Projection of the Sphei'e in Crystallography. 43 



ting either points in a plane, or on the surface of a sphere, enter 

 in the order PQ . RS : RQ . PS are positive, except when one 

 only of the points Q, S lies between P and R. 



13. Let four straight lines 

 passing through K, meet any 

 other straight line in A, B, C, D. 

 Through D draw DE parallel to 

 KA, meeting KB in E, and K C 

 in F. Through E draw EL 

 parallel to KD, meeting KA 

 in L. Draw the straight line 

 D L meeting K B in M, and K C in N. 



Let AB DC _ sinAKB sinDKC 



DB AC "sinDKB sin AKC" 

 By similar triangles, 



KA_AB DF _DC 

 DE~DB' KA~AC" 



DF 



Hence jr^ =X., DF, DE being measured in the same direction 



or in opposite directions, according as A, is positive or negative. 

 Since M is the intersection of the diagonals of a parallelogram, 

 DM = LM, therefore (/3) 



AB DC_LMDN_DN 

 DBAC~DMLN~LN' 



DN 

 Hence :p^=X, D, L being on opposite sides of N, or on the 



same side, according as \ is positive or negative. 



These properties afford constructions by means of which, 

 having given the straight lines KA, KB, KC, KD, or the points 

 A, B, C, D in a straight line, the value of X may be found ; or, 

 having given \ and three of the straight lines KA, KB, KC, KD, 

 or three of the points A, B, C, D, the fom-th may be found. 



14. Let the points D, E, F, G, no three of which are in one 

 straight line, be the gnomouic projections of four poles the 

 symbols of which are known. To find the symbol of any pole P 

 given in position. Let uvw be the symbol of P. Since the 

 symbols of D, E, F, G are given, the symbol of the lines joining 

 them two by two may be found by (■?;). By equating the 

 value of sin EDF sin GDP : sin GDF sin EDP, obtamed by 

 construction, with its value as given by (/x.) in terms of the 

 indices of DE, DG, F, P, an equation is obtained of the form 

 au + bv + cii):=0, where n, h, c are integers. By equating the 

 value of sin DEG sin FEP : sin FEG sin DEP obtained by con- 

 struction, with its value in terms of the indices of DE, FE, 



