The board may by original resolution entered upon its minutes declare its intention to pay for and erect or construct, or acquire by purchase, gift, or condemnation, any bridge, trestle, or other structure upon any county road which crosses any stream, body of water, gulch, navigable water, swamp or other topographical formation constituting a boundary, or to join therein with any other county, city or town, or with this state, or with any other state, or with any county, city or town of any other state, in the erection, or construction, or acquisition of any such structure, and declare that the same is a public necessity, and direct the county road engineer to report upon such project, dividing any just proportional cost thereof.

In the event two counties or any county and any city wish to join in paying for the erection or acquisition of any such structure, the resolution provided in this section shall be a joint resolution of the governing authorities of the counties and cities and they shall further, by such resolution, designate an engineer employed by one county to report upon the proposed erection or acquisition.