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 xii
... ellipsoid ib x2 a2 b2 y2 + 1 , made by a plane lx + my + nz - 0 123-126 . Circular sections 127-128 . Any two circular sections belonging to different series lie on the surface of the same sphere 129-132 . Conditions that the equation ...
... ellipsoid ib x2 a2 b2 y2 + 1 , made by a plane lx + my + nz - 0 123-126 . Circular sections 127-128 . Any two circular sections belonging to different series lie on the surface of the same sphere 129-132 . Conditions that the equation ...
Page xv
... ellipsoid . Another demonstration of the perpendicularity of the prin- cipal sections • 251 252 253 To prove that the curvature of any normal section is equal to the sum of the curvatures of the two principal sec- tions multiplied ...
... ellipsoid . Another demonstration of the perpendicularity of the prin- cipal sections • 251 252 253 To prove that the curvature of any normal section is equal to the sum of the curvatures of the two principal sec- tions multiplied ...
Page 69
... Ellipsoid . Let H , P , Q , R , be all positive , so that the equation is Px2 + Qy2 + Rz2 = H ......... Let the straight line x y 2 = = = n ī m ... ( 1 ) . . ( 2 ) meet the surface in the point x , y , z ; then the combination of ( 1 ) ...
... Ellipsoid . Let H , P , Q , R , be all positive , so that the equation is Px2 + Qy2 + Rz2 = H ......... Let the straight line x y 2 = = = n ī m ... ( 1 ) . . ( 2 ) meet the surface in the point x , y , z ; then the combination of ( 1 ) ...
Page 73
... 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 of x , and put each of them equal to a ; let OB , OB ' , each ...
... 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 of x , and put each of them equal to a ; let OB , OB ' , each ...
Page 75
... ellipsoid , if P , Q , R , be all positive ; an 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 ...
... ellipsoid , if P , Q , R , be all positive ; an 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 ...
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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...