Jerome,
You are correct in that The Ship That Hunted Itself, by Colin Simpson, is about the battle between Carmania and Cap Trafalgar. I have not read it, so cannot comment on its accuracy or how much of the story is fact or fiction. However, the battle took place on 14 September 1914, less than six weeks after the outbreak of WW1 and before Cap Trafalgar had the chance to sink any British shipping. It has been suggested that Cap Trafalgar was disguised as a Union Castle liner (not a Cunarder) , but I don't have any details.
The Carmania spotted Cap Trafalgar's smoke early in the morning and some hours later was able to surprise the German ship with two colliers in the island of Trinidade's only harbour.
Both captains realised that to fight a successful action they required plenty of room, and so separately steamed several miles from the island in order to gain the space required.
The two ships turned towards each other and began to fight, the Carmania (eight 4.7" guns) firing too early and thus allowing the Cap Trafalgar (two 4.1" guns) the first blow. Carmania suffered much the worse of the engagement in the ensuing two hours, being hit 79 times, was holed below the waterline, and had her bridge totally destroyed by shellfire. However, as the range closed her own guns began to tell, and fires broke out on both ships, sailors lining the rails and firing machine guns at their opposite numbers as the ships came within a few hundred yards of one another.
Just as it seemed that the fires on Carmania would burn out of control, Cap Trafalgar veered away, lowering lifeboats as she heeled over to port. A shell below the waterline had ruptured several compartments, and the ship was rapidly sinking, although the colliers were able to pull 279 sailors from the wreck before she sank. 51 were killed in the fighting or the sinking (other reports say 16 or 17 lives were lost), including Captain Wirth. Carmania was equally shattered, listing severely, heavily flooded and burning, with nine men dead and many more wounded. It was at this point that Cap Trafalgar's contemporary, the armed merchant cruiser SMS Kronprinz Wilhelm arrived, seemingly to provide the coup de grace for the shattered ship. However, the Kronprinz Wilhelm's captain feared a trap, since many ships both German and Allied in the area had doubtless been listening to the SOS calls of the Cap Trafalgar, which, though in German code, had been supplemented by messages from the Carmania with the British code. Since multiple warships were on their way to the location, and the Cap Trafalgar had presumably already sunk, the captain of the Kronprinz Wilhelm turned his ship about and sailed away without firing a shot.
The following day the Carmania was rescued and brought into Pernambuco.
Rick