Henc 



normal to Four Faces of a Crystal in one Zone. 



SI- _eh + ik + gl Y>u + qv + rw 

 qr ~ pA + qA: -f r/ CM + fo + gw ' 



99 



(6) 



According as the right-hand side of this equation is positive or 

 negative, s and q lie on the same side, or on opposite sides 

 of r. 



By the above construction, PS can readily be found when PQ, 

 PR, and the symbols of P, Q, R, S are known. The construc- 

 tion may also be used in finding the symbol of S, when PQ, PR, 

 PS, and the symbols of P, Q, R are 

 given. 



Let KP, KQ, KR, KS be four zone- 

 circles intersecting in the point K ; 

 efg, pqr the symbols of KP, KR respect- 

 ively; hkl, uvw the symbols of poles 

 Q, S in the zone-circles KQ, KS re- 

 spectively. Let the zone-circle QS meet 

 KP in P, and KR in R. It is easily 

 proved that 



sin PKQ sin SKR sin PQ sin SR, 



sin QKR sin PKS 

 Hence 



sin QR sin PS 



sin PKQ sin SKR _ eh-\-ik + gl pw + qr; -|- vw 

 sin QKR sin PKS ~~pA-FqA: + r/ eu + iv + gw' 



(7) 



where the lower sign is to be taken when one only of the poles 

 Q, S lies in the lune PKR. Hence also, when PKQ, PKR, PKS 

 are measured all in the same direction from KP, 



±lE±3'fL (cot PKS - cot PKR) = e^ + f^ + g^ cotPKQ-cotPKR). (8) 

 -f-qv + rw^ ' pA-f-q^ + r/' 



ew-f 

 pw-f-qv 



Equations (7) and (8) may be used with advantage in the 

 process of calculating the angles between faces of crystals, espe- 

 cially of those belonging to the oblique and anorthic systems. 



Any edge of a crystal is parallel to the axis of the zone which 

 contains the two faces which, by their intersection, form the edge, 

 and may be denoted by the same symbol as the zone to the axis 

 of which it is parallel. The sides of any face of a crystal are 

 arallel to the axes of zone-circles intersecting in its pole, and 

 make with each other the same angles as the zone-circles to the 

 axes of which they are respectively parallel. Hence equations 

 (7) and (8) may be applied to determine the side of a face of a 



H 2 



