Prudent men also, who had been willing for the sake of peace to submit to any established authority, desired not to see the claims of that family revived; claims which must produce many convulsions at present, and which disjointed for the future the whole system of hereditary right.

Besides, allowing the title of the house of Lancaster to be legal, Henry himself was not the true heir of that family; and nothing but the obstinacy natural to faction, which never without reluctance will submit to an antagonist, could have engaged the Lancastrians to adopt the earl of Richmond as their head.
His mother indeed, Margaret, countess of Richmond, was sole daughter and heir of the duke of Somerset, sprung from John of Gaunt, duke of Lancaster: but the descent of the Somerset line was itself illegitimate, and even adulterous.

And though the duke of Lancaster had obtained the legitimation of his natural children by a patent from Richard II., confirmed in parliament, it might justly be doubted whether this deed could bestow any title to the crown: since in the patent itself all the privileges conferred by it are fully enumerated, and the succession to the kingdom is expressly excluded.[*] In all settlements of the crown made during the reigns of the Lancastrian princes, the line of Somerset had been entirely overlooked; and it was not till the failure of the legitimate branch, that men had paid any attention to their claim.

And to add to the general dissatisfaction against Henry's title, his mother, from whom he derived all his right was still alive; and evidently preceded him in the order of succession.
* Rymer, tom.vii.p.849.

Coke's Inst.iv.Inst.part i.