In $\mathbb{R}^n, $ suppose $u$ is a harmonic function in $B_1 (0)$. If $ 0\leq a\leq 1$ , then how to show that $$\int_{S^{n-1}} u(a^2 \omega)u(\omega)d\sigma_{\omega}=\int_{S^{n-1}}u^2(a\omega) d\omega.\quad (1)$$
It is some corollary of the mean value principle of harmonic functions. I know that if $u$ is harmonic, then $$u(0)=\frac{n}{\omega_n}\int_{B_1(0)} u(y)dy=\frac{1}{\omega_n}\int_{S^{n-1}}u(\omega)d\sigma_\omega$$ equation (1) holds when $a=0, 1$. I tried to take derivative w.r.t $a$ but failed.