Cor. 2. TN.NC=QN2 (cor. 8. 6. and 17. 6.) =CQ-CN2 (47.1.) VC2-CN2= (cor. 1. Art. 67.) VN.NU.

Cor. 3. The sub-tangent NT is greater than 2VN; for since (by the preceding) TN.NC=VN.NU, (16. 6.) NT: VN :: NU: NC; but CU NC, (NC+CU=) NU > 2NC, NT > 2VN.

Cor. 4. If PG be the normal, then (cor. S. 6. and 17. 6.) TN.NG=PN2, and TN.NC: TN.NG:: VC: L (cor, 1. Art. 67. and cor. 1. Art. 72.). NC: NG: VU: L (15.5.).

73. If HP be a diameter and KO its conjugate, then PM being drawn perpendicular to KO cutting the axis VU in G, the rectangle PM.PG=EC2.

For if Cy be drawn parallel to PM, the angle PGN=yCG (29. 1.), but yCG+yCt=(GCt=)

a right angle, and ytC+yCt=a right angle (32. 1.), yCG+y Ct =ytC+yCt; take away the common angle yCt, and the remainder yCG=ytC, PGN=(yCG=) yt C, and PNG Cyt being right angles;

the triangles PGN, Cyt are equiangular (32. 1.); and PG: (PN= by 34. 1.) Cn :: Ct: (Cy=) PM (4. 6.); PM.PG=Cn.Ct (16. 6.) EC2 by cor. 1. Art. 72. Q. E.D.

74. Join PS, then if PG be drawn perpendicular to Tt, and Gk perpendicular to PS, Pk=L.

For the angles at k and M being right angles, and the angle kPM common, the triangles PMR, PKG are equiangular (32. 1.). PR: PM: PG: Pk (4. 6.), and PR.Pk-PM.PG (16.6.) EC2 (Art. 73.), . (PR=by Art. 60.) VC: EC:: EC: Pk (16.6.). But VC: EC:: EC: L (Art. 56.), Pk=L (9.5.). Q. E. D.

75. If PC, CO be semi-conjugate diameters, and PN, Om be perpendicular to the axis, then will CN2+ Cm2=VC2.

For VC2-Cm2: Om2 :: VC2 EC2 (cor. 1. Art. 67.) :: VC2-CN2: PN (Art. 67.) But OC being parallel to tT, and the angles at m and N right angles, . (29. 1.) the triangles COm,

PNT are similar, and (4.6.) Om: Cm :: PN: NT; ·.· (22. 6.) Om2: Cm2 :: PN2 : NT2, ·.· from this and the first analogy (22. 5.) VC2—Cm2 : Cm2 :: VC2—CN2 : NT2. But CN.NT : NT2 :: CN: NT (1.6.) ·.· by inversion Cm2: VCa— Cm2 :: NT : CN, and by composition VC: VC2-Cm2:: CT: CN :: (1.6.) CN CT: CN2. But VC2=CN.CT (cor. 1. Art. 72.), Cm2== CN2 (14.5.), ·.· VC2=CN2 + (m2. Q. E. D.

Cor. 1. Hence VC2—CN2—Cm2, ·:· Cm2 : PN2 :: VC' EC by the first analogy in the proposition, and Cm : PN:: VC : EC (22. 6.). In like manner, because VC3 — Cm3 =CN3, ·: CN2 : 0m2 :: VC2 : EC2, and CN : Om :: VC : EC.

Cor. 2. Hence also Cm : PN :: CN : Om, ·.· (16.6.) Cm.Om =PN.CN.

76. If PN, Om be perpendicular to the axis VU, and PC, CO semi-conjugate diameters, then will PN+Om'=EC'.

For CN: Om2 :: VC2 : EC2 (cor. 1. Art. 75.), :: VC3— CN2 : PN2 (cor. 1. Art. 67.) ·. summing the antecedents and consequents (12. 5.) VC2 : 0m2 +PN2 :: VC-CN3 : PN2 :: (Art 67.) VC2 : EC2, ··· Om2 + PN2=EC2 by 14. 5. Q. E. D.

Cor. 1. Because CP and CO are semi-conjugate diameters to each other, CP will be parallel to a tangent at 0; and Cn2+ Cr2 = (0m2+PN2 34. 1.=) EC2; and hence the same relation subsists between the ordinates and abscissas to the minor axis, that does between those to the major axis.

77. CP+CO2=VC2 + EC.

For VCCN2 + Cm2 (Art. 75.), and EC’=PN3 +0m' (Art. 76.); ·.· VC2+EC2=(CN2 +PN2+Cm2 +0m2=) CP' + CO2 (47.1.). Q. E. D.

78. If Ve a tangent to the major axis, be made equal to the semi-minor axis, and C be joined cutting PN, any ordinate to the major axis in M; then will MN1 +PN2=Ve2.

For the triangles eVC and MNC being similar (2. 6.) Ve: MN :: CV: CN, and Ve2 : MN2 :: CV2 : CN2 (22. 6.), ':: Ve2 : Ve2 MN: CV CV-CN (prop. E. 5.) :: Ve: PN (cor. 1. Art. 67.); ... Ve2 — MN2 = N(14.5.), and consequently MN2 + PN2 =Ve'. Q. E D.

:

Cor. Because MN' + PN=(Ve2=) EC2, see also the figure to Art. 73. and Om2 + PN2 =EC2 (Art. 76.); ·: MN=0m, and MO being joined, it will be parallel to the axis VU (33. 1.). Hence, if a straight line OC be drawn from the extremity O of the parallel

MO, through the centre C, it will be the conjugate diameter to PC; and hence by this proposition, having any diameter of an ellipse given, the position of its conjugate may be readily determined.

78. If PC, CO be semi-conjugate diameters, and PM be drawn perpendicular to CO (see the figure to Art. 73.) then will CO.PM VC.EC.

Because PN, Om are perpendicular to the axis, and Cy perpendicular to the tangent, (cor. 1. Art. 75.) CN: Om :: VC: EC, and (16. 5.) CN : VC :: Om: EC; and the triangles TCy, OCm being similar CT: Cy :: CO: Om (4. 6.), the two latter analogies being compounded (prop. F. 5.) CN.CT: VC.Cy :: CO: EC; but (because CN.CT=VC', cor. 1. Art. 72.) VC2: VC.Cy: VC: Cy: CO EC; (16. 6.) VC.EC=OC.Cy= OC.PM (34.1.) Q. E. D.

:

Cor. 1. Let VC=a, EC=b, PC=x, Cy=y, then (Art. 77.) CO (VC + EC' - PC' =) a2 + b2 — x', '.' y2 = (Cy' = VC.EC

CO'

=)

a2b2
a2 + b2 — x2 °

Cor. 2. Hence, if at the vertices of two diameters which are conjugates to each other, tangents be drawn, a parallelogram will be circumscribed about the ellipse, the area of which is 4C0.PM a constant quantity. See the figure to Art. 58.

79. If CP, CO be semi-conjugate diameters, then will FP.SP =CO'.

P

T

m

M

C

For the triangles SPt, PRM, FPT are similar, because TF, PM, and tS are parallel, the angles at T, M, and t right angles, and TPF=tPS (cor. 3. Art. 57.)=PRM (29. 1.); ·: SP : St :: PR : PM, and ́ FP : FT :: PR : PM (4.6.), these analogies being compounded (prop. F. 5.) SP.FP: St.FT :: PR: PM. But (Art. 78.) VC.EC= OC.PM, ·: (VC=by Art. 60.) PR: PM :: OC: EC (16. 6.); and PR2 : PM2 : : OC2 : EC2 (22. 6.) ; ·.· from above SP.FP : St.FT:: OC': EC; but St.FT=EC2 (Art. 61. B.) SP.FP OC2 (14.5.) Q. E. D.

E

R

S

80. Let OX be the conjugate and Qu an ordinate to the diameter PG, then will Pv.vG: Qv2 :: PC2 : CO2.

Draw PN, vn, QH, and Om perpendicular to the axis VU, and vr parallel to it. Then because PN is parallel to Qr, vr to TN, and Qu to PT, the triangles PTN, Qur are equiangular, and (4. 6.) Qr (rv=by 34. 1.) Hn::

m

CN (22. 6.). But (cor. 1. Art. 67.) VC-CH: QH:: VC: —CN3 : PN2 (being each as VC2 : EC1) ·: ex æquo (22.5.)

CN
NT

2

VC –CH*: Hn+Cn: VC-CN: CN :: (cor. 2.

Art. 72.) CN.NT. CN2:: (15.5.) NT: CN; : (since CN:

VC:: VC: CT by cor. 1. Art. 72.; whence, by cor. 2, 20. 6,

ст

NT

CN

Hn2 (by reduction, and from the figure); ·.· CN2 – Cn2 =

Ст.

NT Hn3 (by dividing by CN), or NT.CN-Cn3=CN.Hn3 ;

(16. 6.) CN2 — Cn2: Hn2 :: CN: NT :: (by inversion in the 7th analogy, above) CN2: VC-CN; (16. 5.) CNCn' : CN: Hn: VC2-CN2; but (2.6.) CN: Cn: CP: Cv, : CN3 - Cn2: CN2 :: CP2 — Cv2 : CP2 (part 4. Art. 69.). Also, (by similar triang. and 22. 6.) rv2=Hn2 (Cm2=by Art. 75.) VC2—CN2 :: Qv2 ; CO2; ·.· (CP2 — Cv2=cor. 5. 2.) Pv.vG : CP2 :: Qv2 : CO2, and (16.5.) Pv.vG : Qv2 :: PC2 : CO2. Q. E. D.

:

Cor. Hence it may likewise be shewn by similar reasoning, that if Qu be produced to meet the curve again in q, Pv.vG : qua :: PC: CX3, ·: Qv : qv :: CO: CX. But CO=CX (cor. Art. 58.), ·.· Qv=qv.

81. The parameter P to any diameter PG is a third proportional to the major axis and conjugate diameter; that is, VU: OX:: OX: P.

Let the ordinate Qv passing through the focus F meet the curve again in q; then will Qq be the parameter to the diameter PG, and (cor. Art. 80) Qv=÷P. Because (Pv.vG=) PC-Cv2: Qv2 :: PC: CO (Art. 80.). Qv2 : PC-Cv CO: PC(prop. B.5.) But because Ce is parallel to vF o (Art. 60.) Pe=VC, PC-Cv : (Pe-Se') Pe-er :: Pu2 : Pr2 ; ; PC : Pe2 ·. ex æquo (22.5.) Qv2

P.

F

S

G