290 Mr. L. Fletcher's Crystallographic Notes. 



twin from parallel orientation of the individuals, there must be, 

 in addition to the rotation through two right angles round a 

 normal to (101), a farther rotation of one of the two crystals 

 either through two right angles about a normal to one of the 

 faces of the prism {110}, or through a single right angle 

 about the vertical axis parallel to the edges of this prism. 

 Though this double rotation may be compounded into a single 

 rotation round the normal to a face of the octahedron -Jl 1 1}, 

 the angle of this single rotation will not be 180°, as is the case 

 in other twins, but 119° 31'. 



If this statement of Sadebeck be accepted as having a satis- 

 factory foundation, the growth must be regarded as up to the 

 present unique in character; for no other regular composi- 

 tion appears to have yet been discovered in which, starting 

 from a parallel orientation, a double rotation is absolutely 

 necessary for the representation of the relative disposition of 

 the two individuals. There exist twin-growths of tetartohedral 

 crystals, it is true, such as those of sodium chlorate and certain 

 regular compositions of quartz, described by Prof. Groth, which 

 are somewhat analogous in character: but they are capable of 

 a more or less satisfactory representation by a simple rotation 

 of one of the individuals through two right angles from a 

 position where corresponding crystallographic lines of the 

 right and left individuals are identical in direction ; they are 

 moreover intimately related to the directions of the crystallo- 

 graphic axes. As, however, it had been impossible for Sade- 

 beck to convince himself, from simple examination of the spe- 

 cimens, that certain faces assumed by Haidinger to be parallel 

 might not be inclined to each other at an angle of 1° 23^, it was 

 possible to entertain a doubt as to the specimens being suffi- 

 ciently well crystallised to allow of an absolute certainty in the 

 distinction of the two tetrahedra; and as the law is so curious 

 from its extreme rarity and simplicity, and so important in 

 its bearing on the general question of twin-growth, about 

 which there has lately been much discussion, it seemed desi- 

 rable to place the law, if possible, beyond all suspicion. 



The accuracy of Sadebeck's inference as to the disposi- 

 tion of the two tetrahedra in this twin-growth is confirmed in 

 the most satisfactory manner by the specimens in this col- 

 lection. 



The Freiberg specimen of fig. 10 shows not only that the 

 individuals are symmetrical to the plane of composition, but 

 also that the differences of the two tetrahedra of each indi- 

 vidual are too marked to allow of this symmetry of physical 

 peculiarities being an accident of the growth. 



The specimen from Pool mines (fig. 11) is even more satis- 

 factory still; for the faces <w &> 2 which give such excellent 



