A Treatise on the Application of Analysis to Solid Geometry |
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a₁ Ax² axis centre chords co-ordinate axes co-ordinate planes coefficients condition cone constant cos² cosines cylinder d³x d³y determined developable surface dF dF diameters diametral plane differential direction-cosines distance ds² dx² eliminating ellipse ellipsoid equal expression find the equation formula geometrical Hence hyperbola hyperboloid infinite Let the equations line of intersection lines of curvature locus Multiplying normal plane normal sections origin osculating circle osculating plane P'y² P'z² P₁ perpendicular plane curve plane of xy planes parallel positive principal sections projections Px² quantities r₁ radii radius of curvature rectangular relation right angles second degree second order sin² singular points straight line substitute tangent plane umbilicus values vanish variables x₁ y₁ z₁ zero
Popular passages
Page 16 - To express the area of a triangle in terms of the coordinates of its angular points.
Page 233 - ... partly on one side, and partly on the other side of the principal railway, and that without reference to the title under which...
Page 34 - The angle between two planes is the same as the angle between two lines drawn perpendicular to them ; that is, it is equal to the angle between their normals.
Page 264 - Differentiating the first of these equations with respect to x, the second with respect to y, and the third with respect to z...
Page 10 - 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 ABCD, the plane of PQMN is also perpendicular to it, and the inclination of PQ to the plane AS CD is measured by the angle between PQ and MN or the equal angle QPR.
Page 50 - ... where a', b', c', are the cosines of the angles which the new axes make with the old axis of y, and a", b'', c", of those which they make with the old axis of z.
Page 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.
Page 182 - ... a = 0 : da this is the equation to a plane perpendicular to the axis of x, and the characteristic is the circle determined by the intersection of this plane with the sphere u = 0. From the equation to the plane it is obvious that whatever a may be, the plane does not pass through the centre of the sphere, and consequently the characteristic is in this case a small circle of the sphere. (230) It is easy to shew that the surface u = 0, in which a is supposed to be constant, is always touched by...
Page 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.
Page 14 - ... the cosines of the angles which a straight line makes with three rectangular axes. Taking the origin O (fig. 8) in the line, let POx = a, POy = /3, POz = y, and let x, y, z be the co-ordinates of any point P in the line; then if the distance OP be r, we have, by Art. (14), r...