Whenever the respective boards of supervisors deem it best for the welfare of a family or in the public interest that an indigent remain in a county not responsible for his support, the county responsible for the support of the indigent may agree to support him in the county not so responsible; but no indigent supported in this manner shall be deemed to have acquired a residence in the nonresponsible county. Such agreement shall be made by the responsible county with the nonresponsible county, and a record or copy thereof shall be sent to and filed in the office of the department.