Number of Fractions contained in any "Farey Series.'''' 251 



We may here observe (though this is a different point) that 

 the expressions "time-integral of force," for impulsion, and 

 "space-integral of force," for potential energy, have in some 

 cases, from frequent use, reacted upon the users of them ; so 

 that we find impulsion called the " total force in a finite time," 

 and potential energy the " sum of the tensions," " force " and 

 " tensions " being force and forces proper. The competent 

 writers now alluded to have inadvertently allowed themselves 

 to use expressions precisely analogous to the statement that 

 the area of a plane curve is the sum of the ordinates. It can 

 be only in some transcendental sense that a force or pressure, 

 supposed constant for simplicity, which has existed statically 

 for a given time, may be said to be the sum of the applications 

 of force in each second of that time ; and this is equally true 

 if the force have been existing kinetically. 



What we may call the mathematical analogy between 

 energy, or Fs, and impulsion, or Ft, is complete, notwith- 

 standing the metaphysical considerations involved in the dif- 

 ference between their factors s and t. Each is of two dimen- 

 sions ; while force, or F, is of one. F being measurable in 

 pounds, Fs is measurable in foot-pounds, and Ft in second- 

 pounds. Each is equally disparate from F ; though very 

 dissimilarly so, since they are disparate from each other. 

 As Fs is the power of performing work, so Ft is the power of 

 producing momentum. Neither had a name until the present 

 century. It was this that gave room to the remarkable contro- 

 versy concerning the true measure of the "moving force" of 

 a body in motion which went on for over forty years in the last 

 century, " to the great scandal of science" as Montucla says. 

 Each needs a distinctive appropriated name, as much as the 

 other. One, viz. Ft, has not yet acquired a universally recog- 

 nized proper name ; but the sooner it does so the better for 

 the learner, and also for the science of dynamics, which then 

 need not be guilty of the rather unscientific proceeding of 

 occasionally giving more than one meaning to " Force." 



Your obedient servant, 



M. H.' Close. 



XXXVII. On the Number of Fractions contained in any u Farey 

 Series'''' of which the Limiting Number is given. By J. J.- 

 Sylvester*. 



AFAREY series ("suite de Farey ") is a system of all 

 the unequal vulgar fractions arranged in order of mag- 

 nitude, the numerator and denominator of which do not exceed 

 a given number. 



The first scientific notice of these series appeared in the 

 * Communicated "by the Author. 



