About the political relationship there could be no such doubt.  William stood, theoretically, if not actually, in much the same position to Henry II, as John Baliol afterwards occupied to Edward I. It was not till the accession of Richard I that William recovered his freedom.  The castles in the south of Scotland which had been delivered to the English were restored, and the independence of Scotland was admitted, on William’s paying Richard the sum of 10,000 marks.  This agreement, dated December, 1189, annulled the terms of the Treaty of Falaise, and left the position of William the Lion exactly what it had been at the death of Malcolm IV.  He remained liegeman for such lands as the Scottish kings had, in times past, done homage to England.  The agreement with Richard I is certainly not incompatible with the Scottish position that the homage, before the Treaty of Falaise, applied only to the earldom of Huntingdon; but the usual vagueness was maintained, and the arrangement in no way determines the question of the homage paid by the earlier Scottish kings.  For a hundred years after this date, the two countries were never at war.  William had difficulties with John; in 1209, an outbreak of hostilities seemed almost certain, but the two kings came to terms.  The long reign of William came to an end in 1214.  His son and successor, Alexander II, joined the French party in England which was defeated at Lincoln in 1216.  Alexander made peace with the regent, resigned all claims to Northumberland, and did homage for his English possessions—­the most important of which was the earldom of Huntingdon, which had, since 1190, been held by his uncle, David, known as David of Huntingdon.  In 1221, he married Joanna, sister of Henry III.  Another marriage, negotiated at the same time, was probably of more real importance.  Margaret, the eldest daughter of William the Lion, became the wife of the Justiciar of England, Hubert de Burgh.  Mr. Hume Brown has pointed out that immediately on the fall of Hubert de Burgh, a dispute arose between Henry and Alexander.  The English king desired Alexander to acknowledge the Treaty of Falaise, and this Alexander refused to do.  The agreement, which averted an appeal to the sword, was, on the whole, favourable to Scotland.  Nothing was said about homage for this kingdom.  David of Huntingdon had died in 1119, and Alexander gave up the southern earldom, but received a fief in the northern counties, always coveted of the kings of Scotland.  This arrangement is known as the Treaty of York (1236).  Some trifling incidents and the second marriage of Alexander, which brought Scotland into closer touch with France (he married Marie, daughter of Enguerand de Coucy), nearly provoked a rupture in 1242, but the domestic troubles of Henry and Alexander alike prevented any breach of the long peace which had subsisted since the capture of William the Lion.  In 1249, the Scottish king died, and his son and successor,[40] Alexander III, was knighted by Henry of England, and,