1353 
EUR NI + [Ri NI doi fia} ae, npt iA) 
where the second integral has to be taken over the domain 2 
enclosed by o. On the left hand side do represents a three-dimensi- 
onal surface-element expressed in natural units and N a vector 
of the magnitude 1 in natural measure conjugate with or per- 
pendicular to that element ($ 7) and directed towards the outside of 
the domain 2. The index 2 shows that the vector [R..N]+[R,.N] 
must be expressed in w-measure. At each point of the surface we 
must resolve the vector along the four directions of the coordinates, 
express each component in z-measure ($ 10) and finally, after multi- 
plication by do, we must add algebraically all 2,-components; 
similarly all 2,-components and so on. ë 
It must be expressly remarked that if an equation like (10) in 
which we are concerned with the composition of vectors at different 
points of the field-figure, shall have a definite meaning we must 
know which components are to be considered as having the same 
direction, so that they can be added. This has been determined by 
the introduction of coordinates. 
On the right hand side of the equation the index « means that 
the vector q must be expressed in wz-measure and the facfor £ had 
to be introduced because d&2 is imaginary. 
One can prove that equation (10) is equivalent to the differential 
equations which in EINSTEIN'S theory serve for the same purpose 
and further that when the equation holds for one choice of coordi- 
nates it will also be true for any other choice. 
§ 14. The prvof for these assertions must be deferred to the 
second part of this communication. For the present we shall only 
add that the part of the principal function referring to the electro- 
magnetic field is given by 
H, = fs (R.2 + Ri?) dQ, s 
where R, and Ry are, expressed in natural units, the two rotations 
that are characteristic of the field. Like the two other parts of the 
principal function, H, is not changed by a deformation of the field- 
figure. In this statement it is to be understood that the parallelo- 
grams by which Re and R, are represented take part in the deforma- 
tion. 
Some remarks on the way in which, starting from the principal 
function, we may obtain the fundamental equations of the theory 
