It is
clear that Gauss co-ordinates are nothing more than an association of
two sets of numbers with the points of the surface considered, of such
a nature that numerical values differing very slightly from each other
are associated with neighbouring points " in space."
So far, these considerations hold for a continuum of two dimensions.
But the Gaussian method can be applied also to a continuum of three,
four or more dimensions. If, for instance, a continuum of four
dimensions be supposed available, we may represent it in the following
way. With every point of the continuum, we associate arbitrarily four
numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates."
Adjacent points correspond to adjacent values of the coordinates. If a
distance ds is associated with the adjacent points P and P1, this
distance being measurable and well defined from a physical point of
view, then the following formula holds:
ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2,
where the magnitudes g[11], etc., have values which vary with the
position in the continuum. Only when the continuum is a Euclidean one
is it possible to associate the co-ordinates x[1] . . x[4].
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