maker to
produce one should there be sufficient demand for it. Sometimes a
combination of graphical work with an integraph will serve the purpose.
This is the case if the variables are separated, hence if the equation

Xdx + Ydy = 0

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.