35] EQUATIONS DEDUCED FROM LEAST ACTION. 107 



Inserting this value of dq gives 



dP _ d / dP \ _ -]/~N_ $W _ _^_ 



= _ ,/^ fdw 

 Accordingly we have 



dW 



In order that this may vanish for arbitrary variations, dx r , dy r , dz rj 

 the coefficient of each variation must vanish, so that we must have 



dW 



= or - 



c 



d*z r dW d*z r dW 



mr ~w+~W r = > mr ~dt^~~ ~ f W f = * Zr \ 



which are the ordinary equations of motion for a free system. 



The variations dx r , dy r , 8z r are arbitrary only if all the particles 

 are free. If there are constraints the variations must be compatible 

 with the equations of condition, 



that is we must have the & linear relations between the <S's, Chapter III 

 equations 14). We may then as in 25 multiply the equations 

 between the <5's by undetermined factors 1; A g , . . . A* and add them 

 to the integrand. We shall then have 



We may now determine the k factors A 1? >1 2 , . . . ^, so that k of 

 the factors multiplying the variations vanish identically. Then the 



