Archive image from page 507 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni09rees Year: 1819 CONSTRUCTION. r ture of the parabola, â / = bx, and ' = â ; fubftituting therefore thefe values for .ir, and it will be'â + ⢠b- b .f , 2 c y' b + y' + 2 dy + c' + da â a' = o. Or, multiplying by I1, b c + b- X y' ± 2 Ji3 X J> + « + <P â <f x b: = o. Which may rcprefent any biquadratic tquation that wants tbe fccond term ; fince fueh values may be found for a, b, c, and d, by comparing this with any
Archive image from page 507 of The cyclopaedia; or, Universal dictionary. The cyclopaedia; or, Universal dictionary of arts, sciences, and literature cyclopaediaoruni09rees Year: 1819 CONSTRUCTION. r ture of the parabola, â / = bx, and ' = â ; fubftituting therefore thefe values for .ir, and it will be'â + ⢠b- b .f , 2 c y' b + y' + 2 dy + c' + da â a' = o. Or, multiplying by I1, b c + b- X y' ± 2 Ji3 X J> + « + <P â <f x b: = o. Which may rcprefent any biquadratic tquation that wants tbe fccond term ; fince fueh values may be found for a, b, c, and d, by comparing this with any propofed biquadratic, as to make them coincide. And then the ordinates from the points, P, P, P, P, on the axis will be equal to the roots of that propofed biquadratic. And this may be done, though the parameter of the parabola (viz. b) be given : that is, if you have a parabola already made or given, by it alone you may refo've all biquadratic equations, and you will only reed to vary the centre of your circle and its radius. J f the circle defcribed from the centre, C, pafs through the Verte s I | then CP' = CA: = CDI+ A D that is, a1 = d' + c' ; and the a(t term of the biquadratic (â'- r a' â a') will vamfh ; therefore, dividing the reft byr, there arifes the cubic, -- b- x y ± 2 db:=o. Let the cubic equation propofed to be refoived -h r o. Compare the terms of thefe two equa- tions, and you will have ±: 2 b c + b2 = + p, and ± 2 db- = 4- /-, or, T c = â + â¢â, and d = ±-tt- From 2 2 5 2U' which you have this of the cubic y3 ;£py ± r = O, by means of any g;ven parabola APE. From the point B tale in the axis (forward, if the equation has âpi but , if p is pofitivej the line BD = -±â ; then raife the perpendicular D C = ââ, and from C 2 0 - b~ defcribe a circle pacing through the -vertex A, meeting the pa- rabola in P, fo fhatl the ordinate P M be one of the roots of the cubic y3 + py ± r = o. The ordinates that ftand
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