[Integral,0:c][phi](x)dx = h/[mu] [Sigma]y = ([lambda]h/[mu])z,
where z is the displacement of the point B which can be measured. The curve
y' = [phi](x) may be supposed cut out as a templet. By putting this under
the points P the area of the curve is thus determined--the instrument is a
simple integrator.
The integral can be made more general by varying the distances NG = y".
These can be set to form another curve y" = f(x). We have now y = [mu]y'y"
= [mu] f(x) [phi](x), and get as before
[Integral,0:c]f(x) [phi](x)dx = ([lambda]h/[mu])z.
These integrals are obtained by the addition of ordinates, and therefore by
an approximate method. But the ordinates are numerous, there being 79 of
them, and the results are in consequence very accurate. The displacement z
of B is small, but it can be magnified by taking the reading of a point T'
on the lever AB. The actual reading is done at point T connected with T' by
a long vertical rod. At T either a scale can be placed or a drawing-board,
on which a pen at T marks the displacement.
If the points G are set so that the distances NG on the different levers
are proportional to the terms of a numerical series
u_0 + u_1 + u_2 + ...
and if all P be moved through the same distance, then z will be
proportional to the sum of this series up to 80 terms. We get an _Addition
Machine_.
The use of the machine can, however, be still further extended. Let a
templet with a curve y' = [phi]([xi]) be set under each point P at right
angles to the axis of x hence parallel to the plane of the figure. Let
these templets form sections of a continuous surface, then each section
parallel to the axis of x will form a curve like the old y' = [phi](x), but
with a variable parameter [xi], or y' = [phi]([xi], x). For each value of
[xi] the displacement of T will give the integral