Q = x{½([Omega]0 + [Omega]_n) + [Omega]1 = [Omega]2 + ... + [Omega](n-1)}.
The areas [Omega]0, [Omega]1 ... are easily ascertained by means of
the polar planimeter. A slight difficulty arises in the part of the
solid lying above the last contour curve. This will have generally a
height which is not exactly x, and a form more rounded than the other
layers and less like a conical frustum. The volume of this may be
estimated separately, and taken to be the area of its base (the area
[Omega]_n) multiplied by 1/3 to ½ its height.
[Illustration: FIG. 151.]
Fig. 151 shows the results of one of Harlacher's gaugings worked out
in this way. The upper figure shows the section of the river and the
positions of the verticals at which the soundings and gaugings were
taken. The lower gives the curves of equal velocity, worked out from
the current meter observations, by the aid of vertical velocity
curves. The vertical scale in this figure is ten times as great as in
the other. The discharge calculated from the contour curves is 14.1087
cubic metres per second. In the lower figure some other interesting
curves are drawn. Thus, the uppermost dotted curve is the curve
through points at which the maximum velocity was found; it shows that
the maximum velocity was always a little below the surface, and at a
greater depth at the centre than at the sides. The next curve shows
the depth at which the mean velocity for each vertical was found. The
next is the curve of equal velocity corresponding to the mean velocity
of the stream; that is, it passes through points in the cross section
where the velocity was identical with the mean velocity of the stream.