114 



CONIC SECTIONS. 



Hyperbola. C, the middle of the line joining them, and perpendicu- 

 *— -v— lar to that line at P and p ; and in the same line, let CA 

 and C a be taken in contrary directions, each a mean 

 proportional between CF and CP ; then, if a point D be 

 conceived to move in the plane of the lines, in such a 

 manner, that DF its distance from one of the given 

 points, has to DE its distance from the given line adja- 

 cent to that point, the given ratio of CF to CA, and 

 tonsequently so that D/'its distance from the other 

 given point has to D e. its distance from the other given 

 line the same given ratio, (Seat. I. Prop. 1.) the point 

 D will describe a curve line called an Hyperbola. 



2. Each of the lines PQ, p q is called a directrix. 



3. Each of the points F, / is called nfrcus. 



4. The given ratio of CF to CA is called the deter- 

 mining ratio. 



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

 Cor. to Def. 1. S. 4. If M?«be drawn through F at 



right angles to Ff, and FM, Fw be taken, so that FM 

 and F m are each to FP in the determining ratio, the 

 points M, m are in the hyperbola, and these are the 

 «nly points in which the line M m can meet the curve. 

 Def. 6. The straight line A a is called the trans- 

 verse axis. 



7. The extremities of the transverse axis are called 

 the vertices. 



8. The distance of either focus from the centre is 

 called the eccentricity. 



Corollaries to Def. 1, 3, 4, 6. 



1. The hyperbola passes through the extremities of 

 the transverse axis. For since CA : CP : : CF : CA, 

 therefore CF— C A : CA— CP : : CF : CA ( 19. 5. E.) that 

 is, FA : AP : : CF : CA. Again, since Ca : CP : : CF : 

 Ca, therefore, C« + CF : CV + Ca : : CF : C a, that is 

 F a : a P : : CF : C a, or CA, therefore A and a are points 

 in the hyperbola. 



2. The distance between the foci has to the trans- 

 verse axis, and the transverse axis has. to the distance 

 between the directrices, the determining ratio. 



Prop. I. 



Fiz. 23,. The difference between two straight lines drawn 



from any point D in the hyperbola to the foci is equal 



to the transverse axis. 



¥o \ E S'Rl :: l p: i U \ Cor. 2. to Def.l. 3. 4. 6. 

 andcD:D/:: Vp : AftJ 



therefore eD— ED : Df— DF : : Vp : Aa. 



Put cD— ED=Pp, therefore Df— DF=Aa. 



Scholium. 

 This property of the hyperbola affords a good defini- 

 tion of the curve, and has been employed as suchby 

 many writers on conies. The following mechanical 

 description is also derived from it : Let one end of a 

 string be fastened at F, (Fig. 28.) either of the foci, 

 and the other to E, the extremity of a ruler /DE, and 

 let the difference between the length of the string and 

 the length of the ruler be equal to A a, the transverse 

 axis. Let the other end of the ruler be fixed at the 

 •ther focus f, and let it revolve about fas a centre in 

 tiie plane of the figure, while the string is stretched by 

 means of a pin at D, so that the part of it between E 

 and D is applied close to the edge of the ruler. By the 

 revolution of the ruler, the lines Df, DF will manifest- 

 ly be equally increased, and therefore their difference 

 will always be the same quantity, .yiz. the line A a; 

 hence the point D will describe a branch of a hyper- 



bola, of which A a is the transverse axis, and F, / the Hyperbola 

 foci. The other branch may be described, by fixing - — v— 

 the end of the ruler to the other focus. 



Prop. II. 



Let FM be a perpendicular to the transverse axis at Fig. 29. 

 either focus, which meets the hyperbola at M, and let No - x i i- 

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

 radius ; then, if from any point in the curve, a line DF 

 be drawn to, or through the focus, meeting the circle 

 in I, so that the points D, I may be on concrary sides, 

 or the same side of the focus, according as the focus 

 and the point D are on the same side or contrary sides 

 of the directrix, and let IN be perpendicular to the 

 axis, then the rectangle contained by the lines FD.PN 

 is equal to the constant rectangle PF.FM. 



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

 DE (Cor. to Def. 1—4.), and (from the position of the 

 lines) the angles FDE, IFP are equal (29. 1. E.) the 

 triangles FDE. IFP are similar (6. fj. E.), and the an- 

 gles DFE, FIP equal; hence FE is parallel to PI (2. 

 I.E.): Therefore the triangles FPE, PNI having the 

 angles at F and P equal (29. 1. E.), and the angles at 

 P and N being right angles, the triangles are equian- 

 gular : And because FD : FE : : IF : IP, and FE : FP : : 

 IP : PN, ex seq. FD : FP : : IF or FM : PN (22. 5. E.), 

 and FD.PN=FP.FM. . 



Cor. Hence it is evident how the point in which a 

 straight line drawn from the focus meets the curve may 

 be found, via. by taking FD a fourth proportional to 

 PN, FM and FP. 



Scholium. 

 From this proposition we may acquire a correct no- 

 tion of the figure of the curve, by considering the 

 changes which take place in the magnitude of tiie 

 line FD, while the angle PFD increases. Let us sup- 

 pose the line FD to come to the position FA on tiie 

 same side of the directrix as the focus (Fig. 29. No. i.), 

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

 diameter farthest from P ; and as PN will then be the 

 greatest possible, FD will be the least. Suppose now 

 the line FD to depart from the position FA, and to re- 

 volve about F, then the angle DFP increasing, the 

 point N will approach to P; and as the rectangle 

 FD.PN has always the same magnitude, the line FD 

 must increase. Now, the determining ratio being a 

 ratio of majority, and FM to FP in that ratio (Cor. to 

 Def. 1—4), FP must be less than FM ; therefore P 

 falls within the circle : hence it follows that PN may 

 become less, and consequently FD greater than any 

 assignable quantity. If we suppose the point I to ar- 

 rive at last at K, one of the intersections ot the direc- 

 trix and circle, then the line PN altogether vanishes ; 

 therefore, corresponding to this position, there can be 

 no intersection of the curve and the revolving line FD. 

 By supposing the line FD to depart from the position 

 FA, and to revolve in the contrary direction, or from 

 A towards m, it will appear that it increases continually, 

 until FI, its prolongation, arrive at the position FI, (/* 

 being the other intersection of the directrix and circle), 

 and here the point in which the line meets the hyper- 

 bola again has no existence. 



If we now suppose the line FD (Fig. 29- No. 2.) 

 which is drawn from the focus F to the part ol the 

 hyperbola on the other side of the directrices, to come 

 to the position Fa, and departing from thence, to re- 



Vg. 29. 

 No. I. 



Fif. 29. 



Nj. & 



