SECTIONAL TRANSACrriONS.— A. 303 



the sample. Practical convenience dictates the exact nature of the symmetric functions 

 which should be calculated, and the derivation of the final result involves the methods 

 of combinatorial analysis, and in particular the theory of separations. The nature of 

 earlier work on this subject, involving knowledge of the products of the elementary 

 symmetric functions, is sketched, and the waj' in which the most modern methods 

 have been evolved is thereafter described, with appropriate illustrations. 



Mr. J. 0. Irwin. — The Approximate Numerical Evaluation of Single and 

 Double Integrals. 



(A) The principal methods of quadrature for one variable : (a) The Euler-McLaurin 

 formulae ; (6) Other formulae dependent on equidistant ordinates ; (c) Formulae 

 based on Everett's central-difference interpolation formulae ; (d) Gauss' method. 



(B) The extension of A (c) to double integrals. Other formulae for cubature. 



Dr. L. J. CoMRiE. — Modern Bahhage Machines. 



Babbage, over 100 years ago, was the pioneer in efforts to build a machine that 

 would calculate a function from its higher order differences. His partially completed 

 difference-engine, as well as that of Scheutz, whom he inspired, is now in the Science 

 Museum at South Kensington. Other spasmodic attempts to build machines for 

 mechanical integration are on record. About three years ago three commercial 

 machines appeared — the Nova Brunsviga Model IVA, the Brunsviga-Dupla, and the 

 Burroughs Class 1 1 machine — all of which are capable of integrating from second finite 

 differences, the latter with printing of argument, function, first and second differences. 

 They will be described and demonstrated. 



Monday, September 8. 



Presidential Address by Dr. F. E. Smith, C.B., C.B.E., Sec. R.S., on 

 The Theories of Terrestrial Magnetism. (See p. 15.) 



Dr. P. A. M. DiRAC— TAe Proton. 



It is believed that all matter is built up from two elementary kinds of particle, 

 the electron and the proton. Recent theoretical work seems to show, however, that 

 these two kinds of particle are not independent, but are connected in such a way 

 that actually there is only one fundamental kind of particle in nature. 



The kinetic energy W of an electron is given in terms of the momentum, according 

 to the principle of relativity, by an equation which is quadratic in W, allowing of 

 negative roots as well as of positive ones. Ordinarily the negative roots are discarded 

 as being unwanted and physically meaningless. This is not permissible in the 

 quantum theory, however, since there transitions can take place from states of 

 positive energy to states of negative energy. It then becomes necessary to give a 

 physical meaning to the negative-energy states. This we can do only by assuming 

 that nearly all the negative-energy states are occupied by electrons, with just one in 

 each state in accordance with the Exclusion Principle of Pauli. We can then interpret 

 the unoccupied negative-energy states as jnotons. Elementary considerations show 

 that they will appear to us as things having a positive energy, and also a positive 

 charge. 



There are certain difficulties in the theory that have not yet been removed. They 

 are (1) the great difference in the masses of the proton and the electron, and (2) the 

 fact that the theory predicts that electrons and protons will annihilate one another 

 at a rate which is much too great to be correct. These difficulties will perhaps be 

 removed by a better understanding of the interaction between ele(;trons. 



Dr. F. W. Aston, F.R.S. — Some New Mass-Spectra. 



Mr. R. Stoneley. — The Identification of the Phases of Earthquake Shocks. 



It has been shown by Jeffreys, Conrad and others that for the different waves 

 recorded in the P and S phases of earthquakes, the distance-time curves approximate 



