Check out Equation 3.6:
$$
^{i-1}_iT = \left[
\begin{array}{c}
c\theta_i & -s\theta_i & 0 & r_{i-1} \\
s\theta_ic\alpha_{i-1} & c\theta_ic\alpha_{i-1} & -s\alpha_{i-1} & -s\alpha_{i-1}d_i \\
s\theta_is\alpha_{i-1} & c\theta_is\alpha_{i-1} & c\alpha_{i-1} & c\alpha_{i-1}d_i \\
0 & 0 & 0 & 1
\end{array}
\right]
$$
and Figure 3.21, "Link parameters of the PUMA 560":
$$
\begin{array}{c}
i & \alpha_{i-1} & r_{i-1} & d_i & \theta_i \\
\\
1 & 0 & 0 & 0 & \theta_1 \\
2 & -90^{\circ} & 0 & 0 & \theta_2 \\
3 & 0 & a_2 & d_3 & \theta_3 \\
4 & -90^{\circ} & a_3 & d_4 & \theta_4 \\
5 & 90^{\circ} & 0 & 0 & \theta_5 \\
6 & -90^{\circ} & 0 & 0 & \theta_6 \\
\end{array}
$$
Here I've used $r$ instead of $a$ to prevent any confusion between $\alpha$ and $a$. In Equation 3.6, $s$ means $\sin$ and $c$ means $\cos$.
To your question, $\alpha$ is more related to the orientation of two joints relative to each other as opposed to the motion of the joints relative to each other. That is, $\alpha$ is generally fixed by physical construction and thus, as a parameter, isn't time-dependent.
So hopefully you can see that, from Figure 3.21, all of the $\alpha$ values are either +/- 90 degrees or zero. Equation 3.6 then uses it only in sine and cosine terms, meaning that they can all be reduced to +/-1 or zero.
I really, really hate when authors jump through simplifications like this without a heads up, so I totally understand the confusion.
