19th century Connecticut newspapers include ads to reclaim those who had escaped indenture. Indenture, on the basic level, meant that someone “bound” themselves to an employer to work for a set period of time in exchange for some form of reward. Why do some of those ads mention the town as the ones doing the binding?
The reasoning is actually defined in Connecticut Statute:
If any person, who has had relief from any town, shall suffer his children to mis-spend their time, and shall neglect to employ them in some honest calling, or if any person does not provide competently for his children, whereby they are exposed to want, or if any poor children, in any town, live idly, or are exposed to want, and there are not to take care of them, the selectmen of such town, with the assent of a justice of the peace, shall in indenture such children […]
The General Statutes of the State of Connecticut, Revision of 1875 (Hartford: The Case, Lockwood & Brainard Co., Printers, 1875), 193.
In short, if the town thought they needed to be involved to protect the child, they could complete the contract themselves.

