Elements of analytical geometry and the differential and integral calculus . A^ sin. m cos. m-^B^ sin. n cos. w=0. (3) A^B^sin.^7i—m)=A^BK (4) The values of m and w must be taken so as to respond to thefollowing equation, because the rectangular axes are in factconjuffate diameters. u4^ |-52 «=0. (5) These equations unfold two very interesting properties. Scholium 1. By adding (1) and (2) Or 4A-\-4B^=4A^-\-4BK Thai is, the sum of the squares of any two conjugate diameters isequal to the sum of the squares of the axes. Scholium 2. Equation (3) or (5) will give us m when n i
Elements of analytical geometry and the differential and integral calculus . A^ sin. m cos. m-^B^ sin. n cos. w=0. (3) A^B^sin.^7i—m)=A^BK (4) The values of m and w must be taken so as to respond to thefollowing equation, because the rectangular axes are in factconjuffate diameters. u4^ |-52 «=0. (5) These equations unfold two very interesting properties. Scholium 1. By adding (1) and (2) Or 4A-\-4B^=4A^-\-4BK Thai is, the sum of the squares of any two conjugate diameters isequal to the sum of the squares of the axes. Scholium 2. Equation (3) or (5) will give us m when n isgiven ; or give us n when m is given. Scholium 3. The square root of (4) gives^^sin. (n—m)=ABy which shows the equality of two surfaces, one of which is ob-viously the rectangle of the two us examine the other. Let n represent the angle^CB, and m the angle the angle NCF will berepresented by (n—m). Since the angle J/lY^is thesupplement of NCP, the twoangles have the same sineNM=^ the right angled triangle NKM, we have\ \ A \\ sm.(n—m) : THE ELLIPSE. 61 MK=zA sin.(7j—7»),But NC=B. Whence MK-NC=^AB ^r[s.,{n—m)= the parallelogramNQPM. Four times this parallelogram is the parallelogramML, and four times the parallelogram -D (7^^ which is measuredby AB, is equal to the parallelogram HF. Hence equation (4)reveals this general truth: The rectangle which is formed hy drawing tangent lines throughthe vertices of the axes is equivalent to any parallelogram whichcan he found hy drawing tangents through the vertices of conju-gate diameters. Note.—The student had better test his knowledge in respect to the truthsembraced in scholiums 1 and 3, by an example: Suppose the semi-major axis of an ellipse is 10, and the semi-minor axis 6,and the inclination of one of the conjugate diameters to the axis of X is takenat 30*^ and designated by m. We are required to find il^ and B^, which together should equalA^-^B^, or 136, and the area NCPM, which should e
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Keywords: ., bookauthorrobinson, bookcentury1800, bookdecade1850, bookyear1856
