There have been at least two brands of "automatic revolvers" built.
The first was the famous Webley-Fosbery Automatic revolver made from 1901 to 1939.
The second is the current Mateba pictured above.
These unique revolvers use the recoil of the gun to automatically rotate the cylinder to the next chamber.
When the gun is fired, the barrel and cylinder move a short distance to the rear, somewhat like the slide of an automatic.
As the assembly moves rearward than back forward, the cylinder is rotated to the next chamber, and the hammer is re-cocked.
The Webley-Fosbery was noted for it's failure to operate in the mud of the trenches in World War One, due to the complicated design.
Since the guns have no advantage over either the revolver or the automatic, but share the weakness of both, the Mateba is mostly an expensive oddity for collectors.