This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1903 Excerpt: ...the corresponding lines of reference x = 0, y = 0 is a double point on the curve; this point is a node, conjugate point, or cusp, according as this quadratic factor has real, imaginary, or equal linear factors. For suppose that the functions flt f2 have the common quadratic factor a a + 6 A + c. The parameters of the ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1903 Excerpt: ...the corresponding lines of reference x = 0, y = 0 is a double point on the curve; this point is a node, conjugate point, or cusp, according as this quadratic factor has real, imaginary, or equal linear factors. For suppose that the functions flt f2 have the common quadratic factor a a + 6 A + c. The parameters of the points of intersection of any straight line Ax + By--0 passing, through the intersection of the lines of reference x = 0 and y = 0 are given by AM) + Bf2() = 0. This equation is equivalent to (a+bk + c)(k-d) = 0, hence all such straight lines meet the curve in two points corresponding to the values of A given by the equation ak + bk + c = 0. The coordinates of these points are the same, viz. (0:0:1); consequently this is a double point on the curve. If the third value of A, viz. d, is equal to either of those given by the quadratic equation above, the corresponding straight line meets the curve in three coincident points, i. e. is a tangent to the curve at the double point. Since one tangent corresponds to each root of the quadratic equation, the point is a. node, conjugate point, or cusp, according as the roots of this equation are real, imaginary, or equal. Example (i). To find the nature of the double point on the Mrve x (y-x) = z(y-2xf. Any point on this curve satisfies the equations x = y _ a (A-l)a-1)2(A + 1) A' Any straight line, Ax + By = 0, meets the curve at points whose parameters are given by (X-l)a + B(X + l) = 0. The common quadratic factor (A--V)1 gives two equal roots 1, 1; hence the point is a cusp. If the third value of A is also 1, then A + 1B = 0, consequently the straight line y--2x = 0 is the tangent at this cusp. Example (ii). (as---)2 (z--y) = 2yz, . The coordinates of any point on this curve satisfy the implicit equ..
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