420 Heviews — S. Hilton's Mathematical Crystallography. 



however, show that there is a most striking resemblance between 

 the recent and fossil shells to which he refers. 



Mr. Moore describes these univalve raollusca, the medusas, and 

 certain more doubtful animals as a " halolimnic fauna " — a stranded 

 relic of an old marine fauna which has been little changed by 

 isolation. He accounts for their presence by supposing that 

 Tanganyika was connected with the sea during some part of the 

 Secondary period by way of the Congo basin. There still remains, 

 however, the difficulty that the halolimnic fauna is not found either 

 in the neighbouring lakes or in the old sediments which have been 

 examined round their margins. We are, indeed, greatly indebted 

 to Mr. Moore for stating the Tanganyika problem more clearly and 

 definitely than it was ever done before, rather than for proposing 

 an adequate solution of it. 



III. — Mathematical Crystallography and the Theory of Groups 

 OF Movebients. By Harold Hilton, M.A. 8vo ; pp. 262. 

 Oxford : at the Clarendon Press, 1903. 



AT the last Oxford Meeting of the British Association, held in 

 the year 1894, the President of the Geological Section attempted 

 to explain in simple language some of the most important recent 

 developments of the theories of crystal-structure, then only on 

 record in isolated memoirs, almost all of them foreign. Mr. Hilton, 

 in the present work, has taken the same problem in hand ; he has 

 collected for the benefit of geologists and others the results obtained 

 by Fedorow, Schonflies, and Barlow, and has arranged them in 

 orderly sequence in twenty-six chapters. Beginning, in Part I, with 

 a brief explanation of the stereographic projection, the author passes 

 at once to the discussion of Homogeneity and the limited number (32) 

 of finite groups of movements which are consistent with the law 

 of rational indices and are therefore applicable to Crystallography ; 

 and then considers the dependence of the physical properties of 

 crystals on the Symmetry. In Part II the author enters into the 

 consideration of the Structure-theory. If a collection of molecules 

 can be brought into self-coincidence by an operation, such an 

 operation is termed a ' symmetry-operation ' ; all the symmetry- 

 operations proper to the collection of molecules are said to form 

 a 'group ' ; the author thereupon proceeds to evolve the 230 distinct 

 groups which are geometrically, not necessarily mechanically, 

 possible in homogeneous matter. Mr. Hilton next considers the 

 regular partitioning of space, and mentions the interesting case in 

 which space is partitioned by Lord Kelvin into 14- walled cells, all of 

 them similar and similarly orientated, the walls being not in general 

 plane. A final chapter gives a history of the Structure-theories. 

 The book is illustrated with no fewer than 188 figures, some of 

 them of a remarkably complicated character. Everyone interested 

 in crystals will feel deep gratitude to Mr. Hilton for having so 

 successfully undertaken a very difficult task, and to the University 

 Press for having in this praiseworthy way made the work immediately 

 accessible to the student. 



