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LXVI. Notices respecting New Books. 
A Course of Modern Analysis. By HE. T. Wuirraxsr, JA. 
Cambridge University Press. 1902. 
HE extended title of this exceptionally excellent work is * An 
Introduction to the General Theory of Infinite Series and of 
Analytic Functions; with an account of the principal Transcendental 
Functions.” When it is stated further that the functions discussed 
are the Gamma, the Legendre, the Hypergeometric, the Bessel, and 
the Elliptic Functions, the general scope of the book will be under- 
stood. The author has shown great discrimination and reserve in 
his treatment ; tor it is only too easy in a subject of this kind to 
deviate into side issues so that the student has the vaguest ideas 
us to the general aim of his wanderings. Here, however, we find 
in the 170 pages which constitute Part I. a lucid, compact, and yet 
sufficiently detailed development of the theory of functions in a 
form necessary and sufficient—to use the familiar phrase-—for a 
sound discussion of the important special types of functions treated 
of in Part II. Each chapter is enriched with an appropriate set 
of examples or exercises, many of which are important theorems 
associated with the names of the discoverers. Had the majority 
of these been treated at length as part of the text—as is the custom 
with some authors—-the book could easily have been made of 
formidable dimensions. In this connexion especially the author 
has shown great wisdom. The working student will probably find 
it profitable to read the text carefully through so as to get a general 
view of the subject, and then turn back and familiarize himself with 
the methods by working a selection of the exercises out in detail. 
The value of the book is further enhanced by a series of short 
historic notes and by constant references to important memoirs by 
the many distinguished mathematicians who have helped to develop 
the modern theory of functions. 
Mathematical Crystallography and the Theory of Groups of Movements, 
By Harotp Hitton, M.A. Oxford: At the Clarendon Press. 
1903. 
Mr. HixTon is to be congratulated on having made an important 
contribution to our mathematical literature.’ Crystallography is a 
science which in its practical aspects concerns the mineralogist 
and the chemist ; but very few of those who are familiar with the 
forms and classification of crystals will find Mr. Hilton’s pages 
easy reading. After a discussion of the geometry of crystals and 
the various laws recognized by crystallographers, the author enters 
upon his real work, namely, the complete mathematical discussion 
and classification of the various groups of moyements possible under 
the limitations suggested by the laws of crystals. The book is in 
fact a treatise on the theory of a set of finite groups of a special 
type, involving certain operations of translation, reflexion, and 
