ON RIFLED GUNS. 25 



K* 



[<^ + Se^=?Py (S) 



T. W "/' OX./' , . 



or 



Sp _ S P. /> 2 



/j 7r" c 1 + 2 c ^ 



The first member is the tensile strain upon a unit of surface of a section through the 

 axis, each element of the surface having the same strain and equal to that on a cross section 

 of the circular filament in the surface of the bore. 



(14)— Solving with respect to P, 



1 M 0?. c ( c + 2 '") 

 V , p* 



r.=i-M.^.-^^ (io) 



which gives the internal pressure just sufficient to produce the tensile strain M. — . 



(15) — Solving with respect to dp, we have 



8 P p 3 



dp =*-M:*+2p-c (11) 



which gives the enlargement of the bore. 



(1G) — Dividing both members of Eq. (9) by P. there will result, 



M '7_s , 



P. tt'c 2 + 2 P c ^ 



which gives a direct relation between the tension of the gas, as measured by its pressure on 

 unit of surface of the bore, and the tensile strain upon the material of the gun on an equal 

 extent of section through the axis. This for the same gun is constant. 

 (17) — Solving Eq. (9) with respect to c, we find 



e^ i+ ---z5) <i3) 



This will give the thickness necessary to resist the pressure P. which develops the 



tensile strain M — -, the radius of the bore being p, and the upper sign being taken to satisfy 



the inequality (2). 



(18)— On the other hand, if the powder be excessively quick, Yt would become com- 

 paratively small. Let us, for illustration, suppose it equal to c, which would bring the wave 

 front only to the outer surface of the gun when the gas has its greatest action; then would 



4 

 ip = 0, and <fi — — 2 



and Eq. (7) would become, writing c x for c, 



M.^.[d-(.+ tl r-n^-i^]='-i > -^ 



or 



44 ( 337 ) 



