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The facts, as the judge found them, were as follows. On June 20, 1943, the Conners Company chartered the barge, ‘Anna C.’ to the Pennsylvania Railroad Company at a stated hire per diem, by a charter of the kind usual in the Harbor, which included the services of a bargee, apparently limited to the hours 8 A.M. to 4 P.M. On January 2, 1944, the barge, which had lifted the cargo of flour, was made fast off the end of Pier 58 on the Manhattan side of the North River, whence she was later shifted to Pier 52. At some time not disclosed, five other barges were moored outside her, extending into the river; her lines to the pier were not then strengthened. At the end of the next pier north (called the Public Pier), lay four barges; and a line had been made fast from the outermost of these to the fourth barge of the tier hanging to Pier 52. The purpose of this line is not entirely apparent, and in any event it obstructed entrance into the slip between the two piers of barges. The Grace Line, which had chartered the tug, ‘Carroll,’ sent her down to the locus in quo to ‘drill’ out one of the barges which lay at the end of the Public Pier; and in order to do so it was necessary to throw off the line between the two tiers. On board the ‘Carroll’ at the time were not only her master, but a ‘harbormaster’ employed by the Grace Line. Before throwing off the line between the two tiers, the ‘Carroll’ nosed up against the outer barge of the tier lying off Pier 52, ran a line from her own stem to the middle bit of that barge, and kept working her engines ‘slow ahead’ against the ebb tide which was making at that time. The captain of the ‘Carroll’ put a deckhand and the ‘harbormaster’ on the barges, told them to throw off the line which barred the entrance to the slip; but, before doing so, to make sure that the tier on Pier 52 was safely moored, as there was a strong northerly wind blowing down the river. The ‘harbormaster’ and the deckhand went aboard the barges and readjusted all the fasts to their satisfaction, including those from the ‘Anna C.’ to the pier.

After doing so, they threw off the line between the two tiers and again boarded the ‘Carroll,’ which backed away from the outside barge, preparatory to ‘drilling’ out the barge she was after in the tier off the Public Pier. She had only got about seventy-five feet away when the tier off Pier 52 broke adrift because the fasts from the ‘Anna C,’ either rendered, or carried away. The tide and wind carried down the six barges, still holding together, until the ‘Anna C’ fetched up against a tanker, lying on the north side of the pier below- Pier 51- whose propeller broke a hole in her at or near her bottom. Shortly thereafter: i.e., at about 2:15 P.M., she careened, dumped her cargo of flour and sank. The tug, ‘Grace,’ owned by the Grace Line, and the ‘Carroll,’ came to the help of the flotilla after it broke loose; and, as both had siphon pumps on board, they could have kept the ‘Anna C’ afloat, had they learned of her condition; but the bargee had left her on the evening before, and nobody was on board to observe that she was leaking. The Grace Line wishes to exonerate itself from all liability because the ‘harbormaster’ was not authorized to pass on the sufficiency of the fasts of the ‘Anna C’ which held the tier to Pier 52; the Carroll Company wishes to charge the Grace Line with the entire liability because the ‘harbormaster’ was given an over-all authority. Both wish to charge the ‘Anna C’ with a share of all her damages, or at least with so much as resulted from her sinking. The Pennsylvania Railroad Company also wishes to hold the barge liable. The Conners Company wishes the decrees to be affirmed.
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[T]he deckhand and the ‘harbormaster’ jointly undertook to pass upon the ‘Anna C’s’ fasts to the pier; and even though we assume that the bargee was responsible for his fasts after the other barges were added outside, there is not the slightest ground for saying that the deckhand and the ‘harbormaster’ would have paid any attention to any protest which he might have made, had he been there. We do not therefore attribute it as in any degree a fault of the ‘Anna C’ that the flotilla broke adrift. Hence she may recover in full against the Carroll Company and the Grace Line for any injury she suffered from the contact with the tanker’s propeller, which we shall speak of as the ‘collision damages.’ On the other hand, if the bargee had been on board, and had done his duty to his employer, he would have gone below at once, examined the injury, and called for help from the ‘Carroll’ and the Grace Line tug. Moreover, it is clear that these tugs could have kept the barge afloat, until they had safely beached her, and saved her cargo. This would have avoided what we shall call the ‘sinking damages.’ Thus, if it was a failure in the Conner Company’s proper care of its own barge, for the bargee to be absent, the company can recover only one third of the ‘sinking’ damages from the Carroll Company and one third from the Grace Line. For this reason the question arises whether a barge owner is slack in the care of his barge if the bargee is absent.
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Since there are occasions when every vessel will break from her moorings, and since, if she does, she becomes a menace to those about her; the owner’s duty, as in other similar situations, to provide against resulting injuries is a function of three variables: (1) The probability that she will break away; (2) the gravity of the resulting injury, if she does; (3) the burden of adequate precautions. Possibly it serves to bring this notion into relief to state it in algebraic terms: if the probability be called P; the injury, L; and the burden, B; liability depends upon whether B is less than L multiplied by P: i.e., whether B less than PL. Applied to the situation at bar, the likelihood that a barge will break from her fasts and the damage she will do, vary with the place and time; for example, if a storm threatens, the danger is greater; so it is, if she is in a crowded harbor where moored barges are constantly being shifted about. On the other hand, the barge must not be the bargee’s prison, even though he lives aboard; he must go ashore at times. We need not say whether, even in such crowded waters as New York Harbor a bargee must be aboard at night at all; it may be that the custom is otherwise[.] * * *. We leave that question open; but we hold that it is not in all cases a sufficient answer to a bargee’s absence without excuse, during working hours, that he has properly made fast his barge to a pier, when he leaves her. In the case at bar the bargee left at five o’clock in the afternoon of January 3rd, and the flotilla broke away at about two o’clock in the afternoon of the following day, twenty-one hours afterwards. The bargee had been away all the time, and we hold that his fabricated story was affirmative evidence that he had no excuse for his absence. At the locus in quo- especially during the short January days and in the full tide of war activity- barges were being constantly ‘drilled’ in and out. Certainly it was not beyond reasonable expectation that, with the inevitable haste and bustle, the work might not be done with adequate care. In such circumstances we hold- and it is all that we do hold- that it was a fair requirement that the Conners Company should have a bargee aboard (unless he had some excuse for his absence), during the working hours of daylight.
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B< PL B: the Burden of precautionary measures. P: the Probability of the harm occurring L: the magnitude of the loss should it occur

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