1349 
sions (1,2) (2,3), (3,1), which do not intersect the indicatrix itself 
but the conjugate indicatrix, this proposition follows from a well- 
known theorem of Gauss in the theory of curvature of surfaces ; 
for the other three (1,4), (2,4), (8,4), which cut the indicatrix itself, 
the proof can be given by direct calculation. The considerations 
necessary for this, and some other calculations with which we shall 
be concerned further on will be communicated in a later paper. 
In considering the three last-mentioned extensions I have confined 
myself to triangles with real sides ($ 7, 0). 
The quotient 
é 
A — Lab 
is now for each extension a definite number, which we may consider 
as a measure of the curvature of the two-dimensional extension 
(a,b); the sum K of the six numbers A,, may be called the cur- 
vature of the field-figure at the point P in question. This quantity 
is the same that has been introduced by HirBerr ; this results from 
the calculation of its value, which at the same time shows K to 
be independent of the special choice of the directions 1, 2, 3, 4 
introduced in the beginning of this §. 
The numbers A, are all real and have a meaning that can be 
indicated without the introduction of coordinates; moreover their 
sum C is not changed by a deformation of the field-figure. 
If now d®2 is an element of the four-dimensional extension 
of the field-figure, expressed in natural measure, the part of the 
principal function belonging to the gravitation field is 
a, == | Kae, . Se Rear cue? delen 
where the integration is extended to the domain considered (§ 6) 
while x is the gravitation constant. HH, too is not changed by a 
deformation of the field-figure. 
The factor 7 has been introduced in order to obtain a real value 
for H,, the element d being represented in natural measure by a 
negative imaginary number ($ 8). 
§ 10. What we have to say of the electromagnetic field must be 
preceded by some considerations belonging to what may be called 
the “vector theory” of the field-figure. 
A line-element PQ, taken in a definite, direction, (indicated by the 
order of the letters), may be called a vector. Such vectors can be 
compounded or decomposed by means of parallelograms or paral- 
lelepipeds. Especially, when coordinates #,,...«, have been chosen, 
86* 
