300 SECTIONAL TRANSACTIONS.— A. 



With the oidiiiarv conception of an ' infinite value,' onh' the crudest algebraic 

 combinations are allowable ; and results proved for finite values have to be discussed 

 separateh' for infinite ones, even when the final statement is formally the same in 

 both cases ; or again cases of non-finiteness are avoided altogether, and the treatment 

 left incomplete. Hence, some more refined and inclusive mode of definition is a 

 desideratum. 



The possibility of such a definition of wider scope appears bound up with the 

 algebra oj finite many -valued limits, in whose light an infinite limit, or complete infinity, 

 is seen to have some of the properties of a many- valued quantity ; and such a ' complete 

 infinity ' may be analj-sed into constituents {simple infinities) which have themselves 

 ' not qxiite ' the character of numbers. 



The ordinary rules of arithmetic are discussed with reference to these various 

 ' infinities ' and illustrations given. 



Prof. W. E. H. Berwick.— TAe Cotnplex MuUijplication of Elliptic 

 Fwictions. 



Application of the addition theorem in trigonometry leads to formulae expressing 

 tan 26, tan 39 ... in terms of tan 6. Similar formulse in elUptic functions express 

 F{»»m) in terms of F(m), F'(m) for all integral values of m. 



■When the modulus is such that the primitive periods are 2c, 2ca/-?i the function 

 F(?t-v/-7i) is also doubh^ periodic in 2c, 2cv'-m and so, by a well-known theorem on 

 functions of a complex variable, is expressible rationally in terms of F(m), F'(m). 

 A method of obtaining this rational expression is given when F(m) is Weierstrass' 

 elliptic function of the second order. Numerical examples are included, one showing 

 F(uV'-5) in terms of F(m). The paper ends with a statement of four arithmetical 

 problems on complex multiplication which are not yet entirelj' settled. 



Friday, September 5. 



Sir E. Rutherford, Pres.R.S., Mr. F. A. B. Ward, and Mr. C. E. Wynn- 



WiLLiAMS. — Neiv Methods of Analysis of a.— Particles. 



Papers dealing with Aspects of the Solid State : — 



Considerable progress has been made recentlj' towards an understanding of the 

 physical properties of the solid state, due, on the one hand, to the advent of the wave 

 mechanics and on the other to the development of X-ray technique. The object of 

 the discussion is to review recent advances from the theoretical and experimental 

 points of view. 



Prof. J. E. Lennard-Jones. — The Nature of Cohesion. 



The object of this discussion is to re^^ew the progress which has been made within 

 the last three or four years towards an understanding of the nature of solid bodies. 

 This has been due to the happy interplay of theory and experiment, for both have 

 made important contributions to our knowledge. This first paper attempts to review 

 some of the principles which have been established recently with regard to the nature 

 of atoms and their bearing on the all-important question of their reaction with one 

 another. We cannot hope to understand or influence the physical properties of solid 

 bodies until we understand the nature of the cohesive forces which hold soUds together. 



There are three main new ideas which have become established within the last 

 few years. Firstly, there is the main idea of the wave mechanics that it is impossible 

 to follow the electron in all its ways. All that we can hope for is a knowledge of the 

 probability that an electron shall do this or do that. The result of this is that we 

 now deal in patterns instead of orbits. An electron in the presence of a positive 

 nucleus may have this or that pattern, each continuous about the nucleus and 

 occupj'ing the whole of space. To each pattern corresponds a definite energy, and 

 the electron normally stays in that pattern which has the lowest energy. 



Secondly, there is the principle, first enunciated by Pauli, that there are never 

 more than two electrons which take up the same pattern, and of these pairs one has 

 one kind of electron spin and the other the opposite kind. 



