And provided also, That the treasurer of such House of Refuge, on or before the second Monday of the preceding month, shall transmit, by the public mail, to the commissioners of such of the counties as may have become indebted for the maintenance and instruction of inmates in such House of Refuge, an account of the expense of maintaining and instructing them during the three previous calendar months, which account shall be signed by the treasurer, and sworn or affirmed to by him, and attested by the superintendent of the department of such House of Refuge in which each of such inmates may be living.