3IO 



NA TURE 



[January 26, 1893 



potash destroys it gradually, forming potassium chloride, fluoride, 

 and carbonate. It was obtained by treating carbon tetrachloride 

 with a mixture of antimony trifluoride and bromine in equal 

 molecular proportions. It is notable that the bromofluoride 

 produced by the mixture acts not as a bromising but a fluorising 

 agent. — On a simplification of some of Tesla's experiments, by 

 H. Schoentjes. Like some recent workers in England, Prof. 

 Schoentjes has found that most of the experiments can be pro- 

 duced, although with lesser intensity, without the bobbin 

 immersed in oil, the discbarge exciter, and the condenser, simply 

 by the first Rhumkorff coil, whose dimensions need not exceed 

 7x17 cm, — On a process of sterilisation of albumin solutions 

 at ,1CK)° C. , by Kmile Marchal. Albumin can be easily sterilised 

 at 100° C, without coagulation, by first adding 0'05 gr. per litre 

 of borax, or o'co5 of ferrous sulphate in a 2 to 5 per cent, 

 solution, or 4 to 5 gr. nitrate of urea per litre of 10 per cent, 

 solution. The "incoagulable albumin " thus obtained is per- 

 fectly suitable for cultivations. 



SOCIETIES AND ACADEMIES. 

 London. 



Royal Society, November 24, 1892. — " Memoir on the 

 Theory of the Compositions of Numbers," by P. A. MacMahon, 

 Major R.A., F.R.S. 



In the theory of the partitions of numbers the order of 

 occurrence of the parts is immaterial. Compositions of num- 

 bers are merely partitions in which the order of the parts is 

 essential. In the nomenclature I have followed H. J. S. 

 Smith and J. W. L. Glaisher. What are called " unipartite " 

 numbers are such as maybe taken to enumerate undistinguished 

 objects. " Multipartite" numbers enumerate objects which are 

 distinguished from one another to any given extent ; and the 

 objects are appropriately enumerated by an ordered assemblage 



of integers, each integer being a unipartite number which speci- 

 fies the number of objects of a particular kind ; and such 

 assemblage constitutes a multipartite number. The ist Section 

 treats of the compositions of unipartite numbers both analyti- 

 cally and graphically. The subject is of great simplicity, and 

 is only given as a suitable introduction to the more difficult 

 theory, connected with multipartite numbers, which is deve- 

 loped in the succeeding sections. 



The investigation arose in an interesting manner. In the 

 theory of the partitions of integers, certain partitions came under 

 view which may be defined as possessing the property of in- 

 volving a partition of every lower integer in a unique manner. 

 These have been termed " perfect partitions," and it was 

 curious that their enumeration proved to be identical with that 

 of certain expressions which were obviously "compositions" of 

 multipartite numbers. 



The generating function which enumerates the composition 

 has the equivalent forms — 



K + Jh + '^3 + ••• 



I - ^1 - ^2 — /%3 - ...' 

 g] - gj + ^3 - ... 



I - 2(gi - gj - gg - ...) 



where h^, a^ represent respectively the sum. of the homogeneous 

 products of order s and the sum of the prod ucts s together of 

 quantities 



«!. oj, ttj, ..., a,„ 

 and the number of compositions of the multipartite 



AA •••/« 



is the coefficient of a/itto^-" ... a.„p " in the development accord- 

 ing to ascending powers. 

 It is established that 



{l - .fi(2ai -I- 02 + ... -h a„)} {l - ^2(201 -f 2a2 + ... + an)) ... {l - J«(2ai + 202 + ... + 2a«) 

 is also a generating function which enumerates the compositions ; the coefficient of 



^/i.f2^- - s„^"a/^a/^- ... a/". 

 being the number of compositions possessed by the multipartite 



/i/« ... />«. 

 The previous generating function may, by the addition of the fraction I and the substitution of jjOi, ^.^a^, &c., for a^, a.^, &c., 

 be thrown into the form 



4 ^ . » . 



I - 2(2^x*'l ~ 2JiJ2«1«2 + •■• (~ )"'^^-^l-^2 ••• •S'"<»1«2 ••• "■") 



and hence these two fractions, in regard to the terms in their expansions which are products of powers of JjOj, s^a», ..., Sva,„ must 

 be identical. This fact is proved by means of the identity — 



4 , - ' : 



{l - s^{2tt^ + 0.2+ ••• + ««)} {l - •f2(2ai + 202 -1- ... + a")} ... {l - .f«(20i -t- 202 + ... + 2o„)|- 



= i 



multiplied by 

 where 



I — 2 (2JiOi - 2v2«i«2 + •••(- )"■*■ V2 ... J«ai««2 



I + ^ 2(Aki + an) ... (Ak< + UkA - (Ak, + 2aKi) ... (Akj + 2aKt) ^^^ ^^ ^^^ 



(I - Ski) (I - S/c/) "' "'' '" ""' 



Sk = iK(2ai + ... + 2aK + Ok+1 + ... + a„) - Sk{Ak + 20k), 

 and the summation is in regard to every selection of i integers from the series 



I, 2, 3, ... n, 

 and i takes all values from i to « — i. 



This remarkable theorem leads to a crowd of results which are interesting in the theory of numbers. 



The geometrical method of "trees " finds a place, and, lastly, there is the fundamental algebraic identity- 



/c {1 - Ji(/^Oi -^ a2 + ... + a„)} {l - s„{ka-^ + ka^ + ... a„)} ... {l - s„{kay+ ka^ + ... + kan)) 



_ I I ___^ 



k 1 - k-S,s^ai + k{k - i)2Ji.f2«i«2 - — + {-)"/i{k - l)"-^Vi — J««i02 ••• «« 

 multiplied by 



T + T '^(^''i + °'^i) — (A/« + ain) - (A^i + katj) .. . (A/,, + kat„) 



(^- i)(i-s/i)(i-s^2)-(i -SM ' '■■■ '" 



which reduces to that formerly obtained when k is given the special value 2. 

 NO. I 2 13, VOL. 47] 



