ON RIFLED GUNS. O 



the origin at A; AB as the axis y and AC as that of x; then will AG =y aud AP = se. Ami 

 because the arc AX and AG must, from Mi'' construction, bear to one another a constant ratio, 

 call this ratio m. and we hive 



V 



*' ■ ' mr. 



0— £)■ w 



xz=r — r cos y> .:= r I — cos 



EQUATION OF A HELIX OF THE GROOVE. 



(3) — Now wrap the developed surface around the cylinder of the bore; denote tho radius 

 of the latter by p. Conceive a plane through the axis of the gun to revolve about that line 

 and to start from a position in which it contains the element of the cylinder upon which the 

 groove starts at the bottom. Denote the variable angle which this plane makes with its 

 initial position by W; then will 



x=p W. 

 and Eq. (1) 



* ,.r=iYi-coB^) (2) 



whence 



(P w \ 

 1 1' 



and denoting the length of the groove in the direction of the axis by I, and recollecting that 

 for y = I we have r == p W, we get, 



l=mr. cos - " (1 — 1) =r J -. mr. 



or 



21 

 mr = —• 



Denote by n the ratio of the circumference of the bore to that portion of the same into which 

 the entire helix is projected, then will 



n 

 and these substituted in Eq. (2) give 



P- ¥ ~-P-\}~™Q*-j)}\ (3) 



// = constant J 



for the equations of a helix of the groove, with the origin on the axis of the piece; or 



j<> — constant. j 



(4) 

 P=t 



(4) — Differentiating Eq. (3), dividing by dt, and making 



dy 



=V= velocity of the projectile, 



(315) 



