350 



Prof. Miller on the Application of 



and if the zoue-axis pqr meet the plane uvw in R, and the plane 

 mno in F, 



v(p»i + q« + ro)OF = n{\m + qv + no) OR. 

 Hence, if a'= (e»j + hi + go)OD, b' = (h»i + kn + lo)OE, 

 c'=(p»i + qrt-t-ro)OF, we shall have 



where 



'4-0P=^0Q=^0R, 

 a c 



u' = eu + iv +gw, 



v' = hu + k?; + \w, 



Since u', v', w' are whole numbers, it follows that the planes may 

 be referred to the zone-axes efg, hid, pqr as axes of the system ; 

 and that the symbol of the face uvw, when referred to these axes, 

 becomes u'v'w'. 



10. It appears from accurate measurements of the mutual incli- 

 nations of the faces of a crystal, and from calculations founded on 

 those measurements, that, if through any point within a crystal 

 planes be drawn parallel to the faces of the crystal, and any three 

 intersections of these planes, not in one plane, be taken for axes, 

 the positions of the faces of the crystal will be found to obey the 

 law which has been assumed for the system of planes which has 

 been the subject of the preceding investigation. Hence all the 

 conclusions arrived at respecting such a system of planes, hold 

 equally for the faces by which a crystal is bounded. 



1 1 . Let P, Q, R, S be four faces in Yio; 6. 

 one zone, of which P, Rpass through 

 the origin 0, and Q, S pass through 

 any point K in the axis OY. Let 

 hkl, uvw be the symbols of the faces 

 Q, S which are projected into KF, 

 KR respectively ; efg, pqr the sym ■ 

 bols of any zone-axes in the faces 

 P, R which are projected into OP, 

 OR respectively. OD, OP are the 

 projections of the portions of the 

 zone-axis efg intercepted between O 



and the faces hkl, uvw ; and OF, OR are the projections of the 

 portions of the zone-axis pqr intercepted between and the faces 

 hkl, uvw. Therefore by (9), 



v{di-^ik + ^l)OJ)=k{QU + h-\-^io)OV, 

 t;(p/i -f- q^ -f r/) OF = A(pM -f qw -f- r«t?) OR. 

 Hence, if . _ p?< + qt> + rw di-\-ik + ^l 



~ p/« -t- qA -I- r/ ew + fw + g2«' 

 OF.OP=i.OD.OR. 



