One or both of the lap rollers on which a lap roll is mounted for unwinding purposes is driven with an increasing speed during the winding-off interval so as to maintain the thickness of the lap layer constant. This, in turn, maintains the count of the sliver to be formed in the comber at a constant number. The increase in the lap roller drive can be made dependent upon the length of the lap layer which is unwound from the roll, on the mass of the lap layer delivered to the comber or on the mass of a sliver produced from the combed lap as measured at a point downstream of the comber. The speed may be increased in a stepwise manner or continuously.