VOL. LXXXVIII.] PHILOSOPHICAL TRANSACTIONS. 353 



ordinate pm may be always a 4th proportional to the arc oq, the rectangle ab X 2 

 fe, and the line cl ; this cycloid shall cut the ellipse in m, so that, if mc be joined, 

 the area acm shall be to cmb :: m : n. 



Demonstr. Let ap = x, pm = y, AC = c, ab = a, and 2ef = b ; then, by the 

 nature of the cycloid gmr, — pm : oq :: 2 fe X ab : cl, and qo = ao — aq = by 



const. — ; — X (ak — agi) ; also, cl = ab — 2ac, since ac = lb. Therefore, 



— pm : — - — X ak - aq :: ab X 2ef : ab — 2ac ; or — y : — - — x arc 00° — 

 arc vers. sin. x :: ab : a -— 2c; therefore — y (a — 2c) or -f y (2c — a) = ab X 

 ( x arc . 90° — arc v. s. x), and by transposition ab X arc . v. s. x + y 



(2c — a) = — ■— X arc 90°. To these equals add 2y (x — x) = 0, and multiply 



by — 1 ; then will ab X arc v. s. x -j- (2x — a) y •— 2y (x — c) = X ab 



arc 90°, of which the 4th parts are also equal ; therefore 



«*""■" + <£±H» - !(« _ c) = ± X -4- X arc 9 0°. Now because 



4 4 2 V y 4M+N T 



1,1 If 



afb is an ellipse, y 2 = - x (ax — a? 2 ), and y = - */ (ax — x' 2 ) ; therefore 

 ab x arc v. s. • . 2x — a v> b ,, ,. y , x ab m v< ^^ 



Multiply both numerator and denominator of the first and last terms by a ; then will 



- X T X arc v. s. x + —— X - •(«*- a?) -> f (a?- c) as - X T X 



4 a v 7 2 v 7 a4M+N 



X arc 90 . Now the fluxion of an arc whose versed sine is x and radius -, is 



2 



equal to -r. which is also the fluxion of the arc whose sine is */ - and 



1 2 -v/( ajt — * ) 



radius unity; therefore - X (— X arc sin \/- -\ X V (ax — x 2 \) — ■¥ 



J a v 4 a 4 v 7/ 2 



(o: — c) is equal to - XrX — — — X arc 9O ; and, by the quadrature of the 

 circle, — - x arc sin. */ — | ^— X \/(ax — x 2 ), is the area whose abscissa is 



4 a 4 v " 



x ; consequently the semicircle's area is — X arc 90 . But the areas of ellipses 

 are to the corresponding areas of the circles described on their transverse axes, as 



h nP 1 x Qx — ■ cl 



the conjugate to the transverse ; therefore - X (— X arc sin. \/ — | — X 



*/ (ax — x 2 )) is the area whose abscissa is x, of a semi-ellipse, whose axes are a 

 and b ; and consequently - X — X arc 9O is the area of the semi-ellipse. There- 

 fore the area apm — — (x — c) is equal to — ■ — of amfb. But ~ (x — c) = -^ x 



2 X ' ~ m+n 2 V y 2 



(ap — ac) = —- X pc, is the triangle cpm ; consequently, apm — cpm, or acm, 



si 



is equal to X amfb ; and acm : amfb :: m : m + n ; or (dividendo) acm : 



n M + N • \ / 



cmfb :: m : n ; and the area of the ellipse is cut in a given ratio by the line drawn 

 through the focus, a. e. d. 



vol. xviii. Z z 



