Elements of Analytical Geometry: Embracing the Equations of the Point, the Straight Line, the Conic Sections, and Surfaces of the First and Second OrderA. S. Barnes & Company, 1867 - 352 páginas |
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Outras edições - Ver todos
Elements of Analytical Geometry: Embracing the Equations of the Point, the ... Charles Davies Visualização completa - 1847 |
Elements of Analytical Geometry: Embracing the Equations of the Point, the ... Charles Davies Visualização completa - 1859 |
Elements of Analytical Geometry: Embracing the Equations of the Point, the ... Charles Davies Visualização completa - 1856 |
Termos e frases comuns
A²B² A²y² Analytical Geometry asymptotes axis of abscissas axis of X B²x² bisect Bx+D circle circumference co-ordinate axes co-ordinate plane YX coefficients conjugate axis conjugate diameters contrary signs cos² curve described designate the co-ordinates determine directrix distance ellipse equa equal equation of condition equation will become find the equation foci formulas for passing generatrix geometrical give given line given point hence hyperbola hyperboloid imaginary imaginary curve indeterminate last equation Mz² nate negative Ny² obtain ordinates origin of co-ordinates parabola parallel perpendicular point of contact point of intersection point of tangency polar equation positive primitive axes PROBLEM Prop PROPOSITION quantities radius radius-vector rectangle represent right line satisfy the equation Scholium secant line second degree sin² square Substituting these values supplementary chords suppose supposition surface tang tangent line transverse axis triangle Trig variables vertex الله
Passagens mais conhecidas
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 256 - 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 278 - ... by*, and hence the co-ordinate z becomes known. Scholium 2. The lines in which a plane intersects the co-ordinate planes, are called the traces of the plane. These traces are found by combining the equation of the plane with the equations of the co-ordinate planes. Thus, if in the equation z + ax + by — c = 0, we make y = 0, which is the characteristic of the co-ordinate plane ZX, the resulting equation, C z + ax — c = 0, B' will designate the trace CD common to the two planes.
Página 305 - Fig. 59, be the centre of the sphere ; prq its horizontal, and p's'q' its vertical projection. Let M be the point assumed, as in Art. (107). Analysis. Since the radius of the sphere, drawn to the point of contact, is perpendicular to the tangent to any great circle at this point, and since these...
Página 276 - A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line which passes through its foot in that plane, and the plane is said to be perpendicular to the line.
Página 257 - C"P" its projection on the co-ordinate plane YZ. Now, since a line is determined in space when two of its projections are known (Des. Geom. Art. 26), it follows that the conditions which fix the projections will determine the...
Página 135 - 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 124 - AB, it may be readily shown, that, a/ 3 : x" * :: (B' + y')(B'- y') : (B ' + y")(B'-y"). 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. IF gate diameters. Turn ED round the centre C, until it becomes perpendicular to AB, and then...
Página 106 - PROPOSITION IV. THEOREM. If through the vertices of the transverse axis two supplementary chords be drawn, the product of the tangents of the angles which they form with it, on the same side, will be negative and equal to the square of the ratio of the semi-axes. The equation of a straight line passing through the point A, of which the co-ordinates are a/= — A...
Página 313 - 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 u. 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 of intersection on the co-ordinate plane YX. It is, however, better to discuss the curve in its own plane, and for this purpose we will refer it to the two axes. AY, AD, which are in the plane of the curve, and at right angles to each other....