French swore disgustedly. He had, of course, realised that the name might be false, but yet he had hoped against hope that he might really have reached the end of at least this portion of his quest. But here he was, as far from the truth as ever! He would now have to make a fresh start to trace this elusive lady—he used another adjective in his mind—and he couldn’t see that he was any better equipped for the search now than when he had started out from Mr. Williams’s office. It was a confoundedly exasperating case—just bristling with promising clues which one after another petered out as he came to follow them up. Being on it was like trying to cross a stream on stepping-stones which invariably gave way when he came to place his weight on them. It was an annoying thought also that that would scarcely be the view his chief would take of the matter. The chief had not been over-complimentary already in his comments on his handling of the case, and French felt that he would view this new check in anything but a sympathetic spirit. However, grousing about it wouldn’t lead anywhere, and with an effort he switched his thoughts back to his problem. As he thought it over a further point occurred to him. Since his first visit to the Savoy he had wondered why the lady had turned up there so much later than the other passengers from the _Olympic_, and now he saw the reason. The episode of the handbag had taken place some four hours after the vessel’s arrival, long after the special boat train had left. Mrs. X—for she was still Mrs. X—must therefore have travelled up by an afternoon train, probably the 5.26 or 6.22 p.m. from the West Station, which got in 6.58 and 8.20 respectively. Now, why this delay? What had she done during these four hours? The answer was not far to seek. Was it not to give her time and opportunity to assume her disguise? He felt it must be so. The lady was her natural self—other than in name—on board the _Olympic_, and having no opportunity to alter her appearance, she had passed through the customs in the same character. Hence the ship’s staff and the customs officer had instantly recognised her photograph. But it was obvious that her impersonation of Mrs. Root must begin before she interviewed the Southampton police, and that accounted for the hesitation of Sergeant McAfee and the people in London in identifying her. She had therefore made herself up between passing through the customs at, say, eleven o’clock, and calling on the Sergeant at three. Where was she during those four hours? He put himself in her place. Confronted with her problem, what would he have done? Gone to a hotel, unquestionably. Taken a room in which to assume the disguise. Had Mrs. X engaged a bedroom in one of the Southampton hotels for that afternoon?