  We present CCD phase-binned light curves at 490 nm for 97 Cepheid variable
stars in the bar of the LMC. The photometry was obtained as part of the French
EROS project and has excellent phase coverage, permitting accurate
decomposition into Fourier components. We identify as `sinusoidal' or
s-Cepheids those stars with periods less than 5.5 d and small second-harmonic
components. These stars comprise $\sim$30% of our sample and most form a
sequence $\sim$1 mag brighter than the LMC classical Cepheids in the
period-luminosity diagram. They are also generally bluer and have
lower-amplitude light curves. We infer that the s-Cepheids are first-overtone
pulsators because, when their periods are converted to expected
fundamental-mode values, they obey a common period-luminosity-colour relation
with classical Cepheids. This also confirms the reality of the colour term in
the Cepheid period-luminosity-colour relation. Further, the blue edge of the
classical Cepheid instability strip agrees well with the theoretical
calculations for the fundamental mode made by Chiosi et al. (1993) for the
Hertzsprung-Russell and period-luminosity diagrams, but we find that our
observed s-Cepheids are $>0.2$ mag brighter and bluer than the Chiosi et al.\
predictions for the first-overtone. We identify a number of features in plots
of our stars' Fourier-component amplitude ratios and phase differences. These
features have been identified with resonances between different pulsation
modes. In the LMC we find these features seem to occur at periods very similar
to Galactic ones for classical Cepheids, but at different periods for
s-Cepheids. We discover a double-mode Cepheid in the LMC, for which $P({\rm
first overtone})/P({\rm fundamental}) = 0.710 \pm 0.001$, very similar to
observed ratios for Galactic double-mode Cepheids.
