348 Prof. Miller on the Application of 



V . DE= — V . 0B= —vb. BM is obviously parallel to OF, the 

 diagonal of a parallelopiped the edges of which are respectively 

 parallel to the axes OX, OY, OZ, and proportional to — ua, —vb, 

 — wc, or to ufl, vb, we. 



Any line parallel to BM will be p. ^ 



denoted by the symbol uvw. The 

 whole numbers u, v, w will be called 

 its indices. 



6. In OY take OB = b, and let 

 hkl, pqr be the symbols of any two 

 edges BM, BS passing through B, 

 and meeting the plane ZOX in M, S. 

 Let MS meet OZ in W, and OX 

 in U. DrawMD,SG paraUel to OZ. 

 Then, by (5), 



k.OD = -ha, k.DM=-lc, q.OG=-pa, q.GS = -rc. 

 Hence 



(kr-lq)OU = (lp-hr)«, (hq-kp)OW=(lp-hr)c. 

 Therefore 



-OU=?OV=-OW, 

 a c 



where 



«=kr— Iq, v=lp — hr, t« = hq— kp. 



Since u, v, tv are integers, the plane UBW is one of the planes 

 of the system. Hence a plane of the system may always exist 

 parallel to the edges formed by the intersections of any two pairs 

 of planes of the system. 



7. When uvw is the symbol of a plane parallel to the edges hkl, 

 pqr, the indices u, v, lo are derived from h, k, 1, p, q, r exactly 

 in the same manner as the indices u, v, w are derived from 

 h, k, 1, p, q, r, when uvw is the symbol of the edge in which the 

 planes hkl, pqr intersect. 



If the two symbols followed by the first and second indices of 

 each, be written one underneath the other, and three letters X 

 in the intervals between every four indices beginning with the 

 second, it will be seen that u= product of indices joined by the 

 thick stroke of the first X— product of indices joined by the 

 thin stroke ; and that v and w are formed in the same manner 

 from the products of the indices joined by the thick and thin 

 strokes of the second and third letters X respectively. 



h k I h k 



XXX w=.kr—lq, v = lp—hr, vf=.hq—kp, 

 p q r p q 



By this rule numerical values of u, v, w may be formed at 



