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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A.S. Barnes & Company, 1856 - 352 páginas
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Página 279 - 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 55 - ... the tangent of the angle which the line makes with the axis of abscissae), was lately employed by M. Crova* for the discussion of experiments relating to the degree of constancy possessed by so-called
Página 254 - 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 153 - 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 187 - 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 321 - Y' + cos. Z cos. Z' cos. U = cos. X cos. X" + cos. Y cos. Y
Página 291 - V" cos, V", and add, we find cos s F + cos 3 7"+ cos s F" = 1 ; that is, the sum of the squares of the cosines of the three angles which a plane forms with the three co-ordinate planes, is equal to radius square or unity.
Página 154 - ... the parameter of any diameter is equal to four times the distance of its vertex from the focus.
Página 139 - ... 2p#, which is the equation of the 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....
Página 122 - 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.

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