I 75 -

( 574 )

lnfinitely cominued) the lcast Term A,'becomes lnfinitely Small, or^rr: q.’A

And conlequently —-i- doth also Vanifh, and thcrcby the Aggregate be-

VR . , . . .

come; — ^- That is, t^as will appear by .Dmdwg V R by R— x ;•)

V

R

V VrI • rT

V R

•+, (Sc. = —-• (fuppofing the Pro-

R — i

gression to begin at V — i.) That is, (dividingalllby R, that so the Pro-

„. , . V i , V V V V

gression may begin at —— = -: ) - =- - 4 - -->- —

0 1 h R m R — i R R R R 5

-t* &c. That is in our prcfcnt Cale, (becaule of V = i, and R —. >«)

i

m m

m 3

-, tfc. -

That is, (putting r. — m — l)

—, of what it Wo uId have been, if there had been no Resistance,n

q. This Infinite Progression is fitly expressed by an Ordinata in the Ex-terior Hypcrbola, Parallel to one of the Asymptoten ; and the feveral Membersof that, by the leveral Members of this, cut in Continual Proportion. As isthere Demonstrated at Prop. 15. For let S H, be an Hyperbolo, between theAsymptote s, AB, AF: And let the Ordinate D H, ("in the Exterior Hyper-bolo, Parallel to A F) reprefeht the Itnprcfled Force undiminissied • «r theLine to be deseribed in liich Time, by a Cclerity anlwerable to such Undi-

milhed Force. And let • B S (a Kke Ordinate) be -— thereof , which

m

therefore, being lesi than DH, fas being Equal to a part of it) will be fur-ther than it from AF. In A B, (which I put = 1,) let B d, be such apart thereof, as is B S of D H. Now becaule (as is well known) all theInscribed Parallelograms, in the Exterior Hyperbolo, A S, A H, isc. are Equal i

and therefore their Sides : 'Reciprocal : Therefore as Ad — 1 -—,

m

(sopposing B d, to be taken from B toward A,) to AB = 1, (or as m —x

to m i) so is B 8 — -—— D H to d b, which is therefore Equal to-—

m m— 1

of DH; that is, (as will appear by Dividing 1, by m — 1,) to --—

m

* 4 * -+ -i- -&c, of DH.

m m m 3

Or