On the Definition of Force as the Cause of Motion. 341 



preted as force producing momentum for unit of time, and t as 

 the mere number of such units, so that when that number is 1, 

 we have / = tiiv. But we must not forget the relation of this 

 equation to that of energy, viz. : js — ^mv'. If we give an in- 

 terpretation of fs corresponding to that just given of ft, we must 

 say that / is here force doing work through unit of space, and 

 that s is the mere number of such units ; thus making f to have a 

 totally different signification in the two equations ; but / must 

 have the very same meaning in both ; since the latter is derived from 

 the former ; and as it does not mean force acting through unit of 

 time in the latter, it must not do so in the former. In both cases it 

 is simple pressure or tension, which is not acting, nor proportional 

 to the amount oi the action, but only to the intensity of the action. 



Connected with this (whether as cause, or effect, or as a collateral 

 illustration of the same tendency) is an occasional ambiguity in 

 the word " acceleration." This usually means now rate of change 

 of velocity ; but it sometimes has its simpler original meaning, 

 viz. : amount of change of velocity. The effect that this will 

 have on the meaning of " force," if joined with it in the same 

 sentence, is obvious. 



(4.) We are constantly told that impulse is "force acting 

 through time." We have already animadverted on force "acting " ; 

 but we have now to object further to this statement that it de- 

 clares that impulse is force in some continuous change of condi- 

 tion. This statement is analogous to saying that the area of a 

 plane curve is the ordinate moving along the axis of x. It comes 

 very near to identifying totidem verbis force and impulse. 



(5.) As a little step further {for the learner) it is said of im- 

 pulse (and of its equivalent momentum) that it is the " time- 

 integral of force." What is meant by this is, of course, that 



impulse is / -fdt. This elliptical and conveniently brief form of 



expression is intelligible enough and quite allowable in other 

 matters ; but this statement read in the light of the definition of 

 force as the cause of motion, is calculated to mislead the learner. 

 There is actually some little danger of its reacting on the dyna- 

 micists themselves as would appear by the fact that some of them, 

 as another step in advance, tell us that the time-integral of force 

 is "what we may call its whole amount during any time," and 

 that another writer calls it "the total force durino' a finite time." 



