CONIC SECTIONS. 



137 



EWrxe. 



3. Each of thepoint9 F,/is called a, focus. 



4. The given ratio of CF to CA is called the dt'er- 

 mining ratio. 



5. The point C is called the centre of the ellipse. 

 Con. to Def. 1. The ellipse passes through the 



points A, a ; and these are the only points in which it 

 can meet the line Aa : For, since CA : CP : : CF : CA, 

 therefore, C\— CF : CP— CA : : CA : CP : : CF : CA, 

 (19. 5. E.), that is, FA : AP :: CF : CA. Also, since 

 C a : CP : : CF : C a, therefore C a + CF : CP + C a : : 

 CF : Ca ; that is, Fa L a P : : CF : C a or C A. The truth 

 of the second part of the corollary is evident from the 

 way in which a line is divided into parts having a given 

 ratio. 



Con to Def. 1, 3, 4. If a straight line MFm be drawn 

 through F either of the foci, at right angles at Art, and 

 FM and Fin be each taken to FP in the determining 

 ratio, the points M, m will be in the ellipse ; and these 

 are the only points in which the line Mm can meet the 

 curve. 



Def. 6. The straight line Aa is called the Transverse 

 Axis. 



7. The points A, a, the extremities of the transverse 

 axis, are named the Vertices. 



8. The distance of either focus from the centre is 

 called the Eccentricity. 



Con. fen Def. 2, 3, 4, 6. The distance between the foci 

 has to the transverse axis, also the transverse axis has 

 to the distance between the directrices, the determin- 

 ing ratio. 



Prop. I. 



The sumof twostraightlines drawn from any point D 

 in the ellipse to the foci, is equal to the transverse axis. 



For ED : DF : : Vp ■ Aa, 



andeD:D/:: Yp: Aa, 



therefore ED -f eD : DF -f- T>f: : Vp : Art : 



But ED + eB=Pp, therefore DF-\-Df=Aa. 



Scholium. 



Some writers on conic sections employ this property 

 in defining the ellipse thus: Let F andjfbe two given 

 points in a plane, and let a point D move on that plane 

 so, that the sum of its distances from the given points 

 may be equal to a given line j then the path of the mo- 

 vim- point will be an ellipse. 



The property in question affords also an easy mecha- 

 nical description of the curve : Let the extremities of a 

 string, equal in length to the transverse axis, be fixed 

 at the foci F,f, and let the string be stretched into the 

 Straight lines DF, T)f by a pin held at D, while the 

 point of the pin is moved upon the plane, then it is 

 manifest that the point will trace an ellipse. 



Prop. II. 



Let FM, a perpendicular to the transverse axis at 

 either focus, terminate in the ellipse at M, and let a 

 circle be described on F as a centre, with FM as a ra- 

 dius, then if, from any point D in the curve, a line DF 

 be drawn to the focus, and produced to meet the circle 

 in I, and IN be drawn perpendicular to the axis, the 

 rectangle contained by the variable lines PN, FD is 

 equal to the constant rectangle PF.FM. 



Draw DE perpendicular to the directrix, and join 

 EF and PI : Then, because FM or FI : FP : : FD : 

 DE (Cor. to Def. 1, "■, 4), and the angles EDF, PFI 

 are equal r "). J. E.), the triangles EDF, PFI are 

 similar (6. 6. E.), therefore FE is parallel to PI (28. 



VOL. VII. PART I. 



1. E.) : Hence the angles EFP, IPN 'are equal, and Ellipse, 

 the angles at P and N being right angles, the tri- 

 angles EPF, PNI are also equiangular, and FD : FI 

 (: : FF : PI) : : FP : PN (4. 6. E.), therefore FD.PN^ 

 FP.FI^FP.FM (16.6. E.) 



Con. From this proposition we may determine D, 

 the point in which a line FD given by position, pass- 

 ing through the focus meets the curve, viz. by taking 

 a point I in DF produced, such that PF may be to FI 

 in the determining ratio, and drawing IV perpendicular 

 to PF, and taking FD a fourth proportional to PN, 

 PF, and IF. 



Scholium. 



From this proposition we may get a correct notioa 

 of the figure of the ellipse, by considering the changes 

 which take place in the magnitude of the line FD, 

 while the angle PFD gradually changes, increasing 

 from nothing to two right angles. For let us first 

 suppose the line FD to come to the position FA, 

 then the point N will recede to L, the extremity of the 

 diameter of the circle farthest from P ; and as in every 

 position of the line FD, the rectangle PN.FD is equal 

 to the constant rectangle PF.FM, when the point N is 

 at L, so that PN is the greatest possible, then FD will 

 be the least possible. Suppose, now, the line FD to 

 depart from the position FA, and to revolve about F ; 

 then, as the angle PFD increases, the point N will ap- 

 proach to L', the extremity of the diameter nearest to 

 P, arriving at it when the revolving line coincides with 

 Pa : the line PN will now be the least, and (because 

 the rectangle PN.FD is equal to the constant rectangle 

 PF.FM) the line FD will be the greatest possible : and 

 as PN has had all degrees of magnitude between the 

 two limits PL and PL', so FD will have had all de- 

 grees of magnitude between FA and Fa. 



If the line FD be supposed to continue its revolu- 

 tion, and to move to the other side of the axis Aa, it 

 will decrease exactly as it had increased, until at last, 

 after a complete revolution, it returns to its least mag- 

 nitude FA. Upon the whole, we may draw the fol- 

 lowing conclusions : 



1. The ellipse is a continuous line, surrounding a 

 space in which the foci and centre are situated. 



2. Of all the lines which can be drawn from either fo- 

 cus to the curve, the shortest is the part of the conju- 

 gate axis between the focus and nearest vertex, and the 

 greatest is the remainder of that axis ; and of the rest, 

 that which is nearer to the least is less than that more 

 remote. 



3. Lines drawn from either focus to the curve, making 

 equal angles with the axis, and lying on opposite sides 

 of it, are equal to one another. 



Definition 8. 

 A straight line Bb passing through the centre, at p; , f & 

 right angles to the transverse axis, and terminating in 

 the curve, is named the Conjugate Axis. 



Prop. III. 

 The distance of either extremity of the conjugate axis 

 from either focus is equal to the semitransverse axis. 



For, draw BF to either of the foci, and BG perpen- Fig. 9. 

 dicular to the directrix. Then FB : BG : : CA : CP or 

 BG, therefore FB=CA. 



Con. 1. The conjugate axis isbisectedat the centre, for, 

 join F6 ; and, because FB=:F/;, the angle FBC is equal 

 to F b C (5. 1. E.), and the angles at C being right an- 

 gles, and the side FC being common to the triangles 

 s 



