By Smith, Charles, 1844-1916
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Are x = 0, y = 0, z= Find the co-ordinates of the centre of the sphere in 25. scribed in the tetrahedron formed by the planes whose equations =- a. are y + z - 0, z + x - 0, x + y = 0, and x + y + z CHAPTER III. SURFACES OF THE SECOND DEGREE. The most general equation of the second degree, viz. 50. 2 ax* + + cz* + 2fyz + 2^,7? + 2htvy -f 2w + 2vy + 2wz + d = 0, contains ten constants. But, since we may multiply or divide the equation by any constant quantity without altering the relation between as, y, and z which it indicates, there are really only nine constants which are fixed for any particular surface, viz.
Find the equations of the straight lines which bisect the angles x y .. ,, , = z- and, x-. = u = -z . between the lines T = m m , I m n V m n = r. Then be two points, one on each line, such that the co-ordinates of P are li\ mr, nr, and of Q are I r, m r, n r; hence the co ordinates of the middle point of PQ are ^(1 + / ) r, ^ (//* + m ) r, % (n + n ) r. Since , Let P, Q S. S. G. . OPOQ 2 T&E STRAIGHT 18 middle the l m+m +l n on is point . By + ri. , 1 the equations ; = required of the equations are of the bisector n- the preceding Article cos 6 therefore sin = A/ therefore sin 26.
Closed polyhedron on any plane is zero. Find the co-ordinates of the centre of the sphere in 24. scribed in the tetrahedron formed by the planes whose equations and x + y + x=l. are x = 0, y = 0, z= Find the co-ordinates of the centre of the sphere in 25. scribed in the tetrahedron formed by the planes whose equations =- a. are y + z - 0, z + x - 0, x + y = 0, and x + y + z CHAPTER III. SURFACES OF THE SECOND DEGREE. The most general equation of the second degree, viz. 50. 2 ax* + + cz* + 2fyz + 2^,7?