Henry Pelham (1694-1754) was the brains and financial acumen in the family, while his older brother Thomas, Duke of Newcastle managed the family’s extensive patronage and foreign policy.
In many ways Pelham was an improved version of Walpole. He pursued the same policy of peace, economy and low taxes, but although he used the same patronage methods as Walpole, he appears not to have been personally corrupt, holding the office of Paymaster of the Forces from 1730-43, the last four of which were war years, without conspicuous self-enrichment. He was a less outgoing personality than Walpole, and not an especially good Parliamentary speaker, but like Walpole was a superb administrator and excellent with finance.
The trajectory of Pelham’s period in office was the opposite of Walpole’s, with him gradually growing in power. Initially, he shared power with Bath and was subordinate to the autocratic Carteret (Earl of Granville from 1744) who once remarked of him: “He was only a chief clerk to Sir Robert Walpole, and why he should expect to be more under me, I can’t imagine; he did his drudgery and he shall do mine.”1
However, the Pelhams forced out Granville and Bath in November 1744 and formed a “Broad Bottom” administration with several opposition Whigs and even a few Tories, although the King vetoed the admission of the rising opposition Whig William Pitt. Then in February 1746 the King dismissed them and attempted to return to Granville/Bath, but the attempt was a failure and the Pelhams on returning signed a memorandum to the King demanding freedom of action, including the ability to include Pitt in the government. This represented the first time a British monarch had been overruled in his choice of government; from then until his death in 1754 Henry Pelham was an exceptionally strong prime minister.
Before achieving full power, Pelham had been faced with the Young Pretender’s (Charles Edward Stuart, 1720-88) rebellion, which from Scotland reached Derby before it turned back. Pelham’s solution was to recall the King’s younger son the Duke of Cumberland (1721-65) and the British force from Flanders. Cumberland finally defeated the Pretender at Culloden in April 1746, but then undertook a policy of suppressing the Highland clans, over Pelham’s opposition.
Pelham established the Whigs with a huge Commons majority at the 1747 election, a position repeated in that of 1754, just after his death. He also ended the War of the Austrian Succession, reduced public spending sharply and reduced the interest on the National Debt, with the help of the financier Sampson Gideon converting it in 1751 into the famous Consols, paying 3½% until 1757, 3% thereafter.
Two reforms were Lord Chancellor Hardwicke’s (Philip Yorke, 1st Earl of Hardwicke, 1690-1764) Marriage Act of 1753, banning elopements and a very unpopular shift to the Gregorian Calendar in 1752, which made tenants pay a full quarter’s rent in that year’s third quarter for only 81 days’ occupancy. On the other hand, a 1753 attempt to give civil rights to Jews met with such opposition that it was repealed in the following year.
Pelham deserves to rank below Walpole only because he wasn’t first and didn’t last as long. Say the mid-teens.
 “A History of British prime ministers” – Dick Leonard, Palgrave Macmillan, 2014, p41.