On the excursions of drifted Brownian motion and the successive passage times of Brownian motion

By using the law of the excursions of Brownian motion with drift, we find the distribution of the $n-$th passage time of Brownian motion through a straight line $S(t)=a+bt\mathrm{.}$ In the special case when $b=0,$ we extend the result to a space-time transformation of Brownian motion.