After the dissolution of a firm the authority of each partner to bind the firm, and the other mutual rights and obligations of the partners, continue notwithstanding the dissolution, so far as may be necessary to wind up the affairs of the firm and to complete transactions begun but unfinished at the time of the dissolution, but not otherwise:
Provided that the firm is in no case bound by the acts of a partner who has been adjudicated insolvent; but this proviso does not affect the liability of any person who has after the adjudication represented himself or knowingly permitted himself to be represented as a partner of the insolvent.