VOL. XTI.] PHILOSOPHICAL TEANSACTIONS. 351 



SO I to m; which m is therefore greater than 1. — 4. Therefore the effective 

 force, and consequently the celerity, as to a first moment, is to be the m* part 



of what it would be, had there been no resistance. — 5. This m"* part, or—, 



' fit 



is also the remaining force after such first moment; and this remaining force is, 

 for the same reason, to be proportionally abated as to a second moment : that 



is, we are to take — of it, that is of the impressed force : and for a 



m mm ' 



third moment, at equal distance of time, ; for a fourth, — ; and so on 



mmm m* 



infinitely. 



• 6. Because the length dispatched, in equal times, is proportional to the 



celerities ; the lines of motion, answering to those equal times, are to be as 



— , — ,, — J, -7, &c. of what they would have been, in the same times, had 



m m*' m' m*' ■> ' ' 



there been no resistance. — 7. This therefore is a geometrical progression, conti- 

 nually decreasing. — 8. This decreasing progression, infinitely continued, is yet of 



a finite magnitude, and equal to of what it would have been in so much 



* m — I 



time, if there had been no resistance. For the sum or aggregate of a geome- 

 trical progression, is ~ , supposing z the greatest term, a the least, and 



r the common ratio ; that is, — — -— . Now in the present case, sup- 

 posing the progression infinitely continued, the least term a becomes infinitely 

 small, or = O; consequently — — - also vanishes, and then the aggregate be- 



comes barely = -^ ; that is, by division, z-j-i- -|--i+-i-f- &c. = 



jr^, supposing the progression to begin at z = 1. That is, dividing all by 



r, that so the progression may begin at — = — , — — = — -|- — -j- 



-j- -f &c. That is, in the present case, (because of z = 1, and r = m) — 



•A 1 — r 4- &C' = ; that is, (putting « = m — l), — of what it 



' mm ' m^ ' m — 1 ' ^" ° '' n 



would have been, if there had been no resistance. 



9. This infinite progression is fitly expressed by an ordinate in the exterior 

 hyperbola, parallel to one of the asymptotes; and the several members of that, 

 by the several members of this, cut in continual proportion. For let S H, fig. 2, 

 be an hyperbola between the asymptotes ^B, AF : and let the ordinate D H, 

 in the exterior hyperbola, parallel to AF, represent the impressed force undi- 



