REVIEWS 695 



sources various statements by Oughtred as to how mathematics should be taught 

 — an interesting conclusion to the work. 



W. W. Rouse Ball. 



Combinatory Analysis. By Major Percy A. MacMahon, F.R.S., D.Sc, 

 LL.D. (late Royal Artillery), of St. John's College, Cambridge. Vol. II. 

 [Pp. xix + 340.] (Cambridge : at the University Press, 1916. Price 

 1 8 s. net.) 

 The issue of the second volume of Major MacMahon's "Combinatory Analysis" 

 follows close upon that of the first. 1 It is welcome for many reasons. We are 

 glad to have this great work complete ; for it has been an unamiable weakness of 

 English mathematicians often to interrupt their tasks for so long that the first 

 parts of their works have become tired of waiting for the companion volume, and 

 settled down into a state of unblessed singleness upon our shelves. This volume 

 redeems, within a year or so, the pledge made by its predecessor, and fully up- 

 holds the high expectations based upon the part first published. In a certain 

 sense the two volumes are so much one book that they might have been bound 

 together within a single cover ; thus treated they would have formed a book of 

 some 650 pages — a size exceeded by some treatises of similar format published by 

 the Cambridge University Press. One disadvantage of this course would, how- 

 ever, have been that we should not have had the brief historical note in the 

 Introduction of the present volume upon Waring's Formula for the sum of the 

 powers of the root of an equation, in which the writer restores to its true author 

 a formula which, on German authority, he had in the first volume attributed to 

 Albert Girard. 



In the contents of Vol. II. Major MacMahon has relied even more than in 

 the earlier part upon his own work. These problems are often so highly special- 

 ised that it is impossible to deal with them as fully as they deserve in these pages ; 

 their solution depends upon analysis which would be intelligible only if presented 

 at very great length. The volume consists of five new sections dealing with par- 

 titions of numbers, partition in two dimensions, and symmetric functions of several 

 systems of quantities. 



In the account given of the partition of numbers Euler's theory naturally claims 

 the first place, but much of the work of Jacobi, Gauss, Cayley, Sylvester, and 

 Glaisher is presented to us, and enriched by the author's own work. The mathe- 

 matician whose turn of mind inclines him to studies in algebra is always attracted 

 to these intractable series, which crop up so unexpectedly in the theory of par- 

 titions and in elliptic functions. Such a series is Jacobi's series, celebrated by our 

 author as "justly considered to be one of the most remarkable in the whole range 

 of pure mathematics." 



(i-.r)3(i--*r3) 3 (i— *3) 3 . . . . = 1 - ye+sx* - 7s* + . + (- 1) x + \ . . 



It is, perhaps, the part which series of this kind play in connection with other 

 branches of mathematics that in the present state of our knowledge justifies their 

 study, if any justification were needed ; they have no special theory of their own 

 apart from the applications in which they play so notable a part. Their mystery 

 has been deepened by a remarkable group of identities (presented in this book 

 perhaps for the first time) discovered by the young Cambridge mathematician, 

 Mr. Ramanujan. These identities have applications in the partitions of numbers, 

 and resemble in form some of Euler's results ; but they are curious because, 

 1 See Science Progress, April 1916. 



