Xdx + Ydy = 0
has to be integrated where X = p(x), Y = φ(y) are given as curves. If we write
au = ∫Xdx, av = ∫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 = y0, for x = x0. This determines c = u0 + v0, where u0 and v0 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 Harmonic analysers. expanded in a series.
y = A0 + A1 cos θ + A2 cos 2θ + ... + An cos nθ + ...
+ B1 sin θ + B2 sin 2θ + ... + Bn sin nθ + ...
where θ = 2πx / c.