w al E E., being the value of E for a density of load ing unity, serves as a measure of the quality of the powder as to force or potential, q being a small fraction.
It is readily seen that the space in the powder chamber, c', unoccupied by the powder substance and lcnown as the "initial air space," is c' I ) in which is the density of the powder sub stance, or its "specific gravity," as it is other wise called. The value of 6 for all powders is about 1.6, being given for smokeless (nitro cellulose) powders, as 6 =--1 .644 .012k by Colonel Kisnemslcy, h being the content of moisture, expressed as a percentage.
When the projectile has advanced along the bore a distance uo, the volume of travel, au., corresponding to which is the initial air space, one "expansion" is said to have occurred.
The travel to maximum pressure is generally accepted as proportional to the travel, uo, corre sponding to the initial air space and called the "reduced length of the initial air space." The value of or of b will then contain 2 a factor A I As the pressure on the base of the bore is actually moving the mass of the projectile with a velocity v and the mass of the powder charge with a velocity less than that of the projectile it is doing work represented by 1 um2 1 w _ _ A:v2 2 g ' 2 g Or I WO + 2 g tv in which K lies between zero and unity. Since b is inversely proportional to the maximum pressure it follows that, if we are to consider pressures on the base of the bore (the ones measured), the factor 1 + aG) will occur in the denominator of b . Hence b will be of the form I b Q 1 K w The quantity Q , constant for a given gun, powder and projectile, is found by experiment to be of the form Q =- The exponent r is nearly constant and _has a mean value ;4 for widely varying conditions.
The coefficient S has a value dependent on the form and size of the grains and on the inherent speed of combustion of the substance of the powder. The value of b , then, talces the form A j b --= S ( ± ") The value of the constants entering a and are found from the values of a and b fol firings conducted with charges of varica weights. The data from such firings shook always include I. The muzzle velocity, V 2. The maximum pressure, )5 3. The length of the rifled bore, U 4. The weight of the projectile, w 5. The weight of the powder charge, 6. The volume of the chamber, c' 7. The total volume of the bore, C 8. The description of the powder includim the composition, content of moisture, grains, etc Then a U V b + U 16 m al p =.._
9 a b or 16 (C.-e) tna2 and b , v-(-- 1) so that 16( ) I b (C--e) U U m V' and I b V 9 P(C-c') b 9 (C-c') u / miTs U32' min U 2 The ratio of (C-c') to that of the work done by a constant pressure equal to the maximum pressure acting over a path equal to the total travel of the projectile, to the actual work done in conferring energy of translation on the projectile over the same path. That is, it is the ratio of the maximum pressure to the mean effective pressure represented by the energy 1 2 which is the muzzle energy.
Representing the mean effective pressure by p, and the velocity corresponding to a constant pressure equalling the maximum pressure by V. this ratio is 7; /V \ 1 p in k sno that = m and I , \ 2 \ b I 3 kr1 \q V) Placing 3 J, 8 V so that 9 P(C-c') 32 in which it follows that 2 + 1) whence , v Ls(1,1_2) and b = u V L2 (E,1_2)) a =-- V LI (L2_2)) Thus b and a are readily calculated for each shot fired. The values of a' and of b are calcu lated and recorded together with the data for the shot, the charge being varied.
To find the constants: 1 w as 2 g C.;and, E being calculated for each shot, E. and ci are found from the relation log E = log E.+ q log A By plotting this relation on cross-section paper log E. is found to be the value of log E sponding to A 1 for which log A =-- O. Then log (log E log Eo) by which means values of to q are found for different values of log A. and a mean value of log q is taken.
As regards b 1 b s + so that S ic \ ( A \ tv b kp ti and 1 -I- k S W = kw k bo The values of the second member are known for each value of (5, the value of 6 being about 1.6, but for nitrocellulose powders it is more exactly given by the formula involving the content of moisture. For powders used in the United States its mean value is about 1.58 corresponding to about 5 per cent moisture and volatiles. As the grain diminishes in size the percentage of moisture is less. The values of (I = S are plotted as a function of 0, and if the graph be produced to a point corre- spending to (:).--=o the value of 1 is the ordinate at that point. Both this graph and the preceding one should be generally taken as straight lines. Having drawn the straight line and knowing the value of S at any point the value of K and hence of c is easily found and the inverse problem is completely solved.