This piece of news had a profound effect on Markham. As was his habit when agitated, he rose and began pacing back and forth, his hands clasped behind him. Heath, too, though slower to grasp the significance of Vance's revelation, puffed vigorously on his cigar—an indication that his mind was busy with a difficult adjustment of facts.

    Before either had formulated any comment the rear door of the hall opened and light footsteps approached the drawing-room. Belle Dillard, returning from Mrs. Drukker's, appeared in the archway. Her face was troubled, and letting her eyes rest on Markham, she asked:

    "What did you say to Adolph this morning? He's in an awful state of funk. He's going about testing all the door-locks and window-catches as if he feared burglars; and he has frightened poor Grete by telling her to be sure to bolt herself in at night."

    "Ah! He has warned Mrs. Menzel, has he?" mused Vance. "Very interestin'."

    The girl's gaze turned swiftly to him.

    "Yes; but he will give me no explanation. He's excited and mysterious. And the strangest thing about his attitude is that he refuses to go near his mother. . . . What does it mean, Mr. Vance? I feel as though something terrible were impending."

    "I don't know just what it does mean." Vance spoke in a low, distressed voice. "And I'm afraid even to try to interpret it. If I should be wrong. . . ." He became silent for a moment. "We must wait and see. To-night perhaps we'll know.—But there's no cause for alarm on your part, Miss Dillard." He smiled comfortingly. "How did you find Mrs. Drukker?"

    "She seemed much better. But there's still something worrying her; and I think it has to do with Adolph, for she talked about him the whole time I was there, and kept asking me if I'd noticed anything unusual in his manner lately."

    "That's quite natural in the circumstances," Vance returned. "But you mustn't let her morbid attitude affect you.—And now, to change the subject: I understand that you were in the library for half an hour or so last night just before you went to the theatre. Tell me, Miss Dillard: where was your hand-bag during that time?"

    The question startled her; but after a momentary hesitation she answered: "When I came into the library I placed it with my wrap on the little table by the door."

    "It was the lizard-skin bag containing the key?"

    "Yes. Sigurd hates evening dress, and when we go out together I always wear my day clothes."

    "So you left the bag on the table during that half-hour, and then kept it with you the rest of the evening.—And what about this morning?"

    "I went out for a walk before breakfast and carried it with me. Later I put it on the hat-rack in the hall for an hour or so; but when I started for Lady Mae's at about ten I took it with me. It was then I discovered that the little pistol had been returned, and I postponed my call. I left the bag down-stairs in the archery-room until you and Mr. Markham came; and I've had it with me ever since."

    Vance thanked her whimsically.

    "And now that the peregrinations of the bag have been thoroughly traced, please try to forget all about it." She was on the point of asking a question, but he anticipated her curiosity and said quickly: "You went to the Plaza for supper last night, your uncle told us. You must have been late in getting home."

    "I never stay out very late when I go anywhere with Sigurd," she answered, with a maternal note of complaint. "He has a constitutional aversion to any kind of night life. I begged him to stay out longer, but he looked so miserable I hadn't the heart to remain. We actually got home at half past twelve."

    Vance rose with a gracious smile.

    "You've been awfully good to bear with our foolish questions so patiently. . . . Now we're going to drop in on Mr. Pardee and see if he has any illuminatin' suggestions to offer. He's generally in at this time, I believe."

    "I'm sure he's in now." The girl walked with us to the hall. "He was here only a little while before you came, and he said he was returning home to attend to some correspondence."

    We were about to go out when Vance paused.

    "Oh, I say, Miss Dillard; there's one point I forgot to ask you about. When you came home last night with Mr. Arnesson, how did you know it was just half past twelve? I notice you don't wear a watch."

    "Sigurd told me," she explained. "I was rather mean to him for bringing me home so early, and as we entered the hall here I asked him spitefully what time it was. He looked at his watch and said it was half past twelve. . . ."

    At that moment the front door opened and Arnesson came in. He stared at us in mock astonishment; then he caught sight of Belle Dillard.

    "Hallo, sis," he called to her pleasantly. "In the hands of the gendarmerie, I see." He flashed us an amused look. "Why the conclave? This house is becoming a regular police station. Hunting for clews of Sprigg's murderer? Ha! Bright youth done away with by his jealous professor, and that sort of thing, eh? . . . Hope you chaps haven't been putting Diana the Huntress through a third degree."

    "Nothing of the kind," the girl spoke up. "They've been most considerate. And I've been telling them what an old fogy you are—bringing me home at half past twelve."

    "I think I was very indulgent," grinned Arnesson. "Much too late for a child like you to be out."

    "It must be terrible to be senile and—and mathematically inclined," she retorted with some heat, and ran up-stairs.

    Arnesson shrugged his shoulders and looked after her until she had disappeared. Then he fixed a cynical eye on Markham.

    "Well, what glad tidings do you bring? Any news about the latest victim?" He led the way back to the drawing-room. "You know, I miss that lad. He'd have gone far. Rotten shame he had to be named Johnny Sprigg. Even 'Peter Piper' would have been safer. Nothing happened to Peter Piper aside from the pepper episode; and you couldn't very well work that up into a murder. . . ."

    "We have nothing to report, Arnesson," Markham broke in, nettled by the man's flippancy. "The situation remains unchanged."

    "Just dropped in for a social call, I presume. Staying for lunch?"

    "We reserve the right," said Markham coldly, "to investigate the case in whatever manner we deem advisable. Nor are we accountable to you for our actions."

    "So! Something has happened that irks you." Arnesson spoke with sarcasm. "I thought I had been accepted as a coadjutor; but I see I am to be turned forth into the darkness." He sighed elaborately and took out his pipe. "Dropping the pilot!—Bismarck and me. Alas!"

    Vance had been smoking dreamily near the archway, apparently oblivious of Arnesson's complaining. Now he stepped into the room.

    "Really, y' know, Markham, Mr. Arnesson is quite right. We agreed to keep him posted; and if he's to be of any help to us he must know all the facts."

    "It was you yourself," protested Markham, "who pointed out the possible danger of mentioning last night's occurrence. . . ."

    "True. But I had forgotten at the time our promise to Mr. Arnesson. And I'm sure his discretion can be relied on." Then Vance related in detail Mrs. Drukker's experience of the night before.

    Arnesson listened with rapt attention. I noticed that his sardonic expression gradually disappeared, and that in its place came a look of calculating sombreness. He sat for several minutes in contemplative silence, his pipe in his hand.

    "That's certainly a vital factor in the problem," he commented at length. "It changes our constant. I can see that this thing has got to be calculated from a new angle. The Bishop, it appears, is in our midst. But why should he come to haunt Lady Mae?"

    "She is reported to have screamed at almost the exact moment of Robin's death."

    "Aha!" Arnesson sat up. "I grasp your implication. She saw the Bishop from her window on the morning of Cock Robin's dissolution, and later he returned and perched on her door-knob as a warning for her to keep mum."

    "Something like that, perhaps. . . . Have you enough integers now to work out your formula?"

    "I'd like to cast an eye on this black bishop. Where is it?"

    Vance reached in his pocket, and held out the chessman. Arnesson took it eagerly. His eyes glittered for a moment. He turned the piece over in his hand, and then gave it back.

    "You seem to recognize this particular bishop," said Vance dulcetly. "You're quite correct. It was borrowed from your chess set in the library."

    Arnesson nodded a slow affirmative.

    "I believe it was." Suddenly he turned to Markham, and an ironic leer came over his lean features. "Was that why I was to be kept in the dark? Under suspicion, am I? Shades of Pythagoras! What penalty attaches to the heinous crime of distributing chessmen among one's neighbors?"

    Markham got up and walked toward the hall.

    "You are not under suspicion, Arnesson," he answered, with no attempt to conceal his ill-humor. "The bishop was left at Mrs. Drukker's at exactly midnight."

    "And I was half an hour too late to qualify. Sorry to have disappointed you."

    "Let us hear if your formula works out," said Vance, as we passed out of the front door. "We've a little visit to pay to Mr. Pardee now."

    "Pardee? Oho! Calling in a chess expert on the subject of bishops, eh? I see your reasoning—it at least has the virtue of being simple and direct. . . ."

    He stood on the little porch and watched us, like a japish gargoyle, as we crossed the street.

    Pardee received us with his customary quiet courtesy. The tragic, frustrated look which was a part of his habitual expression was even more pronounced than usual; and when he drew up chairs for us in his study his manner was that of a man whose interest in life had died, and who was merely going through the mechanical motions of living.

    "We have come here, Mr. Pardee," Vance began, "to learn what we can of Sprigg's murder in Riverside Park yesterday morning. We have excellent reasons for every question we are about to ask you."

    Pardee nodded resignedly.

    "I shall not be offended at any line of interrogation you take. After reading the papers I realize just how unusual a problem you are facing."

    "First, then, please inform us where you were yesterday morning between seven and eight."

    A faint flush overspread Pardee's face, but he answered in a low, even voice.

    "I was in bed. I did not rise until nearly nine."

    "Is it not your habit to take a walk in the park before breakfast?" (I knew this was sheer guesswork on Vance's part, for the subject of Pardee's habits had not come up during the investigation.)

    "That is quite true," the man replied, without a moment's hesitation. "But yesterday I did not go,—I had worked rather late the night before."

    "When did you first hear of Sprigg's death?"

    "At breakfast. My cook repeated the gossip of the neighborhood. I read the official account of the tragedy in the early edition of the evening Sun."

    "And you saw the reproduction of the Bishop note, of course, in this morning's paper.—What is your opinion of the affair, Mr. Pardee?"

    "I hardly know." For the first time his lacklustre eyes showed signs of animation. "It's an incredible situation. The mathematical chances are utterly opposed to such a series of interrelated events being coincidental."

    "Yes," Vance concurred. "And speaking of mathematics: are you at all familiar with the Riemann-Christoffel tensor?"

    "I know of it," the man admitted. "Drukker uses it in his book on world lines. My mathematics, however, are not of the physicist's type. Had I not become enamored of chess"—he smiled sadly—"I would have been an astronomer. Next to manoeuvring the factors in a complicated chess combination, the greatest mental satisfaction one can get, I think, is plotting the heavens and discovering new planets. I even keep a five-inch equatorial telescope in a pent-house on my roof for amateur observations."

    Vance listened to Pardee with close attention; and for several minutes discussed with him Professor Pickering's recent determination of the trans-Neptunean O,[21] much to Markham's bewilderment and to the Sergeant's annoyance. At length he brought the conversation back to the tensor formula.

    "You were, I understand, at the Dillards' last Thursday when Mr. Arnesson was discussing this tensor with Drukker and Sprigg."

    "Yes, I recall that the subject came up then."

    "How well did you know Sprigg?"

    "Only casually. I had met him with Arnesson once or twice."

    "Sprigg, also, it seems, was in the habit of walking in Riverside Park before breakfast," observed Vance negligently. "Ever run into him there, Mr. Pardee?"

    The man's eyelids quivered slightly, and he hesitated before answering.

    "Never," he said finally.

    Vance appeared indifferent to the denial. He rose and, going to the front window, looked out.

    "I thought one might be able to see into the archery range from here. But I note that the angle cuts off the view entirely."

    "Yes, the range is quite private. There's even a vacant lot opposite the wall, so that no one can see over it. . . . Were you thinking of a possible witness to Robin's death?"

    "That, and other things." Vance returned to his chair. "You don't go in for archery, I take it."

    "It's a trifle too strenuous for me. Miss Dillard once tried to interest me in the sport, but I was not a very promising acolyte. I've been to several tournaments with her, however."

    An unusually soft note had crept into Pardee's voice, and for some reason which I could not exactly explain I got the feeling that he was fond of Belle Dillard. Vance, too, must have received the same impression, for after a brief pause he said:

    "You will realize, I trust, that it is not our intention to pry unnecessarily into any one's private affairs; but the question of motive in the two murders we are investigating still remains obscure, and as Robin's death was at first superficially attributed to a rivalry for Miss Dillard's affections, it might help us to know, in a general way, what the true situation is concerning the young lady's preference. . . . As a friend of the family you probably know; and we'd appreciate your confidence in the matter."

    Pardee's gaze travelled out of the window, and the suggestion of a sigh escaped him.

    "I've always had the feeling that she and Arnesson would some day be married. But that is only conjecture. She once told me quite positively that she was not going to consider matrimony until she was thirty." (One could easily guess in what connection Belle Dillard had made this pronouncement to Pardee. His emotional, as well as his intellectual life, had apparently met with failure.)

    "You do not believe then," pursued Vance, "that her heart is seriously concerned with young Sperling?"

    Pardee shook his head. "However," he qualified, "martyrdom such as he is undergoing at present has a tremendous sentimental appeal for women."

    "Miss Dillard tells me you called on her this morning."

    "I generally drop over during the day." He was obviously uncomfortable and, I thought, a little embarrassed.

    "Do you know Mrs. Drukker well?"

    Pardee gave Vance a quick, inquisitive look.

    "Not particularly," he said. "I've naturally met her several times."

    "You've called at her house?"

    "On many occasions, but always to see Drukker. I've been interested for years in the relation of mathematics to chess. . . ."

    Vance nodded.

    "How did your game with Rubinstein come out last night, by the by? I didn't see the papers this morning."

    "I resigned on the forty-fourth move." The man spoke hopelessly. "Rubinstein found a weakness in my attack which I had entirely overlooked when I sealed my move at the adjournment."

    "Drukker, Professor Dillard tells us, foresaw the outcome when you and he were discussing the situation last night."

    I could not understand why Vance referred so pointedly to this episode, knowing as he did how sore a point it was with Pardee. Markham, also, frowned at what appeared to be an unforgivably tactless remark on Vance's part.

    Pardee colored, and shifted in his chair.

    "Drukker talked too much last night." The statement was not without venom. "Though he's not a tournament player, he should know that such discussions are taboo during unfinished games. Frankly, though, I put little stock in his prophecy. I thought my sealed move had taken care of the situation, but Drukker saw farther ahead than I did. His analysis was uncannily profound." There was the jealousy of self-pity in his tone, and I felt that he hated Drukker as bitterly as his seemingly mild nature would permit.

    "How long did the game last?" Vance asked casually.

    "It was over a little after one o'clock. There were only fourteen moves in last night's session."

    "Were there many spectators?"

    "An unusually large number, considering the late hour."

    Vance put out his cigarette and got up. When we were in the lower hall on our way to the front door he halted suddenly and, fixing Pardee with a gaze of sardonic amusement, said:

    "Y' know, the black bishop was at large again last night around midnight."

    His words produced an astonishing effect. Pardee drew himself up as if he had been struck in the face; and his cheeks went chalky white. For a full half-minute he stared at Vance, his eyes like live coals. His lips moved with a slight tremor, but no word came from them. Then, as if with superhuman effort, he turned stiffly away and went to the door. Jerking it open he held it for us to pass out.

    As we walked up Riverside Drive to the District Attorney's car, which had been left in front of the Drukker house in 76th Street, Markham questioned Vance sharply in regard to the final remark he had made to Pardee.

    "I was in hopes," explained Vance, "of surprising some look of recognition or understanding from him. But, 'pon my soul, Markham, I didn't expect any effect like the one I produced. Astonishin' how he reacted. I don't grasp it—I don't at all grasp it. . . ."

    He became engrossed in his thoughts. But as the car swung into Broadway at 72nd Street he roused himself and directed the chauffeur to the Sherman Square Hotel.

    "I have a gaspin' desire to know more of that chess game between Pardee and Rubinstein. No reason for it—sheer vagary on my part. But the idea has been workin' in me ever since the professor mentioned it. . . . From eleven until past one—that's a deuced long time to play off an unfinished game of only forty-four moves."

    We had drawn up to the curb at the corner of Amsterdam Avenue and 71st Street, and Vance disappeared into the Manhattan Chess Club. It was fully five minutes before he returned. In his hand he carried a sheet of paper filled with notations. There was, however, no sign of jubilance in his expression.

    "My far-fetched but charmin' theory," he said, with a grimace, "has run aground on base prosaic facts. I just talked to the secret'ry of the club; and last night's session consumed two hours and nineteen minutes. It seems to have been a coruscatin' battle, full of esoteric quirks and strategical soul-searchin's. Along about half past eleven the onlooking genii had Pardee picked for the winner; but Rubinstein then staged a masterly piece of sustained analysis, and proceeded to tear Pardee's tactics to smithereens—just as Drukker had prognosticated. Astonishin' mind, Drukker's. . . ."

    It was plain that even now he was not entirely satisfied with what he had learned; and his next words voiced his dissatisfaction.

    "I thought while I was at it I'd take a page from the Sergeant's book, so to speak, and indulge in a bit of routine thoroughness. So I borrowed the score sheet of last night's game and copied down the moves. I may run over the game some day when time hangs heavy."

    And, with what I thought unusual care, he folded the score and placed it in his wallet.