Elements of analytical geometry and the differential and integral calculus . and minoraxes, is The formulas for changing rectangular co-ordinates into ob-lique, the origin being the same, are (Prop. IX, Chap. I,)x-=^x COS. m-\-y COS. n. y=ix sm. m-\-y cos, n. Squaring these, and substituting the values of x^ and y^ inthe equation of the ellipse above, we have{A^sm^ri,-\B^cos^ri)y^-\-{A^s^m-\-B^cos^m)x^ ) _^2^j 9.(A^ sin. m sin. n-\B^ cos. m cos. n)yx ) But if we now assume the condition that the new axes shallbe conjugate diameters, then A^ sin. msin. n-\B^ cos. mcos. w=0,which reduces the pre
Elements of analytical geometry and the differential and integral calculus . and minoraxes, is The formulas for changing rectangular co-ordinates into ob-lique, the origin being the same, are (Prop. IX, Chap. I,)x-=^x COS. m-\-y COS. n. y=ix sm. m-\-y cos, n. Squaring these, and substituting the values of x^ and y^ inthe equation of the ellipse above, we have{A^sm^ri,-\B^cos^ri)y^-\-{A^s^m-\-B^cos^m)x^ ) _^2^j 9.(A^ sin. m sin. n-\B^ cos. m cos. n)yx ) But if we now assume the condition that the new axes shallbe conjugate diameters, then A^ sin. msin. n-\B^ cos. mcos. w=0,which reduces the preceding equation to [(^) {A^sm.^n-\-B^Cos.^n)y^-\-{{-B^ cos.^m)x^=zA^B^,which is the equation required. But it can be simplified as fol-lows ! The equation refers to the two di-ameters BB and DD as axes. Forthe point B we must make y=0,then ^-= ^!^ = A^im.^m-\-B^GOS.^m(CBy=A-. (P)Designating CB hy A, and CD by the point D we must make x=0. Then y2= -^^^ =(CI)y=:BK (Q) A^sin.^n+B^cos.^n ^ ^ From (P) we have {A^sm.^m+B^cos.^m)=:^^LFrom (Q) (+B^cos.^n)=:^ THE ELLIPSE. These values put in (F) give x^=:A^Br B^ A^ Whence A^y+Bx^=A^B^. We may omit the accents to x and y\ as they are generalvariables, and then we have for the equation of the ellipse referred to its center and conju-gate diameters. Scholium. In this equation if we assign any value to x lessthan A, there will result two values of y, numerically equal, andto every assumed value of y less than B\ there will result twocorresponding values of Xj numerically equal, differing only insigns, showing that the curve is symmetrical in respect to itsconjugate axes, and that each axis bisects all chords which are par-allel to the other axis. Observation.—As this equation is of the same form as thatof the general equation referred to rectangular co-ordinates onthe major and minor axis, we may infer at once we can findequations for ordinates, tangent lines, &c. referred to conjugateaxes, which will be in the
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