The pathetic image of a goldfinch that dies of starvation in a cage is the inciting incident that causes Elizabeth-Jane to look for her missing "father" at the end of the novel. In Chapter 45, Hardy uses the story of the goldfinch that has died in captivity as a condensed allegory for Henchard's own downfall and imprisonment in his own inescapable ways:
Mrs. Donald Farfrae had discovered in a screened corner a new bird-cage, shrouded in newspaper, and at the bottom of the cage a little ball of feathers—the dead body of a goldfinch. Nobody could tell her how the bird and cage had come there, though that the poor little songster had been starved to death was evident. The sadness of the incident had made an impression on her. She had not been able to forget it for days [...]
The story of the goldfinch, which Henchard himself buys as a wedding gift for Elizabeth-Jane, is a microcosm of the tale of Henchard's own cycles of grace and disgrace. When the bird arrives in Casterbridge in Henchard's hand, having come from a "humble" shop, its fortunes are apparently supposed to change if all goes as planned. Henchard buys it with the intention of reconciliation with his daughter, and for a moment the novel seems to imply such a "little harmony" would be possible.
However, because Henchard and his daughter do not reconcile, events beyond the bird's control conspire to make its ornamental cage its grave. The bird cannot escape its cage, no matter how hard it tries or how beautifully the "poor little songster" chirps. The cruelty of coincidence in this novel means that the limitations set by external events doom the goldfinch regardless. Like the tiny bird, Henchard starts his life in humble beginnings and with few prospects. He has a period where it seems like his fortunes are improving and he can be happy, and then eventually comes to failure and death because he is imprisoned by something beyond his control. For the bird, this is the cage; for Henchard, it is his volatile character and personal weakness. The very tragic language of the "little ball of feathers" and the lonely death of the songbird sets the stage for Henchard's own awful demise, which Hardy describes very shortly after. Both man and bird die alone and forgotten, victims of allegorically similar circumstances.