208 Mr. Mallet on the Physical Conditions 



the gun, though for a great number of rounds, perhaps, quite imperceptibly ; 

 but the injury effected by each successive discharge goes on in an increasing 

 ratio, and after a certain number, more or less, must end in the destruction of 

 the gun; — yet that when the value of the constant Te is high, and the surplus of 

 strength to the impulse large, as in the case of the forged wrought-iron barrels of 

 small arms, this process of gradual destruction by use is inconceivably prolonged, 

 as evidenced by the endurance of old firelocks and fowling pieces, from many of 

 which, thousands of shots have been fired without perceptible deterioration. 



130. In all the foregoing equations, the metallic bar has been supposed 

 a straight imiform prism, fixed at one end, and loaded or otherwise strained (in 

 extension or compression) at the other. But if for L we substitute its value, 

 27r7?, in assuming the bar bent round and united at its extremities into a ring, 

 to form a portion of the cylinder of a gun, all the equations will equally apply 

 to the tangential strains, while they apply directly to the longitudinal ones. 

 If the value of J. be assumed small, i. e. aunit of section of the whole thickness 

 of the gun, then 2i? may be taken = the caliber, for the portion of metal exposed 

 to the greatest strain; but if the whole thickness of the gun be included in A, 

 and for a determinate length along its axis, then the value of R must be taken 

 at an intermediate point between D' and D" (Eq. 1), to be specially ascer- 

 tained, where the mean of the tangential strain, variable in the radius, shall be 

 situated. 



131. In fact it is plain that a unit in length and thickness of the gun may be 

 viewed as an elastic hoop, expanding and contracting under impulse, the maxi- . 

 mum elongation being measured from an imaginary point of origin taken any- 

 where in its circumference ; and, as we found (Eq. 20, 22) that the extension or 

 compression due to P, when suddenly applied, is = 2i, so, in calculating the 

 stress upon the unit of section of the metal of the gun, we must take the exten- 

 sion or compression as double that due to the maximum pressure of the elastic 

 gases per square inch, and only give such a value to R (in Eq. 1) as will satisfy 

 this condition, without crippling the particular metal to which we may apply 

 our calculations. It would appear to be from ignorance of this, and the reliance 

 upon conclusions as to extension or compression of the materials, based upon 

 the assumption of merely statical strains, that many failures of newly projected 

 guns, have latterly occurred. 



