The investigation into the circumstances connected with the riots of the 12th inst. has been pursued during the past week under the direction of Wm. BAYARD, Esq., Coroner and of D. ANSLEY, B.L. PETERS, W.H. NEEDHAM, George A. LOCKHART and Robert PAYNE, Esqrs, examining Justices assisted by Mr. WHEELER,Clerk of the Peace. John HAGGERTY and David HAGGERTY charged as rioters,have been bound in recognizance to appear for trial; Charles HEAGAN and Owen SULLIVAN, as rioters, and for assaulting the Mayor (the latter with a knife) have been committed for trial; Archibald BROWN stands committed for trial for feloniously shooting with intent to kill.; Charles SWEANY, for striking the Mayor with a brick; Patrick CORNEY, Daniel DRISCOLL, Jeremiah McCARTHY and James DONOVAN for shooting into the crowd and throwing bricks, when several persons were killed, attacking Boon's waggon and committing other acts of violence, having been partially examined and remanded. Joseph CORKERY charged as an active ringleader is in custody and will be examined today. The Coroners inquests upon the bodies of John DELAY and Pat ALLAN who were shot have resulted in verdicts of wilful murder against person or persons unknown; and in the case of Richard FOSTER who died of his wounds on Thursday, an inquest is now being held. The number of killed, positively ascertained by name is three, DELAY, ALLAN and FOSTER and of the wounded, six.