This is a Hale 24-pounder war rocket of ca. 1865-1870, a type of "stickless" or "rotary" war rocket developed from 1844 by the Englishman William Hale to eliminate the long and cumbersome wooden guidesticks of other war rockets. Hale rockets evolved over the years until the final pattern of ca. 1865 shown here.
The basic principle was that the exhaust gases propelled the rocket and also caused it to rotate in flight by impinging on curved exhaust vanes in the rear. The spinning helped deflect any wind and kept the rocket on course during its flight, known as spin stabilization. Hale rockets were widely used up to about 1899 but became obsolete due to advances in other artillery. This object was donated to the Smithsonian in 1968 by the Royal Artillery Institute, U.K.
This object is not on display at the National Air and Space Museum. It is either on loan or in storage.