24 ON RIFLED GUNS. 



But Q r and Q p must be equal. Hence after omitting the common factors and multiplying by 

 p, we have 



M. S j [i (p + c)*-i p* + i (p + c). ^. sm (.. ^7) - ( -jy. sin 2 (l *. y,)] = J- f. P (4) 



(9) — Make ^ the variable ratio of the sine to its arc; that is, 



(^) 



c 

 Vt 



(5) 



y '= Y ^V - («) 



v ' v*y 



and we have, 



M. ^ [i (/> + of - i /, 2 + i- (/> +c) c. <p - i u v] = i. , "P. (7) 



(10) — The least value possible for y is zero, and this will occur when 



or when the molecular disturbance only reaches the outer surface of the gun at the instant 

 of greatest action on the bore, in which case the outer layer of molecules will afford no aid 

 whatever. 



If 



c 



Vt> 1 



then the disturbance would fall short of the outer surface at the instant of greatest action, 

 and all that part of the gun beyond the wave front would be useless. 



(11) — The greatest value for <p is unity, and this will happen when the arc is so small that 

 the sine may lie taken equal to the arc; that is, when the molecular disturbance moves with 

 great rapidity and the powder burns very slowly. 



(12) — The velocity of sound through cast iron is about 18,673 feet a second; and it is 



stated in Benton's* Ordnance and Gunnery, page 48, that a grain of powder of a particular 



kind, and having a diameter of 0,056 inches, will burn up in 0,056 of a second. Making, 



therefore, 



V=18673 f 



t = 0,056 s 



Vt = 1045 f .7 

 we find 



Vt = 1045 f .7 



^- = 0,003. 

 Vt 



(13) — So that in ordinary practice <p and <p\ may be taken as equal to unity, and Eq. (7) 



becomes 



(330) 



