Projection of the Sphere in Crystallography. 



49 



But 



sinPOQ sinROS _sinOHL sin OWU 

 sin OLH sin OUW ' 



PL OU 

 OHOW 



(^) 



sinROQ sin POS 

 Therefore 



sin POQ sin ROS _ eA + fA; + g/ pu + qr + xw 

 sin KOQ sin POS " pA + q/t + xl ew + ft; + gw' 



where OP, OQ, OR, OS are four lines in one plane, efg, pqr 

 the symbols of the lines OP, OR, and likl, uvw the symbols of 

 planes parallel to the lines OQ, OS. 



Each of the angles POQ, ROQ, POS, ROS being supposed 

 less than two right angles, sin POQ sin ROS, sin ROQ sin POS 

 will have the same signs except when one only of the lines OQ, 

 OS lies in the angle POR. 



29. Let h, 1, u, w be the orthogonal projections of the points 

 H, L, U, W in the preceding figure, on a plane through 0, per- 

 pendicular to the intersection of the planes hkl, uvw. In this 

 plane draw lines perpendicular to Ou, hi, Ow, uw, meeting a 

 circle described round the centre 0, in the points P, Q, R, S. 

 OQ, OS are respectively perpendicular to the planes hkl, uvw. 

 OP, OR are respectively perpendicular to planes containing 

 the lines efg, pqr, parallel to the intersection of the planes 

 hkl, uvw. 



By similar triangles, 



Oh__Ou 01 _ Ow 



OH~OU' OL~OW 



Therefore 

 sin PQ sin RS 

 sin RQ sin PS 



_ 01 Ou 



~ Oh Ow 

 Hence 



sin Ohl sin Owu 

 sin Olh sin Ouw 



oj. qu 

 "OH ow 



sin PQ sin RS __ eh + ik-\-gl^pu + qv + rw 

 sin RQ sin PS pA + qk -\- rl eu + iv + gw' 



(t) 



where OP, OQ, OP, OS are perpendiculars to four planes inter- 

 secting one another in parallel lines, eig, pqr the symbols of 

 lines in the planes perpendicular to OP, OP, and hkl, uvw the 

 symbols of the planes perpendicular to OQ, OS. 



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

 a semicircle, sin PQ sin RS, sin RQ sin PS will have the same 

 signs except when one only of the points Q, <S lies in the arc PR. 



30. The expressions for the indices of a plane when referred 

 to new axes, are easily dcducible from {p). Let efg, hkl, pqr be 



Phil. May. S. 4. Vol. 18. No. 117. July 1 859. E 



