We present narrowband imagery and long-slit intermediate-resolution spectro
scopy of the multiple-shell planetary nebulae NGC 2438 and NGC 5882. A comp
arative study of the chemical abundances of the main nebulae and their surr
ounding halos shows a moderate N/O enrichment of the central regions. The o
xygen abundances appear constant throughout the nebulae, but there are some
indications of a moderate helium enrichment in the NGC 5882 main nebula. T
his study, along with with data published in the literature for six other m
ultiple-shell planetary nebulae (M1-46, NGC 6543, NGC 6720, NGC 6751, NGC 6
826, and NGC 7662), provides insight into the chemical enrichment and physi
cal conditions of halos. All the nebulae in this sample have chemical abund
ances typical of type II planetary nebulae, thus indicating a small chemica
l enrichment. In addition, we do not find evidence to support a strong chem
ical enrichment of the main nebulae relative to the halos. The conclusion i
s that the third dredge-up, in these nebulae, is of small or null effect. F
inally, we find a clear evidence that the gas in the halos is ionized by st
ellar UV radiation leaking through the material of the main nebula.