26 



OF THE PROPEBTIES OF CURVES. 



a+x); now sin. o+J;— (cos. Jt).(sin. a) + (cos. o,).(sln. ar) 

 and the joint ordinate will be (i+c.(cos. o)).(sin. x) +c. 

 (sin. a). (cos. x) ; if therefore d be the angle of which the 

 .(sin. a) 



tangent is ; 



■ its sine and cosine will be in the ratio 



i+c.(cos.n) 



of c.(sin.o.) to i+c(cos. a), and (cos. (i).(sin. x) + (sin. d). 

 (cos. x), will be to the ordinate in the constant ratio of sin. 

 d to c.(sin. a) ; but (cos. d).(sin. x) + (sin. d).(cos. x) is the 

 sine of rf+x ; consequently the newly formed figure is a 

 harmonic curve. 



The same maybe shown 



^ geometrically, by placing 



two circles, having their 



diameters equal to the 

 greatest ordinates of the 

 separate curves, so as to in- 

 tersect each other in an 

 angle equal to twice the angular distance of the origin of 

 the curves : then a right line revolving round their intersec- 

 tion with an equable velocity will have segments cut off 

 by each circle equal to the corresponding ordinate, and the 



sum or difference of the segments will be the joint ordinate: 

 and if a circle be described through the point of intersec- 

 tion, touching the common chord of the two circles, and 

 having its radius equal to the distance of their centres, this 

 circle will always cut off in the revolving line a portion 

 equal to the ordinate. For if AB be made parallel to CD, 

 and EB toFG,^ABEziCGF=CHK : but EIB is a right 

 angle, as well as HCF, and EI : IB : -. FC : CH : : AE : CH, 

 since AF is equal to twice the distance of the centres, which 

 bisect AH and FH, and therefore to CE, and FC=AE, or 

 EI : AE : : ir : CH ; but EI : AE : : ID : AC, therefore IB : 

 CH : : ID : AC, and the triangles ACH, DIB, are similar, 

 and ii.DBI— CHA=:DKA,and AD is a parallelogram, con- 

 sequently KDzr ABmCG. 



If the circle CG be supposed to revolve round C, the in- 

 tersection H will always show the angular distance of the 

 point in which the curve crosses the axis ; and the distance 

 of the centres will be equal to the greatest ordinate. If 

 therefore the circles are equal, the greatest ordinate wUlalso 

 vary as the chord of an arc increasing equably, or as the 

 ordinatt of the harmonic curve. 



