# Maximizing entropy on a channel

In Shannon's paper "A Mathematical Theory of Communication", in Theorem 8 he states:

Theorem 8: Let the system of constraints considered as channel have a capacity $$C = \log W$$. If we assign $$p^{(s)}_{ij} = \frac{B_j}{B_i}W^{\ell^{(s)}_{ij}}$$ where $$l^{(s)}_{ij}$$ is the duration of the $$s^{th}$$ symbol leading from state i to state j and $$B_i$$ satisfy $$B_i = \Sigma_{s,j}B_jW^{\ell^{(s)}_{ij}}$$ then H is maximized and equal to C.

My question is, what exactly is $$B_i$$ and $$B_j$$? It's not defined anywhere else. In a different section he mentions "a sequence $$B_i$$ of symbols from the source", but that doesn't make much sense here.

A necessary and sufficient condition such that $$\eta_c$$ is maximized and equal to $$1$$ is that the set of symbol lengths $${l_{ij}}^v$$ must satisfy: $${p_{ij}}^v = (\beta_j/\beta_i)\omega_r^{-{l_{ij}}^v},\tag{3}$$ where $$\beta_i$$ and $$\beta_j$$ are respectively the $$(k, i)$$th and $$(k, j)$$th cofactor of the matrix $$A$$ for any $$k$$.
The matrix $$A$$ is defined earlier as:
$$A \triangleq \left[\textstyle\sum\omega^{-{l_{ij}}^v} - \delta_{ij}\right];$$