  Three-body continuum states are investigated with the hyperspherical method
on a Lagrange mesh. The $R$-matrix theory is used to treat the asymptotic
behaviour of scattering wave functions. The formalism is developed for neutral
as well as for charged systems. We point out some specificities of continuum
states in the hyperspherical method. The collision matrix can be determined
with a good accuracy by using propagation techniques. The method is applied to
the $^6$He (=$\alpha$+n+n) and $^6$Be (=$\alpha$+p+p) systems, as well as to
$^{14}$Be (=$^{12}$Be+n+n). For $^6$He, we essentially recover results of the
literature. Application to $^{14}$Be suggests the existence of an excited $2^+$
state below threshold. The calculated B(E2) value should make this state
observable with Coulomb excitation experiments.
