48 Prof. Miller on the employment of the Gnomonic 



\\a Vb \c' 

 £___y__£ 

 p« qfc re' 



Therefore, since the plane contains the lines 



p?< + qw + r7/; = 0, 

 ?/ = kr— Iq, i) = lp— hr, M; = hq--kp, . . (tt) 



28. Through 0, the origin of coordinates, draw the straight 

 lines OP, OR, parallel to the lines efg, pqr, meeting the plane 

 hkl in H, L, and the plane uvw in U, W. Let planes parallel to 

 YOZ, passing through the points H, U, L, W, meet OX in 

 H', U', L', W. Draw OQ, OS parallel to HL, U W respectively. 



The equations to the lines OP, OR are 



ea fh gc' 



X _y _ z 



p« q6 re ' 

 and the equations to the planes hkl, 

 uvw are 



a b c 



X y s _, 



U \-Vt- +1V— =:t. 



a b c 



The distances OH', OU', OL', OW are the values of x at the 

 points in which OP, OR intersect the planes hkl, iwiv. There- 

 fore 



(eA + f^+gOOH' =e«c?, 



(ett + ft) + giv)OV =eat, 

 (pA + qA:+r/)OL' =pad, 

 (pw + qv + rw)0 W = ^at. 

 By similar triangles, 



Hence 



0H'_0U' OL'_OW' 



OH ~ OU ' OL ~ OW 



OL OU _ eh + ik + gl pw + qv + vw 

 OH 0\V "" pA + qA + r7 e«< + h + gw ' 



ip) 



