Mr. L. Fletcher's Crystallo'gmphic Notes. 283 



a study of the stereographic projection of fig. 1 ; for as 

 T c = T 8, ci will be rotated into the position of S, and o\ 

 into the position of e, while 71 will be rotated to d, and d\ 

 to 7. With the other poles it will be different : thus a will 

 rotate into the position a 1} where T« = T« 1 = 89° 18^, and 

 «« 1 = Ta-T« 1 = 90°41| / -89°18i / = P23i / =^ ] = a a 1 =Z) / S 1 . 

 Also T Q 1= =T Q = 89^9', whence Q Q x = 90° 51'— 89° W = 

 1° 42'; and T = T d = 44° 34£', whenceC d = 90 o 51'. 



Next, let M be a point bisecting the arc Q Q x ; the poles of the 

 two octahedra will be symmetrically disposed to the twin-plane, 

 represented in the projection by the line IBM; for TM= 

 T B = 90°, and the arcs T6, T* x T« TZ> T« T/3 Taj Tft are 

 all equal. From this it will follow that Ma=Mj3 = M« 1 =M6 1 ; 

 also that Ma 1 =Mj8i = M& =M«, and that M 7 M 7l MdMd t 

 McM<?! MSMSj, being all right angles, are equal to each 

 other. The poles of the two individuals are thus not only 

 symmetrical to the twin-plane M B M, but also to the plane 

 T B T at right angles with it, represented by an irrational 

 symbol approximating to (100 99), and thus not a crys- 

 talloid plane. The same arrangement of poles might therefore 

 be obtained by rotating the original octahedron about the line 

 M M parallel to the tangent of the circle at T, and therefore 

 to a " terminal edge " of the octahedron {1 11}. It may be 

 remarked that, although the same arrangement of poles will 

 be obtained by rotation about the lines T T MM, the lettering- 

 will not be identical in the two cases ; but as in both cases the 

 planes which are quite or nearly coincident in direction are 

 always represented in the one individual by italic letters and 

 in the other by greek, there will be no crystallographic dif- 

 ference in the results of these two methods of derivation. 

 This proves that, as has been mentioned above, Haidinger's 

 statement, " regular composition often takes place parallel to 

 a plane of {101} or perpendicular to the terminal edges of 

 {111}," indicates only a single case if the reference be to 

 the plane of rotation. 



Haidinger's law might therefore be equally well expressed 

 in the two following ways : — 



I. Twin-plane a face of the form {101}; composition-plane 

 perpendicular to the twin-plane. 



II. Twin-axis a terminal edge of {1 1 1}; composition-plane 

 parallel to the twin-plane. 



We are now in a position to discuss the difference of growth 

 which will be produced by a. variation in the position of the 

 plane of composition. As the line T T is perpendicular to the 



