  Let $L_t$ be the longest gap before time $t$ in an inhomogeneous Poisson
process with rate function $\lambda_t$ proportional to $t^{\alpha-1}$ for some
$\alpha\in(0,1)$. It is shown that $\lambda_tL_t-b_t$ has a limiting Gumbel
distribution for suitable constants $b_t$ and that the distance of this longest
gap from $t$ is asymptotically of the form $(t/\log t)E$ for an exponential
random variable $E$. The analysis is performed via weak convergence of related
point processes. Subject to a weak technical condition, the results are
extended to include a slowly varying term in $\lambda_t$.
