A four-masted jackass barque is square-rigged on the two foremost masts (fore and main masts) and fore-and-aft rigged on the two after masts (the mizzen and spanker or jiggermasts). Some 19th century sailors called such a ship "a fore-and-aft schooner chasing a brig". In general a jackass barque is a sailing ship which is half square-rigged and half fore-and-aft rigged.
A well-known example of a (white wood-hulled) four-masted jackass-barque was the Olympic in her times (she was sayed to be the only one in the world), a 1,402 GRT "Down Easter" (a square-rigged sailing ship from the dockyards of the downeastern ports, preferably made of timber), launched in 1892 at the shipyards of the New England Ship Building Company, Bath, ME, for Capt. W. H. Besse of New Bedford, MA1). Her maiden voyage under her first master, Capt. Stephen Bourne Gibbs, led her from Bath to New York City, South Street Sea Port, with "clean swept holds" and without any ballast, than with a cargo of iron rails and plates around Cape Horn to Portland, Oregon. She carried steel, nitrate and other cargoes. Capt. Gibbs was an experienced square-rigged skipper and took great care of best stowage of the cargo having the stevedores put 2/3 of it in the 'tween decks. For her new owner Hackfield & Co. of Honolulu, managed by Williams, Dimond & Co., the Olympic ran in the sugar trade between Hawaii, California, and Australia from 1901 to 1912. Originally rigged with royal sails over double top and topgallant sails on both square-rigged masts and the main mast equipped with an additional single sky sail, she was re-fitted with royal sails over double top but single topgallant sails on both square-rigged masts and no main sky sail after having been dismasted in 1901 during a voyage from Hawaii to San Francisco under Capt. Gibbs, who retired from sea in the same year. Under her new master Capt. T. H. Evans she sailed between New Zealand, Australia and the US westcoast in the timber trade. Towards the end of World War I she changed hands to Thomas Crowley & Co. In that time she was re-rigged as a four-masted barquentine with a square-rigged foremast and three fore-and-aft rigged other masts. During the slump years cargo space of a sailing ship wasn't needed anymore. After having been laid up for some years the 30-year-old ship was converted into a towing barge. Her last master was Capt. T. J. Halcrow.
A five-masted jackass barque which has probably never been built would be equipped with square-rigged fore and main masts, with a partially square-rigged and partially fore-and-aft rigged mizzen mast, and fore-and-aft rigged jigger and spanker masts.
1) Capt. Alfred Basil Lubbock The Down Easters, American deep-water sailing ships, 1868-1919, first published by Charles E. Lauriat Co, Boston, 1929