TRANSACTIONS OF SECTION C. 637 



for this by the hypothesis that both the natural limiting planes and the cleavage- 

 planes are those planes of a point-system which are most densely sprinkled with 

 points of the system. Curie and Liveing, independently of each other, have been 

 led to the same result from considerations relative to capillary constants. Sohncke, 

 however, pointing out that there are many cases — for instance, calcite — where an 

 excellent cleavage-plane is rarely a limiting plane, suggests that his generalised 

 point-system is more satisfactory than a Bravais system in that not only the density 

 of the sprinkling must be had regard to, but also the tangential cohesion of the- 

 particles in the plane, end that in his system these may be independent of each 

 other; while Wulft" remarks that Sohncke's arrangement is identical with that of 

 Bravais for the anorthic system, where the same objection holds, and he denies the- 

 legitimacy of the reasoning by which the hypothesis of a relation between the 

 density of the sprinkling of points on a plane and the likelihood of the natural 

 occurrence of the plane as a limiting face is supported. 



Complexity of Indices. — Doubtless, however, crystal faces are observed of which 

 the symbols involve indices far exceeding 6 in magnitude — so complex, in fact, that 

 one is tempted to doubt the rigidity of the experimental proof that indices are 

 necessarily rational. Often, though the numbers are high, their ratios differ by 

 only small amounts from simple ones. A most patient and detailed study of such 

 faces was made for danburite by the late Dr. Max Schuster of Vienna, and the 

 results were brought by him some years ago to the notice of this Section. From 

 careful examination of similar faces in the case of quartz, Molengraaft" has been 

 led to conclude that it is extremely probable that such faces are of secondary 

 origin and have been the result of etching : they would in such case correspond, 

 not to original limiting planes, but to directions in which the crystal yields most 

 readily to solvent or decomposing influences. 



Optical Characters. — Passing from the purely geometrical characters of crystals 

 to the optical, we may in the first place remark that the relationship between crys- 

 talline form and circular polarisation discovered by Herschel in the case of quartz 

 has been generalised since the issue of Whewell's Report. We now know that 

 many crystallised substances belonging to different systems give circular polarisa- 

 tion, and that all of them are merosymmetrical in facial development or structure; 

 further, they belong to types of symmetry which have a common feature, though 

 this is only a necessary, not a sufficient, condition. 



The importance of the discovery of the dispersion of the mean lines has 

 already been referred to. 



We may recall attention to the fact noticed by Norremberg, that when cleavage- 

 plates of biaxal mica are crossed in pairs and the pairs are piled one upon another 

 in similar positions, the optical figure yielded by the combination approaches nearer 

 and nearer to that of a uniaxal crystal the thinner the plates and the more nume- 

 rous the pairs : in the same way, by means of triplets of plates, each plate being- 

 turned through one-third of a complete revolution from the position of the preced- 

 ing one, Reusch found it possible to closely imitate the optical figure of a right- 

 handed or a left-handed circularly polarising crystal. 



And it has been observed that repeated combinations of differently orientated 

 parts actually occur in crystals. Large crystals of potassium ferrocyanide, for 

 example, are really composite, and the different parts are differently orientated : 

 on the one hand, a thick slice may give an optical figure which is uniaxal; on 

 the other hand, a thin slice shows two optic axes inclined to each other at a con- 

 siderable angle. 



It has been suggested that the circular polarisation of quartz and other crystals- 

 is due to a spiral molecular arrangement corresponding to that of the mica-tripleta 

 as arranged by Reusch. Such a spiral arrangement is shown by the points of the 

 corresponding Sohnckian system. 



Optical Anomalies. — As already mentioned, we owe to Brewster the establish- 

 ment of the relation between the optical behaviour of crystals and the systems of 

 crystallisation. But in the course of his long research Brewster met with numerous 

 puzzling exceptions, and to the investigation of the origin of their peculiar optical 

 behaviour he devoted much study ; subsequent workers have concurred in express- 



