involved in the Construction of Artillery. 207 



The time of one complete oscillation is 



tJ-;K=2. IC^)=2. /^^^+^)^' 



and 



ky-^"^\ 9 J~ A ff^^^ 



k' being 



9 \_ if 9'^^ 



l' + l")~>i\{P + P')Lp 



-J(''"-?^) 



k' and »•' corresponding to k and r in the equation of preceding cases. The 

 number of oscillations per second is therefore, 



T 27r 27Tyj\l'-l"J 27: y!\{P+P)L/ 



It may be shown that the entire elongation or compression of the bar I at any 

 instant of its motion t is given by the equation 



l^r + l"(l- cos k'T) + ^ sin k'T, 



128. It follows from the foregoing, that the period and time of oscilla- 

 tions produced in elastic bars, by longitudinal extension or compression are 

 independent of the intensity or the velocity of the impulse producing theui, 

 and depend upon the product of the cross section of the bar, multiplied into 

 the coefficient of elasticity of its material, upon the absolute length of the bar, 

 that oscillates, and upon the original strain of compression or tension, under 

 which it oscillates, — in all cases assuming the elasticity to remain perfect, and 

 that the strain remains constant for the whole of the oscillation. 



129. Professor Hodgkinson and others have shown that strains, however 

 feeble, produce some permanent elongation or compression in iron, and the same 

 is probably true for all materials. Perfect elasticity, therefore, is not found in 

 the nature of solids, and investigations founded upon its assumption can only be 

 taken as general guides in practice. Professor Hodgkinson's results (Trans. Brit. 

 Assoc. ) infer, that in the case of cannon, every discharge must permanently injure 



2 E 2 



