. Plane and solid analytic geometry . e is60°; (e) The lower nappe of the cone of (d) ;(/) The cone of (d). 6. What do the following pairs of equations represent ? (a) r = S, cf> = 120°; (d) r = 3, tan2 c^ = 1; (6) r = 3, (9 = |7r; (e) tan (9 = 2, <^ = f tt; # (c) 0 = 30°, <^ = 45°; (/) tan ^ = - 1, cos^ <^ = i. 7. Find the equations in spherical coordinates of the fol-lowing curves: (a) The small circle on the earth of colatitude 47°;(6) The semicircle on the earth of longitude 135°; (c) The complete circle determined by the semicircle of (6); (d) The ray from 0 of colatitude 60°
. Plane and solid analytic geometry . e is60°; (e) The lower nappe of the cone of (d) ;(/) The cone of (d). 6. What do the following pairs of equations represent ? (a) r = S, cf> = 120°; (d) r = 3, tan2 c^ = 1; (6) r = 3, (9 = |7r; (e) tan (9 = 2, <^ = f tt; # (c) 0 = 30°, <^ = 45°; (/) tan ^ = - 1, cos^ <^ = i. 7. Find the equations in spherical coordinates of the fol-lowing curves: (a) The small circle on the earth of colatitude 47°;(6) The semicircle on the earth of longitude 135°; (c) The complete circle determined by the semicircle of (6); (d) The ray from 0 of colatitude 60° and longitude 25°. 8. Determine the locus of each of the following equations:(a) r = 4 cos <^ ; (5) r = 6 sec <^ ; (c) r = 3 esc <^. Find the equations in spherical coordinates of the followingsurfaces. Identify each surface. 9. a;2-f 2/2+^2= 9. 13. 3x-^2y = 0. 10. a;2 + ?/2_A:V = 0. 14. 3^-4 = 0. 11. 0^2 + 2/2+22^4?/. 15. 2a; + 5 = 0. 12. 4(aj2 + 2/^)+922 = 36. 16. x^ = 2yz. SPHERICAL AND CYLINDRICAL COORDINATES 587. Fig. 3 2. Cylindrical Coordinates. Given a point 0, the axis of zthrough 0, and the plane K through 0 perpendicular to theaxis of z. In K introduce a system of polar coordinates, asshown. Let P be any point of space and let N be its projec-tion on K. Then the polar coordinates, r and $, of N and thedirected line-segment NP = z deter-mine the position of P. The threenumbers, taken together, are knownas the cylindrical coordinates (r, 6, z)of P. As in the case of polar coordinatesin the plane (Ch. X, § 1), r is re-stricted to be positive or zero, while0 is unrestricted. The positive direc-tion of rotation for the measurementof 6 is as indicated in the figure. If ro (> 0), 6q, Zq are particular values of r, 0, z, the equationr = ?o represents a circular cylinder whose axis is the axis of2; 6 = Oq defines a half-plane issuing from the axis of z, andz = Zq represents a plane perpendicular to the axis of z. Transformation to and from Rectangular Coo
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