A C172 ON A TRAINING FLT SLIDES SIDEWAYS OFF RWY 24. ICE HAD FORMED ON THE MELTED SNOW AT 9G0; NY.

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WE HAD DEPARTED 9G0 FOR A VFR TRAINING FLT 10 MI TO THE SW. UPON RETURNING TO 9G0; THE BRIGHT AFTERNOON SUN; WHICH HAD PARTIALLY MELTED SOME OF THE PACKED SNOW AND ICE ON THE RWY; WAS NOW MUCH LOWER IN THE SKY; AND TEMPS HAD DROPPED BY SEVERAL DEGS. UPON TOUCHDOWN ON RWY 24; THE ACFT SLID TO THE L SIDE OF THE RWY; APPARENTLY PUSHED BY A SUDDEN XWIND. ALL EFFORTS BY ME AND MY STUDENT TO STRAIGHTEN OUT AND REMAIN ON THE RWY WERE IN VAIN; AND THE ACFT CONTINUED THROUGH THE SNOW BANK AT THE L SIDE OF THE RWY AND CAME TO REST IN THE SNOW COVERED GRASS. AS A RESULT OF THE SUDDEN DECELERATION WHILE PASSING THROUGH THE SNOW BANK; THE ACFT NOSED DOWN AND SUSTAINED A BENT PROP. THERE WAS ALSO MINOR DAMAGE TO ITS R WINGTIP FAIRING DUE TO THE GUST UPSET. I BELIEVE THE ROOT CAUSE OF THIS INCIDENT WAS THE REFREEZING OF PREVIOUSLY MELTED SNOW AND ICE ON THE RWY COMBINED WITH AN UNEXPECTED XWIND AT TOUCHDOWN. (WINDS HAD BEEN LIGHT AND VARIABLE FOR MOST OF THE AFTERNOON). I WOULD ALSO THEORIZE THAT WE MIGHT HAVE UNKNOWINGLY HAD A FROZEN BRAKE INFLT; WHICH WOULD ACCOUNT FOR THE ACFT RESISTING OUR EFFORTS AT KEEPING IT ON THE RWY AFTER TOUCHDOWN.