The Duke of Wellington presents his Compt^s to Miss J. He has only this morning received her letter without date; in which Miss J. apprizes the Duke that she is about to quit Hampstead; but she does not state where she is about to reside. In respect to Miss J.'s wish that the Duke should write to Her, the Duke assures her that he will receive Her Letters with pleasure; and that he will answer them regularly. But that he must be permitted to do so, in a form, which shall not expose him to the accusation of treating a Lady with disrespect. From the perusal of Miss J.'s Letter the Duke does not exactly understand whether Miss J. did or did not send back the Paquet containing the Duke's Letters to her. As the Duke stated upon a former occasion, it is a matter of Indifference whether Miss J. has burnt the Letters; or kept them; or sent them back. The Duke begs leave to inform her however that if she sent them back he has not yet received them. They may however be in His House in London as he has been absent from there since the 19th of last Month. In Miss J.'s next letter she inquires into the truth of a report of the Duke's indisposition that has appeared in the papers. The reply is decidedly testy. WALMER CASTLE, Nov^r 3, 1835. The Duke of Wellington presents his Compliments to Miss J. He received only this morning Miss J.'s Letters of the 28^th and 31^st October. The Duke returns his thanks to Miss J. for her kind Enquiries about His Health. The Duke was unwell for one day; of which the Newspapers obtained Intelligence; and as usual misrepresented and exaggerated the facts. The Duke is much concerned that Miss J. is not satisfied with the formal style of His Notes. She was not satisfied when he wrote to her in a form more consistent with familiarity. Such form was considered disrespectful. The Duke assures Miss J. that he can reply to any letter which she may think proper to address the Duke as fully in one form as the other. The poor Duke's life was frequently made a burden to him by the comments expressed by the papers upon his health and his conduct. His letters abound in impatient references to the inaccuracies of the daily journals in all that pertained to himself. These finally irritated him to such an extent that, against the advice of wise friends, he sued the "Morning Journal" for libel, and succeeded in obtaining the punishment of the editor and printer by fines and imprisonment. His course in this matter was deprecated by those who thought that a man of his acknowledged character and position could afford to disregard such attacks.