 | Isaac Wilber Jackson - Conic sections - 1845 - 116 pages
...: : AC2 : MO'*, , _,.-, __, BC'xAC* and F'MxFM = MQfa = CN2. (Cor. 1.) PROPOSITION XXIII. THEOREM. The difference of the squares of any two conjugate...diameters, is equal to the difference of the squares of the axes. That is, (Fig. 25,) MM'2 — NN"» = AAra — BB". From the triangles MCF and MCF', we have (Geom... | |
 | James Devereux Hustler - Conic sections - 1845 - 85 pages
...tangents are drawn at A and B, CD coincides with CB, and PF with AC. Hence CDxPF=ACxBC. PROP. XVI. The difference of the squares of any two conjugate...diameters is equal to the difference of the squares of the axes. Draw CZX perpendicular to AB and PD, Then CP2-CD* = PX*-DX* = 4Pixi^(Eucl. 8. n.) But LX : OZ... | |
 | Nathan Scholfield - Conic sections - 1845 - 542 pages
...CA) (CM+CA). = CM2— CA" CA3 = CM2— Cm* And similarly, CB3 = dmt—PM'. PROPOSITION XV. THEOREM. The difference of the squares of any two conjugate diameters, is equal to the same constant quantity, namely, the difference of the squares of the two axes. That is, if Pp, Dd,... | |
 | Nathan Scholfield - Conic sections - 1845 - 244 pages
...CA) (CM+CA). = CM'— CA' CA' = CM'— Cm' And similarly, CB' = dm'— PM'. PROPOSITION XV. THEOREM. The difference of the squares of any two conjugate diameters, is equal to the same constant quantity, namely, the difference of the squares of the two axes. That is, if Pp, Dd,... | |
 | Nathan Scholfield - 1845 - 894 pages
...CA) (CM+CA). = CM'— CA' CA'=CM'— Cm' And similarly, CB' = am1— PM1. PROPOSITION XV. THEOREM. The difference of the squares of any two conjugate diameters, is equal to the same constant quantity, namely, the difference of the squares of the two axes. That is, if P/?, T)d,... | |
 | Elias Loomis - Conic sections - 1849 - 252 pages
...equal to CA ' —CH ' or AHxHA'; hence CA ' : CB ' : : CG ' : EH ' . PROPOSITION XV. THEOREM. The sum of the squares of any two conjugate diameters, is equal to the sum of ike squares of the axes. Let DD', EE' be any two conjugate diameters; then we shall have DD''+EE"=AA''+BB''.... | |
 | James Hann - Conic sections - 1850 - 146 pages
...b'2 = a2 - J2 From equation (3), 4a'6'sin(a' — a)=4a5 (1), (2), (3). (5). Equation (4) shews that the difference of the squares of any two conjugate diameters is equal to the difference of the square of the principal axes. Equation (5) shews that the rectangle described on any system of conjugate... | |
 | John Radford Young - Geometry, Analytic - 1850 - 294 pages
...) and (2),' A'2— Bx2 — A2— B2 (4), . that is the difference of t lie squares of any system of conjugate diameters is equal to the difference of the squares of the principal diameters. , From equation (3) there results 4Ax Bx sin. [A' B'] = 4AB. Hence, as in the... | |
 | Elias Loomis - Calculus - 1851 - 300 pages
...Art. 69, Cor. A 2A1 5, and that of the minor axis to -^-. PROPOSITION XIII. — THEOREM. (88.) The sum of the squares of any two conjugate diameters is equal to the sum of the squares of the axes. Let DD', EE' be any two conjugate diameters. Designate the co-ordinates... | |
 | James Haddon - Calculus, Differential - 1851 - 190 pages
...¿QCA=<¡,. - . (1) .dp p ab ab ,_ С'-Г' ~ dp,~ ab .. («*)• Hence But since, in an ellipse, the sum of the squares of any two conjugate diameters is equal to the sum of the squares of the major and minor axes, therefore (2л)2 + (25)2=(2г-)2 or et i% з /^Г 3... | |
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Fig. 83,84. conjugate diameters is equal to the sum of the squares of the axes ; but in an hyperbola the difference of the squares of any two conjugate diameters is equal to the difference of the squares of the axes. 
