I0O 
throughout the book; and it should serve as a con- 
venient guide for leading on students to the intelligent 
study of more complete and elaborate treatises on 
water supply. 
THE MATHEMATICAL THEORY OF 
CRYSTAL STRUCTURE. 
® Mathematical Crystallography and the Theory of 
Groups of Movements. By Harold Hilton, M.A. 
Pp. xii+262; with 188 figures in the text. (Oxford: 
Clarendon Press, 1903.) Price 14s. net. 
Were the fostering care of the energetic pro- 
fessor a small but vigorous school of mineralogy 
is growing up at Oxford. We are not surprised to 
note in the preface that it was due to Prof. Miers’s 
suggestion that Mr. Hilton undertook the task which 
he has so successfully accomplished. Mr. Hilton has 
had a distinguished career at Oxford, and it is with 
pleasure we observe that a mathematician of his 
attainments has turned his attention to a subject which 
receives such scanty consideration in this country. 
Mr. Hilton’s book appears at an opportune moment, 
since it is the generally accepted idea that the geo- 
metrical theory of crystal structure has reached some- 
thing like finality. A good historical account of the 
NA TORE 
development of the subject is contained in the British | 
Association Report, 1901, and the present work supple- | 
ments that survey by supplying the detailed reason- | 
ing. 
the rather wide title would lead a reader to suppose, 
The scope of the book is more restricted than | 
and it is almost wholly concerned with the symmetry | 
and structure of crystals. With the exception of a few 
chapters, it follows closely Schénflies’s ‘ Krystall- 
systeme und Krystallstructur,’’ but some features are 
introduced from the writings of Jordan, Fedorow and 
Barlow. Very slight allusion is made to Sohncke’s 
work. We think it would have added to the value of 
the book had a page or two been devoted to his 
systems. Of course, they appear among the space- 
groups; but at the same time some Sohncke-system 
forms the basis of every space-group. This is Barlow’s 
way of considering the subject, and may be found 
easier of comprehension by many readers, especially 
if the theory of groups be new to them. The book is 
distinguished by the fine series of diagrams of the 
space-groups, which have been drawn independently 
in the way suggested by Fedorow. The explanation 
of the figures, which is given on p. 171, might have 
been made more conspicuous so as to catch the eye 
more readily. The absence of such diagrams appreci- 
ably adds to the difficulty of understanding Schénflies’s 
work. 
Mr. Hilton divides his book into two parts. In the 
first he determines the thirty-two classes of centro- 
symmetry, which obey the law of rational indices and 
are therefore alone applicable to crystals. Another, 
and perhaps more logical method, is to assume that 
crystalline structure is cross-grained; that is to say, 
that a lower limit can be found to the distances 
between the elementary parts, whatever they may be. 
With this assumption, it may be shown that the only 
possible axes of symmetry have respectively 2-, 3-, 4- 
and 6-fold symmetry. The law of rational indices 
NO. 1779, VOL. 69] 
[DECEMBER 3, 1903 
alone is not entirely satisfactory on account of a 
peculiar case of pseudotrigonal symmetry which in 
that way arises. This part includes an elaborate 
chapter on the coordinates of equivalent points, and 
a chapter of considerable interest on the growth of 
crystals. 
The second part corresponds very closely to the 
second part of Schénflies’s book. After determining 
the fourteen varieties of lattices, the author discusses 
the properties of geometrical operations and the in- 
finite groups of movements. The dynamical flavour 
which unavoidably clings to the subject is unfortunate, 
and without a note of warning the reader may be mis- 
led into the idea that something in the nature of a 
movement does actually occur. On p. 159 the author 
proves the fundamental proposition connecting the 
space-groups with the corresponding point- and trans- 
lation-groups, and in the succeeding six chapters he 
deduces the 230 space-groups belonging to the six 
systems. A chapter follows on the partitioning of 
space, with special reference to Schénflies’s elementary 
cell. 
In the next chapter, on crystal-molecules, mention 
is made of attempts that have been made to assign 
arrangements to particular substances. It may be 
noted that at the present day there is a tendency to 
regard the molecules which compose a crystal, or 
rather their spheres of influence, as being in contact. 
In that case Barlow’s theory of closest-packing would 
have some justification. A brief historical sketch 
brings the book to a close. 
Mr. Hilton has prepared a masterly exposition of 
a difficult subject, and we can heartily commend the 
book to the attention of crystallographers. 
OUR BOOK SHELF. 
Das Haar, die Haarkrankheiten, ihre Behandlung und 
die Haarpflege. By Dr. J. Pohl. Fifth revised 
and enlarged edition. Pp. 178. (Stuttgart: Deutsch 
Verlags Anst., 1902.) 
Tuis is a popular treatise upon a subject which has 
received too little attention from scientific observers. 
The first part of the work deals with the structure and - 
development of the human hair, with a brief account 
of the methods of investigation. The author, with 
the painstaking thoroughness of the German savant, 
has made a large number of researches into the rate 
of growth and the normal fall of the hair. The 
common belief that cutting the hair promotes its 
growth is shown to be erroneous. Each hair has, on 
the average, a normal life of about seven years, at the 
end of which time it falls out and is replaced by a 
new one. In health there is a normal fall of hair 
which varies somewhat with the age of the individual. 
The second part of the work is devoted to the dis- 
eases of the hair and their treatment, and to the care 
of the hair. The author considers that in most cases 
oil or pomade is beneficial, but he insists that the 
quantity applied must be small. The vexed question 
of washing the hair is discussed. In individuals in 
whom the scalp is healthy, Dr. Pohl is of opinion that 
too frequent washing is inadvisable. He advocates 
the use of bran, yolk of egg, and other demulcents in 
the water used. Rapid and thorough drying of the 
hair after washing is insisted upon, especially for 
ladies. 
