the Angles formed by any Planes of Crystals. 315 



This plane may be represented by (p ; q ; r). And another 

 plane for which the corresponding quantities are p 1 , q\ r\ 

 will be represented by {p 1 ; 5 1 ; r 1 ). Let the dihedral angles at 

 the lines, x, y, z, respectively, be a, /3, 7 ; and let 6 be the 

 angle contained by the planes (p ; q ; r) and (p 1 ; g 1 ,• r 1 ), then 

 we shall have, in all cases, 



. cog & _ PP l +qf+rr l — (pq l +p>q) cos 7— Q'+pV) cos/3— (qr l +q*r) cos« 

 ^{(l^+f+r- — 2pq cos 7 — 2pr cos j3—2qrcos a) fp' a + &c.)| 



The second factor of the denominator differing from the first, 

 in having p\ q\ r 1 instead of p,q,r. 



Thus, if the primary form be a rhomb, we shall have 

 y—^—a : and if planes be derived from the same law operating 

 upon different edges of the rhomb, the planes will be (p; q; r), 

 (P ; r; q), (q ; r ; p), & c . the result will be a bipyramidal do- 

 decahedron, and the alternate dihedral angles &, 6\ at the 

 edges of the pyramids, will be given by the formula: 



coc 0_ P * + <2qr — (q 2 + r2 + 2p<l+2pr) cos a 

 p2_j_^2_(. r 2 — <2,{pq-\-j>r + qr) cos a ' 



_ cog f _ 2pq+r*—(p* + q*+2pr + 2qr) cos a 

 P~ + q 2 +r 2 — c 2(pq +pr -f qr) cos a " 



If the secondary plane, instead of being derived by trun- 

 cating the angle which is the origin, be parallel to the trun- 

 cation of some other solid primary form, some of the quantities 

 p, q, r will be negative, and the formulae will still be appli- 

 cable. 



If the secondary plane be parallel to the truncation of an 

 edge of the primary, as, for instance, the edge x, the cor- 

 responding index p will be 0. Thus, (0 ,• q ; r) represents a 

 plane replacing one of the superior edges of the rhomb ; and 

 (0 ,• q ; — r) represents a plane replacing one of the lateral 

 edges. 



The same formula is equally applicable to the other pri- 

 mary forms besides the rhomb; and the reference of the 

 secondary planes of these forms to one of the angles as the 

 origin, is capable of being rendered very simple. 



