A treatise on the application of analysis to solid geometry, commenced by D.F. Gregory, concluded by W. WaltonJ. Deighton, 1852 - 310 pages |
From inside the book
Results 1-5 of 69
Page 41
... dy + ( z - z ' ) dz = 0 , ( x − x ' ) dx + ( y — y ' ) dy ' + ( z — z ' ) dz ' But , from the equations to the lines , dx ī = dy dz da ' m = — - dy ' dz ' and = n ľ m n = 0 . ; hence , eliminating the differentials by dividing each ...
... dy + ( z - z ' ) dz = 0 , ( x − x ' ) dx + ( y — y ' ) dy ' + ( z — z ' ) dz ' But , from the equations to the lines , dx ī = dy dz da ' m = — - dy ' dz ' and = n ľ m n = 0 . ; hence , eliminating the differentials by dividing each ...
Page 84
... dx ' + m df dy ' df + n r + Rr2 = 0 ..... ( 3 ) , dz ' where the terms after the third vanish , because the ( 84 ) CHAPTER VI Theorems relating to Surfaces of the Second Degree 5788 67 95-99 Diametral planes.
... dx ' + m df dy ' df + n r + Rr2 = 0 ..... ( 3 ) , dz ' where the terms after the third vanish , because the ( 84 ) CHAPTER VI Theorems relating to Surfaces of the Second Degree 5788 67 95-99 Diametral planes.
Page 85
... dx ' dy ' dz ' Writing now the equation to the surface at full length , it is Ax2 + By2 + Cz2 + 2A'yz + 2B'zx + 2C'xy + 2A ′′ x + 2B " y + 2C " z + E = 0 ; and therefore f ( x , y ' , z ' ) is a function of the same form , which , it ...
... dx ' dy ' dz ' Writing now the equation to the surface at full length , it is Ax2 + By2 + Cz2 + 2A'yz + 2B'zx + 2C'xy + 2A ′′ x + 2B " y + 2C " z + E = 0 ; and therefore f ( x , y ' , z ' ) is a function of the same form , which , it ...
Page 129
... ( dy d'z - dz d'y ) + d3y ( dz d3x - dx d3z ) + d3z ( dx d3y - dyd3x ) = 0 as the required condition . If x be considered as the indepen- dent variable of which y and z are functions , d3x = 0 , d3x = 0 , and the preceding relation ...
... ( dy d'z - dz d'y ) + d3y ( dz d3x - dx d3z ) + d3z ( dx d3y - dyd3x ) = 0 as the required condition . If x be considered as the indepen- dent variable of which y and z are functions , d3x = 0 , d3x = 0 , and the preceding relation ...
Page 139
... dy , dz , to which they are ultimately proportional , we have , from ( 3 ) , dx dy - Τ m = dz n Differentiating the equation ( 1 ) , we have .. ( 4 ) . dF dx dx + dy + dy dF dF dz = 0 ....... ( 5 ) . dz Eliminating da , dy , dz ...
... dy , dz , to which they are ultimately proportional , we have , from ( 3 ) , dx dy - Τ m = dz n Differentiating the equation ( 1 ) , we have .. ( 4 ) . dF dx dx + dy + dy dF dF dz = 0 ....... ( 5 ) . dz Eliminating da , dy , dz ...
Other editions - View all
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...