If bonds have been authorized at an election held pursuant to this chapter prior to January 1, 1960, a portion of which bonds have not yet been issued, and the board determines that the sale of such unissued portion would be insufficient to finance the completion of the improvement for which the bonds were authorized, the board by resolution adopted prior to January 1, 1972, may modify the improvement and may authorize the issuance of bonds in an amount not exceeding the amount of such unissued portion for the purpose of paying the cost of the improvement as modified, without any further election, but only after notice and hearing in the same manner as provided in Article 2 (commencing with Section 71880) of Chapter 3 of this part and Article 6 (commencing with Section 53520) of Chapter 3, Part 1, Division 2, Title 5 of the Government Code, and provided that at the conclusion of the hearing the board shall by resolution determine that the territory within the district or improvement district will be benefited by the improvement as modified. Bonds so authorized may be issued and sold in the same manner as bonds authorized by an election under this chapter.