18 SECTIONAL ADDRESSES. 



in space where matter of density p is present. In general relativity, 

 when the field is statical, these are replaced by an equation 



where A 2 V is the Beltrami's second differential parameter for the form 

 ds 2 = %a ilc dxi dx k which specifies the line-element in the three-dimen- 

 sional space, Tj fc is the energy-tensor, and N is the velocity of light at 

 the point. This equation reduces to Laplace's equation in one extreme 

 case (when no matter or energy is present at the point) and to Poisson's 

 equation in another extreme case (when the energy is entirely in the 

 form of ordinary matter), but it offers an infinite variety of possibilities 

 intermediate between the two, in which energy is present but not in the 

 form of ordinary matter. It is possible that this equation, which evidently 

 suggests an approach to the new wave-mechanics, may play as important 

 a part in the microphysics and astrophysics of the future as the equations 

 of Laplace and Poisson have played in the ordinary physics of the 

 past. 



Let us take another consequence of the new theory. Consider the 

 field due to a single gravitating particle. Take any plane through the 

 particle, and in this plane draw the family of concentric circles, whose 

 centre is at the particle. The length of the circumference of these circles 

 will, of course, diminish as we take circles nearer to the centre : and at 

 one place we shall have a circle whose circumference is of length 



4:rpM/c 2 



where (3 is the Newtonian constant of attraction, M is the mass of 

 the particle in grams, and c is the velocity of light in empty space. When 

 we arrive at this circle we find that the element of length directed radially 

 towards the centre is infinite : that is to say, the space within the circle 

 is impenetrable. Every gravitating particle has a ring-fence around it, 

 within which no other body can approach. 



It will be noticed that in all that I have said I have used the ordinary 

 language of three-dimensional physical space, and have avoided mention 

 of that four-dimensional world of space-time which looms so largely in 

 most expositions of relativity. The reason is that I have been speaking 

 only of phenomena belonging to the statical class, i.e. those for which the 

 field does not vary with the time : and for such phenomena, as Levi-Civita 

 showed in a famous paper on the Rendiconti dei Lincei of 1917, the four- 

 dimensional problem can be reduced to a three-dimensional one of the same 

 kind as physicists have been accustomed to deal with. It may be consoling 

 to those who distrust their own powers of doing research in four dimensions 

 to know that in general relativity there are enough important unsolved 

 problems of the statical type, for which capacity in three dimensions is 

 sufficient, to keep all the investigators of the world busy for at least 

 another generation. 



It is interesting to see how these new three-dimensional problems 

 differ from those of the older Physics. Taking as an example a small 



