In scientific researches in the domain of physics we often meet with the
following problem: Being given any function whatever, y = f(x), to find
a curve whose equation shall be
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y = | f(x)dx + C.
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[TEX: y = \int f(x) dx + C.]
Let us take an example that touches us more closely; let us suppose that
we know an induced current, and that we can represent it by a curve
y=f(x). The question is to find the inductive current, that is to say,
the curve represented by the equation
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y = | f(x)dx + C.
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_/
[TEX: y = \int f(x) dx + C.]
The apparatus called an integraph, constructed by Messrs. Napoli and
Abdank-Abakanowicz, is designed for solving this problem mechanically,
by tracing the curve sought. Let us take another example from the domain
of electricity, in order to better show the utility of the apparatus;
let us suppose that we have a curve representing the discharge of a pile
or of an accumulator. The abscisses represent the times, and the
ordinates the amperes. The question is to know at every moment the
quantity of coulombs produced by the pile. The apparatus traces a curve
whose ordinates give the number of coulombs sought. We might find a
large number of analogous applications.
[Illustration: THE INTEGRAPH.]
The apparatus is represented in the accompanying figure. An iron ruler,
I, parallel with the axis of the X's, is fixed upon a drawing-board, and
is provided with a longitudinal groove in its upper surface. In this
groove move two rollers, which, in the center of the piece that connects
them, carry two brass T-squares that are parallel with each other and at
right angles with the first, or parallel with the axis of the Y's.
Between these two rulers move two carriages, the first of which (nearest
the axis of the X's) carries a point, A, designed to follow the contour
of the curve to be integrated, while the second, which is placed further
away, is provided at the center with a drawing-pen, A', whose point is
guided by two equidistant wheels, R, R', that roll over the paper in
such a way as to have their plane parallel with a given straight line,
and that have always a direction such that the tangent of the point's
angle with the axes of the X's is constantly proportional to the
ordinate of the primitive curve.
The carriages are rendered very movable by substituting rolling for a
sliding friction of the axes. To this effect, the extremities of the
axes of the wheels that support and guide them are made thin, and roll
over the plane surface of recesses formed for the purpose in the lateral
steel surfaces of the carriages, while the circumference of the wheels
rolls in grooves along the two T-squares.
These latter are, on the one hand, carried by rollers that run in the
groove of the iron, I, and, on the other, by a single roller that runs
over the paper. At right angles with one of these bars is fixed a
divided ruler, through one point of which continually passes a third
ruler, whose extremity pivots upon the point, A, of the first carriage.
When the divided ruler is placed upon the axis of the X's, and the
point, A, of this carriage is following the contours of the figure to be
integrated, the tangent of the angle made by the inclined ruler with the
axis of the X's will be proportional to the ordinate of the figure. The
wheels, R and R', of the drawing-pen, A', of the second carriage must
move parallel with this ruler. In order to obtain such parallelism, we
employ a parallelogram formed as follows: Two gear-wheels of the same
diameter are fixed upon the ruler that ends at the point, A, of the
first carriage, and their line of centers is parallel with the latter.
The second carriage likewise carries two drums equal in diameter to
those of the toothed wheels. These are fixed, and their line of centers
must remain constantly parallel with the line of centers of the
gear-wheels, and consequently with the straight line which passes
through the point, A. This parallelism is obtained by means of a weak
steel spring, or of a silken thread passing over the four wheels, the
two first of which (the gear-wheels) hold it taut by means of a barrel
and spring placed in the center of one of them.
The edge of the wheels, R, R', of the second carriage prevents the
latter from giving way to the traction of the threads, permitting it
thus to move only in the direction of their plane.
It will be seen that by this system two of the sides of the
parallelogram are capable of elongating or contracting through the
unwinding and winding of the silken thread on the drums of the two cog
wheels, which latter, gearing with each other, allow of the escape of
but the same length of the two threads.
It will be observed that in this system integration is effected by
forcing the pen to follow a certain direction, and that consequently the
curve does not depend upon the dimensions of the different parts of the
apparatus.--_La_ _Lumiere Electrique_.
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