OF THE PROPERTIES OF CUUVES. 



23 



vature is simply as the second fluxion of tlie 

 ordinate. 



For the fluxion of the curve becomes equal to that of the 

 absciss, and the perpendicular chord to the diameter. 



196. Definition. If the sum of two 

 right lines drawn from each point of a curve 

 to two given points, is constant, the curve is 

 an ellipsis, and the two points are its foci. 



197. Definition. The right line pass- 

 ing through the foci, and terminated by the 

 curve, is the greater axis, and the line bisect- 

 ing it at right angles, the lesser axis. 



198. Theorem. A right lino passing 

 through any point of an ellipsis, and making 

 equal angles with the right lines drawn to the 

 foci, is a tangent to the ellipsis. 



1''===^ — — — '' Let AB make equal 



angles with BC and 

 BD, then it will touch 

 the ellipsis in B. Let 

 E be any other point 

 in AB. Produce DB, 

 take BF=;BC,and join 

 CF, then AB bisects the angle CBF, and CAB is a right 

 angle. Join EC, ED, EF, GD, then EC=EF, and EC+ 

 ED=EF+ED, and is greater than DF (79), or BC+BD, 

 or GC+GD, therefore E is not in the ellipsis, and AB 

 touches it. 



199. Theorem. The right lines drawn 

 from any point of the ellipsis to the foci, are 

 to each other as the square of half the lesser 

 axis to the square of the perpendicular from 

 either focus, on the tangent at that point. 



Let A and B be the 

 foci, C the point of 

 contact, and AD the 

 perpendicular to the 

 tangent CD, draw BE 

 and BF parallel to AD 

 and CD, produce AD 

 each way, and let it 

 meet BF and BC in F and G. Then sincez.ACD=BCE 

 =DCG, CG=AC;andBG=:AC+BC. And BFq^TBGq 



— FGqZ:BAq_FAq (lis), therefore BGq — BAq=:FGq 



— FAq; but (FG+FA).(FG-FA)=FGq-rAq; and 



FG4.FA=aFD=2BE, and FG-FA=AG=:2AD; also 

 BG=2BH, and BA=2BI, whence BGq— BAq=:4HIq, 



therefore BE.AD=HIq, and BE=:— 3, but BE : BC : : 



AD : AC, and BE^AD.l'^^lUS, or ^=1^ 

 AC AD' AC ADq 



200. Theorem. The chord of the circle 

 of equal curvature 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 pro- 

 duct of the distances divided by one fourth 

 of the greater axis. 



Let AB be an eva- 

 nescent arc of the el- 

 lipsis coinciding with 

 the tangent, then the 

 radius of curvature bi- 

 secting always the an- 

 gle CAD or CBD, the 

 point E where the radii 

 AEand BE meet will ul- 

 timately be the centre of the circle Of equal curvature. Let 

 BF, BG, be parallel to AC, AD, then BH, bisecting FBG, 

 will be parallel to AE: but EBH=CBF+FBH— CBEr: 

 CBF-(-^FBG— iCBD = CBF — iCBF -f-iDBG= | (CBF-)- 

 DBG)=:L(ACB-f-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 of BDA - 

 but the sines of ABC and ABD are ultimately equal ; con- 

 sequently 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 reciprocals : 

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

 mean of AC and AD, then the angle AIBziAEB ; for the 

 evanescent angles ACB, AIB, or their sines, are recipro- 

 cally as AC, AI, beini; opposite to the same angle BAE 

 and having AB opposite to them ; 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 per- 

 pendicular falling on the tangent of an ellipsis 

 from its focus, is to the square of the distance 

 of the point of contact from the focus, as a 

 third proportional to the axes is to the focal 

 chord of curvature. 



