250 Rev. M. H. Close on the Meaning of "Force" 



tion of momentum, just as potential energy, or Fs, is ex- 

 pended in the performance of work. 



(d) It is sometimes said that Newton has proved the com- 

 position of the effects of conjoint forces in Corollary 1 to the 

 Laws of Motion. " Forces " here means impulsions; for it is 

 only with these that that corollary deals. This appears not 

 only a priori, because force proper does not produce motion, 

 but also from the reference in that corollary to Law II., which 

 must be read in the light of Newton's own exposition of it 

 quoted above. If the times of the two impulsions be equal, 

 then (but not otherwise) are the sides of the parallelogram 

 proportional to the forces proper concerned; and the com- 

 position of these follows from Cor. 1. Though this additional 

 step was so easy to Newton that he has unconsciously taken it 

 for granted in Cor. 2, which, in the discussion, though not 

 in the enunciation, deals with forces proper, yet it is an 

 additional step, and should not be ignored; for it has, not un- 

 reasonably though perhaps unnecessarily, presented difficulty 

 to distinguished dynamicists. The composition of forces 

 proper does not follow immediately from that of velocities. 

 Newton's argument, as clearly drawn out, consists of three 

 stages, viz. : — 1, the composition of velocities (kinematical) ; 

 2, the composition of impulsions (kinetical) ; 3, the composi- 

 tion of forces proper (statical). The second is involved in the 

 first; but the third has to be deduced from the second. 



(e) Sometimes F, as in ¥t = mv, is spoken of as though it 

 were impulsion of unit of time, and not merely numerically 

 equivalent thereto and measurable thereby. Similarly v, or 

 velocity, is sometimes spoken of as the space described in unit 

 of time. In this and other cases there is no danger of any 

 misapprehension on the part of the learner; but it is otherwise 

 in the case of force. From frequent repetition the above form 

 of speech has sometimes solidified, as it were, into a quasi- 

 ontological principle, by which force and impulsion of unit of 

 time are identified — these two being totally disparate, as much 

 so as force proper and energy of unit of space. It is not easy 

 to criticise satisfactorily the phrase "time-integral of force," 

 which may have originated only as a brief and convenient, 

 though acknowledgedly loose, expression. But, considering 

 the insufficient recognition of the complete disparateness of 

 force and impulsion, we may be allowed to suspect that in this 

 expression it is not that dt is understood, but that " force " 

 originally meant, and still means, the differential of impulse, 

 or F dt ; which would quite fall in with what we have just 

 been complaining of. At any rate this expression tends to 

 keep up the obscurity of distinction between force and impul- 

 sion or its element or its differential. 



