An explicit expression is defined to be a Hermite polynomial of degree $n$, $n=1,2,\dots,$ in variables $\xi(t_1),\xi(t_2),\dots,\xi(t_n)\in\{\xi(t),\ t\in T\}$ where $\{\xi(t),\ t\in T\}$ is a real Gaussian process. Some properties of these polynomials are investigated. Especially, $E^sH_n(\xi_1,\xi_2,\dots,\xi_n)=H_n(\hat{\xi_1},\hat{\xi_2},\dots,\hat{\xi_n})$, where $\hat{\xi_k}=E^s\xi_k$.