VOL. XVI.] rHILOSOPHICAL TRANSACTIONS. 353 



parts unequal in Fj4 or hd. — 17. But what is the proportion of r to 1, or of 

 1 — r to 1 , or 1 to m, remains to be inquired by experiment. 



18. If the progression be not infinitely continued; but end suppose at JV", 



and its least term be a = MN: then, out of — - = - H 1 — r, &c. is 



r — 1 m mm ' m' 



to be subducted ——, that is, by division, — + ^ + 4» ^^''^ '■hat is, in the 



present case, — | — -. 1 — j, &c. And so the aggregate will be — — |- 



r-, &c. = . 



mm ' mi ' n 



And thus much as to the line of projection, in which, secluding the resist- 

 ance, the motion is accounted uniform ; dispatching equal lengths in equal 

 times. Consider we next the line of descent. 



19. In the descent of heavy bodies, it is supposed, that to each moment of 

 time, there is superadded a new impulse of gravity to what was before ; and 

 each of these, without considering the air's resistance, to proceed equally, 

 from their several beginnings, through the succeeding moments. Hence arises, 

 for the first moment 1, for the second I + 1, for the third 1 + 1 + 1, and so 

 on, in arithmetical progression ; as are the ordinates, in a triangle, at equal 

 distances ; and such are also the continual increments of the diameter, or of 

 the ordinates in the exterior parabola, answering to the interior ordinates, or 

 segments of the tangent, equally increasing. As is known^ and commonly 

 admitted. 



20. If we take in the consideration of the air's resistance ; we are then for 

 each of these equal progressions, to substitute a decreasing geometrical pro- 

 gression, in like manner, as in the line of projection. — 21. Hence arises, for 



the first moment -; for the second — I — -; for the third — I 1 — ■,, 



m m ' m' m ' m'^ ' m'' 



&c. And such is therefore the descent of a heavy body falling by its own 

 weight ; the several impulses of gravity being supposed equal. That is, as 

 FL, FM, FN, &c. in the line of descent, answering io FL, LM, MN, 

 &c. in the line of projection. 



22. But though the progressions for the line of projection, are like to each 

 of those in the line of descent, it is not to be thence inferred, that therefore 



— in the one, is equal to — in the other ; but in the line of projection, 

 suppose, — /, such a part of the force impressed, and a celerity answerable, 

 in the line of descent, —g, such a part of the impulse of gravity. — 23. Those 



VOL. III. Z z 



