Elements of Analytical Geometry: Embracing the Equations of the Point, the Straight Line, the Conic Sections, and Surfaces of the First and Second Order

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Wiley & Long, 1836 - 352 páginas
 

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Página 258 - P'p, drawn perpendicular to the co-ordinate planes, may be regarded as the three edges of a parallelopipedon, of which the line drawn to the origin is the diagonal. We have therefore verified a proposition of geometry, viz : the sum. of the squares of the three edges of a rectangular parallelopipedon is equal to the square of its diagonal. Scholium 4. This last result offers an easy method of determining a relation that exists between the cosines of the angles which a straight line makes with the...
Página 283 - A plane is a surface, in which, if two points be assumed at pleasure, and connected by a straight line, that line will lie wholly in the surface.
Página 40 - The equation of a line is the equation which expresses the relation between the co-ordinates of every point of the line. 4. In the equation y...
Página 191 - That the difference of the squares of any two conjugate diameters is equal to the difference of the squares of the axes. Hence, there can be no equal conjugate diameters unless A=B, and then every diameter will be equal to its conjugate : that is, A'=B'.
Página 140 - I —e cos v Scholium 6. It should be remarked, that the common numerator in these values of r, is equal to half the parameter of the transverse axis.
Página 325 - Y' + cos. Z cos. Z' cos. U = cos. X cos. X" + cos. Y cos. Y
Página 315 - Let the surface of this cone be now intersected by a plane passing through the axis of Y, and consequently perpendicular to the co-ordinate plane ZX; and designate by u the angle DAX, which the secant plane makes with the co-ordinate plane YX. The equation of this plane will be the same as that of its trace AD (Bk. VIII, Prop. II, Sch. 3) : that is, z = x tang M. If we combine this equation with the equation of the surface, and eliminate z, we shall obtain the equation of the projection of the curve...
Página 137 - MRS^ at a point on the indifference curve we can do so by drawing tangent at the point on the indifference curve and then measuring the slope by estimating the value of the tangent of the angle which the tangent line makes with the X-axis.
Página 126 - Hence, the squares of the ordinates to either one of two conjugate diameters, are to each other as the rectangles of the segments into which they divide the diameter. Scholium 4. This property enables us to describe an ellipse by points when we know two conjugate diameters and the angle which they form with each other. Let AB, ED, be two conjuif' J gate diameters. Turn ED round the centre C, until it becomes perpendicular to AB, and then describe an ellipse on AB and E'D
Página 143 - ... parabola referred to the rectangular axes of which A is the origin. Scholium 1 . The axis of abscissas AX is called the axis of the parabola, and the origin A is called the vertex of the axis, or principal vertex; and the constant quantity 2p is called the parameter . The equation of the parabola gives from which we see, that for every value of x there will be two equal values of y with contrary signs. Hence, the parabola is symmetrical with respect to its axis. We see further, that y will increase...

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