On the Definition of Force as the Cause of Motion. 339 



applied by a spring wliicli keeps up with the body in its accele- 

 tion, it is, as in the statical case, qu'te unaltered during the 

 period of the action — it has not been in any degree expended. 

 The force might diminish during the action ; but this would be 

 because of other circumstances than expenditure. It might even 

 diminish at the very rate at which the momentum increases ; 

 but this would be solely because, from the nature of the con- 

 ditions, it is always proportional to e~\ 



Otherwise thus — The same persons who speak of the " ex- 

 penditure " of force tell us that force is the rate of change 

 of momentum. But surely a rate cannot be expended. 

 It may be diminished, it may sink to zero, but not by expenditure. 

 Even though we decline to assert that force is the rate mentioned, 

 and cautiously restrict ourselves to saying that it is proportional 

 to that rate, this argument still applies. Returning to the equa- 

 tion of area, we shall see that it is utterly impossible to regard 

 y (which corresponds to /) as having been expended when we 

 have reached the farthest point on the axis of x consistent with 

 the conditions given. This is so even if y there becomes 0, but 

 y may just as well be there at its maximum ; that which really 

 has been expended is the whole length along the axis of x, the 

 independent variable (which corresponds to t), allowed us by the 

 conditions. 



The expressions which we are now deprecating when applied 

 to force are quite easily avoidable. There are others connected 

 with the idea that force is the cause of motion which are, in 

 reality, equally incorrect ; but it must be acknowledged that it 

 would be almost impossible to avoid them without the appearance 

 of pedantry, and, in some cases, without very inconvenient cir- 

 cumlocution also. Force is constantly spoken of as " acting," 

 even when statical pressure is meant ; but that which is incapable 

 of expending itself cannot act in the true sense of the word, 

 even when concerned in action. Force only exists, or obtains ; it 

 is a circumstance, not an agent ; it is essentially un-related with 

 time, though it can be brought into connexion therewith ; it may 

 be a necessary condition, or an unavoidable accompaniment, of 

 the action which does take place ; but that is all. There are 

 other expressions, such as " attractive force," " repulsive force," &c., 

 which it were hard to call incorrect ; they are rather elli23tical 

 expressions, which it would be excessively inconvenient to give 



