1053 
where the summation has to be extended over 4,/=1, 2, 3. When 
in the following an index can assume one of the values 1, 2, 3, 
we will denote this index by a Latin letter. If on the contrary an 
index can assume one of the values 0, 1, 2, 3, it will always be 
denoted by a Greek letter. In case of summation over an index, 
occurring twice in a product, the sign of summation will be omitted 
in both cases. If now we perform the transformation (1), the 
expression Gy; dz, dx, becomes again a quadratic form of the diffe- 
1 
rentials of the space coordinates, and — (go, de)’ becomes again the 
60 
square of a linear differential form. Since the separation in two 
parts of the expression of the line-element given by (2) is only 
possible in one way, we may conclude, that the expressions 
Joo 
are invariants with regard to the transformation (1). Consequently 
the quantities go./Vgoo possess the character of a vector, and from 
this we conclude again, that the bilinear differentialform 
g Eken EE APP att LAG 
Ow, gs Ox, 
is also invariant with regard to the transformation (1). The constant 
s may be chosen arbitrarily, because the quantity g,, appears only 
multiplied by a constant factor after the transformation. Choosing 
the special value s=1, we see that all terms for which u = 0 
or »=O become equal to zero, so that in this case we may omit 
the index O under the summation, and we obtain the result, that 
the expression 
gol Ò (gok 
4 hs En id 0x 5) 
is invariant. As the coefficients of this differential form are anti- 
symmetrical with regard to the indices & and /, we may consider the 
expression (4) as a linear form of the differentials dj, —= derde dvd. 
Gy der der and 
s—} 
00 
IRE OG buds Ns tell) 
Now for a threedimensional extension, the expression EWG Dandy, 
remains invariant for an arbitrary transformation of coordinates, 
where G represents the determinant of the coefficients Gj, in the 
expression do? = Gj derde, for the invariant line-element, and where 
under the summation the indices 4,/,m assume the sets of values 
1,2,3 and 2,3,1 and 3,1,2. Consequently the quantities V Gdzyz 
are transformed as the components of a covariant vector (if we, in 
the usual way, call the transformation of the components de, of a 
