OF THE PROPERTIES OF CURVES. 



25 



is an eqiial cycloid, and the length of its arc 

 is double that of the portion of the tangent 

 cut off by the vertical tangent. 



Let two equal cir- 

 cles AB, BC, rolling 

 on the parallel bases 

 DA and EB, at the 

 distance of a diame- 

 ter of the circles, de- 

 scribe with the points 

 F and G the equal cy- 

 cloids EF and EG. 

 Draw the diameter FH ; then H will be the point that 

 coincided with D, and HA=DA=EB= arc BG, and the 

 remainders AF and GC^re equal, therefore ./.ABF:^CBG 

 (133), and FBG is a right line («a). But FG is perpendi- 

 cular to AF (134), therefore it touches EF (206), and it is 

 always perpendicular to EG (2oa) ; therefore EG will coin- 

 cide with the involute of EF, for they set out together from 

 E, and are always perpendicular to the same line FG (193), 

 which they could not be if they ever separated. Conse- 

 •quently the curve EF is always equal to FG (192), or 2FB, 

 twice the portion of the tangent cut off by EB. 



209. Theorem. The fluxion of the cy- 

 cloidal arc is to that of the basis, as tlie 

 evolved radius to the dianieler of the g«ne- 



rating circle. 



p For the increment GI=sBK, and BK : 

 BL : : BG : BC, and 2BK. : BL : : FG : BC, 

 which is therefore the ratio of the flux- 

 ions. 



ScHOiiuM. If the fluxion of the base 

 be constant, that of the curve will v^ry as 

 the distance of the describing point from 

 the point of contact. 



210. Definition. If the absciss be 

 «qual to the arc of a given circle, and the 

 perpendicular ordinate to the corresponding 

 sine, the curve will be a figure of sines. 



211. Definition. If a second figure of 

 sines be added, by taking ordinates equal to 

 the cosines, the pair may be called conju- 

 gate figures of sines. 



212. Theorem. The radius of curvature 

 of the figure of sines at the vertex is equal to 

 the ordinate. 



VOL. 11. < 



For the fluxion of the base becoming ultimately equal to 

 that of the absciss in the corresponding circle, while the 

 ordinates are also equal, the curve ultimately coincides 

 with a portion of that circle. 



213. Theorem. The area of each half 

 of the figure of sines is equal to the square 

 of the vertical ordinate. 



For the fluxion of the absciss 

 being constant, tliat of the sine 

 .varies as the cosine (142), there- 

 fore the fluxion of the ordinate of 

 the figure of sines may always be 

 represented by the corresponding 

 ordinate of the conjugate figure. Let AB, CD, be the con- 

 jugate figures, then EF will represent the fluxion of EG, 

 and, since the arcand sine are ultimately equal, the fluxion 

 of EG at C will be equal to that of the absciss, therefore BC 

 will always represent the constant fluxion of the abscisi. 

 But the fluxion of the area AEF, is the rectangle under the 

 fluxion of the absciss AE and the ordinate EF ; that is, the 

 rectangle under BC and the fluxion of EG, and the fluent 

 BC.(AD— EG) is therefore equal to the area, which at C 

 becomes BCq. 



214. Definition. Each ordinate of the 

 figure of sines being diminished in a given 

 ratio, the curve becomes the harmonic curve. 



Scholium. The ordinates being diminished in a con- 

 stant proportion, their increments and fluxions are dimi- 

 nished in the same proportion, the fluxion of the base re- 

 maining constam. 



215. Theorem. The radius of curvature 

 at the vertex of the harmonic curve is to that 

 of the figure of sines, on the same base, as the 

 greatest ordinate of the figure of sines to tliat 

 of the harmonic curve. 



For taking any equal evanescent portions of the vertical 

 tangents the radii will be inversely as the sagittae, which are 

 similar portions of the corresponding ordinates, and ate 

 •therefore to each other in the ratio of those ordinates. 



216. Theorem. The figure, of which 

 the ordinates are the sums of the correspond- 

 ing ordinates of any two harmonic curves, oa 

 equal bases, but crossing the absciss at dilfer- 

 ent points, is also a harmonic curve. 



The absciss of the one curve being x, that of the other 

 will be a-\-x, and the ordinates will be 2i.(sin. x) and c. (sin . 



