Expanding on Brigette's work, I found four lines from Edward III to William Lamb, 2d Viscount Melbourne:

.. 1 King Edward III of England (1312 - 1377)
.. + Philippa de Hainault (1314 - 1369)
.... 2 Prince Lionel Plantagenet (1338 - 1368)
.... + Elizabeth de Burgh (1332 - 1363)
...... 3 Philippa Plantagenet (1355 - 1382)
...... + Sir Edmund de Mortimer (1352 - 1381)
........ 4 Elizabeth Mortimer (1371 - 1417)
........ + Sir Henry Percy (1364 - 1403)
.......... 5 Sir Henry Percy (1393 - 1455) [ 1 ]
.......... + Alainor Neville (1407 - 1472) [ 2 ]


.... 2 Sir John of Gaunt (1340 - 1399)
.... + Katherine de Roet (1350 - 1403)
...... 3 Sir John Beaufort (1372 - 1410)
...... + Margaret de Holand (1385 - 1429)
........ 4 Sir Edmund Beaufort (1406 - 1455)
........ + Alianor Beauchamp (1407 - 1467)
.......... 5 Eleanor Beaufort ( - 1501)
.......... + Sir Robert Spencer (1435 - 1501)
............ 6 Margaret Spencer (1472 - )
............ + Thomas Cary (1460 - 1536)
.............. 7 Mary Cary
.............. + Sir John Delaval (1498 - 1562)
................ 8 Sir John Delaval (1512 - 1572) [ 3 ]
................ + Anne Ogle (1509 - ) [ 4 ]
.................. 9 Sir Robert Delaval (1542 - 1607)
.................. + Dorothy Grey (1554 - 1591)
.................... 10 Sir John Delaval (1590 - 1652)
.................... + Elizabeth Selby (1590 - )
...................... 11 George Delaval (1613 - 1694)
...................... + Margaret Grey (1633 - 1709)
........................ 12 Edward Delaval (1664 - 1744)
........................ + Mary Blake (1664 - 1711)
............................ 13 Anne Delaval (1692 - 1765)
............................ + Sir Ralph Milbanke (1685 - 1745)
.............................. 14 Sir Ralph Milbanke (1725 - 1793)
.............................. + Elizabeth Hedworth (1726 - 1767)
................................ 15 Elizabeth Milbanke (1751 - 1818)
................................ + Sir Peniston Lamb (1745 - 1828)
.................................. 16 William Lamb (1779 - 1848)
.................................. + Caroline Ponsonby (1785 - 1828)


...... 3 Joan Beaufort (1379 - 1440)
...... + Sir Robert de Ferrers (1373 - 1396)
........ 4 Mary de Ferrers ( - 1458)
........ + Sir Ralph Neville (1392 - 1458)
.......... 5 John Neville (1416 - 1482)
.......... + Elizabeth Newmarch (1420 - )
............ 6 Joan Neville
............ + William Gascoigne ( - 1463)
.............. 7 William Gascoigne ( - 1487) [ 5 ]
.............. + Margaret Percy [ 6 ]


...... 3 Joan Beaufort (1379 - 1440) [again]
...... + Sir Ralph de Neville (1364 - 1425)
........ 4 Alainor Neville (1407 - 1472) [ 2 ]
........ + Sir Henry Percy (1393 - 1455) [ 1 ]
.......... 5 Sir Henry Percy (1421 - 1461)
.......... + Alianor Poynings (1422 - 1484)
............ 6 Margaret Percy [ 6 ]
............ + William Gascoigne ( - 1487) [ 5 ]
.............. 7 Margaret Gascoigne
.............. + Sir Ralph Ogle ( - 1512)
................ 8 Anne Ogle (1509 - ) [ 4 ]
................ + Sir John Delaval (1512 - 1572) [ 3 ]