480 Dynamical Theory of Currents [OH. xvi 



coincide with those in the actual motion at the instants t = and t r. 

 Thus the values of 80! , 80 2 , ... at every instant may be any we please which 

 are permitted by the mechanism of the system, except that they must be 

 continuous functions of t and must vanish when t = and when t = r. Whatever 

 series of values we assign to 80 l} &0 2 , ..., we have seen that the equation 



Idt = 



o 



is true. Hence the value of / must vanish at every instant, and we must 

 have 



(499). 



, dt\d0 



549. At this stage there are two alternatives to be considered. It may 

 be that whatever values are assigned to &0 l} 80 2 , ... 80 n , the new configura- 

 tion 0! + &6 l , 2 + $2> @n + 80 n , will be a possible configuration that is to 

 say, will be one in which the system can be placed without violating the 

 constraints imposed by the mechanism of the system. In this case equation 

 (499) must be true for all values of 8^, e># 2 , 80 n , so that each term must 

 vanish separately, and we have the system of equations 



" ............ (5M) - 



There are n equations between the n variables 1} 6^ ... 6 n and the time. 

 Hence these equations enable us to trace the changes in l} 2 , ... 6 n and to 

 express their values as functions of the time and of the initial values of 



550. Next, suppose that certain constraints are imposed on the values of 

 OH Z , @n by the mechanism of the system. Let these be m in number, 

 and let them be such that the small increments BB lt 863, ... 80 n are connected 

 by equations of the form 



0^ + o 2 80 a +...+a B S0 n = .......... ........ (501), 



M#i + M# 2 + ... + &80 W = ................... (502), 



etc. 



Then equation (499) must be true for all values of 80 lf 80 2 , ... which are 

 such as also to satisfy equations (501), (502), etc. Let us multiply equations 

 (501), (502), ... by X, p, ... and add to equation (499). 



We obtain an equation of the form 



-..H = ............ (503). 



at Wj/ j 



Let us assign arbitrary values to 80 m+l , S0 m+2 , ... 80 n , and then assign to 

 the m quantities B0 lt S0 2 , ... 80 m the values given by the m equations (501), 



