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