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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[1]

§ 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))


We shall particularly denote by the values of these quantities belonging to the angle from which the edges issue

  1. Comp. (28) l. c.