dy_2 = - 3l cos [theta] {4 cos² [theta] - 3} d[theta] = - 3 {4 cos²
[theta] - 3} dy
= - 3 {(4/l²) y² - 3} dy,
which gives
dA_2 = - (12/l²)y²dA + 9dA,
and
A_2 = - (12/l²) [Integral]y²dA + 9A.
But the integral gives the moment of inertia I of the area A about the axis
XX. As A_2 is proportional to the roll of W_2, A to that of W, we can write
I = Cw - C_2 w_2,
A[=y] = C_1 w_1,
A = C_c w.
If a line be drawn parallel to the axis XX at the distance [=y], it will
pass through the mass-centre of the given figure. If this represents the
section of a beam subject to bending, this line gives for a proper choice
of XX the neutral fibre. The moment of inertia for it will be I + A[=y]².
Thus the instrument gives at once all those quantities which are required
for calculating the strength of the beam under bending. One chief use of
this integrator is for the calculation of the displacement and stability of
a ship from the drawings of a number of sections. It will be noticed that
the length of the figure in the direction of XX is only limited by the
length of the rail.
This integrator is also made in a simplified form without the wheel W_2. It
then gives the area and first moment of any figure.
While an integrator determines the value of a definite integral, hence a
[Sidenote: Integraphs.] mere constant, an integraph gives the value of an
indefinite integral, which is a function of x. Analytically if y is a given
function f(x) of x and
Y = [Integral,c:x]ydx or Y = [Integral]ydx + const.
the function Y has to be determined from the condition
dY/dx = y.
Graphically y = f(x) is either given by a curve, or the graph of the
equation is drawn: y, therefore, and similarly Y, is a length. But dY/dx is
in this case a mere number, and cannot equal a length y. Hence we introduce
an arbitrary constant length a, the unit to which the integraph draws the
curve, and write
dY/dx = y/a and aY = [Integral]ydx.
Now for the Y-curve dY/dx = tan [phi], where [phi] is the angle between the
tangent to the curve, and the axis of x. Our condition therefore becomes
tan [phi] = y / a.
[Illustration: FIG. 21.]
This [phi] is easily constructed for any given point on the y-curve:--From
the foot B' (fig. 21) of the ordinate y = B'B set off, as in the figure,
B'D = a, then angle BDB' = [phi]. Let now DB' with a perpendicular B'B move
along the axis of x, whilst B follows the y-curve, then a pen P on B'B will
describe the Y-curve provided it moves at every moment in a direction
parallel to BD. The object of the integraph is to draw this new curve when
the tracer of the instrument is guided along the y-curve.
The first to describe such instruments was Abdank-Abakanowicz, who in 1889
published a book in which a variety of mechanisms to obtain the object in
question are described. Some years later G. Coradi, in Zürich, carried out
his ideas. Before this was done, C.V. Boys, without knowing of
Abdank-Abakanowicz's work, actually made an integraph which was exhibited
at the Physical Society in 1881. Both make use of a sharp edge wheel. Such
a wheel will not slip sideways; it will roll forwards along the line in
which its plane intersects the plane of the paper, and while rolling will
be able to turn gradually about its point of contact. If then the angle
between its direction of rolling and the x-axis be always equal to [phi],
the wheel will roll along the Y-curve required. The axis of x is fixed only
in direction; shifting it parallel to itself adds a constant to Y, and this
gives the arbitrary constant of integration.