62 INTRODUCTION. 



215. Theorem. The radios 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 that 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 tlie corresponding ordinates, and are therefore to each other in 

 the ratio of those ordinates. 



216. Theorem. Tlie figure, of which the ordinates 

 are the sums of the corresponding ordinates of any two 

 harmonic curves, on equal bases, but crossing the absciss 

 at different points, is also a harmonic curve. 



The absciss of the one curve being x, that of the other will be «+ 

 X, and the ordinates will be Z».(sin. x) and c (sin. a-\-x) ; now sin. a-f 

 am(cos. a:).(sin. a) +(cos. a).(sin. x) and the joint ordinate will be 

 (6+c.(cos. a)).(sin. rr).+c.(sin. a).(cos. x) ; if, therefore, dhe the angle 



of which the tangent is - — — — '—^ — its sine and cosine will be in the 



«4-c.(cos. a) 



ratio of c.(sin. a.) to fi+c(cos. a), and (cos. c?).(sin. a;)-{-(sin. d).(cos.x\ 

 will be to the ordinate in the constant ratio of sin. d to c.(sin. a) ; 

 but (cos. rf).(sin. a-)+(sin. rf).(cos. x) is the sine of d-^x; conse- 

 quently the newly formed figure is a harmonic curve. 



jy The same may be shown geometri- 



es cally, by placing two circles, having 

 their diameters equal to the greatest 

 ordinates of the separate curves, so as 

 to intersect each other in an angle equal 

 to the angular distance of the origin of 

 the curves : then a right line revolving 

 round their intersection, 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 intersection, 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 re- 

 volving line a portion equal to the ordinate. For if AB be made 



