has to be integrated where X = p(x), Y = [phi](y) are given as curves. If
we write
au = [Integral]Xdx, av = [Integral]Ydy,
then u as a function of x, and v as a function of y can be graphically
found by the integraph. The general solution is then
u + v = c
with the condition, for the determination for c, that y = y_0, for x = x_0.
This determines c = u_0 + v_0, where u_0 and v_0 are known from the graphs
of u and v. From this the solution as a curve giving y a function of x can
be drawn:--For any x take u from its graph, and find the y for which v = c
- u, plotting these y against their x gives the curve required.
If a periodic function y of x is given by its graph for one period c, it
can, according to the theory of Fourier's Series, be [Sidenote: Harmonic
analysers.] expanded in a series.
y = A_0 + A_1 cos [theta] + A_2 cos 2[theta] + ... + A_n cos n[theta] +
...
+ B_1 sin [theta] + B_2 sin 2[theta] + ... + B_n sin n[theta] +
...
where [theta] = 2[pi]x / c.
The absolute term A_0 equals the mean ordinate of the curve, and can
therefore be determined by any planimeter. The other co-efficients are
A_n = 1/[pi] [Integral,0:2[pi]] y cos n[theta].d[theta];
B_n = 1/[pi] [Integral,0:2[pi]] y sin n[theta].d[theta].
A harmonic analyser is an instrument which determines these integrals, and
is therefore an integrator. The first instrument of this kind is due to
Lord Kelvin (_Proc. Roy Soc._, vol xxiv., 1876). Since then several others
have been invented (see Dyck's _Catalogue_; Henrici, _Phil. Mag._, July
1894; _Phys. Soc._, 9th March; Sharp, _Phil. Mag._, July 1894; _Phys.
Soc._, 13th April). In Lord Kelvin's instrument the curve to be analysed is
drawn on a cylinder whose circumference equals the period _c_, and the sine
and cosine terms of the integral are introduced by aid of simple harmonic
motion. Sommerfeld and Wiechert, of Konigsberg, avoid this motion by
turning the cylinder about an axis perpendicular to that of the cylinder.
Both these machines are large, and practically fixtures in the room where
they are used. The first has done good work in the Meteorological Office in
London in the analysis of meteorological curves. Quite different and
simpler constructions can be used, if the integrals determining A_n and B_n
be integrated by parts. This gives
nA_n = - 1/[pi] [Integral,0:2[pi]] sin n[theta].dy;
nB_n = 1/[pi] [Integral,0:2[pi]] cos n[theta].dy.
An analyser presently to be described, based on these forms, has been
constructed by Coradi in Zurich (1894). Lastly, a most powerful analyser
has been invented by Michelson and Stratton (U.S.A.) (_Phil Mag._, 1898),
which will also be described.
[Illustration: FIG. 23.]
The _Henrici-Coradi_ analyser has to add up the values of dy.sin n[theta]
and dy.cos n[theta]. But these are the components of dy in two directions
perpendicular to each other, of which one makes an angle n[theta] with the
axis of x or of [theta]. This decomposition can be performed by Amsler's
registering wheels. Let two of these be mounted, perpendicular to each
other, in one horizontal frame which can be turned about a vertical axis,
the wheels resting on the paper on which the curve is drawn. When the
tracer is placed on the curve at the point [theta] = 0 the one axis is
parallel to the axis of [theta]. As the tracer follows the curve the frame
is made to turn through an angle n[theta]. At the same time the frame moves
with the tracer in the direction of y. For a small motion the two wheels
will then register just the components required, and during the continued
motion of the tracer along the curve the wheels will add these components,
and thus give the values of nA_n and nB_n. The factors 1/[pi] and -1/[pi]
are taken account of in the graduation of the wheels. The readings have
then to be divided by n to give the coefficients required. Coradi's
realization of this idea will be understood from fig. 23. The frame PP' of
the instrument rests on three rollers E, E', and D. The first two drive an
axis with a disk C on it. It is placed parallel to the axis of x of the
curve. The tracer is attached to a carriage WW which runs on the rail P. As
it follows the curve this carriage moves through a distance x whilst the
whole instrument runs forward through a distance y. The wheel C turns
through an angle proportional, during each small motion, to dy. On it rests
a glass sphere which will therefore also turn about its horizontal axis
proportionally, to dy. The registering frame is suspended by aid of a
spindle S, having a disk H. It is turned by aid of a wire connected with
the carriage WW, and turns n times round as the tracer describes the whole
length of the curve. The registering wheels R, R' rest against the glass
sphere and give the values nA_n and nB_n. The value of n can be altered by
changing the disk H into one of different diameter. It is also possible to
mount on the same frame a number of spindles with registering wheels and
glass spheres, each of the latter resting on a separate disk C. As many as
five have been introduced. One guiding of the tracer over the curve gives
then at once the ten coefficients A_n and B_n for n = 1 to 5.
All the calculating machines and integrators considered so far have been
kinematic. We have now to describe a most remarkable instrument based on
the equilibrium of a rigid body under the action of springs. The body
itself for rigidity's sake is made a hollow [v.04 p.0981] [Sidenote:
Michelson and Stratton analyzer] cylinder H, shown in fig. 24 in end view.
It can turn about its axis, being supported on knife-edges O. To it springs
are attached at the prolongation of a horizontal diameter; to the left a
series of n small springs s, all alike, side by side at equal intervals at
a distance a from the axis of the knife-edges; to the right a single spring
S at distance b. These springs are supposed to follow Hooke's law. If the
elongation beyond the natural length of a spring is [lambda], the force
asserted by it is p = k[lambda]. Let for the position of equilibrium l, L
be respectively the elongation of a small and the large spring, k, K their
constants, then
[Illustration: FIG. 24.]
nkla = KLb.
The position now obtained will be called the _normal_ one. Now let the top
ends C of the small springs be raised through distances y_1, y_2, ... y_n.
Then the body H will turn; B will move down through a distance z and A up
through a distance (a/b)z. The new forces thus introduced will be in
equilibrium if
ak([Sigma]y - n (a/b) z) = bKz.