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



less for the natural motion than for the other. The new principle, 

 on the other hand, compares the integrals always taken for a natural 

 motion satisfying the differential equations , but the terminal con- 

 figurations are varied from one motion to another. The principle is 

 therefore known as the Principle of Varying Action. 



In the process of 34 equation 2) we cannot now put the 

 integrated part equal to zero,, but instead of 2) we shall have 



* 



The integrated part, which is the sum of the geometric products of 

 the momenta and the variations of the corresponding positions at 

 the end of the motion minus the corresponding sum at the begin- 

 ning, may now be transformed into generalized coordinates. The 

 integral 



S=f(T-W)dt, 



where T and W are expressed as functions of the time, appropriate 

 to any given motion (whether natural or not) depends upon the 

 terminal configurations, and is called by Hamilton the Principal 

 Function. The terminal configurations being given we had dS = 0. 

 Let us now find an expression for dS in generalized coordinates 

 corresponding to the expression above in rectangular coordinates. 

 Proceeding as in 3|^ equation 43) we obtain 



89 ^.'ll 



Since the various motions are all natural ones satisfying the differ- 



ential equations of motion, the factor of every dq in the. integrand 



vanishes, so that the integral vanishes of itself, and dS is accord- 



ingly expressed as a linear function of the variations of the initial 



and terminal coordinates. Since W is independent of the 0"s and 



cT 



fi^i =pr, making use of the affixes and 1 for the limits t Q and t v 



we may write 



90) 98 = Sr&1t-Z r &a, 



an equation which could have been obtained from the considerations 

 regarding geometric products at the beginning of 37. This 



