CONIC SECTIONS. 



J 59 



Gurvsture. j n the a?ch HP A, which lies on each side of the point 

 """"""V '"*"' of contact, is wholly without the parabola. 



If the segment cut off by the circle be less than the 

 parameter (Fig. 82.), and therefore KR greater than 

 either KM or Km, then, reasoning as before, it will 

 appear that DE 2 is greater than HE 2 , and d E 2 great- 

 er than h E 2 , so that the points H, h are within the pa- 

 rabola ; and as the same will hold for every other posi- 

 tion of the ordinate nearer to the tangent, the arch 

 HP/*, which lies on each side of the point of contact, 

 is wholly within the parabola. 



Case 2. Next, let the section be an ellipse, or an hy- 

 perbola, (Fig. SI, S3.)* Take V a point in KR, so 

 that p P : pE : : KR : KV, and therefore Pp : KR : : 

 »E : KV : : pE.EP: KV.EP. But Pp: KR : : pE.EP: 

 DE 2 or dE 1 ( 18. Sect. II. and 26. Sect. III.); therefore 

 DE 2 , also dE*=KV.EP. Now HE*=KM.EP, and 

 hE I =Km.EP (Lemma), therefore DE 3 : HE 2 :: KV : 

 KM, and dE 1 : hE* : : KV : Km. Now, as Pp and RK 

 are similarly divided at E and V, if E approach to P, 

 the point V will approach to R, and as E may come 

 nearer to P than any assignable line, so V may come 

 nearer to R than any assignable line ; but as then HG 

 and kg will approach to PO, and M and m to P, it is 

 evident that the ordinate D d may have such a position, 

 that the points M, m and the vertex P may be all on 

 the same side of V, and the same thing may have place 

 for every other position of the ordinate nearer to the 

 tangent : therefore, in these circumstances, when KP, 

 the segment cut off from the diameter, is greater than 

 KR the parameter (Fig. 81.), KV will be less than ei- 

 ther KM or Km, and consequently DE 2 less than HE 2 , 

 and d E 2 less than h E 2 ; thus the points H, h, as well 

 as every other point in the arch HP h, which lies on 

 both sides of the vertex, are without the ellipse or hy- 

 perbola. On the contrary, when KP is less than KR, 

 the parameter (Fig. 83.), KV will be greater than ei- 

 ther KM or Kw, and therefore DE 2 greater than HE 2 , 

 also dE 2 greater than hE 2 , and therefore the points H, 

 h, as well as every other point in the arch HP/;, are 

 within the ellipse or hyperbola. 



Cor. 1. If a circle touch a conic section, and cut off 

 from the diameter that passes through the point of con- 

 tact a segment equal to its parameter, it will have the 

 same curvature with the conic section in the point of 

 contact. For if a greater circle be described, it will 

 cut off from the diameter a segment greater than its 

 parameter, therefore a part of its circumference on each 

 side of the point of contact will be wholly without the 

 come section ; and as it will also be without the former 

 circle, it will not pass between that circle and the conic 

 section at the point of contact. If a less circle be de- 

 scribed, it will cut off from the diameter a segment less 

 than its parameter ; therefore a part of its circumfe- 

 rence on each side of the point of contact will fall with- 

 in the conic section ; and as it will be within the for- 

 mer circle, it will not pass between that circle and the 

 conic section at the point of contact. Hence (Def. 2.) 

 the circle which cuts off a segment equal to the para- 

 meter, is the circle of curvature. 



Cor. 2. Only one circle can have the same curva- 

 ture with a conic section in a given point. 



Prop. II. 



The circle of curvature at the vertex of the axis of a 

 parabola, or at the vertex of the transverse axis of an 

 ellipse or hyperbola, falls wholly within the conic sec- 



tion; but the circle of curvature at the vertex of the Curvature, 

 conjugate axis of an ellipse, falls wholly without the ~=-y~— r 

 ellipse. 



Let Ppbe the axis of a parabola (Fig. S4.), and PHKA Pi^s. si, 

 the circle of curvature at its vertex, which therefore cuts 85, 86, 87. 

 off from the axis a segment FK equal to the parameter 

 of the axis ; because the tangent at the vertex is com., 

 mon to the parabola and circle, the centre of the circle 

 is in Pp. Let DEc?, an ordinate to the axis, meet the 

 circle in H and h ; it may be shewn, as in the last pro- 

 position, that DE 2 : HE 2 : : KP : KE. But in every po- 

 sition of the ordinate, KP is greater than KE, there- 

 fore DE 2 is always greater than HE 2 , and d E 2 greater 

 than hE 1 ; therefore the circle is wholly within the pa- 

 rabola. Next let Pp be the transverse axis of an el- 

 lipse or hyperbola (Fig. 85, S6*.), or the conjugate axis 

 of an ellipse (Fig. 87.), and PHKA the circle of curva- 

 ture, then, as in the parabola, the centre of the circle 

 will be in the axis. Draw Drf an ordinate to the axis, 

 meeting the circle in H, h, and take a point V in PK, so 

 that pP : pE: : KP : KV, then it will appear as in last 

 Prop, that DE 2 : HE 2 : : KV : KE. Now, when Pp is 

 the transverse axis of an ellipse (Fig. 85.), as Pp is great- 

 er than KP, and Pp : PK : : PE : PV ; therefore PE is 

 greater than PV, and hence KV is always greater than 

 KE. Therefore DE 2 is greater than I IE 2 , also a?E 2 

 greater than hE 1 , so that the circle falls wholly within 

 the ellipse. Again, when Pp is the transverse axis of 

 an hyperbola (Fig. 86.) as pE is greater than pP, there- 

 fore KV is greater than KP, and consequently greater 

 also than KE ; hence DE 2 is greater than HE 2 , and 

 </E 2 is greater than kE z , and the circle is wholly within 

 the hyperbola. Lastly, when Pp is the conjugate axis 

 of an ellipse (Fig. 87.), as Pp is less than KP, and Pp 

 : KP: : PE : PV, therefore PE is less than PV ; hence 

 KV is less than KE, and consequently DE 2 is less than 

 HE- ; also dE z less than hE z , therefore the circle is 

 wholly without the ellipse. 



Prop. III. 



The circle of curvature at the vertex of any diameter 

 of a conic section which is not an axis, meets the conic 

 section again in one point only, and between that point 

 and the vertex of the diameter the circle falls wholly 

 within the conic section on the one side, and wholly 

 without it on the other. 



Case I. Let the section be a parabola, of which Pp pjg, gg. 

 is a diameter (Fig. 88.), and PKL the circle of curva- 

 ture at the vertex, cutting off from the diameter a seg- 

 ment PK equal to its parameter. Draw KL a diame- 

 ter of the circle, and draw PO perpendicular to KL, 

 this line will necessarily meet the circle again, let it 

 meet the circle in I ; draw IS parallel to the tangent 

 at P, meeting the chord PK in S ; then because IP is 

 perpendicular to KL; IS l zrPS.PK (Lemma); hence 

 (Prop. 13. Sect. IV.) I is a point in the parabola. Let 

 DE d, an ordinate to the diameter Pp, meet the arch 

 PKI any where in H ; draw HG perpendicular to KL, 

 meeting PK in M, then because KP is equal to the pa- 

 rameter, as in Prop. I. Case 1. DE 2 : HE 2 : : KP: KM 

 : : KO : KG. But wherever the point H be taken in the 

 arch PKI, KO is greater than KG, therefore DE 2 is 

 also greater than HE 2 ; thus the arch PHKI falls 

 wholly within the parabola. 



" As the reasoning applies alike to the ellipse and the hyperbola, to avoid a number of figures, those for the hyperbola ara omitted. 



