If it appears by the certificate that the petition has not the required number of signatures, the elections official shall so notify the person designated as the person to whom notification of the insufficiency of the petition is to be sent. The petitioners shall have 30 days from and after the date of receiving the notice of insufficiency to present and file a supplement bearing additional signatures. Upon the receipt of the supplement, the elections official shall proceed forthwith to examine it. He or she shall complete his or her examination within 10 days from the date of its receipt by him or her. If it appears that the additional signatures and those which have not been legally rejected upon the original petition total the requisite number, the county elections official shall forthwith attach to the petition his certificate, duly dated, showing that the petition has been signed by the requisite number of voters, and shall immediately present it to the board of supervisors if it is in session, otherwise at its next meeting.