A Treatise on the Application of Analysis to Solid GeometryDeightons, 1845 - 276 páginas |
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Términos y frases comunes
Ax² axis centre chords co-ordinate axes co-ordinate planes coefficients condition cone constant cos² cosines cylinder d²z d³x d³y determined developable surface dF dF dF dF dx diameters diametral plane differential equation direction-cosines ds² dx² eliminate ellipse ellipsoid equa equal expression find the equation formula geometrical given Hence homogeneous function hyperbola hyperbolic paraboloid hyperboloid Let the equations line of intersection lines of curvature locus multiply normal plane normal sections origin osculating circle osculating plane P'y² P'z² P₁ parameters perpendicular plane curve plane of xy planes parallel positive projection Px² quantities r₁ radii radius of curvature rectangular second degree second order sin² singular points sphere straight line substitute tangent plane values vanish variables x₁ x² y² y₁ zero
Pasajes populares
Página 16 - To express the area of a triangle in terms of the coordinates of its angular points.
Página 112 - As k admits of an indefinite number of values, and to each value of k there corresponds a position of the line in each system, we may, by assigning a proper series of values to k, cause the line represented by either (A) or (B) to trace out the surface (1). Hence there are two ways in which the hyperboloid of one sheet may be generated by the motion of a straight line, the onecorresponding to equations (A), the other to (B). (135) It is easy to find the condition to which must be subject the direction-cosines...
Página 233 - ... partly on one side, and partly on the other side of the principal railway, and that without reference to the title under which...
Página 34 - The angle between two planes is the same as the angle between their normal vectors, as calculated from the following equation...
Página 38 - Find the length of the perpendicular from a given point on a given straight line in space.
Página 267 - ... the coefficient of friction when the whole pressure upon the axis takes place at the upper ring. 21. The sum of the squares of the projections of any three conjugate diameters of an ellipsoid (whose semi-axes are a, b, c) upon a given principal diameter is constant ; and the tangent planes at the extremities of three conjugate diameters intersect in an ellipsoid whose equation is r2 I/2 i* JL tJ e* a* b* c2 22.
Página 9 - Fundamental Theorems. (12) Theory of Projections. When a point is referred to a plane by means of a straight line drawn parallel to a fixed axis, the point where the line meets the plane is called the projection of the point on the plane. Thus in fig. (1) A is the projection of P on the plane of yz, В is the projection on the plane of xz, and (7 that on xy.
Página 10 - PQ on ABCD. Since PM and QN are both perpendicular to the same plane, they are parallel to each other ; in the plane therefore in which they lie draw PR parallel to MN, and meeting QN in R, 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...
Página 14 - This last result offers an easy method of determining a relation that exists between the cosines of the angles which a straight line makes with the co-ordinate axes.
Página 256 - Q or the radius of curvature of an oblique section is the projection, upon the plane of this curve, of the radius of curvature of the normal section which passes through the same tangent line. (298) A line of curvature in any surface is the locus of a series of its consecutive points, such that the normals at each point shall meet the normal at the consecutive one. The equations to the normal are x' - x = y - y ^ z - z UVW ' Let each member of this equation be represented by Q.