6 9 6 SCIENCE PROGRESS 



although they have been verified to a large number of terms, they have not been 

 established generally. They thus resemble Fermat's last theorem, which has 

 withstood all attempts to solve it for two hundred years ; how long Ramanujan's 

 identities will remain unproved it were idle to speculate upon ; but there is 

 one difference between the two cases besides that of date. We know that 

 Ramanujan cannot at present prove his identities, but whether Fermat had a 

 method of solving his theorem will probably never be known. It will be enough, 

 perhaps, to quote here the first of the identities. It is written : 



T J- X J. ^ + + ^ + 



I+ F=* (i -x)(i-x>y ' ' + (i -*) (i -*').. (i-;rO 



- (i - x)- 1 (i - **r (i - *«)" .... (i - **r (i - * 9 )~ l (i - ^y i .... 



the indices of x in the last product increasing by 5 in the factors. The identity 

 expresses that the partitions whose parts are limited to one of the two forms 

 tm -f 1, 5?« + 4 are equi-numerous with those which involve neither repetitions 

 nor sequences. This section of the book abounds with deductions from the theory 

 of partitions of strange identities ; thus the author establishes in this way the 

 equality of the two series : 



r = » 2-!2lL ™ - *> y*-d + 4*- + *~) (m odd) 



r - I * 1 - *» » m - 1 -^ (1- x m Y 



Hitherto a partition of n has been defined as a collection of positive numbers 

 whose sum is n, but in the seventh section the author restricts the collection by 

 prescribing an order amongst the parts, namely the descending (or rather non- 

 ascending) order of magnitude. The generating function of Euler — 



(1 -x)"(l -**)"" • . . (I-*T 



gives as coefficient of x" the number of partitions of x" when the function is ex- 

 panded in powers of x. Major MacMahon substitutes for this function a crude 

 generating function— 



(1 - a,*)- 1 (1 - x^A,)" (1 - x 3 *A 3 )- J .•••('- M*,-:)" 1 



in the expansion of which all negative powers of X's are omitted, and then the X's 

 are all put equal to unity ; the coefficient of x" gives the number of partitions in 

 which the parts have a descending order of magnitude. Beyond this an ultra- 

 crude generating function is introduced, but thither a review can scarcely follow. 

 Such partitions are regarded as one-dimensional. 



In the following sections the author considers two-dimensional partitions of a 

 number the parts of which are arranged in descending order whether we pass 

 along a row from the left in the direction of Ox or along a column in the direction 

 of Oy. The complete solution of the problem is given, having yielded to the 

 lattice function and its derivatives which the author invented for its subjugations. 

 The question of partitions in solido is broached, but the complete solution has not 

 been reached ; we are told that the subject bristles with difficulties. Partitions in 

 hyper-space have not yet been introduced. 



The book contains ideas which are new, and which may well prove the starting- 

 point of future investigations. In this particular branch of analysis the English 

 mind has proved itself skilful and adventurous before. But the subject is one of 

 very great difficulty, and it is not given to every one to bend the bow of Achilles. 



The University Press of Cambridge has a reputation for printing which it 

 would be idle to deny. In mathematical printing, however, the printer has, 



