PBINCIPLE OF LEAST ACTION 327 



266. Before proving this, however, we may attempt to interpret 

 equation (156). Let us denote T Why L. Then 



C*S(T-W)dt= C &Ld$ 



Jti Jt! 



= f \L' - L) dt 



Jt! 



X<2 /2 



L'dt- I Ldt 

 Jti 



-(***). 



/*<2 



If we denote / Ldt 



Jh 



by S, the equation becomes SS = 0, or 



S' = S. 



Thus the value of the function S for the actual motion is the 

 same, except for small quantities of the second and higher orders, 

 as the corresponding function S r for any slightly different motion, 

 which begins and ends with the same configuration at the same 

 instants. In other words, the function S is either a maximum or a 

 minimum when the series of configurations is that which actually 

 occurs in nature. 



PRINCIPLE OF LEAST ACTION 



267. The total energy will, by the theorem of 143, remain 

 constant during the actual motion, say equal to E, so that at every 

 instant we shall have 



In the slightly varied series of configurations it is not true that 

 the total energy remains constant throughout the motion, but out 

 of the infinite number of slightly varied series of configurations. 

 there will still be an infinite number for which the conditions 

 already postulated are satisfied, together with the condition that 

 the total energy at every instant shall have the value E. For such 



