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axis centre centric surfaces chords circle coefficients condition cone consecutive considered constant coordinate axes coordinate planes cosines cosv curve of contact cutting plane cylinder determined developable surface dF dF diametral plane differential equation direction-cosines director dv du dv dx dy dz eliminate ellipse ellipsoid equa equal expression find the equation formula geometrical given plane given point gives Hence homogeneous function hyperbolic paraboloid hyperboloid infinite number Let the equations line of intersection lines of curvature locus Multiplying negative normal plane normal sections origin osculating circle osculating plane parabola parameters plane curve plane of xy plane passing planes parallel positive principal sections projection quantities radii radius of curvature ratios rectangular relation represent right angles ruled surfaces second degree second order shews sphere straight line substitute suppose surfaces of revolution tangent plane three equations umbilicus values vanish variables zero
Page 16 - To express the area of a triangle in terms of the coordinates of its angular points.
Page 120 - Ie 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 one corresponding to the equations (A), the other to the equations (B). 135. It is easy to find the condition to which...
Page 51 - ... 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. These nine quantities are connected by certain conditions: for, since Ox' is a line, of which the direction_cosines are a, a', a", we have, by Art.
Page 35 - The angle between two planes is the same as the angle between their normal vectors, as calculated from the following equation...
Page 17 - To express the volume of a tetrahedron in terms of the co-ordinates of its angular points.
Page 288 - ... 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 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 149 - ... plane of polarization of the wave is perpendicular to this axis; the corresponding ray is parallel to the line of intersection of the tangent plane at the end of the axis and the plane containing the axis and the wave-normal; the ray-velocity is the reciprocal of the length of the perpendicular from the centre on the tangent plane. By reciprocating with respect to a sphere of unit radius concentric with the ellipsoid, we obtain a similar proposition in which the ray takes the place of the wave-normal,...
Page 308 - A cone is the surface generated by the motion of a straight line which always passes through a fixed point and a given curved line.