636 REPOKT— 189-1. 



It now only remained to discover that Professor Hessel had already arrived at 

 the thirty-two types of crystallographic symmetry by mathematical reasoning 

 more than sixty years ago ; his \vork, being far in advance of his time, appears to 

 have attracted no attention, and the memoir remained unnoticed till more than half 

 a century after its publication. 



Starting from Sohneke s definition of a regular point-system, and proceeding, 

 though independently, by a method wbich closely resembles that of the regular 

 partitioning of space by Schunflies, Mr. William Barlow has given in a paper just 

 issued a general definition applicable to all homogeneous structuies whatever, and 

 has shown that every such homogeneous structure falls into one or other of thirty- 

 two types of symmetry, coinciding exactly with the tbirty-two types of crystal- 

 symmetry, lie points out that each of those homogeneous structures which possess 

 planes of symmetry or centres of symmetry does so by reason of its having an 

 additional property beyond mere homogeneity, namely, that if we disregard mere 

 orientation, it is identical with its own image in a mirror. Mr. Barlow further 

 discovers that every one of the Sohuckian point-systems can be geometrically con- 

 structed by finite repetition of some one of a certain ten of them. 



Lord Kelvin, who, with characteristic versatility, has lately enlightened us with 

 his researches on Molecular Tactics, has quite recently attacked another problem 

 of the same group, and has sought to discover the most general form of cell which 

 shall be such that each cell encloses a single point of a Bravais system, while all the 

 cells resemble the parallelepipeds, of which we have already spoken, in being equal, 

 similar, similarly orientated, and in completely tilling up space. He finds that in 

 the general case the cell can have at most fourteen walls, which may be themselves 

 either plane or curved, and m.ay meet in edges either plane or curved. Having 

 regard, however, to the limited time at our disposal, we may hesitate before 

 foliowiug Lord Kelvin into his curious and many-walled cell. 



The deduction of the thirty-two types of symmetry by mathematical reasoning 

 was also made independently by both Gadolin and Viktor von Lang thirty years 

 ago from the law of rationality of indices ; while Fedorow points out that the 

 method of deduction recorded in the recent German treatise of Schonflies is 

 remarkably similar to the one independently published by himself in Russia. 

 Both Curie and Minnigerode have also lately given comparatively brief solutions 

 of the problem. 



Nor must I omit to mention to you the elaborate memoir dealing with the 

 .symmetry of parallelepipedal point-systems which was written by the late Pro- 

 fessor Henry Stephen Smith, whose too early death this University has so much 

 reason to deplore. To the outer world he was perhaps best known as one of the 

 most perfect mathematicians of the age, but those who had the good fortune to find 

 themselves among his pupils will always treasure up in their memory rather the 

 kindly courtesy, the warm sympathy of the man, than the genius, however tran- 

 scendent, of the mathematician. 



To sum up this part of the subject — it is now established that a definition of the 

 regularity of a point-system can be so framed that thirty-two, and only thirty-two, 

 types of symmetry are mathematically possible in a regular system, and that these 

 are identical with the types of symmetry that have been actually observed in 

 -crystals, or are inferred by analogy to be crystallographically possible. 



It remains for subsequent investigators to determine what the points of the 

 .system really correspond to in the crystal ; according to Schonflies, the physicist 

 and the chemist can be allowed in each crystal absolute control within a definite 

 elementary region of space, and the crystallographer is only entitled to demand 

 that the features of this region are repeated throughout space according to one or 

 other of the thirty-two types of symmetry already referred to ; or, what appears 

 to be the same thing, the crystallographer requires mere homogeneity of structure. 



Simplicity of Indices. — We have seen that the planes containing points of a 

 regular point-system have rational indices. But there still remains unaccounted 

 .for the remarkable fact that the indices of the natural limiting faces, and also of 

 the cleavage-planes of a crystal, are not merely whole numbers, but are in general 

 extremely simple whole numbers. Bravais and his followers have sought to account 



