OF THE PROPERTIES OF CURVES. 51 



210. Definition. If the absciss be equal 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 conjugate figures of sines. 



212. Theorem. The radius of curvature of the figure 

 of sines at the vertex is equal to the ordinate. 



For the fluxion of the base becoming ultimately equal to that of 

 tlie absciss in the corresponding circle, while the ordinates are also 

 equal, tlic curve ultimately coincides witli a portion of tliat 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 con- 

 stant, that of the sine varies as the cosine 

 (142), therefore the fluxion of the ordinate of 

 tlie figure of sines may always be represented 

 by tlic corresponding ordinate of the conju- 

 gate figure. Let AB, CD, be the conjugate 

 figures, then EF will represent the fluxion of EG, and, since the arc 

 and sine are ultimately equal, the fluxion of EG at C will be equal 

 to that of tlie absciss, therefore BC will always represent the con- 

 stant fluxion of the absciss. 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 constant pro- 

 portion, their increments and fluxions are diminished in the same 

 proportion, the fluxion of the base remaining constant. 



E 2 



