involved in the Construction of Artillery. 201 



equal to the work done in the opposite direction by the elastic resistance of the 

 bar, or 



P ^ -V PI" = \AeLr\ (28) 



V being the initial velocity. 



While the elasticity is perfect, the law of the oscillation is the same as in 

 the former case, but the extent of their excursions is greater. 



The static extension or compression, V, is also that at the moment of maxi- 

 mum velocity, 



V = jX, 



and the maximum extension or compression 



l" = l'+ h'^+Y^\ (29) 



or, replacing I' and k by their values, 



or for the unit of length, 



If P' be the effort capable of producing statically or permanently the maxi- 

 mum extension or compression (within the limit of perfect elasticity), we 

 have 



so that the excess of this over that of P, as well as the excess of the extension 



or compression jil'^ + p")' ^°*^ increase with the initial velocity % but more 



gradually in proportion as the actual length L of the bar is greater. 



In the second half of each complete oscillation, or that in the opposite direc- 

 tion to the commencing movement (whether due to extension or compression), 

 there will in this case be a contraction of the bar, or an extension beyond its 

 primary limits due to the reaction of the moving strain, acting in the reverse 

 direction at the termination of the oscillation ; this is given by the expression, 



