Why do we say "Fills the bill (or fits the bill)"?

Meaning:

Meets a need/serves a purpose.

Background:

The bill in this context is a theatrical poster/leaflet describing the acts to be presented. If the bill had too few acts, more acts would be sought to 'fill the bill'. This expression appears to have originated in the USA in the 19th Century. The expression can also sometimes be used when a particular act or event takes up the entire show (or at least dominates).

Note: Although when I set this particular week's quiz I had in mind 'Fill the Bill', F T Bill could also translate as 'Foot The Bill' - meaning to pay the bill - e.g. two people in a restaurant with one offering to foot the bill for the other. All three options - fit the bill, fill the bill and foot the bill were all considered correct when selecting this week's winner.

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