380 Mr. W. P. Browne on Action at a Distance. 



by the law of the Conservation of Energy, the two "Works," 

 or the two products of force and distance moved through, 

 must be equal and opposite. It then proceeds : — " Since the 

 forces (F) are equal and opposite, and the works (Fs) are also 

 equal and opposite, the distances (Y) must be equal but not 

 opposite ; that is, the two bodies must move over precisely the 

 same distance and in the same sense ; which practically asserts 

 that they move together and are in contact, as long as the 

 action is groins on." 



This argument may be confuted in several ways. 



(a) Force is measured by the acceleration, or change of 

 velocity, which it generates ; and if the force F, with which A 

 acts on B, is equal and opposite to the force — F, with which 

 B acts on A, it follows (assuming the masses to be equal) that 

 the acceleration of B will be equal and opposite to the accele- 

 ration of A. Hence, even if A and B are moving "together 

 and in contact" at the beginning of the space s, they will not 

 be so at the end, because their velocities will have changed. 



(b) The argument is built on the assumption that if one of 

 the works is taken as positive, say W, the other will be negative, 

 say — W. But Work, on Dr. Lodge's definition, or Energy, 

 on Rankine's definition, may be measured by vis viva, or by 



■5- r 2 ,and therefore is an essentially positive quantity (as pointed 



out by Rankine, 'Applied Mechanics/ art. 548, p. 499). We 

 can no more have negative work than we can have negative 

 mass. 



(c) There is no difficulty in adducing a case in which the 

 proposition is obviously untrue. This is the simple one of two 

 equal particles meeting each other with equal and opposite 

 velocities. It will be admitted that both will be reduced to 

 rest, and that, until they have been reduced to rest, they are 

 not moving " over the same distance and in the same sense." 

 Therefore, according to Dr. Lodge, until they have been 

 reduced to rest they cannot exercise any action upon each 

 other. What, then, is the action which reduces them to rest ? 

 To say they are reduced to rest instantaneously on coming 

 into absolute contact, does not remove the difficulty ; and if it 

 did, it would be by assuming that the force acting between 

 them was literally infinite, since it would stop a finite velocity 

 in no space at all. 



(d) Finally, I may point out what is the real and intrinsic 

 vice of Dr. Lodge's argument, viz. that he assumes that the 

 Conservation of Energy is universally true of Potential Energy 

 only, whereas (as is shown in any text-book) it is only uni- 

 versally true of Kinetic and Potential Energy together. This 



