82 



NATURE 



[March 29, 19 17 



tionately so numerous as males. Ag-ain, in the 

 life of salmon at sea, while g^rowth is more rapid 

 in summer than in winter, it is erroneous to 

 say that feeding- is practically confined to the 

 summer. 



Regarding purely marine species, the general 

 information is voluminous and authoritative. 

 The author should, however, revise his impres- 

 sions as to the relative sizes of the two sexes of 

 the conger ; the species of the dog-fish pest of 

 the English Channel, which is oredominantly 

 Acanthias vulgaris, and not ScylUum canictda; 

 and the adequacy of the scale-reading method for 

 the determination of the age of haddock. 



Notwithstanding what we have criticised as 

 faults of commission or omission in this sub- 

 stantial work, it is one which no one interested 

 in fishery science or desirous of an up-to-date 

 grasp of some of the phenomena underlying prac- 

 tical fishery questions can afford to overlook. 



THE PARTITIONS OF NUMBERS. 



Combinatory Analysis. By Major P. A. Mac- 

 Mahon. Vol. ii. Pp. xix + 340. (Cambridge: 

 At the University Press, 1916.) Price 185. net. 

 AXT^HEN the first volume of this work was 

 » • noticed in these columns, the reviewer of 

 that volume expressed the hope that the second 

 would not be long delayed. This hope has been 

 fulfilled, and the reader can now obtain, for the 

 first time, a connected account of all the modern 

 work — so largely due to Major MacMahon him- 

 self — which has been done in connection with the 

 partitions of numbers and with allied problems. 

 For the more historical side of the subject the 

 author refers the reader to Netto's "Combina- 

 torik," and he is more concerned to present the 

 newer processes and ideas which lie at the root 

 of the present rapid development of the subject, 

 and have not hitherto found a place in any 

 book. Thus only the earlier sections of the pre- 

 sent work overlap that of Netto, and certain in- 

 vestigations which are arithmetical rather than 

 algebraical are dismissed briefly. 



It is not possible, in a short space, even to 

 enumerate the many problems of interest which 

 are discussed in this volume — whether problems 

 of analytical development of functions, or problems 

 of a general interest to the non-mathematician, but 

 the solution of which depends on the partitions of 

 numbers. We must perforce confine attention to 

 certain outstanding features, and make no attempt 

 at a summary of the contents of the work. 



The introduction contains a list of the memoirs 

 to which reference is made, and the index to both 

 volumes is at the end of the book. Chap. i. 

 begins with Euler's "intuitive" theory of parti- 

 tions, and gives an account also of the powerful 

 graphical method devised by Ferrers, and used 

 so much by Sylvester. In the next chapter more 

 special attention is given to Durfee's method of 

 studying the graph of a partition, and a very 

 complete set of expansions of generating func- 

 tions is a notable feature. 



NO. 2474, VOL. 99] 



The most remarkable of such expansions are 

 the pair discovered intuitively by Ramanujan. 

 Only a few months ago it was found, by 

 Ramanujan himself, that all the arithmetical 

 labour of many mathematicians who have tested 

 one of these expansions, for example, to eighty- 

 nine terms in the supposed default of a real proof, 

 has been wasted, for the theorem was, in fact, 

 proved by Rogers more than twenty years ago, 

 and at the time attracted little attention. Major 

 MacMahon has made some very significant appli- 

 cations of these theorems to a branch of the 

 theory of partitions on which work was scarcely 

 possible before their discovery. 



The author has succeeded in basing the theory 

 of partitions upon the theory of Diophantine in- 

 equalities. This method is much more funda- 

 mental than that of Euler, and its use has ren- 

 dered the theory of partitions highly general, so 

 that it has now quite lost its earlier character — 

 undoubtedly hitherto the cause of its comparative 

 neglect by mathematicians^ — of a set of somewhat 

 isolated, though elegant, solutions of special 

 problems. It seems fair to claim, in fact, that we 

 are indebted to the author for a new branch of 

 mathematics, and a branch which must dominate 

 future treatises which make a prominent use of 

 algebraical processes. 



Many chapters must be passed over without 

 specific reference, but special mention must be 

 made of those on magic squares, partitions in two 

 dimensions, and the further theory of the Latin 

 square. It is probable that many mathematicians 

 are not aware of the extent to which these sub- 

 jects have developed, and of the field of work 

 which is still unexplored, and capable of yield- 

 ing results which are fundamental not alone in 

 connection with the partitions of numbers. By 

 collecting these researches, which are so very 

 much his own, from their hiding-places in scien- 

 tific memoirs into these two volumes, the author 

 has done much towards the promotion of a more 

 general outlook on the whole range of analytical 

 work usuallv classed somewhat vaguelv as 

 "algebra." 



OUR BOOKSHELF. 



Nature Study Lessons Seasonally Arranged. By 

 J. B. Philip. Pp. ix+147. (Cambridge: At 

 the University Press, 1916.) Price 2s. 6d. net. 

 It is generally agreed that in the early stages 

 science teaching should consist of ' ' Nature 

 study." But it is seldom that teachers realise 

 that this should afford a sound foundation on 

 which later, more serious, study can securely rest. 

 Mr. Philip's little book is a bright and outstand- 

 ing exception to the general rule. It concerns 

 itself solely with botanical material in its twelve 

 chapters; but the child of from twelve to fourteen 

 years of age who works through its pages in the 

 course of a year will not merely have learnt to 

 observe, but also will have gained a firm grasp of 

 fundamental botanical principles. From the outset 

 the author is at pains to impress the fact that the 



