. Plane and solid analytic geometry . FACES 583 20. The locus of a line which so moves that it always inter-sects three fixed skew lines, not parallel to a plane, is a hyper-boloid of one sheet. Prove this theorem in the case that thefixed lines are I x = c, j X — — c, \ x = — z cot 6, 12/= 2 cos ^; [y = — zcosej \y = csindy where c ^ 0 and 6 =^ 0, n-. 21. The locus of a line which so moves that it always inter-sects three fixed skew lines, parallel to a plane, is a hyperbolicparaboloid. Prove this theorem in the case that one of thefixed lines is the axis of z and the others have the equation
. Plane and solid analytic geometry . FACES 583 20. The locus of a line which so moves that it always inter-sects three fixed skew lines, not parallel to a plane, is a hyper-boloid of one sheet. Prove this theorem in the case that thefixed lines are I x = c, j X — — c, \ x = — z cot 6, 12/= 2 cos ^; [y = — zcosej \y = csindy where c ^ 0 and 6 =^ 0, n-. 21. The locus of a line which so moves that it always inter-sects three fixed skew lines, parallel to a plane, is a hyperbolicparaboloid. Prove this theorem in the case that one of thefixed lines is the axis of z and the others have the equationsa; = c, 2; = my; x-=— c^z = — my, where cm 4^ 0. 22. A line moving so that it is always parallel to a fixedplane, 3/, and always intersects two fixed skew lines, neitherof which is parallel to 3f, generates a hyperbolic this theorem when M is the (x, 2;)-plane and the two fixedlines are the last two of the three in Ex. 21. CHAPTER XXIV SPHERICAL AND CYLINDRICAL OF COORDINATES TRANS-. 1. Spherical Coordinates. Griven a point 0, a ray OA issu-ing from 0, and a half-plane m bounded by the line of the ray OA. Let P be any point of space. Join Pto 0 and construct the half-plane, v, deter-mined by OA and OP. Denote the distanceOP by r, the angle AOP by <^, and the anglefrom the half-plane m to the half-plane p by (?•, <^, 0) are the spherical coordinates ofthe point P. For a given value, ?o) of the radius vector r,the point P lies on a sphere whose center isat 0 and whose radius is r = ro. The angle0 is the longitude of P, measured from theprime meridian m, and the angle <^ is thecolatitude (complement of the latitude), at least for a point Pon the upper half of the sphere. The radius vector r is, by definition, positive or zero. Thecolatitude shall be restricted to values between 0 and tt in-clusive : 0 < <^ < TT. The longitude 9 shall be unrestricted ; itshall be taken as positive if measured in the dire
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