A Treatise on the Analytic Geometry of Three DimensionsHodges, Smith, 1862 - 465 pàgines |
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Frases i termes més freqüents
ax² axes axis centre circle co-ordinates coefficients coincident points condition cone confocal conjugate consecutive points constant corresponding cos² cosẞ cosy cubic cuspidal edge denote determine diameter differential equation direction-cosines double point double tangent drawn eliminating ellipsoid envelope equal evidently expressed fixed plane fixed point geodesic given line given point given surface Hence Hessian hyperbola hyperboloid infinity inflexional tangents last article line joining line of curvature locus meets the surface number of points origin osculating plane parallel perpendicular plane at infinity Plane Curves plane meets plane of xy point of contact point x'y'z polar plane quadric radii radius of curvature radius vector reciprocal represents right angles right line ruled surface second degree sphere substituting surface of revolution tangent plane theorem touch the surface triangle umbilic values vanish vertex
Passatges populars
Pàgina 19 - Art. 12, and must have 4-4* equal to the sum of the squares of the coefficients of x, y, and z, in the equation of the last article. 32. To find the length of the perpendicular from a given point x'y'z on a given plane. If we draw through x'y'z...
Pàgina 199 - From the expressions in this article we deduce at once, as in the theory of central conies, that the sum of the reciprocals of the radii of curvature of two normal sections at right angles to each other is constant ; and again, if normal sections be made through, a pair of conjugate tangents (see Art.
Pàgina 65 - Shew that the sum of the squares of the projections of three conjugate diameters of a conicoid on any line, or on any plane, is constant.
Pàgina 226 - ... intersection. If however the two surfaces should touch, every plane through the point of contact meets the surfaces in two curves which touch. Every such plane therefore passes through two coincident points of the curve of intersection : we arrive then at the important result, that " if two surfaces touch, the point of contact is a double point on their curve of intersection.
Pàgina 285 - The constant pd has the same value for all geodesies which touch the same line of curvature.
Pàgina 81 - ... length as infinite. This seems unjustifiable a priori. The present investigation takes the dimensions of the slits into account. 2. Suppose we have an aperture or diaphragm of any shape in a screen placed just in front of the object-glass of a telescope (fig. 1). Fig. 1. Let the axis of the telescope be the axis of z. Let the axes of x and y be taken in the plane of the diaphragm OQ perpendicular to and in the plane of the paper respectively. Let S be a source of light whose coordinates are U,...
Pàgina 4 - ... point where the line is met by a plane drawn through the point perpendicular to the line. Thus, in figure, p. 2, if the axes be rectangular, D, E, F are the projections of the point P on the three axes. The projection of a finite right line upon another right line is equal to the first line multiplied by the cosine of the angle between the lines. Let PP be the given line, and DD
Pàgina 229 - This problem will have a definite number of solutions, and the number will plainly be the number of tangents which can be drawn to the curve from an arbitrary point ; that is to say, the class of the curve.
Pàgina 17 - The angle between the planes is the same as the angle between the perpendiculars on them from the origin. By the last article we have the angles these perpendiculars make with the axes, and thence, Arts. 13, 14, we have AA...
Pàgina 213 - ... equation of the projection of the lines of curvature on the plane of xy. Thus, in the present case, multiplying by -5 and reducing by the equation of the ellipsoid and its differential, we have tf(V-<?) , a" (a" -J") ., or writing ,„ ,