Projection of the Sphere in Crystallography. 47 



where a, b, c arc constants, and d is any positive quantity, will 

 be denoted by the symbol hkl. 



Let the plane hkl meet the axes OX, OY, OZ in the points 

 H, K, L respectively. Then 



0H=?<^, OK=yd, OL=yrf. 

 hkl 



Therefore 



/,05=*WC^^0L 



a c 



The three letters h, k, I will be called the indices of the 

 plane HKL. The plane HKL is equally well denoted by the 

 symbol nh nk nl, where n is any positive quantity. 



26. The equations to planes parallel to the planes hkl, pqr, 

 passing through the origin, are 



a c 



X y s r. 



^ a ^ c 



and the equations to the line in which they intersect are 



x _ y _ z 

 u« ~~ vb wc' 

 where 



n = kr — lq, v = lp — hr, w=hq — kp. . . . (o) 



Any straight line parallel to the straight hue having for its 

 equations 



x__y__s_ 



ua vb wc' 



will be denoted by the symbol uvw. The three letters u, v, w 

 will be called the indices of the line. Any three quantities pro- 

 portional respectively to u, v, w, may be taken for the indices of 

 the line. 



27. Let uvw be the symbol of a plane parallel to the lines 

 hkl, pqr. If a plane be drawn through the origin of coordinates 

 parallel to the plane uviv, and if lines be drawn through the 

 origin parallel to the lines hkl, pqi', the lines will be contained 

 in the plane. The equation to a plane through the origin, par- 

 allel to the plane uvw, is 



X y z n 

 u- +Vy + w-=0, 

 a c 



and the equations to lines through the origin, parallel to the 

 lines hkl, pqr, are 



