204 Mr. Mallet on the Physical Conditions 



V v' vf 



F= M - = M' — and M — , and Mv being always = Mv', during the range 



of compression, 



F-P = M''"-^, F + P-eAi = M- (43) 



t z 



Where the bodies are very compressible, so as considerably to diminish 



the intensity of F in the directions + or — according as P acts with or against 



v' 

 P' substituting the value oiF=P' + M' - as above, we have, 



PArM-^P-eAi = M-, or {P + P~eAi)t = Mv - M'v'. (44) 



If on the contrary the masses il/, M' are such that their mutual compressions 

 during the time of impulse, may be considered insensible with relation to I, 

 which will always be the case when the masses are large, and their range of 

 elasticity small (hardness great), and the length and extensibility of the bar 

 relatively great, — then, as 



F=Mv = M'v', 



M and M' assume the common velocity F", and 



F" = p^F', (45) 



at the first instant of extension or compression of their bar, and their work upon 

 the bar is {M+M') V"-M'V\ or the initial vis viva, 



M' T" 9 7"' 77' 



^M^M')V'-^j^,Mr^ = p^M'V- = f^^-, (46) 



and the corresponding loss of eifect (as M and M') are supposed to remain in 

 contact, 



^ ■,M!V'^ = -;^MV'\ (47) 



M-\-M' P+F 



Lastly, if the mass M, instead of being at rest, had a previous velocity F'", 

 either in the same or in the opposite direction to V (and to which a given 



