The expenses of such asylum shall be met as follows, to wit: at the end of each fiscal year, each of the boards of managers shall make a statement of the financial condition of the asylum under its control, which shall especially show how much the necessary expenses of the asylum exceeded its income, and this excess shall be paid by the several counties that had a child or children in the asylum for any part of the fiscal year; each county shall pay so much per capita according to the number of children from that county in the asylum and the length of time that they were there. Of all these particulars, strict, accurate, and systematic accounts shall be kept by the officers of the board, and when the pro rata of any county shall be thus ascertained and a statement of the pro rata has been brought to the notice of the county mayor or financial agent of any such county, it shall be that person's duty forthwith to draw that person's warrant on the county trustee of the county in payment of the same. No county shall be liable to pay at a greater rate than fifty dollars ($50.00) per capita per annum for each child it had in the asylum, and no county shall pay any expenses of any children of that county unless the child or children be apprenticed to or sent to the asylum by the proper authorities, and by proper orders of the county legislative body.