A new map of the SNR G78.2+2.1 at 232 MHz is presented. This map is us
ed with other images at 408, 1420, 2695 and 4800 MHz to investigate th
e total spectral index of the SNR and the variation of spectral index
across it. The integrated spectral index is alpha = 0.54 +/- 0.02 (whe
re S-nu proportional to nu(-alpha)). Maps of the spectral-index variat
ion across the source are derived using a method which calculates a di
fferential spectral index. This method is independent of offset errors
and large-scale background variations. Systematic spectral-index vari
ations of similar to +/- 0.15 are found, consistently in the many inde
pendent pairs of data sets. Regions of higher spectral index (steeper
spectra) appear around the periphery of the remnant and in its central
area. Regions of lower spectral index (flatter spectra) are also foun
d within the remnant. The latter may result from a change in the natur
e of the non-thermal emission or may represent ''contamination'' from
underlying patches of thermal emission. There is no clear correlation
of spectral index with intensity across the SNR. Theoretical reasons f
or spectral-index variations have been explored but no clear interpret
ation of the apparent spectral steepening near the shock front has bee
n found nor has an explanation of the steep-spectrum central region.