It is pointed out in the paper that the Galileo formula is not self - consistent in the framework of the special theory of relativity via the investigation of reviewing and reconsidering the pole's uniform rotation within the special relativity. A new formula for the rotate velocity of the pole，which is rotating uniformly，is derived in the special relativistic limits by using the special relativistic delivery formula in the inertial system，and by continued fraction and recurrence relation. In the new relative rotation formula，the rotational linear velocity is always less than the velocity of light. Therefore，the problem of horizon in the Galileo rotation formula can be solved. The self - consistence of the new relative rotation formula is verified，and the comparison of the transverse Doppler shift between the relativistic rotating pole and classical rotating one is made. The result can be used to test the relative rotational formula.