SPACE.
Space is tri-dimensional in points, three independent data being necessary and sufficient to determine the position of a point. Of such data the simplest are the distances, Cartesian rectan gular co-ordinates, x, y, 2, of a point P from three fixed planes XOY, YOZ, ZOX, each per pendicular to the other two. These co-ordinate planes determine three lines, called axes, having a common point 0, the Positive directions are indicated by the arrow heads. The cylindrical co-ordinates of a point I' are z and a set of ordinary polar co-ordinates p and a taking the place of x and y in the XY plane. The polar co-ordinates of a point P, Fig. 12, are the radius vector p, and the vecto rial angles 9, and 0, reckoned respectively from the pole 0 and the polar axes OE and ON. If the pole coincide with the origin, and the polar axes with OX and OZ, then x= p cos 0 sin 0, y = p sin 0 sin 4, s = p cos 8, P Vx' yi + tan 0 =y :x, cos 0 =--- s :p, formulae of transformation from one system to the other. If a, p, y, are the direction angles
(made respectively with OX, OY, OZ) of the radius vector p of any point P, Fig. 11, then x=p cos a, y= P cos P, z=p cos y. and 1---cos'a cos'i3+ cosy, sum of squares of direction-cosines of p. Any linear equation As+By+Cs-t-DO represents a plane. The x y c symmetric equation of a plane is a b s a, b, c being the axial intercepts extending from O to the plane. In normal form the equation of the plane is x cos a + y cos /3 +z cos y= p where p is the length, and cos a, cos /3, cos y are the direction-cosines, of the perpendicular from 0 to the plane. To convert Ax + By + Cs + D = 0 to the normal form it suffices to multiply it by the normalizing factor, 1: V A' + B' 0, the new coefficients A: V , B:V , C:V , being cos a, cos 0, cos y, and D: V being — p. The angle 0 between two planes Ax ± By + Cz ± D=0 and A'x B'y + Cs + D'=0 is determined by the relation cos ((AA' ± BB' ± CC):