242 operation, Whence, substituting for cos II, yz, cos П, xz, and cos II, xy, their values in (305), we have, on completing the NN' PP MM' MN' NM' + MP' PM' sin3x,yz"sin3y,xz"sin z,xy +xyz siny,xz/ NP' PN' cos yz,xz+ (sin x,yz + sin x,xy) cos yz,xy + (sin y,zz + sin z,xy, cos xz,xy, second volume of the Cambridge Philosophical Transactions. For a very elegant investigation of this formula, the reader is referred to a paper by Mr. Whewell in the 310. COR. 1. When the planes are at right angles to each other, the numerator of the expression just deduced =0. 311. COR. 2. Let the co-ordinate planes be rectangular; then cos II, II' = MM'+NN'+PP' or MM'+NN'+PP' = ± √(M2+N2 + P2) (M22+N”2 + P′2) 312. PROP. 4. To find the inclination of a straight line to a plane. We shall denote the proposed line by ę, in order to distinguish it from the line p, which is appropriated to signify the perpendicular to the plane ПI. Hence, being any line, and II any plane to which it is inclined, we have sin ę, II=mM+nN+pP to radius 1....(1), sin2 z, xy .(2). ON OBLIQUE CO-ORDINATES. 243 313. COR. 1. When the line is perpendicular to the plane, mM+nN+pP = 1; 314. COR. 2. Let the axes be rectangular; mM+nN+pP then, sin e̱, II= √(m2+n2+p3) √(M3 +N2 + P2) CHAP. XI. ON THE TRANSFORMATION OF CO-ORDINATES IN SPACE. 315. PROP. 1. To pass from one system of oblique axes to another, the origin being the same. Let x, y, z and x, y, z' be the co-ordinates of a point referred to the primitive and new axes respectively. If the same reasoning be applied to the plane, which was used (72) in the case of the straight line, it may be proved that the primitive co-ordinates are linear functions of the new ones; we shall assume, therefore, Suppose that y' and z' each=0, or that the point is on AX', In like manner, by supposing the point to be successively on the axes AY', AZ', we may obtain the values of n, n', n", and of P, P', p". We have, therefore, by substitution, From the symmetrical form of these expressions, they may easily be remembered. 316. COR. 1. Let the primitive axes be rectangular, and the new ones oblique. Then the denominator in each of the foregoing formulas becomes = 1; also sin x', yz = cos x', x, and similarly for the remaining terms; we therefore have x = x' cos x', x+y' cos y', x+z' cos z', Ꮖ y = x' cos x', y+y' cos y', y+z' cos z',y (2), z = x cos x', z+y' cos y', z + cos z', z and since the primitive system is rectangular, we have the following equations : cos x, x + cos x', y+cos2 x', z= cos2 y', x + cos2 y', y + cos2 y', ; y, z= (a). cos2 z, x+cos2 z, y + cos2 z, z= 0 Hence, of the nine angles involved in (3), six only are inde pendent, since equation (a) gives three of them. 317. COR. 2. Let both systems be rectangular. |