In the case of a tie, the candidates receiving the tie vote shall cast lots before the Supervisor of Elections or his deputy, as the case may be, in the manner prescribed by the Supervisor of Elections at 12o'clock noon on the third Friday following the primary, and the one to whom the lot shall fall shall be entitled to the nomination. In any case where the fact of a tie vote is not authoritatively determined until after the third Wednesday following the primary, the time for casting lots shall be at 12 o'clock noon on the second day after the fact of such tie vote authoritatively determined. If any candidate, receiving a tie vote, fails to appear before 12 o'clock noon on such day, the Supervisor of Elections or his deputy, as the case may be, shall cast lots for him. For the purpose of casting lots, any candidate may appear in person, or by proxy appointed by him in writing.