316 SECTIONAL TRANSACTIONS.— A, B. 



into primes again unique, and the general lines on which a solution was ultimately 

 obtained were outlined about a hundred years ago. It was Richard Dedekind who 

 finally consolidated the whole theory some fifty years ago. Dedekind showed that, 

 when the integral elements of a field of algebraic numbers are suitably defined, these 

 integers can be combined (in accordance with rules advanced by him) into ' ideals.' 

 and that every ideal is uniquely expressible as a product of prime ideal factors. These 

 ideals can be arranged in a finite number of classes, the determination of which is 

 one of the fundamental problems connected with the field. In some fields there is 

 only one class of ideals, and every integer can then be f actorised uniquely into primes, 

 just as in the case of ordinary integers. A field of algebraic numbers possesses units, 

 integers which divide every integer contained therein, and an important problem 

 is to determine all the units of the field. There exist only three types of number-field 

 in which as yet the class-number and all the units can be infallibly determined in a 

 finite number of arithmetical operations. 



Mr. B. M. Wilson. — Ramanujan's Work on Congruence Properties of the 



Number of Partitions of n. 



By numerical evidence, based upon calculations made by MacMahon, Ramanujan 

 was led to conjecture the truth of the following theorem : 



If 8 = 5* 7 h 11% 24X = 1 (mod 8), and w = X (mod 8), then p(n) = (mod 8). 



The truth of this conjecture has been established only in five independent special 

 cases and in the few other special cases which can be deduced immediately from 

 these five. In addition to Ramanujan's own publications on the problem and a 

 memoir posthumously edited for publication by Hardy, there exists a long but 

 very incomplete manuscript of Ramanujan, now in the hands of his editors, which 

 contains the results he had obtained up to his death. He there considers also con- 

 gruences with moduli 13, 17, 19, 23, 29 and 31. 



The paper is a report on the methods and scope of Ramanujan's researches, pub- 

 lished and unpublished, on this subject. 



Mr. A.. E. Ingham. — The Analytical Method in the Theory of Numbers. 



A general account of the method of Hardy and Littlewood as applied to Waring's 

 problem, including some discussion of the more recent developments based on the 

 analysis of the ' singular series.' 



Department of Cosmical Physics. 



Joint Discussion with Sections G and K on The Climates of the Past- 

 (See p. 386.) 



SECTION B.— CHEMISTRY. 



(Communications on Textiles, received at special sessions, will be found onp.411seg.) 



(For references to the publication elsewhere of communications entered in the 

 following list of transactions, see p. 430.) 



Thursday, September 1. 



Presidential Address by Dr. N. V. Sidgwick, F.R.S., on Co-ordination 

 Compounds. Followed by Discussion — Prof. G. T. Morgan, F.R.S., 

 Prof. C. K. Ingold, F.R.S., Dr. S. Stjgden, Dr. F. G. Mann. (For 

 Address, see p. 27.) 



Dr. Sugden. — The physical reality of residual valencies shown by the optical 

 activity of co-ordinated compounds of beryllium, copper, platinum, &c, is discussed. 

 A definite picture of residual affinity is offered by the singlet linkage. Application 

 of the rules of the extended electron valency theory to (a) acetylacetone and its metallic 

 derivatives, (6) compounds of co-ordination number 6, discussing the isomerides 

 possible on (i) the octahedral theory and on (ii) the singlet theory, (c) odd co-ordina- 

 tion numbers, (d) ' molecular compounds.' 



