56S 



GUNNERY. 



GUNNERY. 



670 



It appears from the latter table that up to about 1400 yards the 

 ranges with different degrees of elevation might be calculated by the 

 parabolic theory from the ranges for 45, the initial velocities in the 

 above tables having been computed from the formula 

 2 h= range with an elevation of 45, 

 the times of flight from the formula 



Range 



T _ Range _ 



The theory of the motions of projectiles would therefore be very 

 simple if we might neglect the effects produced by the resistance of the 

 air during the flight of the shot ; but, in fact, when high charges of 

 powder are employed, the trajectory bears no resemblance to a para- 

 bolic curve, and can only be expressed by equations of a transcendental 

 nature. 



From the time of Galileo to that of Newton, though the subject of 

 the movement of projectiles occupied the attention of nearly every 

 mathematician in Europe, it seems to have been taken for granted that 

 the resistance of the air was too small to deserve much consideration. 

 Even Dr. Halley, while he admitted that its effects might become sensible 

 when the projectile was light, conceived that it would be of no 

 importance when the heavier kinds of shot were employed ; and it is 

 easy to conceive that the ideas then entertained of the form of the 

 trajectory were very wide of the truth. Huygens himself, from an 

 unfounded opinion that the resistance was proportional to the velocity 

 simply, asserted that the path of a shot through the air was a 

 logarithmic curve. 



It was reserved for Newton to develop the true laws of the resistance 

 experienced by bodies moving in fluid media, and to make a near 

 approach to the form of the curve described by a projectile in the air. 

 In a scholium to prop. 4 (' Principia," lib. ii.), he shows that such 

 resistance is proportional to the square of the velocity ; and elsewhere 

 he proves that, aeteris paribus, the resistance to globular bodies varies 

 as the squares of their diameters and as the density of the medium. 

 He also takes notice of the retardation which would be caused by 

 the condensation of the fluid in front of the body when the motion is 

 rapid, and of that produced in consequence of the air not being able 

 to fill up immediately the partial vacuum which exists behind the 

 ball during its flight. And, hi a scholium to prop. 10, he explains 

 that the curve described in a uniformly resitting medium is a species 

 of hyperbola having the asymptote of the descending branch in a 

 vertical position. 



The evidence afforded by the investigations and experiments of 

 Newton, concerning the effects produced by the resistance of the air, 

 induced a few mathematicians to adopt in their researches the principles 

 which he had established. Daniel Bernoulli appears to have been the 

 first who did so ; but, from an example in which he compares the 

 ascent of a cannon-ball in the ah- when projected vertically upwards, 

 with the height to which it would rise in vacuo with the same initial 

 velocity, he has manifestly estimated the resistance much too low. And 

 it was not till 1746, when Mr. Robins submitted a paper on the resist- 

 ance of the atmosphere to the Royal Society, that any correct idea was 

 formed of the enormous effect produced by this resistance, or sufficient 

 experiments made to arrive at any correct theory on the subject. The 

 methods employed by Mr. Robins were, for high velocities, the ballistic 

 pendulum, and for low velocities the whirling machine, which he 

 invented. As with low velocities the shot rebounds from the pendulum 

 block, which would therefore, from the elasticity of the shot and block, 

 not give correct results. For a description of this machine, see Robins' 

 ' New Principles of Gunnery,' and also Button's ' Tracts, No. 36,' Dr. 

 Hutton having employed the machine made by Robins. With the 

 ballistic pendulum, the gun being placed at different distances from 

 the pendulum block, and the velocities at the several distances deter- 

 mined by the result of the impact, the loss of velocity by passing 

 through the spaces was immediately determined, and from this the 

 amount of pressure due to the various velocities was determined. 



In the following investigation respecting the trajectory of a shot hi 

 ir, the line of motion is supposed to be hi a vertical plane, and the 



Fig. 3. 



resistance of the medium is supposed to vary proportionally to the 

 quare of the velocity at every point of the curve. 



Let A B' B fy. 3, be the curve, of which let M m be an indefinitely 

 small arc described in the unit of time (as one second) in consequence 

 of the projectile force ; then, if the force of gravity and the resistance 

 of the air were not to act on the shot, the latter might in the next 

 equal portion of time be supposed to describe the line mn in the 

 direction of a tangent to the curve at M. But, during this portion of 

 time, let the diminution of motion caused by the resistance of the air 

 be represented by n n' and the deflection produced by gravity be 

 represented by n' m' ; then m' will be the place of the shot at the end 

 of that portion of time. Draw the vertical lines M p, mp, n'p', nq ; 

 and the horizontal lines MS, n't. Let A.t=x, PM=)/, and the arc 

 =S / let also R represent the force of the air's resistance and g the 

 force of gravity (both forces being measured by the velocities which 

 they would produce, at the end of one second, in a body moving by 

 their impulses). 



Then, by the laws of motion, the velocities varying proportionally 

 to the forces and times of motion, we have B d t and g d t for the 

 resistance n n' and the force of descent n t during the evanescent 

 portion of time expressed by d t. And by the resolution of motions 

 nt 

 ~ R dt will express the diminution of velocity vertically in con- 



n't 

 sequence of the resistance, while K dt will express the horizontal 



diminution on the same account. But from the similar triangles Msm, 

 n'tn, we have 



nt dy 

 n'n : nt : : am : m : : as : au ; whence r~ = ~r. 



n n az 



n't dx 



Also n'n : n't : : iim : M : : dt ; dx : whence r~ = T- 



n it az 



Therefore the vertical diminution becomes 



R dy dt 



dz ' 



and the dimi- 



R dx dt 

 nution horizontally, -r . To the former adding gdt for the action 



of gravity, as above, we have, for the whole vertical diminution of 



R 1 1 ii dt 

 velocity, rfz + gdt. 



Now the vertical and horizontal velocities of the shot in vacuo, at M, 



dy dx 

 being represented by m and Ms ; that is, by -r and rr respectively : 



and, in the ascending branch of the trajectory the forces arising both 

 from gravity and the resistance of the air being retardative, the 

 velocities in the next second of time will be ; in the horizontal 



dx /dx\ dti / c fy\ 



direction jr d I -77 ), and in the vertical direction, -jr d( -37 ) : 



that is, the diminutions of velocity are, in the former direction, 



(dx \ / 'dy \ 



jj7 1 and, in the latter, d (^")- Consequently, 



B dx dt _ fdx-\ 



H dy dt 



(II). 



But the resistance experienced by a shot moving through the air is, 

 agreeably to the laws of hydrodynamics, represented by some part of 

 the weight of a column of the fluid, whose base is a section through 

 the shot perpendicular to the line of its motion, and whose height is 

 that space through which a body would descend in vacuo to acquire 

 the actual velocity of the shot. Therefore let A be the area of such 

 section, v the velocity of the shot, and h the height due to that velocity ; 



Tp u a 



and let D be the density of the ah- ; then h = -r- and 5- A . D = the 



weight of the column. Putting 2gp to represent some number which 

 is to be determined by experiment we shall have pv* A . D for the 

 resistance, or for the motion destroyed hi one second, arulpv* A . D dt 

 for the motion destroyed in the time dt. But, by dynamics, 

 momentum 



= velocity ; therefore, if M represent the mass of the 



shot, 



mass 

 pv* A . D dt _ 



is the velocity destroyed by the resistance in the time 



dt ; and this is what is expressed above by R dt : consequently we 



mp A . D A . D 1 



have B = ; or, representing by , and for v putting 



its value ^ we have R = ig. 

 (II) wUl become 

 dx dz 



Then the' general equations (I) and 



ldx\ dy dz fdy\ 



~ ^ \d*J an< ^ ~H~df*y ~ ^\"dt) ' k ut <& being considered 



sdt' 



as constant, they may be put in the form 



dx dz 



= d'x, and 



