42 Prof. Miller on the employment of the Gnomonic 



where P, Q, R, S are four points in a great circle, efg, pqr are 

 the symbols of any great circles through P, R, neither of which 

 coincides with QS, and hkl, uvw are the symbols of Q, S. 



Each of the arcs PQ, RQ, RS, PS being supposed less than a 

 semicircle, sin PQ sin RS, sin RQ sin PS have the same signs, 

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



10. By (7), 



sin PKQ sinRKS _ sin PQ sinRS 



sin RKQ ^kTKS ~ smRQ smPS ' 

 Therefore 



sin PKQ sin RKS _ eh + {k + gl ipu + qv + vw , > 



sinRKQsinPKS'~pA + qA + r/eM + fi; + gj<;' ' ^ ' 



where KP, KQ, KR, KS are four great circles passing through 

 the point K, efg, pqr are the symbols of KP, KR, and hkl, 

 uvw are the symbols of any points in KQ, KS, neither of which 

 coincides with K. 



Each of the angles PKQ, RKQ, RKS, PKS being supposed 

 less than two right angles, sin PKQ sin RKS, sin RKQ sin PKS 

 have the same signs, except when one only of the points Q, S 

 lies in the lune PKR. 



11. When the points upon the surface of a sphere referred to 

 in the preceding pages are the poles of the faces of a crystal, it 

 is found that the ratio sin PKQ sin RKS : sin RKQ sin PKS is 

 always that of two whole numbers. Hence, it readily follows 

 that the indices of the poles, and of the great circles passing 

 through any two of them, are whole numbers, either positive or 

 negative, one or two of which may be zero. 



12. Let the straight lines, 

 four of which pass through the 

 point K, intersecting the fifth 

 in the points P, Q, R, S be the 

 gnomonic projections of great 

 circles of a sphere, the centre of 



which is 0. Let K, P, Q, R, S be the originals of K, P, Q, R, S; 

 and let efg, pqr be the symbols of the great circles KP, KR ; 

 hkl, uvw the symbols of Q, S. 



By W, 



sin PKQ sinRKS _ PQ RS _ sin POQ sinRQS _ sin PQ sin RS 

 sin RKQ sin PKS ~ RQ PS ~ sin ROQ sin POS ~ sm RQ sin PS' 

 Hence by (a), («), 



sinPKQ sinRKS _PQ B.S _ch + fk + gl i>u + qv + i'w 

 sinRKQ sin PKS ~ RQ PS ~ pA + qk + il eu + h + giv' ^'^' 



The anharmonic ratios in which the letters P, Q, R, S, dcuo- 



