C172 STUDENT AND INSTRUCTOR LAND OFF THE LEFT EDGE OF RWY DURING NIGHT CROSSWIND LANDING PRACTICE.

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MY STUDENT AND I WERE ON A NIGHT XCOUNTRY. WE ENTERED THE TFC PATTERN ON A R DOWNWIND. THE WX AT THE TIME OF THE LNDG WAS VFR WITH A WIND OUT OF THE SSE. ON DOWNWIND THE RWY WAS IDENTED BY THE LIGHTS; AND THE DECISION TO GO AHEAD AND LAND WAS MADE. UPON TURNING FINAL THE LIGHTS OF THE RWY WERE STILL VISIBLE AND THE ACFT WAS ON THE GS DEPICTED BY THE VASI. WHILE DSNDING INTO THE ARPT THE ACFT WAS BEING CRABBED INTO THE WIND TO COMPENSATE FOR A WIND VELOCITY THAT SEEMED HIGHER THAN WAS RPTED. ALSO DURING THIS TIME; THE ACFT WAS LINED UP TO THE L OF THE CTRLINE OF THE RWY. WHEN THE ACFT WAS OVER THE END OF THE RWY; I LOOKED DOWN AND COULD SEE THE RWY MARKINGS AND CLRED THE STUDENT TO REDUCE PWR AND LAND. DURING THIS PHASE OF THE LNDG THE ACFT WAS ROTATED TO A HIGH PITCH ATTITUDE FOR THE TOUCHDOWN. BECAUSE OF THIS HIGH PITCH ATTITUDE THE RWY LIGHTS AT THE END OF THE RWY WERE OBSCURED BY THE COWLING OF THE ACFT; SO NO DRIFT FROM THE WIND COULD BE NOTICED. WHILE IN THIS ATTITUDE THE ACFT WAS NOT PROPERLY CORRECTED FOR THE INCREASED WIND VELOCITY AND WAS PUSHED OFF OF THE EDGE OF THE RWY BY THE TIME THAT THE ACFT MADE ITS TOUCHDOWN. AS THE ACFT CAME IN CONTACT WITH THE GND IT WAS OBVIOUS THAT THE ACFT WAS NOT ON THE RWY; SO I IMMEDIATELY ADDED FULL PWR AND PITCHED THE ACFT UP TO RELIEVE THE WT OF THE NOSEWHEEL. AFTER A FEW MOMENTS OF THE ACFT RIDING ROUGH THROUGH THE GRASS; THE ACFT LIFTED OFF OF THE GND; AND A GRADUAL CLBOUT WAS ESTABLISHED. UPON REACHING A SAFE ALT; I MADE THE DECISION TO RETURN TO THE ARPT OF ORIGIN. AT NO TIME DURING THIS ENTIRE SERIES OF EVENTS; NOR ON THE FLT TO OUR DEST; DID THE ACFT DISPLAY ANY INDICATION THAT THE PROP WAS DAMAGED NOR THAT IT HAD STRUCK SOMETHING ON THE GND. NOT UNTIL SECURING THE ACFT AT OUR DEST OR ORIGIN WAS IT DISCOVERED THAT THE PROP HAD BEEN DAMAGED DURING THE LNDG.