Focus on The Third-Class Carriage and analyze the Background.
Daumier had drawn and painted images of rail travel since the 1840s, focussing on the people travelling rather than the conveyances.  His series of lithographs, Les Chemins de Fer ("the railway") was published in the French magazine Le Charivari from 1843 to 1858, including prints published in December 1856 with the captions "Voyageurs appréciant de moins en moins les wagons de troisième classe, pendant l'hiver" ("Travellers showing less and less appreciation for travelling in third class in the winter") and "Intérieur d'un wagon de troisième classe pendent l'hiver" ("Interior of a third-class railway carriage in winter").
The paintings relate to Daumier's three watercolors with ink and charcoal, now in the Walters Art Museum in Baltimore - one for each of the first, second and third class carriages - which were commissioned in 1864 by George A. Lucas for William Thompson Walters.  Three working drawings of the same subject have also survived, perhaps tracings, including one in the Bibliothèque nationale de France.  It seems two of the oil paintings were worked on concurrently, but the sequence of the paintings is not clear and the one in New York was left unfinished.  
Daumier was best known as an illustrator, and his paintings remained unknown until an exhibition held by Paul Durand-Ruel in Paris in 1878, the year before Daumier's death.
William Thompson Walters Le Charivari Walters Art Museum third class George A. Lucas New York Paul Durand-Ruel Bibliothèque nationale de France