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Large-scale dust heating and cooling in the diffuse medium of M31 is studiedusing the HiRes IRAS maps in conjunction with UV, optical (UBV) and the HImaps. A dust heating/cooling model is developed based on a radiative transfermodel which assumes a `Sandwich' configuration of dust and stars and takesfully into account the effect of scattering of dust grains. The model isapplied to a complete sample of 'cells' (small areas of size $2'\times 2'$),generated from the above maps. The sample covers the M31 disk in thegalactocentric radius range 2 --- 14 kpc, and includes only the cells for whichthe contribution of the discrete sources to the 60$\mu m$ surface brightness isnegligible ($< 20$\%). This effectively excludes most of the bright arm regionsfrom our analysis. We find that: (1) The mean optical depth (viewed from theinclination angle of 77$^\circ$) increases with radius from $\tau_V\sim 0.7$ atr=2kpc outwards, reaches a peak of ~1.6 near 10kpc, and stays quite flat out to14kpc, where the signal falls below the 5sigma level. (2) A correlation between$\tau_V$ and HI surface density is suggested by the similarity between theirradial profiles. Significant differences are found between the radial profilesof the H_2 gas (estimated from CO) and of the dust (from $\tau_V$), which aremost probably due to the large uncertainty in the CO-to-H_2 conversion factor,and to the under-representation of H_2-rich regions in the sample of cells ofdiffuse regions. (3) The $\tau_V / N(HI)$ ratio decreases with increasingradius in the disk of M31, with an exponential law fit yielding an e-foldingscale length of $9.6\pm 0.4$ kpc. (4) The optical depth adjusted for thisgradient, $\tau_{V,c}$, is strongly and linearly correlated with N(HI) over oneand a half order of magnitude of column density, indicating that at a givenComment: 32 pages uuencoded postcript. To appear in ApJ (the issue of January  1, 1996). 10 figures available on reques

High-Resolution IRAS Maps and Infrared Emission of M31. II. Diffuse Component and Interstellar Dust