If school children in one county reside closer to schools in an adjacent county, they may attend such schools upon the school authorities of the county of their residence arranging with the school officials of the adjacent county for such admission and upon payment of appropriate charges as herein authorized. The board of trustees in the school district in which the pupils reside shall make written application through its county board of education to the board of trustees of the district in which the school is located for the admission of such children, giving full information as to ages, residence and school attainment, and the board of trustees in the school district, agreeing to accept such pupils, shall give a written statement of agreement. Upon receipt of such application the board of trustees of the school and its county board of education shall determine the monthly per pupil cost of all overhead expenses of the school, which will include all expenses of the school not paid by the State. Upon proper arrangement being made for the payment monthly of such overhead per pupil cost for each such child the same shall be admitted to the schools of the adjacent county.