This appeal is the second time that the parties have appeared before this court. In their first visit, see Schexnider v. McDermott Int'l, Inc., 817 F.2d 1159 (5th Cir.), cert. denied, --- U.S. ----, 108 S. Ct. 488, 98 L. Ed. 2d 486 (1987), John A. Schexnider ("Schexnider") appealed from the district court's dismissal of his suit against McDermott International, Inc., a Panama corporation, and McDermott, Inc. on forum non conveniens grounds. Schexnider was partly successful, and we reversed the district court's dismissal of his suit. We affirmed, however, the district court's determination that Australian law governed Schexnider's action and remanded the case to the district court for trial.1 Following a bench trial, the district court entered judgment in favor of the defendants.

Schexnider appeals, arguing that the district court's findings of fact are clearly erroneous and that the district court should not have applied Australian law. Addressing the latter argument first, our earlier decision concerning this case decided the question of what law was to be applied at trial. The decision of a legal issue by an appellate court establishes the "law of the case" and must be followed in all subsequent proceedings in the same case at both the trial and appellate levels unless the evidence at a subsequent trial was substantially different, the controlling authority has since made a contrary decision of law applicable to such issues, or the decision was clearly erroneous and would work a manifest injustice. See White v. Murtha, 377 F.2d 428, 431-32 (5th Cir.1967). In support of his argument that the district court erred in applying Australian law, Schexnider cites no reason that was not fully considered by the panel that directed the district court to apply Australian law. We have, therefore, no basis for reversing the district court in its decision to apply Australian law.2

We turn next to Schexnider's argument that the district court was clearly erroneous in its findings of fact relating to Schexnider's accident. Schexnider's complaint alleges that on April 12, 1981, he slipped in grease or hydraulic fluid that had been allowed to accumulate on a step of a stairway on the Derrick Barge 21 (the "barge"), on which he was employed as a storeman. At trial, the substance of Schexnider's testimony was that both the accumulated grease or hydraulic fluid and the dented treads in the center of the stairs3 caused his accident. The district court's opinion carefully reviewed Schexnider's testimony, as well as the testimony of various other witnesses to the accident itself and to the respective conditions of the barge generally and the stairway specifically. The district court found that Schexnider's method of descending the stairs--slowly, deliberately, and one stair at a time while holding the rail--was the safest method of forward descent and would minimize any possibility that the angle of the stairs and the slight indentations which existed in the treads would have caused his fall. The district court concluded that:

The court is not persuaded that any condition of the stairs or any other appurtenance of the vessel contributed to the plaintiff's accident. Nor is the court persuaded that any foreign substance was on the stairs or played a part in causing the fall. The cause of the fall remains a puzzlement, as the court cannot discern how, after both of the plaintiff's feet were on the top step and he was, for a split second, stopped, his right foot came to fly out from under him when he stepped off with his left. The plaintiff bore the burden of proof in this action, and he has not sustained it. The court finds that no fault or condition attributable to any of the defendants or to any of INA's insureds contributed to the fall.

On appeal, Schexnider focuses on discrepancies between the deposition testimony of the chief engineer and the captain of the barge and their testimony at trial. At the trial, in Schexnider's view, these two witnesses "were repeatedly impeached with their deposition testimony and the written accident report." Schexnider argues, very simply, that "either the truth was told in the depositions or at trial." Clearly, the district court credited the testimony of the chief engineer and the captain at trial. When confronted with his prior, apparently conflicting deposition testimony at trial, each man explained that at the time of his deposition, which was three years after the accident, he was not given an opportunity to review the barge's records to determine the work being performed on the barge at the time of the accident. Further, deposition testimony concerning the accumulations of fluid and oil on the floor and stairway resulting from the installation of a tension machine did not conflict with trial testimony about the fluid- and oil-free nature of the floor and stairway on the date of the accident because the barge's records reflected that the installation of the tension machine had been completed on April 10th and the accident occurred on April 12th. The district court credited that explanation for the discrepancies between the deposition testimony of the two men and their trial testimony. The district court based its decision on its assessment of the credibility of the chief engineer, the captain and other witnesses. Determinations as to the credibility of witnesses are peculiarly within the province of the district court. Even if we were convinced, which we are not, that had we been sitting as the trier of fact, we would have weighed the evidence differently, our ability to reverse the district court's judgment on the basis that its findings of fact are clearly erroneous is extremely limited. As the Supreme Court said in Anderson v. Bessemer City, 470 U.S. 564, 105 S. Ct. 1504, 84 L. Ed. 2d 518 (1985), "when a trial judge's finding is based on his decision to credit the testimony of one of two or more witnesses, each of whom has told a coherent and facially plausible story that is not contradicted by extrinsic evidence, that finding, if not internally inconsistent, can virtually never be clear error." Id. 470 U.S. at 575, 105 S. Ct. at 1513.

We conclude that the district court's findings of fact are not clearly erroneous.

AFFIRMED.