THE EINSTEIN THEORY OF RELATIVITY 643 



turbance in the space of the world is matter, and the more matter there exists 

 the more space there is to accommodate it. Thus, because space depends on 

 matter, this matter is a necessity of the world. Bearing in mind the fourfold 

 nature of our present world, it can be seen that there is greater continuity 

 in this geodesic structure of space than in the older material structure of the 

 world. 



Point-events are used in the theory because they represent the most 

 fundamental concepts of nature, and the aggregate of all such point-events 

 represents the universe. Therefore, certain coefficients in the equations of 

 space must be identified with natural forces in order to correlate the new 

 dynamics with the old. To illustrate this process, let us take the simple case 

 as given on p. 189 in Science Progress, October 1921. 



Consider a unit gauge of interval length as being / at point P and let it be 

 moved from P in space, whose co-ordinates may be defined as the origin, to a 

 point Q whose co-ordinates are defined by the infinitesimals bxi, bxz, bx^, bx4,. 

 Let the change in its length due to this movement, in terms of the displace- 

 ments, be X.l. 



X is thus made a function of the displacements bxi, bx^, bx^, bx^, since the 

 change in length must be proportional to the initial length, and independent 

 of the direction of motion because the interval length is independent of direc- 

 tion. 



Then X = cf)ibxi + (ft^bx^ + ^3*^3 + (fabx^ .... (i) 



Now consider a frame of co-ordinates x, y, z, t in uniform motion ; let 

 F, G,H, — ^ stand for <3!)i, ^2. ^3. ^4. respectively, in (i) 



dl 

 then X = -7- 



= Fdx + Gdy + Hdz —-^dt . . . . (ii) 



where dl represent the change in length of the rod due to its movement 

 from P to Q. 



On integrating (ii), we get 



log / + const ^[{Fdx + Gdy + Hdz — *t?0 .... (iii) 



From an examination of (iii), it will be seen that if / is to be independent 

 of the path taken from P to Q, the expression 



Fdx + Gdy + Hdz — ^dt 



must be a perfect differential. 

 If this is true we have 







S* bF_ 

 'bx~ bt 

 S^ bG 

 by~ bt 

 ^_bH _ 

 bz bt ~° 



= . . . (iv) 



On examining these forms of (iv) it will be seen that these are the expres- 

 sions for the three components of magnetic force and the three components 

 of electric force, if we let F, G, H, — ^ represent the usual potentials of the 

 electromagnetic theory. Thus we may state that, in the given case, the con- 

 dition for distant intervals to be directly compared, independently of the 

 paths between the points, is that the magnetic and electric forces have a 

 null efiect in the neighbouring four-fold space. 



