  The ultra-long period Cepheids (ULPCs) are classical Cepheids with pulsation
periods exceeding $\approx 80$ days. The intrinsic brightness of ULPCs are ~1
to ~3 mag brighter than their shorter period counterparts. This makes them
attractive in future distance scale work to derive distances beyond the limit
set by the shorter period Cepheids. We have initiated a program to search for
ULPCs in M31, using the single-band data taken from the Palomar Transient
Factory, and identified eight possible candidates. In this work, we presented
the VI-band follow-up observations of these eight candidates. Based on our
VI-band light curves of these candidates and their locations in the
color-magnitude diagram and the Period-Wesenheit diagram, we verify two
candidates as being truly ULPCs. The six other candidates are most likely other
kinds of long-period variables. With the two confirmed M31 ULPCs, we tested the
applicability of ULPCs in distance scale work by deriving the distance modulus
of M31. It was found to be $\mu_{M31,ULPC}=24.30\pm0.76$ mag. The large error
in the derived distance modulus, together with the large intrinsic dispersion
of the Period-Wesenheit (PW) relation and the small number of ULPCs in a given
host galaxy, means that the question of the suitability of ULPCs as standard
candles is still open. Further work is needed to enlarge the sample of
calibrating ULPCs and reduce the intrinsic dispersion of the PW relation before
re-considering ULPCs as suitable distance indicators.
