336 SECTIONAL TRANSACTIONS.— A*. 



A simple type of approximation consists in reducing the many-body 

 problem to a set of one-body problems, by considering the motion of each 

 particle in some average of the fields of the others. The ' self-consistent 

 field ' approximation to the structure of an atom. 



The variation principle and the relation of the ' self-consistent field ' 

 approximation to it. The inclusion of ' exchange ' terms in the ' self- 

 consistent field ' approximation. Survey of results of calculations of self- 

 consistent field, without and with exchange. 



More accurate treatment of simple atomic systems, including dependence 

 of wave-function on mutual distances between electrons, as well on their 

 distances from the nucleus. 



Dr. H. S. W. Massey.— Laws of interaction between particles. 



Dr. Bertha Swirles. — The relativistic self-consistent field method. 



The development of a strict relativistic theory of an atom with many 

 electrons presents much difficulty. It is, however, possible to extend the 

 self-consistent field method, using Dirac's Hamiltonian for the separate 

 electrons in the field of the nucleus and taking account not only of the 

 Coulomb interaction but of the interaction of the spins and of retardation. 



Tables have been constructed from which the total energy of an atom con- 

 taining s-, p-, ^-electrons can be calculated, taking account of ' exchange.' 

 From this the relativistic self-consistent field equations can be derived by 

 a variation method. 



The method has been applied to the evaluation of the separation of the 

 components of the 2 3 P term of helium. Slater's ' method of diagonal 

 sums,' although not as powerful as in the non-relativistic case, shortens the 

 work considerably. A comparison of the method and results is made with 

 those of Breit, who used the relativistic wave equation in its second order 

 form. 



General Discussion on The theory of complex atoms. 



Prof. M. R. Siddiqi. — The theory of non-linear partial differential equations 

 (12.20). 



Non-linear partial differential equations have lately acquired considerable 

 importance owing to the fact that they arise in various modern physical 

 problems, such as the conduction of heat in crystals, and in deep seas, the 

 field theory of Born, etc. Previous investigations dealing with such equations 

 are mostly of a function-theoretical character, requiring a knowledge of the 

 so-called ' Green's function.' 



The present paper is devoted to developing a ' Fourier method ' for the 

 non-linear parabolic and hyperbolic equations : 



8 2 u 8u "^ f8u\s 



&?~*i %, p "- six ' t)ur \8x) ' 



8 f ., N 8u) 8u v . / ^ , 



ix\ PK) hx\ 



So CO 



U V w \ , 



^ = j:Pr(X,t)u r . 



