The signatures to the petition shall be verified as follows:
(a) If the district is wholly in one county, the county elections official shall examine the signatures and from the records of registration ascertain whether or not the petition is signed by the requisite number of registered voters.
(b) If the district is situated in more than one county, each county elections official shall examine that portion of the petition bearing signatures obtained in that county and from the records of registration ascertain the number of signatures of registered voters in that portion of the district lying in that county. The county elections official of a county other than the principal county shall prepare a certificate, and file the same with the county elections official of the principal county, showing the number of valid signatures of registered voters in that county who signed the petition, and shall also show the total number of registered voters in that county within the district on the date of filing the petition.
(c) The county elections official of the principal county shall attach to the petition a certificate showing the total number of valid signatures thereon, and the total number of registered voters within the district as of the date of filing the petition.