4 ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 



is subject are such that certain functions of the coordinates cannot 

 exceed certain limits, either constant or variable with the time. If 

 certain values of Sx, Si/, Sz (with unvaried values of x, y, z } and x, y, z) 

 are simultaneously possible at a given instant, equal or proportional 

 values with the same signs must be possible for Sx, Sy, Sz immediately 

 after the instant considered, and must satisfy formula (1), and there- 

 fore (6), in connection with the values of x, y, z, X, Y, Z immediately 

 after that instant. The values of x, y, z, thus determined, are of 

 course the very quantities which we wish to obtain, since the accelera- 

 tion of a point at a given instant does not denote anything different 

 from its acceleration immediately after that instant. 



For an example of a somewhat different class of cases, we may 

 suppose that in a system, otherwise free, x cannot have a negative 

 value. Such a condition does not seem to affect the possible values 

 of Sx, as naturally interpreted in a dynamical problem. Yet, if we 

 should regard the value of Sx in (7) as arbitrary, we should obtain 



X 



x = , 

 m 



which might be erroneous. But if we regard Sx as expressing a 

 velocity of which the system, if at rest, would be capable (which is 

 not a natural signification of the expression), we should have Sx ~ 0, 

 which, with (7), gives v 



*==-. 



~m 



This is not incorrect, but it leaves the acceleration undetermined. 

 If we should regard Sx as denoting such a variation of the velocity 

 as is possible for the system when it has its given velocity (this also 

 is not a natural signification of the expression), formula (7) would 

 give the correct value of x except when ae = 0. In this case (which 

 cannot be regarded as exceptional in a problem of this kind), we 

 should have Sx ^ 0, which will leave x undetermined, as before. 



The application of formula (6), in problems of this kind, presents 

 no difficulty. From the condition 



z^O, 



we obtain, first, if x = 0, x = 0, 



then, if x = and x = 0, Sx ^ 0, 



which is the only limitation on the value of Sx. With this condition, 



we deduce from (6) that either 



X 



x = , x = , and x > ; 



~m 



X 



or x = . 



m 



