A very stringent law for the prevention of accidents in vessels navigated in whole or in part by steam, went into operation on the 1st of last January. When commenting on the benefits likely to be derived from said law, we stated that ll everything depended on ths capacity and laitbfalness of the inspectors appointed under its provisions." Accidents have not ceased since it went into operation; on the contrary a great number have taken place. This calls tor increased vigilence and ear* on the part of those appointed by government to carry out and enforce the law's requirements. On the 1st of this month, the splendid steamboat New World, running on the North River, burst one of her flues, by which no less than seven persons lost their lives. The engineer was one of the most experienced and cautious men in our country; recklessness was not apart of his character. His word with the owners was law ; it he said that any part was defective, it was at once repaired. We hope the Chief Inspector for this district will see to it that the new steamboat law is rigidly entorced ; this never will be done if he trusts wholly to the men whom he appoints under him, however respectable they may be. We are not positive but from a partial personal observation we are led to believe that the law is not strictly enforced on the steamboats in this district,so far as it relates to the supply of lilfe boats, life preservers, &c. There is perhaps more danger of life on the ierry boats plying between New York City and her suburb cities around, than on any steamboats in the world, and yet so well have these boats been managed that very few accidents have taken place. We speak of the danger as it exists in proportion to the number of passengers carried, and the number of boats and vessels crossing the track of one another every day and night. Every ferry boat should be fully provided with life preservers, because the decks are oftentimes so loaded that lite boats would be of little avail.