OF ARTS AND SCIENCES. 227 



membered that these are only three out of a very large number of 

 apparent twins, iu which no such relation could be established. The 

 following will serve as an illustration. 



A group of crystals was found differing from the previous ones in 

 no essential respects, in which R and R' were in a zone with the prin- 

 cipal axes, making an angle with each other of 4° 8'. This would 

 correspond to a twinning plane with an axial ratio of 0.G32G7, which 

 bears no assignable relation to the fundamental rhombohedron. In 

 other cases the approximation to possible ratios was much closer than 

 this ; but iu most instances the planes on the two crystals were not 

 even iu the same zone, and every variation was noticed, from examples 

 in which the reflected images would fall within the field of the tele- 

 scope of the goniometer to those in which there was a much wider 

 divergence, the crystals beiug displaced to a greater or less extent on 

 the plane of twinning. 



It will be seen by the above examples that in some cases the direct 

 pyramid is iu the same zone with the direct pyramid on the twin 

 crystal, and in other cases it is in the same zone with the indirect; 

 but intermediate positions were noticed, wliich would result from a 

 greater or less rotation of one of the crystals on its own axis. 



