136 IV. PRINCIPLE OF LEAST ACTION. GENERALIZED EQUAT. OF MOTION, 

 and putting t = 0, 



Then equation 90) becomes 



100) 68= Z r pr $<lr- Zr C r 6 C r , 



where ~ - 



Then comparing with 



we have corresponding to equations 96) and 97) 



101) H--PT, 



102) W,-- C " 



Thus we may differentiate S with respect to the m arbitrary 

 constants, no matter how they may appear in the solution of 99), 

 putting the result equal to other arbitrary constants. 



Hamilton's equation 99) assumes a somewhat simpler form when 

 the force -function and consequently H are independent of the time, 

 that is when the system is conservative. We may then advantage- 

 ously replace the principal function S by another function called by 

 Hamilton the Characteristic Function, which represents the action A, 

 35. Making use of the equation of energy, T-\-W=h, to 

 eliminate W 9 we have 



S 

 or 



t t 



f ( T - W ) d t = fa T d t - li (t - Q = A - li ( t - 



103) A = fafdt = S 4- h (t - J ). 



If now the function A does not contain the time explicitly we 

 have differentiating partially 



and our partial differential equation 99) becomes merely 



105) 



