In my former paper [3] I defined an almost periodicity of weakly sationary random processes (a.p.w.s.p.) and presented some basic results of it. In this paper I shall present some notes on the Fourier series of an a.p.w.s.p., resulting from [3]. All the conditions at the introduction of [3] are assumed to hold without repreating them here. The essential facts are as follows : The weakly stationary process $X(t,omega), tin(-infty,infty), omegainOmega$, defined on a probability space $(Omega,a,P)$, has a spectral representation $$X(t,omega)=int_{-infty}^{infty}{e^{itlambdaxi}(dlambda,omega)},$$ where $xi(lambda)$ is a random measure. Then, the continuous covariance $ ho(mu) = E(X(t+u) X(t))$ has the form $$ ho(u)=int_{-infty}^{infty}{e^{iulambda}F(dlambda)},$$ $E$mid$xi(lambda+0)-xi(lambda-0)$mid$^2 = F(lambda+0) - F(lambda-0) lambdain(-infty,infty)$</TEX>, assumimg that $ ho(u)$ is a uniformly almost periodic function.