673 



GUNNERY. 



GUNNERY. 



574 



power may be represented by g, the force of gravity on a body in 

 vacuo. 



By differentiating the equation (XI), making dy=o, and, from the 

 resulting equation, obtaining the value of x ; then, on substituting this 

 value in equation (XI), the resulting value of y would be that of the 

 greatest vertical ordinate of the curve, while the said value of x is the 

 corresponding abscissa. 



What is called the point blank range is the distance from a point on 

 the ground, vertically under the chamber of the gun or howitzer, to 

 the point at which the shot strikes the ground after the discharge, the 

 axis of the bore being supposed to be in a horizontal position, or parallel 

 to the ground if the latter should be inclined to the horizon. The 

 extent of such range may be determined from the equation (XI), by 

 making E = o and considering y as negative. In this case the said 

 equation becomes, b being equal to unity, 



where y is the height of the gun above the level of the Bpot on which 

 the shot falls. And the equation, after reduction, 



Zx 



log. e = log. - 



in which, substituting for x different assumed values, that which renders 

 the two member^equal to each other will be the required range. 



The but en blanc of the French is frequently called the line of metal 

 range, and signifies the distance from the chamber of the gun to the 

 point where the trajectory of the shot crosses (the second time) a line 

 joining the tops of the base, and muzzle rings, and produced. Here 

 the axis of the gun, which is always a tangent to the trajectory at the 

 nearest extremity, makes a small angle with the said line, depending on 

 the dispart, or the difference between the semi-diameters of the gun at 

 the base and muzzle. 



Now, to find the time of flight : from the equation (IV) ; by re- 

 ducing the logarithm to a series and proceeding as before, we shall 

 have 



2bt 

 and having found, from the equation (X), that c . , is equal to 



2Kr 



4hc H os> E e H i ** bx 



Erf* 1 



dxe 



b 



H 



which being integrated gives t = e ' b eog g ~^ h +C, where c is an 



arbitrary constant. 



Now, to obtain c; since t (the time of flight) = o when;t = o, by 

 substituting those values in the equation, the latter becomes 



bcos E 



0; whencel - 

 bx 



-1). 



and consequently^ b cog 



From which equation, on substituting the value of x (the horizontal 

 range), which is supposed to be given, the time ( of the flight of the 

 projectile will be obtained. 



It may he observed here that a knowledge of the time during which 

 a (hell will describe ita trajectory is of great importance, since it enables 

 the gunner so to regulate the length of the fuse, that the shell may 

 explode nearly at the moment that it has reached the object which it 

 is intended to destroy. 



Note. In determing the trajectory, the range, and the time of flight, 

 by the above formulae, the operations must be performed by the aid of 

 logarithms. 



It is of the first importance, in obtaining from the above formulae a 

 near approximation to the required values, that a correct measure of 

 the velocity with which a shot issues from the mouth of a gun should 

 be obtained ; and the determination of such velocity, when the charge 

 of powder is given, is the object of the following investigation, which, 

 ting the differential notation, ia taken from the third volume of Dr. 

 Button's Tract*. 



Let r = the semi-diameter of the shot, or of the bore. 

 D'= the specific gravity of the shot. 

 = 3'1416 (the ratio of the circumference of a circle to its- 



diameter). 

 0=32Jfeet. 



m = 33120 oz. (the prefigure of the atmosphere on 1 square foot). 

 ic = the weight of the shot. 



o=the distance from the bottom of the chamber to the hinder 

 part of the ball. 



b = the length of the bore. 



n the ratio of the expansive force of fired gunpowder to the 



pressure of the atmosphere. 

 v = the velocity of the shot on leaving the gun. 

 x= any variable distance of the shot, in the barrel, from the 



bottom of the chamber. 

 Then r = ir=the area of a transverse section through the bore, or of a 



section through the shot ; and 

 mi'%=the force of the powder on the hinder part of the ball. 



But the expansive force of powder being supposed to be inversely 

 proportional to its density, or to the space which it occupies in the 

 barrel, we have 



: : : mnr^ir : ( = the motive force of the powder at any 



ax x 



point, in the barrel, whose distance from the bottom of the chamber = x). 

 Consequently, dividing this term by the weight of the shot, we have 



for the accelerative force of the powder on the shot at that 



point ; from this term subtracting mr T , which expresses the retarda- 



w 



tion arising from the pressure of the atmosphere against the front of 

 the ball while moving in the barrel, the accelerative force becomes 



n ^(_ A. Jet this be represented by /. Now, by the theory of 



w ^x S 

 forces, we have 



vdv [=>jfdx] ^'i^C^te-dx)-, and the integral of this equation is 



10 ^* X S 



gmr** 



(no hyp. log. xx) + u ; 



where c is an arbitrary constant. 



To find this constant ; since v = o when x = a, on substituting 

 these values the equation becomes 



(na hyp. log. a a) + c ; 



from which c being found and substituted in the preceding equation, 

 the complete integral becomes after reduction, and substituting b 

 for x, 



b 



or substituting for w its value, namely, $ r>ir D', we obtain for the 

 velocity of the shot on leaving the gun, 



1783 // b 



-y-Z'tJ (na hyp.log. - + - 



On comparing the results of the formula with those obtained from 

 experiments made with the Ballistic pendulum, Dr. Hutton found 

 that the expansive force of powder varies, with the quantity employed, 

 from 1778 times to 2300 times the pressure of the atmosphere ; allow- 

 ance being made for the loss of force occasioned by the vent and 

 by windage. Those numbers express the values of n in the formula. 



From the numerous experiments made with the machine above 

 mentioned between the years 1784 and 1791, Dr. Hutton concludes 

 that the initial velocities of shot are directly proportional to the square 

 roots of the weights of the charges, and inversely proportional to the 

 square roots of the weights of the shot (the guns being similar to each 

 other) ; and he gives for the initial velocity in feet the formula 



V2c 

 ; where c is the weight of the charge, and w that 



of the shot. Dr. Gregory's formula, founded on more recent experi- 



V3c 

 , which with reduced windage is more 



nearly correct. It must be admitted however that some uncertainty 

 still exists respecting the value of v, partly on account of variations 

 in the quality of the powder, and partly in consequence of the 

 different degrees of windage ; and these are the chief causes of the 

 want of agreement between the experimented and calculated ranges of 

 shot. This however is not in general greater than that which has been 

 observed between ranges obtained from different trials when made in 

 like circumstances. 



The resistances actually experienced by a shot in passing through 

 the air were, in 1789, made the subjects of experiments with the 

 Ballistic pendulum and the whirling machine as before mentioned, and 

 are described in Hutton's Tracts. The resistances were determined 

 from the general formula, 



cv' 



r = W^' 



Where r = the required resistance in pounds or ounces. 



v = the mean between two velocities, namely, the first 

 velocity and the velocity with which the ball strikes the 

 pendulum. 



v 1 = the difference between these velocities. 

 = space passed through. 

 w = weight of the body in pounds or ounces. 



