values of K are deduced which differ but little from each other, and which give a remarkable verification of formula (18). 14. These values may be calculated in two ways: 1st. By supposing, as in Art. 9, a1 for the standard powder. We find K to be: 2d. By supposing, as in Art. 10, a=1331 for the standard powder. Under this hypothesis, we are led to divide by the square of a, the preceding value of K, which gives for the new value of K 15. For reasons given in Art. 10, we are led to prefer and to adopt formula (22). In the preceding formulæ for pressures as in those for velocities the units chosen are the decimeter, the kilogram, and the second. It is necessary then to divide the results of formula (22) by 100, if we wish, as is customary, to express the pressure in kilograms per square centimeter. CHAPTER II. APPLICATIONS AND NUMERICAL VERIFICATIONS. 16. In this chapter we shall show how by the preceding formula we may determine the characteristics of various powders, and make use of these determinations to calculate à priori the velocities and pressures in any arm whatever. The results will be compared with the results of direct measurement. 17. Experimental determination of the characteristics.-For the same powder, characterized by a system of values a, P, the initial velocity depends upon the five quantities, c, u, p, t, s, and is connected with them by (15). It is evident from the form of this equation, that a and will become known from two velocities determined by experiment with different systems of particular values of the variables, provided that one at least of the first three shall have different values in the two experiments. Put for shortness, ய ய I (23) 1.569 (*) (pu) } с • (# ) ' ( * ) s31⁄23 = &' 0.00795 =Y; equation (15) may then be written X Xva-aß Y. Calling X, Y; X, Y,, two systems of values of X and Y, and v1 and v, the corresponding measured velocities, we have from which aẞ and a may be found by the relations, 18. On the error committed in the determination of the characteristics. It is easy to determine the effect of an error in the measurement of the velocities upon the determination of the characteristics. Consider the values of aß and ɑ in the form Let be the absolute value of the greatest error which is to be apprehended in the measurement of the velocities. The error corresponding in aß and a is a maximum when the error affects v, and v, in opposite senses. We have, consequently, the two limits, It is evident from these The first concerns aẞ, and the second a. values that Y-Y, should be made as large as possible by the choice of proper conditions of fire. 19. Determination of the characteristics by firing guns of 14 and 24 centimeters.-Firing these guns, in the conditions specified in No. 9, will answer very well for this determination. The numerical values of X and Y, calculated by (23), making use of the elements in the table of No. 9, are as follows: in which e, as well as the velocities, should be expressed in decimeters. The following table shows the results of determinations made by this method upon various lots of powder tried by the committee at Gavre. These powders were all mixed in the proportions used in England (75 saltpetre, 10 sulphur, and 15 charcoal); and were fabricated with meules et presses. 20. Determination of the characteristics by firing with different projectiles. In the absence of guns of different calibres the characteristics may be determined by firing different projectiles in the same piece. We might, for example, measure the velocities of the 96 and 144 kilo. projectiles from the 24 cent. gun, the other elements being as defined in No. 9. We have in this case, these are less favorable to accuracy than in the first method. 21. Approximate determination of the force and velocity of combustion of powder.-When the form of the grain is well defined, a and À may be determined from (9). We may then determine ƒ and τ, after the determination of a and B, by the formulæ The time of combustion is the time in free air. We have for the corresponding velocity of combustion, e being the least dimension of the grain. Let us apply this to the powders treated of in No. 19. Their grain is very nearly a parallelopipedon with square base. Their least dimension (thickness) is exactly regulated by the caking, and can be directly measured. The ratio of the other dimensions to the last can be calculated as a function: Ist. Of the thickness e; 2d. Of the number of grains in a kilogram N; and the coefficients a and by the relations (Part III, No. 41), The quantities e, N, and 6 for the powders considered are known, and the characteristics having been determined, we may calculate f and w. The following table gives the elements and results of the calculation: A,B .128 105 1.810 .631 2.262 .734 534 424000 .120 A,S,, No. 1 .130 116 1.776 .673 2.346 .767 .524 457000 .124 No. 2 .130 114 1.794 .670 2.340 .765 537 451000 .121 No. 3 .130 113 1.805 .670 2.340 .765 .560 442000 .116 No. 4 .130 107 1.807 .652 2.304 .750 .545 410000 .119 No. 5 .130 104 1.813 .644 2.228 739 .846 424000 .076 22. Calculation of the characteristics of a powder from its physical properties.-The force of a powder and the velocity of its combustion for a given density depend exclusively upon the process of its manufacture. If, then, these have been determined for a particular powder, they may be used with any other powder of the same fabrication, however it may differ as to form and dimensions of grain. When the density of the powder changes, we may calculate the velocity of its combustion approximately by assuming, as indicated by Piobert's experiments, that the velocity varies inversely as the density. We may thus, for all powders of a given fabrication, determine à priori the approximate values of ƒ and w. From an examination of the form and dimensions of the grain, the values of 7, a and À, may then be determined by (27) and (29) of the preceding paragraph. We have thus all the elements upon which the characteristics depend (9). 22. Characteristics of some common powders.-The following table gives the results of the application of the methods stated to some common powders : W(20-25) 110 1.800 .0972 431000 .172 1.000 3.000 1.000 .885 1209 1.130 SP1 .792 2.584 734 534 1340 1.376 .856 .412 1630 2.076 .750 2.500 .825 .324 1809 2.545 .943 .261 2147 3.612 Remarks. The powders marked W were from the Wetteren factory. The value f=431000 is adopted for them; they are described in No. 9. The velocities of combustion for them are calculated by the formula the constant being so taken as to give for 1.750 the value w=.100. The grains of the powder W(20-25) are considered as cubes; those of the three others being very irregular, were taken as spheres. The fabrication of the four other powders in the table is the same as that of the powder A,B in No. 21. Consequently, the value. ƒ=424000 was taken for them; and their velocities of combustion were calculated by the formula |