A Treatise on the Application of Analysis to Solid GeometryDeighton, 1852 - 310 pàgines |
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A treatise on the application of analysis to solid geometry, commenced by D ... Duncan Farquharson Gregory Visualització completa - 1845 |
A Treatise on the Application of Analysis to Solid Geometry Duncan Farquharson Gregory,William Walton Visualització completa - 1852 |
A Treatise on the Application of Analysis to Solid Geometry Duncan Farquharson Gregory,William Walton Visualització completa - 1845 |
Frases i termes més freqüents
axis centre chords coefficients condition cone constant coordinate planes cos² cosines cosß cosv cosy curve of contact cutting plane cylinder determined developable surface dF dx diametral plane direction-cosines dv du dv eliminating ellipse ellipsoid equa equal expression find the equation formula geometrical given line given point Hence homogeneous function hyperbolic hyperbolic paraboloid hyperboloid Let the equations line of intersection line passing lines of curvature locus Multiplying negative normal plane origin osculating circle osculating plane P₁ parameters perpendicular plane curve plane of yz planes parallel positive projection Px² quantities Qy² r₁ radii of curvature ratios rectangular ruled surfaces Rz² second degree second order shew singular points sphere straight line substitute suppose tangent plane three equations values vanish variables x₁ x²² y₁ y²² zero
Passatges populars
Pàgina 16 - To express the area of a triangle in terms of the coordinates of its angular points.
Pàgina 120 - 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 one corresponding to the equations (A), the other to the equations (B). 135. It is easy to find the condition to which the direction-cosines...
Pàgina 283 - Differentiating the first of these equations with respect to x, the second with respect to y, and the third with respect to z...
Pàgina 14 - ... 1 , a very important relation, to which we shall frequently refer. The cosines of the angles which a straight line makes with the co-ordinate axes are quantities which we shall often have occasion to use, and as they serve to determine the direction of the line, we shall call them the direction-cosines of the line ; and when we wish to speak of a straight line with reference to its direction-cosines, which we may call I, m, n, we shall name it the line [I, m, n\.
Pàgina 14 - ... -y) cosy. It is obvious that this gives us the expression for the length of a diagonal of a parallelepiped in terms of the sides and the angles which they make with each other. 16. To find the relation between 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 = 7, and let a;, y, z, be the coordinates of any point P in the line ; then, if the distance OP be r, we have, by Art. (14), r2 = xL +...
Pàgina 51 - J where a', 6', 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. These nine quantities are connected by certain conditions: for, since Ox...
Pàgina 35 - The angle between two planes is the same as the angle between their normal vectors, as calculated from the following equation...
Pàgina 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.