Prof. Miller's Crystallographic Notices. 513 



the indices which enter into any one of the numerators, refer to 

 the extremities of the arc which appears in the corresponding 

 denominator. 



Supposing PR to be less than a semicircle, sin PQ : sin RQ is 

 negative, except when Qhes between P and R, and sin PS : sinRS 

 is negative, except when S lies between P and R. 



When PQ, RQ, the symbols of the poles Q, S, and of the 

 zones containing P, R, are known, the preceding equation gives 

 the value of the ratio sin RS : sin PS. The ratio sin RS : sin PS 

 having been found, PS is given by the equations 



tan^= i^, tan (PS-iPR) = taniPRtang -O). 



, ,. , , , . . p sinPQsinRS. 

 PS may be found directly by substitutmg tor- pQ ■ pg "s 



. , , cot PS — cot PR ^ 



^q"^^"^^°* cotPQ-cotPR - 



When PQ is less than a quadrant, cot PQ is positive or nega- 

 tive according as PQ, PR are measured from P in the same or 

 in opposite directions. The sign of PS is determined by a 

 similar rule. 



Let KP, KQ, KR, KS be any four zone-circles passing through 

 the pole K ; efg, pqr the symbols of KP, KR ; hkl, uvw the sym- 

 bols of any poles Q, S, except K, in the zone-circles KQ, KS. 

 Then 



sin PKQ sinRKS ~ sin RKQ sinPKS * " ^'^' 

 When PKR is less than two right angles, sin PKQ : sin RKQ 

 is negative, except when Q lies in the lune PKR, and sin PKS : 

 sin RKS is negative, except when S lies in the lune PKR. 

 The angle PKS may be found by means of the equations 



tan^=ji^|^, tan(PKS-|PKR) = taniPKRtan(^-6j). 



, , . ,, • cotPKS-cotPKR . ^ , 



PKSniay also be found by wntmg ^^^^^^^^^p g-^ mstead 



„ smPKQsmRKS . .^x 



sin RKQ sin PKb 



The signs of PKS, cot PKS, cot PKQ are determined by rules 

 similar to those already given for the signs of PS, cot PS, cot PQ. 



Let D, E, G, H be four poles, no three of which arc in one zone- 

 circle, P any other pole. Having given the symbols of the poles, 

 and five of the six arcs joining every two of the poles 1), E, P, G, 

 to find the position of P with respect to D, E. 



From five of the six arcs joining every two of the poles D, E, 

 F, G, compute DE, if not already given, and the angles GDE, 



