by means of a number. The Curves _u_ = 1, _u_ = 2 and _u_ = 3 are drawn
in the diagram. Between the curves _u_ = 1 and _u_ = 2 we must imagine
an infinitely large number to be drawn, all of which correspond to real
numbers lying between 1 and 2. We have then a system of _u_-curves, and
this infinitely dense system covers the whole surface of the table.
These _u_-curves must not intersect each other, and through each point
of the surface one and only one curve must pass. Thus a perfectly
definite value of _u_ belongs to every point on the surface of the
marble slab. In like manner we imagine a system of _v_-curves drawn on
the surface. These satisfy the same conditions as the _u_-curves, they
are provided with numbers in a corresponding manner, and they may
likewise be of arbitrary shape. It follows that a value of _u_ and a
value of _v_ belong to every point on the surface of the table. We call
these two numbers the co-ordinates of the surface of the table
(Gaussian co-ordinates). For example, the point _P_ in the diagram has
the Gaussian co-ordinates _u_ = 3, _v_ = 1. Two neighbouring points _P_
and _P _ on the surface then correspond to the co-ordinates
_P_: _u, v_
_P _: _u_ + _du, v_ + _dv_,
where _du_ and _dv_ signify very small numbers. In a similar manner we
may indicate the distance (line-interval) between _P_ and _P _, as
measured with a little rod, by means of the very small number _ds_.
Then according to Gauss we have
_ds_2 = _g_11_du_2 + 2_g_12_du dv_ + _g_22_dv_2,
where _g_11, _g_12, _g_22, are magnitudes which depend in a perfectly
definite way on _u_ and _v_. The magnitudes _g_11, _g_12 and _g_22,
determine the behaviour of the rods relative to the _u_-curves and
_v_-curves, and thus also relative to the surface of the table. For the
case in which the points of the surface considered form a Euclidean
continuum with reference to the measuring-rods, but only in this case,
it is possible to draw the _u_-curves and _v_-curves and to attach
numbers to them, in such a manner, that we simply have:
_ds_2 = _du_2 + _dv_2
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