1346 i: 
tion so far as it belongs to the space 2 enclosed by that surface. Then 
the quantity H, is the sum, taken with the negative ‘sign, of the 
lengths of all world-lines of material points so far as they lie 
within 2, each length multiplied by a constant m, characteristic of 
the point in question and to be called its mass. *) 
lt must be remarked that the elements of the world-lines of 
material points intersect the corresponding indicatrices themselves. 
The lengths of these lines are therefore real positive quantities. 
A deformation of the field-figure leaves H, unchanged. 
$ 7. We shall now pass on to the part of the principal function 
belonging to the gravitation field. The mathematical expression for 
this part was communicated to me by Einstein in our correspondence. 
It is also to be found in HrrBerT’s paper in which it is remarked 
that the quantity in question may be regarded as the measure of 
the curvature of the four-dimensional extension to which (1) relates. 
Here we have to speak only of the interpretation of this quantity. 
To find this the following geometrical considerations may be used. 
Let PQ and PR be two line-elements starting from a point P 
of the field-figure, QR the line-element joining the extremities Q and 
R. If then the lengths of these elements in natural measure are 
PQ=ds', PR=ds", QR=ds, : 
we define the angle (s',s") between PQ and PF by the well known 
trigonometric formula 
ds? = ds? + ds"* — 2ds'‘ds" cos (s', s") , 
ds'? +- ds''* — ds? 
> . ‚f " —_——=—— nn 
COS (s 78 ) Das'ds" (4) 
from which one can derive 
i é dà dz" 
cos (eye p= 2 (allante (5) 
ds’ ds" 
By means of this formula we are able to determine the angle 
between any two intersecting lines. Of course the two other angles 
of the triangle PQR can be calculated in the same way. 
Now two cases must be distinguished. 
a. The plane of the triangle PQR cuts the conjugate indicatrix, 
but not the indicatrix itself. Then the three sides have positive 
imaginary values. Moreover each of them proves to be smaller than 
1) This agrees with the value of the LaAGRANGIAN function, which is to be found 
e.g. in my paper on “Hamitron’s principle in Einstein's theory of gravitation.” 
These Proceedings 19 (1916), p. 751. 
Ke eT 
