time t. €, is given by (63), z, is given by (32), u, being the reduced length of the chamber, 4 the density of loading, the density of the grains, P and Q are given by (64), and 0= N-I 2 ›n being the ratio of the two specific heats of the products of combustion. 34. Influence of the mode of combustion.—For each particular mode of combustion there is a corresponding form of (35), and consequently system of values of a, λ, .., which we have only to substitute in (65) to derive the theoretical results. We shall return to this subject, only examining now the case in which the grain is a sphere, or a polyhedron in which a sphere may be inscribed. 35. Case of spherical grains.—When the grain is a sphere, or a polyhedron which may be inscribed in a sphere, the combustion is represented by the formula T being the time of burning of a grain. We have consequently and the various formulæ deduced in this chapter are reduced to the following: ... Ist. Coefficients b, c, . . . of u1. The coefficients (47) of the series (41) become These values are also necessary to calculate the coefficients of the series (50) and (53). 2d. Formula of velocity.-In this case, (65) becomes it is this form that we shall frequently apply in our researches. But before discussing the results which we have obtained, and examining how far they are in accord with experimental facts, it is necessary to correct them for the cooling effect of the walls of the gun. The next chapter will be devoted to this point. CHAPTER IV. ON THE EFFECT OF THE COOLING OF THE POWDER GASES BY THE INTERIor Wall of the Gun. 36. Hitherto, works on interior ballistics have not taken into account, to our knowledge, the effects of the cooling of the powder gases by the walls of the bore (1875). This cooling cannot be neglected; in fact, M. de Saint Robert shows experimentally, that by firing a gun the heat absorbed by the piece is a very notable fraction, a fourth for example, of the heat developed by the combustion of the charge. It is then inexact to suppose that the total heat of combustion is transformed into work, and we shall proceed to show how we must, in consequence, modify the general equation for the movement of the projectile which we have established in the first chapter. 37. The fundamental relation of this analysis is equation (2), which is applied, during an infinitely small period of the expansion, to the thermodynamic transformation of the gases already formed. It expresses that the heat dq absorbed by these gases is equivalent to the alteration of their sensible heat, increased by the exterior work accomplished. When there is no heat lost the quantity dq is equal to the heat set free by the gaseous products during the infinitely small period dt of the transformation. It is then given by formula (1). But if the walls of the bore absorb heat, it becomes necessary to deduce from (1) the quantity absorbed. This quantity may be represented by hadt, a being the enveloping surface of the volume occupied by the gas, and h the velocity of flowing of the heat for a unit of surface of the walls of the bore. The following value then follows: (69) dq=c(T-T)dy—hodt, which should be substituted for formula (1), and equation (2) then becomes and integrating, since to, we obtain the equation: 38. Introducing in (71) the pressure and the volume V of the gas, we obtain the new equation: (72) R p V + (n−1) ¤ = ƒy — hadt, which corresponds to equation (7) of the first chapter. We now show how the equation of the movement of the projectile is deduced. The surface is the total surface of a cylinder having for a height the distance u+u, of the projectile from the bottom of the bore. The surfaces of the two bases of this cylinder may be neglected, especially in the case of very long guns, in which the cooling of the gas has most influence; calling 2r the diameter of bore or the calibre of the gun, we have: in cases in which the capacity of the bore is great with respect to that of the chamber. The coefficient h is a function of the excess of the temperature of the gases over that of the wall of the bore. It varies also with the movement of the projectile, but we can substitute for it a constant approximate mean value, consequently equation (72) should be: and it suffices to operate as in No. 5 to obtain the equation of the movement of the projectile. 39. By putting as heretofore R = с f q being heat of combustion of the powder; in fact, calling T, and the temperature and the heat of combustion of the powder, 9 and ƒ the force of the powder, we have f= RT, and q=c T,, so that In order to integrate this equation (73) we will neglect z, following the method of approximation of No. 23 we will obtain also, corrected by taking account of the cooling, the auxiliary integral designated by u1, and we will afterwards deduce by the formulas of No. 30 the definite integral of the problem. The equation (73) contains the unknown function u under the integral sign. In order to evade this difficulty, it would suffice to differentiate with respect to f, which would give a differential equation of the third order; but it is more simple to operate by successive approximation and to integrate (73) after having replaced u under the integral sign by the value u1, which may be integrated because we neglect the additional term. We have then to integrate the equation 2=rhf Sudt in which the second term of the second member is still an unknown which is linear with respect to u and its derivatives. This equation is easily integrated when we ta the first term of its development, (41). We have t |