78
THE MEANING OF RELATIVITY
character, that they must depend linearly and homogeneously upon the
and the
. We therefore put
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(67)
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In addition, we can state that the
must be symmetrical with respect to the indices
and
. For we can assume from a representation by the aid of a Euclidean system of local co-ordinates that the same parallelogram will be described by the displacement of an element
along a second element
as by a displacement of
along
. We must therefore have
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The statement made above follows from this, after interchanging the indices of summation,
and
, on the right-hand side.
Since the quantities
determine all the metrical properties of the continuum, they must also determine the
. If we consider the invariant of the vector
, that is, the square of its magnitude,
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which is an invariant, this cannot change in a parallel displacement. We therefore have
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or, by (67),
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