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< 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
Y = [Integral,0:c] f(x) [phi]([xi]x) dx = F([xi]), . . . (1)
where Y equals the displacement of T to some scale dependent on the
constants of the instrument.
If the whole block of templets be now pushed under the points P and if the
drawing-board be moved at the same rate, then the pen T will draw the curve
Y = F([xi]). The instrument now is an _integraph_ giving the value of a
definite integral as function of a _variable parameter_.
Having thus shown how the lever with its springs can be made to serve a
variety of purposes, we return to the description of the actual instrument
constructed. The machine serves first of all to sum up a series of harmonic
motions or to draw the curve
Y = a_1 cos x + a_2 cos 2x + a_3 cos 3x + . . . (2)
The motion of the points P_1P_2 ... is here made harmonic by aid of a
series of excentric disks arranged so that for one revolution of the first
the other disks complete 2, 3, ... revolutions. They are all driven by one
handle. These disks take the place of the templets described before. The
distances NG are made equal to the amplitudes a_1, a_2, a_3, ... The
drawing-board, moved forward by the turning of the handle, now receives a
curve of which (2) is the equation. If all excentrics are turned through a
right angle a sine-series can be added up.
It is a remarkable fact that the same machine can be used as a harmonic
analyser of a given curve. Let the curve to be analysed be set off along
the levers NG so that in the old notation it is
y" = f(x),
whilst the curves y' = [phi](x[xi]) are replaced by the excentrics, hence
[xi] by the angle [theta] through which the first excentric is turned, so
that y'_k = cos k[theta]. But kh = x and nh = [pi], n being the number of
springs s, and [pi] taking the place of c. This makes
k[theta] = (]]>
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