TRACT 34<-

IN GUNNERY.

339

35. Taking this for granted then in the mean time,namely, that the effect of the charge of powder on therecoil of the gun, is the same either with or without ahall, it will be proper here to investigate a formula forcomputing the velocity of the ball from that recoil. Now,on the foregoing principle, if the chord of vibration befound for any charge without a ball, and then for the samecharge with a ball, the difference of those chords will beequal to the chord which is due to the motion of the ball.This follows front the property of a circle and a body de-scending along it,jriamely, that the velocity is always as thechord of the arc described in a semivibration.

Let then c denote this difference of the two chords, that isc = the chord of arc due to the ball’s velocity,

G = weight of the gun and iron stem, See.b — weight of the ball,

g ~ distance of the centre of gravity of G,o = distance of its centre of oscillation,

. n — its n*. of oscillations per minute,i — distance of the gun’s axis, or point of impact,r = radius of arc or chord c,v = velocity of the ball,

v = velocity of the gun, or of the axis of its bore.

Then, because bitv is the sum of the momenta of theball, and g gov the sum of the momenta of the gun ; andbecause action and re-action are equal, these two mustbe equal to each other, that is biiv= g gey. But, be-cause v is the velocity of the distance i, therefore by similarfigures i : o :: v : ~ the velocity of the centre of oscillation.And because tlie velocity of this centre, is equal to thevelocity generated by gravity, in descending perpendicu-larly through the height or versed sine of the arc describedby it,

and because 2 r : c :: c : ~ x= versed sine to radius r,and r ■■ o ”= vers.sine to radiiisu,

therefore vh : s/ :: 2 h : 2 ho, the velocity

z2