A treatise on the application of analysis to solid geometry, commenced by D.F. Gregory, concluded by W. WaltonJ. Deighton, 1852 - 310 pages |
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Page 71
... Hyperboloid of one sheet . Let one of the coefficients , as R , be negative , so that the general equation ( a ) becomes Px2 + Qy2 - Rz2 = H ..................... . ( 1 ) . If we seek the points where this surface is cut by the line we ...
... Hyperboloid of one sheet . Let one of the coefficients , as R , be negative , so that the general equation ( a ) becomes Px2 + Qy2 - Rz2 = H ..................... . ( 1 ) . If we seek the points where this surface is cut by the line we ...
Page 72
... Hyperboloid of one sheet . 84. The equation to the surface which limits the surface towards the axis of z may be found by eliminating l , m , n , between and = х y ī m = 2 -- n Pl2 + Qm2 - Rn ” = 0 . On dividing each term of the latter ...
... Hyperboloid of one sheet . 84. The equation to the surface which limits the surface towards the axis of z may be found by eliminating l , m , n , between and = х y ī m = 2 -- n Pl2 + Qm2 - Rn ” = 0 . On dividing each term of the latter ...
Page 73
... hyperboloid may be put in a form similar to the second one of the ellipsoid by introducing corre- sponding geometrical quantities . Let OA , OA ' , ( fig . 20 ) , be the distances from the origin at which the surface is cut by the axis ...
... hyperboloid may be put in a form similar to the second one of the ellipsoid by introducing corre- sponding geometrical quantities . Let OA , OA ' , ( fig . 20 ) , be the distances from the origin at which the surface is cut by the axis ...
Page 74
Duncan Farquharson Gregory. 86. Hyperboloid of two sheets . Let two of the coefficients of the equation ( a ) be ... Hyperboloid of two sheets : see fig . ( 21 ) . If we assume a2 = H P , we see , as in the previous cases , that a is ...
Duncan Farquharson Gregory. 86. Hyperboloid of two sheets . Let two of the coefficients of the equation ( a ) be ... Hyperboloid of two sheets : see fig . ( 21 ) . If we assume a2 = H P , we see , as in the previous cases , that a is ...
Page 75
... hyperboloid of one sheet if one of them . only be negative ; and an hyperboloid of two sheets if two of them be negative . If all three be negative , the surface will be imaginary . Hence , directing our attention to the gene- ral ...
... hyperboloid of one sheet if one of them . only be negative ; and an hyperboloid of two sheets if two of them be negative . If all three be negative , the surface will be imaginary . Hence , directing our attention to the gene- ral ...
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Common terms and phrases
axis centre chords coefficients condition cone constant coordinate planes cosines cosn cosß cosv cosy curvature curve of contact cylinder determined developable surface dF dF dF dx diametral plane direction-cosines dv du dv dx dy dz dx² eliminating ellipse ellipsoid equa equal expression find the equation formulæ geometrical given line Hence homogeneous function hyperbolic hyperbolic paraboloid hyperboloid infinite number Let the equations line of intersection lines of curvature locus Multiplying normal plane origin osculating circle osculating plane parameters perpendicular plane curve plane of yz planes parallel positive projection Px² quantities Qy² ratios rectangular ruled surfaces Rz² second degree second order sections shew singular points sphere straight line substitute tangent plane three equations values vanish variables x₁ x²² Y₁ y²² zero
Popular passages
Page 16 - To express the area of a triangle in terms of the coordinates of its angular points.
Page 307 - A Treatise on the Application of Analysis to Solid Geometry. Commenced by DF GREGORY, MA, late Fellow and Assistant Tutor of Trinity College, Cambridge ; Concluded by W. WALTON, MA, Trinity College, Cambridge.
Page 10 - It, so that PR is equal to MN. Now the inclination of a straight line to a plane is the angle which the line makes with the intersection of the plane and a plane perpendicular to it passing through the line. Since, then, PM and QN are perpendicular to ABCD, the plane of PQMN is also perpendicular to it, and the inclination of PQ to the plane AB CD is measured by the angle between PQ and MN or the equal angle QPR.
Page 280 - The sum of the squares of the projections of any three conjugate diameters on a fixed line is constant. Instead of projecting the diameters on the line directly, it is better to project the coordinates of the extremities of each diameter, and add them. Now, if X...