A follow-up survey using the Submillimetre High-Angular Resolution Camera ( SHARC-II ) at 350 \mathrm { \mu m } has been carried out to map the regions around several 850 \mathrm { \mu m } -selected sources from the Submillimetre HAlf Degree Extragalactic Survey ( SHADES ) . These observations probe the infrared luminosities and hence star-formation rates in the largest existing , most robust sample of submillimetre galaxies ( SMGs ) . We measure 350 \mathrm { \mu m } flux densities for 24 850 \mathrm { \mu m } sources , seven of which are detected at \geq 2.5 \sigma within a 10 arcsec search radius of the 850 \mathrm { \mu m } positions . When results from the literature are included the total number of 350 \mathrm { \mu m } flux density constraints of SHADES SMGs is 31 , with 15 detections . We fit a modified blackbody to the far-infrared ( FIR ) photometry of each SMG , and confirm that typical SMGs are dust-rich ( M _ { \mathrm { dust } } \simeq 9 \times 10 ^ { 8 } \mathrm { M } _ { \odot } ) , luminous ( L _ { \mathrm { FIR } } \simeq 2 \times 10 ^ { 12 } \mathrm { L } _ { \odot } ) star-forming galaxies with intrinsic dust temperatures of \simeq 35 \mathrm { K } and star-formation rates of \simeq 400 \mathrm { M } _ { \odot } \mathrm { yr } ^ { -1 } . We have measured the temperature distribution of SMGs and find that the underlying distribution is slightly broader than implied by the error bars , and that most SMGs are at 28 K with a few hotter . We also place new constraints on the 350 \mathrm { \mu m } source counts , N _ { \mathrm { 350 } } ( > 25 \mathrm { mJy } ) \sim 200 – 500 \mathrm { deg } ^ { -2 } .