of the irregularities of the rail heads before grinding in this preferred embodiment of the device in accordance with the invention is constituted by a circuit for correcting control values determined from the measurements taken at the front end as a function of the magnitude of the residual amplitude of the irregularities of the rail heads after grinding in fig3 which schematically represents this correcting circuit the same numerical references have been used to designate the elements constituting the rear measuring device as in fig1 but to which a prime sign has been added these elements feelers sensor processing device and adjusting devices have the same functions as those already described in connection with fig1 the output signals from the rear measuring device representative of the residual amplitude a 2 a 2 and u03c0 2 of the same values measured at the front end of the grinding vehicle are each directed to a comparator element designated 68 for the signal a 2 69 for the signal a 2 and 70 for the signal u03c0 2 to each of these comparator elements there is also connected a device for controlling the maximum acceptable amplitude values a 0 a 0 and u03c0 0 of said values which are considered or deemed acceptable for the ground rail section the control devices 71 for the value a 0 72 for.