  We denote by $\pi_k(R_n)$ the $k$-th homotopy group of the $n$-th rotation
group $R_n$ and $\pi_k(R_n:2)$ the 2-primary components of it. We determine the
group structures of $\pi_k(R_n:2)$ for $k = 23$ and $24$ by use of the
fibration $R_{n+1}\overset{R_n}{\longrightarrow}S^n$. The method is based on
Toda's composition methods.
