AnJ-wcrs io Mathematical ^eflioni. 137 



MATHEMATICAL CORRESPONDENCE. 

 Question I, Anf-wered by the Propofer. 

 i_/ET a be the length of the whole line, and x,y, z, v, w, &c. the parts into which the 

 half of it is to be divided. Then — =: = ~ ^ = " ~ ■ ~ y l&c. bythe 



queftion : Or y = - (a — x), z — -^ (a — x — y) = — (a — x — — x a — .v) = — 



(a —A- XI )= — {ti — X X ) = —^ (a — xy, v = -^{a — X — y— z)-= 



la — x X a — X X a — »') = X a — x (i '■ — X a—x) 



,a X a — X — X X a — *:, 



= X a — X ( Xfl-s;) = X a—x { '. ) = 



a a a^ a a^ ' 



— T ^ " — .vl^ X (a — -v) = — r- X a — x?, iv = (a — x — v — z — tf\ = • 



fl 'a' a ^ ■' 'a 



(a — x X a — x ^ x'a — x\^ ,- X a — x\^) = ^ X 7^^^*, &c. But 



a a a a* 



A' + J' + z + V + TO, &c. = — , by the queftion. Whence, alfo, x + -^- (a — x) + —^ 

 [a — x)' H ^ [a — A.-)' + -^ (a — x)* Sec. to « terms = ; which laft being a geo- 

 metrical feries, whofe firft term is x, and the ratio '-, its fum; by the common rule, is 



a — xy 

 X (— — -1) 



— , from whence x is found = a X ( i — )j and confequently tlie 



feveral divifions of the half line are x (i — ^7— )j ~7~{i— ~77)> "T"(i— )> ~T" 



2" 2° 2" 2° 2° 2" 



(i — "T")* ^'-' where n is any number whatever. 

 2" 



Question II. Anfwered by the Propofer. See Fig. 12. Plate VI. 



LET AF = *•, AC = r, f = celerity, or velocity in the circumference, and F = va- 

 riable force in the fine EF. 



Then CE : FC : : EG : EK by fin. A^. or r : >• - *• : : c : — [r - x) = v = velocity 

 in FE. But, by the laws of variable forces, Vt = v = — — i, and t = !! 1 =; 



dux. ^2rx — x^ rx — xx r , F (r — .r) .v 



^ ^^1 whence -7 ^ — X 



7 (»■-*) 



Vol. I. — June 1797 



•^Zrx — x"- c {r — .-t)' 



