478 



NATURE 



[December 30, 1915 



main body of the work concerning the variations 

 in the volumes of the atoms, more particularly 

 in the cases of oxygen, sulphur, nitrogen, and 

 chlorine. He traces the effect of constitutive in- 

 fluences, e.g., the influence of the homologous 

 increment, unsaturation, and ring structure; 

 valency, and groups; and lastly he discusses the 

 special conception embodied in molecular volume 

 and its relation to other physical properties, such 

 as boiling point, surface tension, and viscosity. 



The book is by no means easy reading, but it 

 bears on every page the evidence of thoughtful, 

 critical insight, and may be unreservedly com- 

 mended as a faithful and accurate digest of the 

 state of present knowledge upon what is con- 

 fessedly a complicated and intricate subject. 



T. E. Thorpe. 



THE PARTITIONS OF NUMBERS. 

 Combinatory Analysis. By Major Percy A. Mac- 

 Mahon. Vol. i. , pp. 300. (Cambridge : At the 

 University Press.) Price 155. net. 



THOSE who, like the present writer, have 

 been privileged to hear Major MacMahon 

 give an account of his methods of solving the 

 problems of which the well-known Latin Square 

 is typical — problems of interest to many who are 

 not professed mathematicians — have been waiting 

 for this treatise with eagerness. Volume i. has 

 now appeared, and it amply fulfils the expecta- 

 tions which have been formed. If we may quote 

 from the preface, the object of the work is to 

 present to mathematicians an account of 

 theorems in combinatory analysis which are of a 

 perfectly general character, and to show their 

 connection as parts of a general doctrine. The 

 modesty of the author forbids him to mention 

 that the greater part of the work is his own, as 

 well as most of the important theorems which are 

 treated. It is fair to say that Major MacMahon 

 has developed a new line of mathematical work, 

 and that many of the main theorems, rescued here 

 for the first time from the author's papers in 

 scientific periodicals, must form an essential part 

 of text-books on higher algebra in the future. 



The author enters a justifiable protest against 

 the relegation of combinatory analysis to a part 

 of the theory of numbers, for the theory is alge- 

 braical up to the point of determining the enu- 

 merating generating functions. He traces his 

 method back to Laplace, who used these func- 

 tions for the theory of probability, but he has 

 greatly simplified Laplace's method by the ex- 

 tended use of symmetric functions. 



The work is divided into six sections, each 

 subdivided into a series of chapters. , As a detail 

 NO. 2409, VOL. 96] 



of arrangement, it would surely have been more, 

 convenient for a reader to find the chapters num- 

 bered consecutively throughout, instead of begin- 

 ning again with each section. Section i. contains 

 the theory of symmetric functions, and of distri- 

 butions into parcels and groups in general, and it 

 is especially notable in its clear account of the 

 uses of Hammond's operators. The theory of 

 separations is taken up in section ii., where an 

 important generalisation of Girard's well-known 

 formula is obtained. Permutations are taken up 

 in section iii., where is proved what the author 

 has ventured to call the "master theorem." This 

 theorem really deserves such a title in a subject 

 of this nature, and this section is, in fact, one of 

 the most interesting in the book. The applications 

 of the theorem to such problems as the sum of 

 the nth powers of binomial coefficients are very 

 elegant, and the chapter on lattice permutations 

 is a very valuable piece of work. The theory of 

 the compositions of numbers appears in section 

 iv., and further applications of the master theorem 

 are developed in this connection. Simon New- 

 comb's celebrated problem suggested by a game 

 of "patience" is treateti in a very attractive man- 

 ner. Section V. takes up the question of distribu- 

 tions on a chess board, preceded by a discussion 

 of the perfect partitions of numbers. This sec- 

 tion will completely displace any other account 

 of such problems, and shows the power of the 

 analytical method very strongly. The sixth and 

 last section is concerned with the enumeration of 

 the partitions of multipartite numbers. 



These brief references will serve to indicate 

 the main outlines of the work, but they neces- 

 sarily miss many of its characteristic features. 

 The book is interesting even when the analysis 

 becomes somewhat cumbrous, which the author 

 allows it to do as little as possible. It is pub- 

 lished by the Cambridge University Press, and 

 maintains the tradition of excellence of this series. 

 While congratulating the author, we hope that 

 the second volume will not be long delayed. 



THE TINTOMETER. 

 Light and Colour Theories and their Relation 

 to Light and Colour Standardisation. By 

 J. W. Lovibond. Pp. xii + 90. (London: 

 E. and F. N. Spon, Ltd., 1915.) Price 65. 

 net. 



MR. J. LOVIBOND is known as the inven- 

 tor of the tintometer. He has written 

 no preface to the present book in the ordinary 

 acceptation of the word, but commences with a 

 chapter on "Purpose," which is largely devoted 

 to enumerating the awards he has received from 

 international juries and various scientific socie- 



