Question: Princess Frederica Dorothea "Louise" Philippine of Prussia, was a member of the House of Hohenzollern, and was a niece of which King of Prussia from 1740 until 1786, the longest reign of any Hohenzollern king?

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Princess Louise of Prussia (1770–1836): Princess Frederica Dorothea "Louise" Philippine of Prussia (24 May 1770 - 7 December 1836) was a member of the House of Hohenzollern.  She was a niece of Frederick the Great, being the second daughter and third child of Prince Augustus Ferdinand of Prussia by his wife Margravine Elisabeth Louise of Brandenburg-Schwedt.

Princess Charlotte of Prussia (1831–1855): Princess Frederica Louise Wilhelmina Marianne "Charlotte" of Prussia (21 June 1831 – 30 March 1855), was by birth a Princess of Prussia and member of the House of Hohenzollern and by marriage Hereditary Princess of Saxe-Meiningen.

William I, German Emperor: William I, or in German Wilhelm I (full name: "William Frederick Louis of Hohenzollern", German: "Wilhelm Friedrich Ludwig von Hohenzollern" , 22 March 1797 – 9 March 1888), of the House of Hohenzollern was King of Prussia from 2 January 1861 and the first German Emperor from 1 January 1871 to his death, the first Head of State of a united Germany.  Under the leadership of William and his Minister President Otto von Bismarck, Prussia achieved the unification of Germany and the establishment of the German Empire.  Despite his long support of Otto von Bismarck as Minister President, William held strong reservations about some of Bismarck's more reactionary policies, including his anti-Catholicism and tough handling of subordinates.  Contrary to the domineering Bismarck, William was described as polite, gentlemanly and, while a staunch conservative, more open to certain classical liberal ideas than his grandson Wilhelm II.

List of monarchs of Prussia: The monarchs of Prussia were members of the House of Hohenzollern who were the hereditary rulers of the former German state of Prussia from its founding in 1525 as the Duchy of Prussia.  The Duchy had evolved out of the Teutonic Order, a Roman Catholic crusader state and theocracy located along the eastern coast of the Baltic Sea.  The Teutonic Knights were under the leadership of a Grand Master, the last of whom, Albert, converted to Protestantism and secularized the lands, which then became the Duchy of Prussia.  The Duchy was initially a vassal of the Kingdom of Poland, as a result of the terms of the Prussian Homage whereby Albert was granted the Duchy as part of the terms of peace following the Prussian War.  When the main line of Prussian Hohenzollerns died out in 1618, the Duchy passed to a different branch of the family, who also reigned as Electors of Brandenburg in the Holy Roman Empire.  While still nominally two different territories, Prussia under the suzerainty of Poland and Brandenburg under the suzerainty of the Holy Roman Empire, the two states are known together historiographically as Brandenburg-Prussia.  Following the Second Northern War, a series of treaties freed the Duchy of Prussia from any vassalage to any other state, making it a fully sovereign Duchy in its own right.  This complex situation (where the Hohenzollern ruler of the independent Duchy of Prussia was also a subject of the Holy Roman Emperor as Elector of Brandenburg) laid the eventual groundwork for the establishment of the Kingdom of Prussia in 1701.  For diplomatic reasons, the rulers of the state were known as the King in Prussia from 1701 to 1772; largely because they still owed fealty to the Emperor as Electors of Brandenburg, the "King in Prussia" title (as opposed to "King of Prussia") avoided offending the Emperor.  As the Prussian state grew through several wars and diplomatic moves throughout the 18th century, it became apparent that Prussia had become a Great Power that did not need to submit meekly to the Holy Roman Empire.  By 1772, the pretense was dropped, and the style "King of Prussia" was adopted.  Thus it remained until 1871, when in the aftermath of the Franco-Prussian War, the King of Prussia Wilhelm I was crowned German Emperor.  From that point forward, though the Kingdom of Prussia retained its status as a constituent state of the German Empire, all remaining Kings of Prussia also served as German Emperor, and that title took precedence.

Princess Alexandrine of Prussia (1842–1906): Princess Frederica Wilhelmina Louise Elisabeth "Alexandrine" of Prussia (1 February 1842 – 26 March 1906) was a member of the House of Hohenzollern as the daughter of Prince Albert of Prussia and his wife Princess Marianne of the Netherlands.

Flag of Prussia: The state of Prussia had its origins in the separate lands of the Margraviate of Brandenburg and of the Duchy of Prussia.  The Margraviate of Brandenburg developed from the medieval Northern March of the Holy Roman Empire, passing to the House of Hohenzollern in 1415.  The Duchy of Prussia originated in 1525 when Albert of Brandenburg-Ansbach, a member of a cadet branch of the Hohenzollerns, secularized the eastern lands of the Teutonic Knights as a Polish fief.  Prince-elector John Sigismund, Elector of Brandenburg, inherited the Duchy of Prussia in 1618, thus uniting Brandenburg and Prussia under one ruler in a personal union; the Elector's state became known as Brandenburg-Prussia.  The Kingdom of Prussia formed when Elector Frederick III assumed the title of Frederick I, King in Prussia, on 18 January 1701.

Princess Frederica Wilhelmina of Prussia: Princess Frederica Wilhelmina Louise Amalia of Prussia (30 September 1796 – 1 January 1850) was a daughter of Prince Louis Charles of Prussia and Frederica of Mecklenburg-Strelitz.  She was a member of the House of Hohenzollern.  By her marriage to Leopold IV, Duke of Anhalt-Dessau, she would become Duchess consort of Anhalt-Dessau.

Province of Hohenzollern: The Province of Hohenzollern (German: "Provinz Hohenzollern" ) or the Hohenzollern Lands (German: "Hohenzollernsche Lande" ) was a "de facto" province of the Kingdom of Prussia.  It was created in 1850 by joining the principalities of Hohenzollern-Sigmaringen and Hohenzollern-Hechingen after both formerly independently ruling Catholic princely lines of the House of Hohenzollern had handed over their sovereignty to Prussia, ruled by the Protestant Hohenzollern branch.  Both used the same dynastic coat of arms as the one used by the Prussian royal family.

Princess Désirée of Hohenzollern: Princess Désirée of Hohenzollern (German: "Désirée Margarethe Victoria Louise Sibylla Katharina Maria Prinzessin von Hohenzollern" ; born 27 November 1963) was born in Munich, Germany.  She is a princess of the House of Hohenzollern and the second child of Princess Birgitta of Sweden and Prince Johann Georg of Hohenzollern.

Frederick the Great: Frederick II (German: "Friedrich" ; 24 January 171217 August 1786) was King of Prussia from 1740 until 1786, the longest reign of any Hohenzollern king.  His most significant accomplishments during his reign included his military victories, his reorganization of Prussian armies, his patronage of the arts and the Enlightenment in Prussia, and his final success against great odds in the Seven Years' War.  Frederick was the last titled King in Prussia and declared himself King of Prussia after achieving full sovereignty for all historical Prussian lands.  Prussia had greatly increased its territories and became a leading military power in Europe under his rule.  He became known as Frederick the Great ("Friedrich der Große ") and was affectionately nicknamed "Der Alte Fritz " ("Old Fritz") by the Prussian and later by all German people.

Supporting paragraphs: Princess Louise of Prussia (1770–1836) Frederick the Great
Answer: Frederick the Great