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 67
... Cones . 1st , Let H = 0 ; then some one at least of the quantities P must be of a different sign from the others , in order that the equation Px2 + Qy2 + Rz2 = 0 may represent a surface : for , if all be of the same sign , the only ...
... Cones . 1st , Let H = 0 ; then some one at least of the quantities P must be of a different sign from the others , in order that the equation Px2 + Qy2 + Rz2 = 0 may represent a surface : for , if all be of the same sign , the only ...
Page 68
... cones , the common right cone being a particular case of them . If we put z = h in ( 1 ) , it becomes Px2 + Qy2 = Rh2 , which is the equation to an ellipse parallel to the plane of ( x , y ) , the origin being in the axis of z , and the ...
... cones , the common right cone being a particular case of them . If we put z = h in ( 1 ) , it becomes Px2 + Qy2 = Rh2 , which is the equation to an ellipse parallel to the plane of ( x , y ) , the origin being in the axis of z , and the ...
Page 69
... cone , the vertex of which is in the origin , unless 1st , The coefficients of the transformed equation are all of the same sign , when it represents a point . 2nd , One of the coefficients of the transformed equation is zero , the ...
... cone , the vertex of which is in the origin , unless 1st , The coefficients of the transformed equation are all of the same sign , when it represents a point . 2nd , One of the coefficients of the transformed equation is zero , the ...
Page 72
... cone of the second degree , by Art . ( 79 ) . It is easily shewn that this cone is an asymptote to the hyperboloid . For , if z ' and z be coordinates of points in the cone and the surface corresponding to the same values of x and y ...
... cone of the second degree , by Art . ( 79 ) . It is easily shewn that this cone is an asymptote to the hyperboloid . For , if z ' and z be coordinates of points in the cone and the surface corresponding to the same values of x and y ...
Page 73
... cone lies between the axis of z and the surface . 85. The equation to the 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 ) ...
... cone lies between the axis of z and the surface . 85. The equation to the 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 ) ...
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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...