Then since CD, PT, and vX are parallel (Art. 96, 101.), TP is therefore parallel to Xv a side of the triangle XCv, : (2. 6.) Cv: vX:: CP: PT, and (22. 6.) Co? : vX' :: CP: : PT°; :: (19. 6.) (02-CP2 : vX-PT2 :: CP2 : PT2. But 1. Co?CP (cor. 5.2.) Co---CP. Co+CP=Pv.vG. 2. (cor. Art. 138.) v X-Qv=-PT or vX-PT=Qv3. (Art. 140.) PT=CD; substituting these results for their equals in the above ana. logy, it becomes Pv.vG : Qu2 :: CP? : CD2. Q. E. D. Cor. Hence Pv.v G o Qv2. 142. The parameter P to any diameter PG is a third proportional to the major axis VU, and the conjugate DO to the diameter PG; that is, P: DO :: DO: VU. Let Mm be the ordinate to the diameter PG which passes through the focus F, which is therefore the parameter P (Art. 102.); then will Mv=<P (Art. 138.). Then because CD, FM are parallel, Cr : CP :: Fe : Pe (2. 6.), and Cr2 : CP2 : : Fe : Pe? (22. 6.), :: dividendo Cyl-CP2 : Cp2 Fe-Pel : Pe. But (Art. 141.) Pr.rG: Mr* :: CP: CD'; - alternando (Pr.rG=) Cr? - CP2 : CP2 :: Mr? : CD', ::: Mr2 : CD! :: Fee - Per : Per. But Fe’ - Pe= Fe Pe. Fe + Pe (cor. 5. 2.) = FP. PS (Art. 109.) =CD’ (Art. 132.); : Mrs : CD: :: CD: : Pe; and (22. 6.) Mr : CD :: CD :(Pe=by Art. 169.) VC; :: (15. 5.) 2Mr or P: D0 :: DO: VU. Q. E. D. 143. If two hyperbolas PQq, PWw be described on the same diameter GP and from any point N in it the ordinates NQ, NW be drawn, NQ shall have a given ratio to NW. In GP produced take any other point n, and from it draw the ordinates ng, nw ; then (cor. Art. 141.). PN.NG : Pn.nG :: NQʻ: nq? :: NW? : nw? ; :-NQ:nq :: NW : nw (22. 6.), and NQ : NW :: ng : nw (16.5.). Q. E. D. Cor. 1. Hence, as in the parabola (Art. 29, and cor.) and the ellipse (Art. 69. cor. 2.) the area NQP: area NWP in a given ratio. Also, if any point v be taken in the axis and vQ, vW be joined, the area PQv ; the area PWv in a given ratio. N Cor. 2. Hence, if PQq be an hyperbola, and from every point N, n, &c. in the diameter, ordinates NQ, nq, &c. be drawn, and if straight lines NW, nw, &c. be drawn from the points N, n, &c. making a given angle with NQ, nq, &c. and having a given ratio to each other, the curve PWw passing through P, and the extremities of those lines, will be an hyperbola, having PG for its diameter. For NQ2 : NW? :: nq2 : nw? :: PN.NG : Pn.nG, that is, nq« PN.NG- (cor. Art. 141.) which is the property of the hyperbola. 144. If two hyperbolas PQq, PWw be described on the same diameter PG, and NQ, NW an ordinate to each be drawn from the same point N, tangents at Q and W will intersect the diameter PG in the same point T. Let QT be a tangent at Q, and join TW; TW is a tangent ; for if not, let it meet the hyperbola again in w, draw the ordinates nw, nq, and produce nq to meet the tangent TQ produced in s. Then because the triangles QTN, STN are similar, as also TNW, Tnw, :;: (4. 6.) NQ : ns (:: TN: Tn) :: NW: nw. But (Art. 143.) NQ : nq :: NW: nw, :: NQ : ns :: NQ : nq ::- (9.5.) ns=nq, the greater equal to the less, which is absurd; :: TW which meets the hyperbola, cannot cut it; TW is therefore a tangent. Q. E. D. Cor. Hence, if GP be the major axis of the hyperbola PQp, since (cor. 1. Art. 117.) tangents at Q and W will in like manner meet the axis GP in the same point T, ::: (Art. 121.) CN.CT =CP, :; (17.6.) CN: CP :: CP: CT. 145. If PM be the diameter of curvature at the point P, and PL, PR chords of curvature, the former passing through the centre C, and the latter through the focus F, then will MP produced be perpendicular to the semi-conjugate diameter EC, and PC:CE:: CE: 4 PL VC: CE :: CE: – PR First. Let PQ be a nascent arc common to the hyperbola and circle of curvature, draw Qv parallel to the tangent PT, join VOL. II. Kk G с н E F .. R CF, and draw the chords PQ, QL, LM, MR. Then the triangles QPv, QPL having the angle QPv common, and (29. 1.) PQv= TPQ=(32.3.) QLP, are equiangular, ::: (4.6.) Pv: PQ :: PQ: PL, ::: (17.6.) Pv.PL =PQ?; but since the arc PQ is indefinitely small, Qu and PQ will be indefinitely near a coincidence, and therefore may be considered as equal, :: Po.PL= PQ:= Qu', also for the same reason vC= PC. But (Art. 141.) Pv.vG:(Qv?=) Pv.PL :: PC: : CE, (15.5.) (VG=) 2 PC: PL :: PC : PL:: PC? : CE?, ::: (cor. 2, 20. 6.) PC: CE:: CE L : ;PL. Secondly. The triangles PCH, PML having the vertical angles at P equal (15. 1.) and likewise the angles at H and L right angles (31. 3. and construction), are equiangular, and PH: PC :: PL:PM:: 1 PL : į PM; but by the former case PC : CE :: CE: PL, ::: ex æquo PH : CE :: CE: PM. Thirdly. The triangles PKH, PMR are similar (15. I, 31.3. and construction) :: PK : PH :: PM : PR (4. 6.) :: PM : PR (15.5.). But, as in the preceding case PH : CE :: CE : PM, : ex æquo (PK=by Art. 109.) VC: CE :: CE:+PR. Q. E. D. Cor. Hence, because 2VC: 2CE:: 2CE : PR by the above, and 2VC: 2CE :: 2CE: the parameter (Art. 142.) :: the chord of curvature PR, passing through the focus, is equal to the parameter. IT U A С 146. If a cone ABD be cut by a plane PVp which meets the opposite cone Abd in any point U except the vertex, the section PVp will be an hyperbola. Let dHbKA be the opposite cone, let BD d be perpendicular to pP; bisect UV in C, draw VL, CF, US, and bd parallel to the diameter BD of the base, then will the section passing through VL, CF, US, and bd parallel to the base be circles (13. 12.) and HK, Pp the intersections of the cutting e plane with the planes of the circles HbKd, pBPD will be parallel (16. 11.). Draw CT a tangent to the circle TFs, then (36.3.) BN.ND=PN2 and bn.nd=Kna, also sC.CF=CT_. Now the triangles VNB, sCV are similar, as are UND, UCF, :;(4. 6.) VN:NB :: VC: Cs and UN: ND :: UC : CF, :: (compounding these analogies) VN.UN: BN.ND :: VC.UC: Cs.CF; that is, VN.NU : PN? :: VCR : CT2.: (Art. 116.) the figure PVp is an hyperbola, C the centre, CV the semi-major axis, and CT the semi-minor axis. Q. E. D. Cor. Hence the section HUK will be the opposite hyperbola to PVp and similar to it; for Vn: nd :: VC : Cs and Un : nb :: UC : CF, (compoundiny) Vn.nU: dn.nb :: UC.VC: Cs.CF, or (as above) Vn.nU : nK? :: VC2: CT2. N P The foregoing are the principal and most useful properties of the Conic Sections ; a branch of knowledge, which is absolutely necessary to prepare the Student for the Physico Mathematical Sciences; many more properties of these celebrated curves might have been added, if our prescribed limits had permitted; but it would require a large volume, to treat the subject in that comprehensive and circumstantial manner, which its importance demands; we must therefore refer the reader, for a more ample detail, to the writings of Apollonius, De l'Hôpital, Hamilton, Emerson, &c. observing in conclusion, that what is here given will, as far as relates to this subject. be fully sufficient to enable him to understand Sir Isaac Newton's Principia, or any other work usually read by Students, on Mathematical Philosophy and Astronomy. THE END. Printed by Bartlett and Newman, Oxford. |