This chapter may be made effective by a resolution adopted by a majority of the governing board or committee of any institution operated by two or more counties, or by a majority of the governing board or committee of any district as defined in subdivision (c) of Section 31468, if one of the counties involved in the operation of the institution has adopted its provisions, or, in the case of a district as defined in subdivision (c) of Section 31468, if one of the counties comprising the organization or association has adopted its provisions. Upon the adoption of the resolution the employees of the institution shall become members of the retirement system of the county designated by the governing board or committee of the institution, and all contributions made by the employees and by the institution shall be paid into the county retirement system of the county selected. The governing board or committee shall cause to be paid to the county operating the retirement system, the institution’s proportionate share of the cost of operation of the system.