REVIEWS 311 



large field. In the introductory chapter use is made of Brodetsky's graphical 

 method, which was expounded last year at the January meeting of the 

 Mathematical Association. In the chapter dealing with linear equations 

 with constant coefiicients, the use of operators is explained in a clear and 

 interesting way. 



In the chapter on partial differential equations, Fourier's sines and cosines 

 series are introduced to show how certain initial and boundary conditions 

 may be satisfied. The conditions under which the expansion of a function 

 into a set of sines, and into a set of cosines, is valid within the range (o, n) 

 are stated. In the chapter dealing with Lagrange's linear partial differential 

 equations, itwo examples are given from a recent paper of Prof. M. J. M. Hill 

 to illustrate his methods in obtaining special integrals. The method of 

 Frobenius is given great prominence. It is first exhibited in use : then the 

 assumptions involved are investigated. There are a large number of ex- 

 amples in the book, and in many cases notes are appended to the differential 

 equations, mentioning the physical problems in which they occur. 



Dorothy Wrinch. 



Introductory Mathematical Analysis. By W. P. Webber, Ph.D., Assistant 

 Professor of Mathematics in the University of Pittsburgh, and L. C. 

 Plant, M.Sc, Professor of Mathematics in Michigan Agricultural 

 College. [Pp. V + 300.] (New York : John Wiley & Sons. Price 

 9s. 6d. net.) 



This book is another of the 19 19 harvest of American primers for the use 

 of pass students of mathematics. It deals with elementary algebra and 

 trigonometry, and gives as well a certain amount of the theory of conic 

 sections and calculus. It is not to be differentiated in any important respect 

 from the other primers, since it retains that view of complex numbers which 

 relies upon an undiscriminating mixture of geometry and algebra, and is 

 without distinction in its treatment of other topics. 



Dorothy Wrinch, 



ASTROM-OMY 



Report on the Quantum Theory of Spectra. By L. Silberstein, Ph.D. 

 [Pp. iv + 42.] (London: Adam Hilger, 1920. Price 5s. net.) 



The quantum theory of spectra was first put forward by Niels Bohr in 

 191 3, and at once attracted considerable attention owing to the facility 

 with which it led to the formula giving the wave-lengths of the lines in the 

 Balmer series of hydrogen, and to a value of the Rydberg constant occurring 

 in that formula which agreed very closely with the value obtained experi- 

 mentally. The simple theory advanced by Bohr in his first paper has 

 since been considerably elaborated in a series of papers by Bohr, Epstein, 

 Paschen, Sommerfeld, and others. These developments have provided 

 further triumphs for the theory ; in particular, they have enabled an ex- 

 planation of the fine structure of spectral lines to be given and results to be 

 predicted which subsequently experiment has verified. The papers of Bohr 

 himself have appeared in the Phil. Mag., but the majority of the other 

 papers are contained in the Annalen der Physik, and have not been readily 

 accessible to readers in this country. Dr. Silberstein's concise report will 

 therefore be found of great value by spectroscopists and others interested 

 in the theory, presenting as it does a clear and readable account of its pre- 

 sent state. The various assumptions which underlie it are plainly stated, 

 and it is carefully pointed out that their only justification at present is to 

 be found in the remarkable agreement with experiment. 



The report was written originally for the private use of the optical firm 

 of Messrs. Adam Hilger, Ltd., of 75A, Camden Road, N.W.i, in whose re- 



