969 
the deductions; and that in a way which fully appreciates the fact that 
the tensor of the ten gravitation potentials and the tensor of the four 
electrodynamical potentials, being directed quantities, have a geometrical 
character (§ 12 etc.). Moreover the tensors of stress, momentum 
and energy appear in a new way from the variation calculation. 
In the following paragraphs this will be shown. Thanks to the 
cited papers and to some others, a short indication will often suffice. 
The variation principle. 
1. For a material particle, falling under the influence of a force, 
Hamiuton’s principle takes the form: 
2 
2 
0=d] — mds + = (p) ky JrP ds, 
1 1: 
where m is the coefficient of mass of the particle, ds the arc-length 
of the world-line run by the particle in the world referred to a system 
of four space-time parameters ‚zr. Further k,(p=1, 2, 3, 4) 
represents the four-vector of the force acting on the particle, while 
Or? (p =1, 2, 3, 4) denote the components of the virtual displacements. 
In the variation of the motion there corresponds to each point- 
instant 2, (m= 1,2,3,4) of the unvaried path a point-instant 
En + Or" (m = 1, 2,3, 4) of the varied path. The final points of the 
path remain unvaried. As usually we assume 
der == > (ab) gar Gon 0a, 
where ga (a, 6 = 1, 2, 3, 4, gas = Joa) are the gravitation potentials. 
If the particle has an electric charge, so that it is influenced by 
an electro-magnetic field this may be taken into consideration by 
writing 
2 2 
0 =d | (— mds + AL (I) epida) + | Xk, drvds. 
1 1 
Here gy; (/=1, 2, 3,4) represent the electro-dynamic potentials, 
four quantities changing from point to point and determining the 
field. 2 is a constant determined by the choice of the units of mass 
and charge in which m and e are expressed. Now 4, no longer 
contains the electric forces. 
2. Applied to a limited extension of the four-dimensional world 
HamiLton’s principle is represented by the equation: 
