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PROCEEDINGS OF THE AMERICAN ACADEMY. 



tively a and b before collision, and a' and b' after collision. Several 

 important laws are subsumed under a law which we may call the law 

 of conservation of extended momentum, namely, 



a + b = a' + b'. (17) 



Assume any set of space-time axes, and write 



a = aiki + a4k4, b = fe]ki + b^, 



a' = fl/ki + a^'ki, b' = fe/ki + Wi^. 



Then the law states that 



(a, + b:) ki + (a, + h,) k, = (a,' + 6/) ki + {a^' + b,') k,, 

 or 



ai + 6i = a/ + 6/, (18) 



a4 + &4 = cii + W- (19) 



Now (by § 21) tti and 64 are the masses of the two particles before 

 collisioh, a/, b/ the masses after collision, and equation (19) expresses 

 the law of conservation of mass or energy. The components ai, b\, 

 tti, bi, are the respective momenta (in the ordinary sense), and equa- 

 tion (18) is the law of conservation of momentum. 



To assume that the impact is elastic is equivalent to assuming that 

 the value of viq for each particle is unchanged by the collision; and 

 since each value of m^ is the magnitude of the corresponding vector 

 of extended momentum, the assumption may be expressed in the 

 equations 



a — a , b — b'. 



The condition that the extended momentum 

 is unchanged gives 



(a+ b).(a+b) = (a' + b')-(a'+ b'), 



or a-b = a''b' 



by the above relations. Hence it follows 

 (Figure 17) that 



cosh 4> = cosh ^', or </> = 4>', 



■n, -,„ as is evident from the rules of projection 



Figure 1/. .,,, it-i , 



previously deduced. It is thus seen that 



the relative velocity is the same before and after collision, and tliereby 



a rule which has been found very useful in the discussion of simple 



