involved in the Construction of Artillery. 20.') 



extension, or compression I, of the bar was due) : then the common velocity, 



as before, 



y„ _ M'V'±3IV"' _ P'V'± PV"' 



and the initial vis viva, 



(M+ M') V- - iMV'±MV"J _ {Fv^r vy 



126. We have now to determine the conditions of the first oscillation, for if 

 the elasticity of the bar then remain uninjured, it is subjected to no greater strain 

 by any subsequent one. Half the vis viva., plus tlie work done by {M-\- M'), 

 acting through the height due to the extension of the bar from the moment of 

 impulse, /', must be equal to 



PT 

 \Pl' = ^P-^=leAM; (50) 



and this, if the elasticity of the bar remain uninjured, must not exceed the 

 value of 



T, = TJAL, Z being = ^eALP, (51) 



or if it be just not broken, must not exceed that of 



Tr=T;AL. (52) 



The maximum extension, assuming the former, and that P has no initial velocity, 

 L' being the length of the bar when most elongated or compressed, and / the 

 proportional extension or compression corresponding, is 



(P + F')~ + {P+P') {I-i')L + yALv = leAP, (53) 



2L 



which may take a simpler form, for, multiplying all the terms by — -r-, we 



have 



V P I' PL 



L'^LI, P^eAi'^eAj^, or-^ = ^' = -^, -^^l'; 



and if i" be the proportional extension or compression, and I" the actual, pro- 

 duced in the bar, by a static strain equal to P', the impulsive body, by analogy, 



eA~^ ~ r eA ~ * ' 



VOL. XXIII. 2 E 



