  The Stokes phenomenon is the apparent discontinuous change in the form of the
asymptotic expansion of a function across certain rays in the complex plane,
known as Stokes lines, as additional expansions, pre-factored by exponentially
small terms, appear in its representation. It was first observed by G. G.
Stokes while studying the asymptotic behaviour of the Airy function. R. B.
Dingle proposed a set of rules for locating Stokes lines and continuing
asymptotic expansions across them. Included among these rules is the "final
main rule" stating that half the discontinuity in form occurs on reaching the
Stokes line, and half on leaving it the other side. M. V. Berry demonstrated
that, if an asymptotic expansion is terminated just before its numerically
least term, the transition between two different asymptotic forms across a
Stokes line is effected smoothly and not discontinuously as in the conventional
interpretation of the Stokes phenomenon. On a Stokes line, in accordance with
Dingle's final main rule, Berry's law predicts a multiplier of $\frac{1}{2}$
for the emerging small exponentials. In this paper, we consider two closely
related asymptotic expansions in which the multipliers of exponentially small
contributions may no longer obey Dingle's rule: their values can differ from
$\frac{1}{2}$ on a Stokes line and can be non-zero only on the line itself.
This unusual behaviour of the multipliers is a result of a sequence of
higher-order Stokes phenomena. We show that these phenomena are rapid but
smooth transitions in the remainder terms of a series of optimally truncated
hyperasymptotic re-expansions. To this end, we verify a conjecture due to C. J.
Howls.
