is related to a similar minor of the determinant of the coefficients . If corresponds to in the way mentioned in § 25, and in the same way to , we have
so that (31) becomes
According to (27) this becomes
for which we may write
Interchanging and in the second of the two parts into which the sum on the right hand side can be decomposed, and taking into consideration that
as is evident from (26) and (27), we find
§ 29. Finally it can be proved that if equation (10) holds for one system of coordinates , it will also be true for every other system , so that
To show this we shall first assume that the extension , which is understood to be the same in the two cases, is the element .
For the four equations taken together in (10) we may then write
and in the same way for the four equations (32)
We have now to deduce these last equations from (33). In doing so we must keep in mind that are the -components and the -components of one definite vector and that the same may be said of and .
Hence, at a definite point (comp. (30))
- Comp. (28) l. c.