VOL. XXV.] PHILOSOPHICAL TRANSACTIONS. 305 



-^'■' ■ ■- " — Q,(ZX ^" XX • • XX • ■ , 



is the rectangle cabB = a — x X V = a — x X -,;— ^^ = cc.v —- — i'. The 



o -^ 2aa — 2ax 2a 



flowing quantity of a:.r h^xx; and the flowing quantity of — — i is — -^. And 



hence also it follows, that the whole area, continued on infinitely towards e, is 

 one-third of the square fikh ; or 4- aa. For supposing x = a, the area above 



, aa aa aa 



becomes — — -r = — . 



2 6 3 



While I was considering the other properties of this curve, and had given 

 some account of them to my ingenious friend Mr. John Colson, he returned 

 me a letter with the addition of the quadrature of the curve's area, which I had 

 not then inquired into. 



V. Supposing still the same things : I say that the solid made by the conver- 

 sion of the area vahnF about the line hZ?, as an axis, is equal to a cylinder 



vx "pi '1*4 



whose radius is fh = a, and height equal to V- -—. And the whole 



^ ^ 2a 2aa ' 8a^ 



solid made by conversion of the whole figure infinitely continued, is equal to 

 an 8th part of a cylinder, whose radius and height are each equal to ph or a. 



Let ^ express the proportion of of the periphery and diameter of a circle. Then 



is^aZj* the area of a circle whose radius is ab. And because ca = ^r = 



-— .V, the fluxion or the solid is^X a — x X ;r-, or -, (ax — ;,-{-ocx-)t. 



2aa — 2ax ' d 2aa — 2ox' d^ 2 ' 2a' ' 



whose flowing quantity is ^ (-^ — \- — ); which solid being divided by -, aa 



• • • "SX TIC r TC^ 



(the area of a circle whose radius is a) gives V~ + ri» ^^^ ^^® height of a 



cylinder on the said circular base, and equal to the solid made by the conversion 

 of the area fq^hf about the line h^ as an axis. When a; = a, that is, when 



XX X^ X4 



2a 2aa ''" So^" 



• « XX x^ ^^ 



the whole figure is turned about its asymptote, the height 1- —^ be- 



come -fa. 



VI. The curve surface of the solid generated by the conversion of the figure 

 Fabnr about hb, is equal to the curve surface of a cylinder whose radius is a, 



and height equal to ^ — 1- —~, And the whole curve surface of the solid 



infinitely continued, is equal to one- third part of the curve surface of a cylinder 

 whose radius and height are equal to fh or a. Which may be demonstrated 

 after the manner of the preceding proposition. 



VII. The radius of the curvature of any particle of the quadratrix is — — , and 



thus found geometrically. In fig. 10, pae is the quadratrix, hd the asymptote, 

 AD the tangent, bd the subtangent to the given point a. Make bv = ad ; on 



VOL. V. R R 



