OF THE PROPERTIES OF CURVES. 47 



200. Theorem. The chord of the circle of equal cur- 

 Yature with an ellipsis at any point, passing through the 

 focus, is equal to twice the harmonic mean of the distances 

 of the foci from the given point, or to the product of the 

 distances divided by one fourth of the greater axis. 



Let A B be an evanescent arc of 

 the ellipsis, coinciding with the 

 tangent, then the radius of curvar 

 ture bisecting always the angle 

 CAD or CBD, tlie point E, in which 

 tlie radii AE and BE meet, will ul- 

 timately be the centre of the circle 

 of equal curvature. Let BF, BG, 

 be parallel to AC, AD ; then BIJ, 



bisecting FBG, will be parallel to AE: but EBHnCBF+FBH— 

 CBE=CBF+iFBG — ^CBD=rCBF--^CBF4-pBG=: | (CBF4- 

 DBG)=:KACB+ADB). Now, in the triangles ABC, ABD, as AC 

 is to the sine of ABC, so is AB to the sine of ACB, and as AD is to 

 the sine of ABD, so is AB to the sine df BDA ; but the sines of ABC 

 and ABD are ultimately equal ; consequently ACB and ADB are 

 inversely as AC and AD, or as their reciprocals, and EBH or AEB, 

 which is the half sum of ACB and ADB, is as the mean of those re- 

 ciprocals : let BI be the reciprocal of that mean, or the harmonic 

 mean of AC and AD, then the angle AIBrr^^EB ; for the evanes- 

 cent angle:* ACB, AIB, or their sines, are reciprocally as AC, AI, 

 since these angles have the same side AB opposite to them in the 

 triangles ABC, ABI, and their equals BC, BI are opposite to the 

 same angle BAC ; for the same reason, taking BK=:2BI, AKB is 

 half of AEB ; consequently K is in the circle of curvature, and BK 

 is its chord. 



201. Theorem. The square of the perpendicular, fall- 

 ing on the tangent of an ellipsis from its focus, is to the 

 square of the distance of the point of contact from the fo- 

 cus, as a third proportional to the axes is to the focal chord 

 of curvature. 



