334 GENERALIZED COORDINATES 



In general let us suppose that we have certain relations imposed 

 by the mechanism, these being of the form 



a^ + a z W,+ . . . + a n W n = 0, (166) 



b^ + b^ + . - - + b n S0 n = ' 0, . . .. (167) - - 



Then equation (165), namely 



.=. 



is true only if 80 V S0 2 , - , $0 n satisfy relations (166), (167), 



For a possible displacement, however, S0 lt '80 z , - , 80 n will be 

 such that equations (166), (167) , and (168) are all true. Let 

 us multiply by X, //., and unity, and add, X, JJL, being quan- 

 tities as yet undetermined undetermined multipliers, we may call 

 them. Then we have the equation 



The quantities W^ S0 2 , . , W n are not at our disposal. If, how- 

 ever, the relations of the type (166) are m in number, we may say 

 that of the quantities 80 lf S0 2 , . ., 80 n all except m are at our dis- 

 posal, and after arbitrary values have been assigned to n m of 

 these quantities, the remaining m quantities must be obtained by 

 solving equations (166), (167) . The configuration obtained in 

 this way must necessarily be a possible one. 



Let us assign arbitrary values to 



