involved in the Construction of Artillery. 203 



that we extend or compress it by a further static strain, and then suddenly 

 relieve it of this latter, tlie train of phenomena produced will be precisely the 

 same as in this last case, and the same formuloi will represent them. 



And it is further obvious, that if the inertia of the bar itself be so consider- 

 able as to be taken into account, similar phenomena will, upon its being slowly 

 extended or compressed and then at once let go, present themselves. To express 

 these, and many other varieties of condition of this problem, that will at once 

 suggest themselves, some modifications of the formula would be needed. These 

 questions, however, are of less practical value. 



125. Case III. — Where there is a permanent strain upon the bar, and it 

 is also subjected to an impulse from a new strain having an initial velocity, it 

 follows from the laws of impact, that if the strain and the impulse be both due to 

 solid bodies, the first, forming part of the bur (or even being the bar itself), and 

 the second a solid striking it with a determinate velocity, then the common 

 velocity after contact will depend upon the elasticity and range of the striking 

 bodies. After several oscillations, and the system has come to rest, w^ have 

 for equilibrium, 



I'-^ + h^ (41) 



p being the weight of the unit in length of the bar. 



Let P be the permanent strain, and P' the impulsive, with the velocity V 

 due to H', so that V = y(2cill'). Then, on the principles already established 

 the maximum extension would be that due to the descent of P + P' through 

 S', and the equation of work done by the strain, and by the bar, would be, 



l^V^eAi, (42) 



if no change of form occur in P and P", and the inertia of the bar be ne- 

 glected. 



The loss of vis viva will depend upon the degrees of compression of P and 

 P' and upon the extensibility of the bar L, the compression being greatest if 

 the extensibility was = 0, and least, if P was free to move under the effect of P' 

 without restraint from the bar, or its resistance = 0. 



If V and v' be the velocities lost and gained in the infinitely short time f, 

 and F= the variable force of reaction, Mand M' being the masses of P, P', 



