ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 5 



x 



That is, if x = 0, x has the greater of the values -- and 0; otherwise, 

 X 



<JU 



m 



In cases of this kind also, in which the function which cannot 

 exceed a certain value involves the velocities (with or without the 

 coordinates), one may easily convince himself that formula (6) is 

 always valid, and always sufficient to determine the accelerations 

 with the aid of the conditions afforded by the constraints of the 

 system. 



But instead of examining such cases in detail, we shall proceed to 

 consider the subject from a more general point of view. 



Comparison of the New Formula with the Statical Principle of 

 Virtual Velocities. Case of Discontinuous Changes of Velocity. 



Formula (1) has so far served as a point of departure. The general 

 validity of this, the received form of the indeterminate equation of 

 motion, being assumed, it has been shown that formula (6) will be 

 valid and sufficient, even in cases in which both (1) and (7) fail. We 

 now proceed to show that the statical principle of virtual velocities, 

 when its real signification is carefully considered, leads directly to 

 formula (6), or to an analogous formula for the determination of the 

 discontinuous changes of velocity, when such occur. This will be the 

 case even if we start with the usual analytical expression of the 

 principle 2 (Z ^ + Y Sy + Z8z) ^ 0, (8) 



to which, at first sight, formula (6) appears less closely related than 

 (7). For the variations of the coordinates in this formula must be 

 regarded as relating to differences between the configuration which 

 the system has at a certain time, and which it will continue to have 

 in case of equilibrium, and some other configuration which the system 

 might be supposed to have at some subsequent time. These temporal 

 relations are not indicated explicitly in the notation, and should not 

 be, since the statical problem does not involve the time in any 

 quantitative manner. But in a dynamical problem, in which we take 

 account of the time, it is hardly natural to use Sx, Sy, Sz in the same 

 sense. In any problem in which x, y, z are regarded as functions 

 of the time, Sx, Sy, Sz are naturally understood to relate to differences 

 between the configuration which the system has at a certain time, 

 and some other configuration which it might (conceivably) have had 

 at that time instead of that which it actually had. 



Now when we suppose a point to have a certain position, specified 

 by x, y, 0, at a certain time, its position at that time is no longer a 

 subject of hypothesis or of question. It is its future positions which 



