Crystallographic Projections. 87 



lines Vjtf-i-Wi^s^O &c, which is 



5. It is interesting to extend these theorems to the pro- 

 jection from any point Q onto a plane perpendicular to Q. 

 The extension is as follows : — 



The projection of the line of intersection of a plane through 

 parallel to the face (u r w) with the plane A B C is the 

 line ug + vy + w£=Q, and the projection of the zonal axis 



£ v K 

 [U V W] passes through ==■= = — ; where 



a(d+ cos«)f=X, b(d + cos <3)t7 = Y, c(d + cos7)f=Z, 



and 0Q = «f; 



the vertices A 1} B l3 Cj of the triangle of reference being the 

 projections of A, B. C, and X, Y, Z being areal coordinates. 



This may be proved thus: — 



Let Q be the point (0, 0, —d), and project from Q onto 

 the plane s=0 (any parallel plane would do). The plane 

 ABC is D 1 .+Doy + D^ + A = 0, where D 1? D 2 , D 3 , A are 

 the determinants |_»h ti 2 1], [n 2 / 2 1]? [^i w 2 1]? [^i ?n 2 %]• 

 The plane joining the intersection of lx + my + nz=0 with 

 the plane ABC to Q is 



( A - rfD 3 ) (£z + my -f /is) + rf«(D lt « + D*y + D 3 ~ + A) = 0, 

 which meets 2=0 in the line 



(A-<ZD,)(fc + my) + ^(D^ + D 2 y + A) = 0. 



Now A! is the point ( 1 t . _ — —- 7J ) &c. : and therefore 



if the areal coordinates of the point (#, 3/, 0) referred to 

 A^C as the triangle of reference in the plane 2=0 are 

 X. Y, Z. 



cZDX = (w 1 + rf)(L 1 ^+M 1 ^ + N 1 ^) &c. ; 



where L„ M l3 X x are the cofactors of Z l5 m l7 >? a + <:/ in the 

 determinant [/, m 2 n z -\-d\ =D and so on. 

 Solving, we have 



d = 



Li 



«! + d 



+ ^ 



/1,, + d 



+ 



/ :; Z 

 w 3 +<f' 







m 2 Y 



//. 2 + d 



+ 



m{L 

 n 3 + d 



Sul 



istituting in 



















(A- 



-,/!>;)(/. 



f + 



/////) + 



d,,(D v 



v+T)iy 



+ A) = 



:.». 





we ge 



r 



















AriiPJ^ft+wO'f ••• + •••) 



X 



+ n( | (tJ/ 1 D 1 + <Z/>»,b, + // 1 + </A) + ... + ... )=0, 



