476 Dynamical Theory of Currents [OH. xvi 



attached. He will know that whatever this mechanism may be, certain laws 

 must govern the manipulation of the ropes, provided that the mechanism is 

 itself subject to the ordinary laws of mechanics. 



To take the simplest illustration, suppose that there are two ropes only, A and B, and 

 that when rope A is pulled down a distance of one inch, it is found that rope B rises 

 through two inches. The mechanism connecting A and B may be a lever or an arrange- 

 ment of pulleys or of clockwork, or something different from any of these. But whatever 

 it is, provided that it is subject to the laws of dynamics, the experimenter will know, 

 from the mechanical principle of " virtual work," that the downward motion of rope A 

 can be restrained on applying to B a force equal to half of that applied to A. 



544. The branch of dynamics of which we are now going to make use 

 enables us to predict what relation there ought to be between the motions of 

 the accessible parts of the mechanism. If these predictions are borne out by 

 experiment, then there will be a presumption that the concealed mechanism 

 is subject to the laws of dynamics. If the predictions are not confirmed by 

 experiment, we shall know that the concealed mechanism is not governed by 

 the laws of dynamics. 



Hamilton s Principle. 



545. Suppose, first, that we have a dynamical system composed of dis- 

 crete particles, each of which moves in accordance with Newton's Laws of 

 Motion. Let any typical particle of mass m x have at any instant t coordi- 

 nates #!, y lt Z-L and components of velocity z/ 1} v lt w lt and let it be acted on by 

 forces of which the resultant has components X lt Y 1} Z : . Then, since the 

 motion of the particle is assumed to be governed by Newton's Laws, we have 



m i7fc = Xi .............................. (490), 



Let us compare this motion with a slightly different motion, in which 

 Newton's Laws are not obeyed. At the instant t let the coordinates of this 

 same particle be ^-fS^, y l + Sy lt z l + Zi and let its components of velocity 

 be MJ + SMJ, v l + Svj, w 1 -\-8w l . Let us multiply equations (490), (491) and 

 (492) by &PJ, &/!, Bzj_ respectively, and add. We obtain 



Now ?*-s 



