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 xv
... respectively by the squares of the cosines of the angles which the principal planes make with the normal plane Condition that two of the coordinate planes may coincide with the principal sections at any point of a surface 286-287 ...
... respectively by the squares of the cosines of the angles which the principal planes make with the normal plane Condition that two of the coordinate planes may coincide with the principal sections at any point of a surface 286-287 ...
Page 2
... respectively , and meeting the coordinate planes in A , B , C ; then the position of the point P is known if we know the lengths of the lines PA , PB , and PC , which are called the coordinates of P. For if along the line Ox we mea ...
... respectively , and meeting the coordinate planes in A , B , C ; then the position of the point P is known if we know the lengths of the lines PA , PB , and PC , which are called the coordinates of P. For if along the line Ox we mea ...
Page 13
... respectively , and adding , we have , by the previous equation , - r2 = ( x , − x ) 2 + ( y , − y ) " + ( z , - z ) " , which is the required expression . If the point be the origin , then x1 = 0 , y1 = 0 , z1 = 0 , and we have for ...
... respectively , and adding , we have , by the previous equation , - r2 = ( x , − x ) 2 + ( y , − y ) " + ( z , - z ) " , which is the required expression . If the point be the origin , then x1 = 0 , y1 = 0 , z1 = 0 , and we have for ...
Page 14
... respectively , and adding , we have , in consequence of the preceding equation , 22 = ( x , − x ) 2 + ( y , − y ) 2 + ( ≈ , − 2 ) 2 + 2 ( y , − y ) ( 2 , − 2 ) cosa - - - - +2 ( 2 , -2 ) ( x , x ) cosẞ + 2 ( x , x ) ( y , -y ) ...
... respectively , and adding , we have , in consequence of the preceding equation , 22 = ( x , − x ) 2 + ( y , − y ) 2 + ( ≈ , − 2 ) 2 + 2 ( y , − y ) ( 2 , − 2 ) cosa - - - - +2 ( 2 , -2 ) ( x , x ) cosẞ + 2 ( x , x ) ( y , -y ) ...
Page 15
... respectively . There- fore if the magnitude of the area be denoted by A , and those of its projections on yz , zx , and xy , by A , A , A , we shall have A = A cosα , A = A cos ß , A = A cosy . x Squaring and adding these , and ...
... respectively . There- fore if the magnitude of the area be denoted by A , and those of its projections on yz , zx , and xy , by A , A , A , we shall have A = A cosα , A = A cos ß , A = A cosy . x Squaring and adding these , and ...
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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...