106 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION. 

 19) iA 



'io 



Now the term 



/; 



2o 



may be integrated by parts, giving 



9 



dP . /*. d /oP\ 7 



o-l OX r / I X r -y- ( 5? ) # 



0a?; r / j r >?3 V' 7 ^/ 



The terms in d^ and 60' r are to be treated in like manner. Since 

 the variations of the coordinates vanish at the limits the integrated 

 terms disappear, leaving 



Now in virtue of 18) since N does not contain the coordinates, 

 dP 



^ ~" Y M~Wx~ r ~ M 



Also since M. does not contain #/., 



cP 1 T fM cN T fM. 



^T == " I/ -*r a~^ =1/^7 



^^ 2 V N dx r V N 

 and consequently 



dP d fdP\ -*/~N dW d 



5~ -^-(^-i) : =--l/irF^ 



^a; dcx' M ox d 



M dx r \ 



^F- 

 N dqj 



The equation of energy, 



gives 



or according to 18), 

 from which we get 



