A Treatise on the Analytic Geometry of Three Dimensions
Hodges, Smith, 1862 - Geometry, Analytic - 465 pages
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Common terms and phrases
angle axes axis becomes called centre circle co-ordinates coefficients common condition cone confocal conic conjugate consecutive consider constant contain corresponding cosẞ cosy cubic curve denote determine developable diameters differential direction distance double point drawn edges eliminating envelope equal equation evident expressed fixed focal focus follows four function geodesic give given given line Hence hyperboloid intersection length line joining locus manner meets normal obtain origin parallel pass perpendicular point of contact polar plane principal projections properties proved quadric quantity radius vector ratio reciprocal regard relation represents respect result right line roots satisfied seen sheet sides sphere square substituting surface tangent plane theorem touch transformed umbilic values vanish vertex written
Page 281 - That is, the sines of the sides of a spherical triangle are proportional to the sines of the opposite angles.
Page 160 - The locus of the foot of the perpendicular from the focus of a conic on a tangent is the auxiliary circle.
Page 185 - 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.
Page 212 - ... 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.
Page 414 - ... the commutative law a + b = b + a, and the associative law (a 3.
Page 93 - It is, that the distance of any point on the quadric from such a focus is in a constant ratio to its distance from the corresponding directrix, the latter distance being measured parallel to either of the planes of circular section.
Page 262 - ... is impossible. Cor. 3. The centre of a sphere, and the centre of any small circle of that sphere, are in a straight line perpendicular to the plane of the circle. Cor. 4. The square of the radius of any small circle is equal to the square of the radius of the sphere diminished by the square of the distance from the centre of the sphere to the plane of the circle (B. IV., P. XI., C. 1): hence, circles which are equally distant from the centre, are equal ; and of two circles which are unequally...
Page 254 - Then, if x', y, z, be the coordinates of a point on the curve of intersection of the plane and the helicoid at a distance r from the new axis of y, we have x' = r cos 6, z = r sin 0, and thus, by (32), y = r(rcos0+p) r sin 0.
Page 215 - 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.
Page 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...