CHAPTER XXIV With the going of Black Roger also went the oppressive loneliness which had gripped Kerrigan, and as he stood listening to the low voices outside, the undeniable truth came to him that he did not hate this man as he had wanted to hate him. He was a murderer and a scoundrel in another way, but he felt irresistibly the impulse to like him and to feel sorry for him. He made an effort to shake off the feeling, but a small voice which he could not quiet persisted in telling him that more than one good man had committed what the law called murder and that perhaps he didn't fully understand what he had seen through the cabin window on the raft. And yet when unstirred by this impulse he knew the evidence was damning. But his loneliness was gone. With Odomar's visit had come an unexpected thrill, the revival of an almost feverish anticipation, the promising of impending things that stirred his blood as he thought of them. "'You will understand strange things then,' Roger Odomar had said, and something in his voice had been like a key unlocking mysterious doors for the first time. And then, wait as patiently as you can!' Out of the basket on the table seemed to come to him a whispering echo of that same word, wait. He laid his hands upon it, and a pulse of life came with the imagined whispering. It was from Marie Anne. It seemed as though the warmth of her hands were still there, and as he removed the cloth the sweet breath of her came to him. And then, in the next instant, he was trying to laugh at himself and trying equally hard to call himself a fool, for it was the breath of newly baked things which her fingers had made. Yet never had he felt the warmth of her presence more strangely in his heart. He did not try to explain to himself why Roger Odomar's visit had broken down things which had seemed insurmountable an hour ago. Analysis was impossible, because he knew the transformation within himself was without a shred of reason. But it had come, and with it his imprisonment took on another form. Where before there had been thought of escape and a scheming to jail Black Roger, there filled him now an intense desire to reach the yellow knife and the Chateau Boulin. It was after midnight when he went to bed, and he was up with the early dawn. With the first break of day the Bateau men were preparing their breakfast. David was glad. He was eager for the day's work to begin, and in that eagerness he pounded on the door and called out to Joe Clamart that he was ready for his breakfast, with the rest of them, but that he wanted only hot coffee to go with what Black Roger had brought to him in the basket. That afternoon the Bateau passed Fort McMurray, and before the sun was well down in the west, Kerrigan saw the green slopes of thick wood hills and the rising peaks of birch mountains. He left outright, as he thought of Corporal Anderson and Constable Fraser at Fort McMurray, whose chief duty was to watch the big waterway. How their eyes would pop if they could see through the padlocked door of his prison. But he had no inclination to be discovered now. He wanted to go on, and with a growing exaltation he saw there was no intention in the part of the Bateau's crew to loiter on the way. There was no stop at noon, and the tie-up did not come until the last glow of day was darkening into the gloom of night in the sky. For sixteen hours the Bateau had traveled steadily, and it could not have made less than sixty miles as the river ran. The raft, David figured, had not traveled a third of the distance. The fact that the Bateau's progress would bring him to Chateau Boulin many days, and perhaps weeks, before Black Roger and Marie Anne could arrive on the raft did not check his enthusiasm. It was this interval between their arrivals which held a great speculative promise for him. In that time, if his efficiency had not entirely deserted him, he would surely make discoveries of importance. Day after day the journey continued without rest. On the fourth day, after leaving Fort McMurray, it was Joe Clamart who brought in David's supper, and he grunted a protest at his long hours of muscle-breaking labor at the sweeps. When David questioned him, he shrugged his shoulders, and his mouth closed tight as a clam. On the fifth the Bateau crossed the narrow western neck of Lake Athabasca, slipping past Chippewine in the night, and on the sixth that entered the Slave River. It was the fourteenth day when the Bateau entered Great Slave Lake, and the second night after that, as dust gathered thickly between the forest walls of the Yellow Knife, David knew that at last they had reached the mouth of the dark and mysterious stream which led to the still more mysterious domain of Black Roger Audemard. That night the rejoicing of the Bateau men ashore was that of men who had come out from under a strain and were throwing off its tension for the first time in many days. A great fire was built, and the men sang and laughed and shouted as they piled wood upon it. In the flare of this fire a smaller one was built, and kettles and pans were soon bubbling and sizzling over it, and a great coffee pot that held two gallons sent out at steam laden with an aroma that mingled joyously with the balsam and cedar smells in the air. David could see the whole thing from his window, and when Joe Clamart came in with supper he found the meat they were cooking over the fire was fresh mousse-stake. As there had been no trading or firing of guns coming down, he was puzzled, and when he asked where the meat had come from, Joe Clamart only shrugged his shoulders and winked an eye, and went out singing about the alouette bird that had everything plucked from it one by one. But David noticed there were never more than four men ashore at the same time. At least one was always aboard the Bateau, watching his doors and windows. And he, too, felt the thrill of an excitement working subtly within him, and this thrill pounded in swifter-running blood when he saw the men about the fire jump to their feet suddenly and go to meet new and shadowy figures that came up indistinctly just in the edge of the forest gloom. There they mingled and were lost in identity for a long time, and David wondered if the newcomers were of the people of Chateauboulin. After that Bateis and Joe Clamart and two others stamped out the fires and came over the plank to the Bateau to sleep. David followed their example and went to bed. The cook fires were burning again before the gray dawn was broken by a tint of the sun, and when the voices of many men roused David, he went to his window and saw a dozen figures where last night there had been only four. When it grew lighter he recognized none of them. All were strangers. Then he realized the significance of their presence. The Bateau had been traveling north, but downstream. Now it would still travel north, but the water of the yellow knife flowed south into Great Slave Lake, and the Bateau must be towed. He caught a glimpse of the two big York boats a little later and six rowers to a boat, and after that the Bateau set out slowly but steadily upstream. For hours David was at one window or the other, with something of awe working inside him as he saw what they were passing through and between. He fancied the water-trail was like an entrance into a forbidden land, a region of vast and unbroken mystery, a country of enchantment, possibly of death, shut out from the world he had known. For the stream narrowed and the forest along the shores was so dense he could not see into it. The treetops hung in a tangled canopy overhead, and a gloom of twilight filled the channel below so that where the sun shot through it was like filtered moonlight shining on black oil. There was no sound except the dull, steady beat of the rowers' oars and the ripple of water along the sides of the Bateau. The men did not sing or laugh, and if they talked it must have been in whispers. There was no cry of birds from a shore. And once David saw Joe Clamart's face as he passed the window, and it was set and hard and filled with the superstition of a man who was passing through a devil-country. And then suddenly the end of it came. A flood of sunlight burst in at the windows, and all at once voices came from ahead, a laugh, a shout, and a yell of rejoicing from the Bateau, and Joe Clamart started again the everlasting song of the ala-wet bird that was plucked of everything it had. Clamart found himself grinning.  Yet he conceded that the awesome deadness of the forest passage had put strange thoughts into his own heart. Before nightfall Bateis and Joe Clamart came in and tied his arms behind him, and he was taken ashore with the rumble of a waterfall in his ears. For two hours he watched the labors of the men as they beached the Bateau on long rollers of smooth birch, and rolled it foot by foot over cleared trail until it was launched again above the waterfall. Then he was led back into the cabin and his arms freed. That night he went to sleep with the music of the waterfall in his ears. The second day the yellow knife seemed to be no longer a river but a narrow lake, and the third day the rowers came into the nine lake country at noon, and until another dusk the Bateau threaded its way through twisting channels and impenetrable forests and beached at last at the edge of a great open where the timber had been cut. There was more excitement here, but it was too dark for David to understand the meaning of it. There were many voices, dogs barked. Then voices were at his door, a key rattled in the lock, and it opened. David saw Bates and Joe Clamart first, and then, to his amazement, Black Roger Audemard stood there, smiling at him and nodding good evening. It was impossible for David to repress his astonishment. "'Welcome to Chateau-Boulain,' greeted Black Roger.  "'It is only courtesy that I should be here to give you welcome.' Behind him Bates and Joe Clamart were grinning widely, and then both came in, and Joe Clamart picked up his donage sack and threw it over his shoulder. "'If you'll come with us, monsieur,' David followed, and when he stepped ashore there were Bates and Joe Clamart and one other behind him, and three or four shadowy figures ahead, with Black Roger walking at his side. There were no more voices, and the dog had ceased barking. Ahead was a wall of darkness which was the deep black forest beyond the clearing, and into it led a trail which they followed. It was a path worn smooth by the travel of many feet, and for a mile not a star broke through the treetops overhead, nor did a flash of light break the utter chaos of the way but once when Joe Clamart lighted his pipe. No one spoke. Even Black Roger was silent, and David found no word to say. At the end of the mile the trees began to open above their heads, and they soon came to the edge of the timber. In the darkness David caught his breath. Dead ahead, not a rifle shot away, was the Chateau Boulin. He knew it before Black Roger had said a word.  He could see nothing but these lights, yet they measured off a mighty place to be built of logs in the heart of a wilderness, and at his side he heard Black Roger chuckling in low exultation. Our home, monsieur, he said. Tomorrow, when you see it in the light of day, you will say it is the finest Chateau in the north, all built of sweet cedar where birch is not used, so that even in the deep snows it gives us the perfume of springtime and flowers. David did not answer, and in a moment Audemars said, Only on Christmas and New Year, and at birthdays and wedding feasts, is it lighted up like that. Tonight it is in your honour, monsieur David. Again he laughed softly, and under his breath he added, And there is someone waiting for you there whom you will be surprised to see. David's heart gave a jump. There was meaning in Black Roger's words, and no double twist to what he meant. Marie Anne had come ahead with her husband. Now as they passed on to the brilliantly lighted Chateau, David made out the indistinct outlines of other buildings almost hidden in the out creeping shadows of the forest edges, with now and then a ray of light to show people were in them. But there was a brooding silence over it all which made him wonder, there was no voice, no bark of dog, not even the opening or closing of a door. As they drew nearer he saw a great veranda reaching the length of the Chateau, with screening to keep out the summer pests of mosquitoes and flies and the night-prowing insects attracted by light. Into this they went, up wide birch steps, and ahead of them was a door so heavy it looked like the post-turn gate of a castle. Black Roger opened it, and in a moment David stood beside him in a dimly lighted hall where the mounted heads of wild beasts looked down like startled things from the gloom of the walls. And then David heard the low, sweet notes of a piano coming to them very faintly. He looked at Black Roger. A smile was on the lips of the Chateau master. His head was up, and his eyes glowed with pride and joy as the music came to him. He spoke no word, but laid a hand on David's arm and led him toward it, while Baptiste and Joe Clamart remained standing at the entrance to the hall. David's feet trod in thick rugs of fur. He saw the dim luster of polished birch and cedar in the walls, and over his head the ceiling was rich and matched as in the Bateau cabin. They drew nearer to the music and came to a closed door. This Black Roger opened very quietly as if anxious not to disturb the one who was playing. They entered, and David held his breath. It was a great room he stood in, thirty feet or more from end to end, scarcely less in width. A room brilliant with light, sumptuous in its comfort, sweet with the perfume of wildflowers and with a great black fireplace at the end of it, from over which there stared at him the glass eyes of a monster moose. Then he saw the figure at the piano, and something rose up quickly and choked him when his eyes told him it was not Marie Anne. It was a slim, beautiful figure in a soft and shimmering white gown, and its head was glowing gold in the lamp-light. Roger Odomar'd spoke. Carmen! The woman at the piano turned about, a little startled at the unexpectedness of the voice, and then rose quickly to her feet, and David Kerrigan found himself looking into the eyes of Carmen Fauchet.  Yes, she was smiling at him, this woman whose brother he had brought to the hangman, this woman who had stolen Black Roger from another. She knew him, he was sure of that. She knew him as the man who had believed her a criminal along with her brother, and who had fought to the last against her freedom. Yet from her lips and her eyes and her face the old hatred was gone. She was coming toward him slowly. She was reaching out her hand, and half blindly his own went out, and he felt the warmth of her fingers for a moment, and he heard her voice saying softly, Welcome to Chateau Boulin, Monsieur Kerrigan. He bowed and mumbled something, and Black Roger gently pressed his arm, drawing him back to the door. As he went he saw again that Carmen Fauchet was very beautiful as she stood there, and that her lips were very red, but her face was white, whiter than he had ever seen the face of a woman before. As they went up a winding stair to the second floor, Roger Audemard said, I am proud of my Carmen, Monsieur David. Would any other woman in the world have given her hand like that to the man who had helped to kill her brother? They stopped at another door. Black Roger opened it. There were lights with him, and David knew it was to be his room. Audemard did not follow him inside, but there was a flashing humor in his eyes. I say, is there another woman like her in the world, Monsieur? What have you done to Marianne, your wife? asked David. It was hard for him to get the words out. A terrible thing was gripping at his throat, and the clutch of it drew tighter as he saw the wild light in Black Roger's eyes. Tomorrow you will know, Monsieur, but not tonight. You must wait until tomorrow. He nodded and stepped back, and the door closed, and in the same instance came the harsh grating of a key in the lock. End of Chapter XXIV There was no other door, except one opening into a closet, and but two windows. Curtains were drawn at these windows, and he raised them. A grim smile came to his lips when he saw the white bars of tough birch nailed across each of them outside the glass. He could see the birch had been freshly stripped of bark, and had probably been nailed there that day. Carmen Fanché and Black Roger had welcomed him to Chateau Boulin, but they were evidently taking no chances with their prisoner. And where was Marie Anne? The question was insistent, and with it remained that cold grip of something in his heart that had come with the sight of Carmen Fanché below. Was it possible that Carmen's hatred still lived deadlier than ever, and that with Black Roger she had plotted to bring him here so that her vengeance might be more complete, and a greater torture to him? Were they smiling and offering him their hands even as they knew he was about to die? And if that was conceivable, what had they done with Marie Anne? He looked about the room. It was singularly bare in an unusual sort of way, he thought. There were rich rugs on the floor, three magnificent black bear skins and two wolfs. The heads of two bucks and a splendid caribou hung against the walls. He could see, from marks on the door, where a bed had stood, but this bed was now replaced by a couch made up comfortably for one inclined to sleep. The significance of the thing was clear. Nowhere in the room could he lay his hand upon an object that might be used as a weapon. His eyes again sought the white birch bars of his prison, and he raised the two windows so that the cool, sweet breath of the forests reached into him. It was then that he noticed the mosquito-proof screening nailed outside the bars. It was rather odd, this thinking of his comfort, even as they planned to kill him. If there was truth to this new suspicion that Black Roger and his mistress were plotting both vengeance and murder, their plans must also involve Marie Anne. Suddenly his mind shot back to the raft. Had Black Roger turned to clever coup by leaving his wife there while he came on ahead of the bateau with Carmen Fanché? It would be several weeks before the raft reached the yellow knife, and in that time many things might happen. The thought worried him. He was not afraid for himself. The danger, the combating of physical forces, was his business. His fear was for Marie Anne. He had seen enough to know that Black Roger was hopelessly infatuated with Carmen Fanché, and several things might happen aboard the raft, planned by agents as Black sold as himself.  And as he stood there, every muscle in his body ready to fight, there came to him faintly the sound of music. He heard the piano first, and then a woman's voice singing. Soon a man's voice joined the woman's, and he knew it was Black Roger, singing with Carmen Fanché. Suddenly the mad impulse in his heart went out, and he leaned his head nearer to the crack of the door and strained his ears to hear. He could make out no words of the song, yet the singing came to him with a thrill that set his lips apart and brought a staring wonder into his eyes. In the room below him, fifteen hundred miles from civilization, Black Roger and Carmen Fanché were singing Home Sweet Home. An hour later David looked through one of the barred windows upon a world lighted by a splendid moon. He could see the dark edge of the distant forest that rimmed in the chateau, and about him seemed to be a level meadow, with here and there the shadow of a building in which the lights were out. Stars were thick in the sky, and a strange quietness hovered over the world he looked upon. From below him floated up now and then a perfume of tobacco smoke. The guard under his window was awake, but he made no sound. A little later he undressed, put out the two lights in his room, and stretched himself between the cool white sheets on the couch. After a time he slept, but it was a restless slumber filled with troubled dreams. Once he was half awake, and the second time it seemed to him his nostrils sensed a sharper tang of smoke than that of burning tobacco. Yet he did not fully rouse himself, and the hours passed, and new sounds and smells that rose in the night impinge themselves upon him only as a part of the troublest fabric of his dreams. But at last there came a shock, something which beat over these things which chained him, and seized upon his consciousness, demanding that he rouse himself, open his eyes, and get up. He obeyed the command, and before he was fully awake, found himself on his feet. It was still dark, but he heard voices, voices no longer subdued, but filled with a wild note of excitement and command. And what he smelled was not the smell of tobacco smoke. It was heavy in his room. It filled his lungs. His eyes were smarting with the sting of it. Then came vision, and with a startled cry he leaped to a window. To the north and east he looked out upon a flaming world. With his fist he rubbed his smarting eyes. The moon was gone. The gray he saw outside must be the coming of dawn, ghostly with that mist of smoke that had come into his room. He could see shadowy figures of men running swiftly in and out and disappearing, and he could hear the voices of women and children, and from beyond the edge of the forest to the west came the howling of many dogs. One voice rose above the others. It was Black Rogers, and at its commands little groups of figures shot out into the gray smoke gloom and did not appear again. North and east the sky was flaming, sullen red, and a breath of air blowing gently in David's face told him the direction of the wind. The chateau lay almost in the center of the growing line of conflagration. He dressed himself and went again to the window. Quite distinctly now he could make out Joe Clamart under his window, running toward the edge of the forest at the head of half a dozen men and boys who carried axes and cross-cut saws over their shoulders. It was the last of Black Rogers' people that he saw for some time in the open meadow, but from the front of the chateau he could hear many voices, chiefly of women and children, and guessed it was from there that the final operations against the fire were being directed. The wind was blowing stronger in his face. With it came a sharper tang of smoke, and the widening light of day was fighting to hold its own against the deepening pall of flamelit gloom advancing with the wind. There seemed to come a low and distant sound with that wind, so indistinct that to David's ears it was like a murmur a thousand miles away. He strained his ears to hear, and as he listened there came another sound, a moaning, sobbing voice below his window. It was grief, he heard now, something that went to his heart and held him cold and still. The voice was sobbing like that of a child, yet he knew it was not a child's. Nor was it a woman's. A figure came out slowly in his view, humped over, twisted in its shape, and he recognized Andre, the broken man. David could see that he was crying like a child, and he was facing the flaming forests, with his arms reaching out to them in his moaning. Then of a sudden he gave a strange cry as if defiance had taken the place of grief, and he hurried across the meadow and disappeared into the timber where a great lightning-riven spruce gleamed dully white through the settling veil of smoke mist. For a space David looked after him, a strange beating in his heart. It was as if he had seen a little child going into the face of a deadly peril, and at last he shouted out for someone to bring back the broken man. But there was no answer from under his window. The guard was gone. Nothing lay between him and escape if he could force the white birch bars from the window. He thrust himself against them, using his shoulder as a battering ram. Not the thousandth part of an inch could he feel them give, yet he worked until his shoulder was sore. Then he paused and studied the bars more carefully. Only one thing would avail him, and that was some object which he might use as a lever. He looked about him, and not a thing was there in the room to answer the purpose. Then his eyes fell on the splendid horns of the caribou head. Black Roger's discretion had failed him there, and eagerly David pulled the head down from the wall. He knew the woodsman's trick of breaking off a horn from the skull, yet in this room without log or route to help him the task was difficult, and it was a quarter of an hour after he had last seen the broken man before he stood again at the window with the caribou horn in his hands.  David thrust the big end of the caribou horn between two of the white birch bars, but before he had put his weight to the lever he heard a great voice coming around the end of the chateau, and it was calling for André the broken man. In a moment it was followed by Black Roger Audemard, who ran under the window and faced the lightning-struck spruce as he shouted André's name again. Finally David called down to him, and Black Roger turned and looked up through the smoke-gloom, his head bare, his arms naked, and his eyes gleaming wildly as he listened. "'He went that way twenty minutes ago,' David shouted. "'He disappeared into the forest where you see the dead spruce yonder. And he was crying, Black Roger. He was crying like a child.' If there had been other words to finish, Black Roger would not have heard them. He was running toward the old spruce, and David saw him disappear where the broken man had gone. Then he put his weight on the horn and one of the tough birch bars gave way slowly, and after that a second was wrenched loose, and a third, until the lower half of the window was free of them entirely. He thrust out his head and found no one within the range of his vision. Then he worked his way through the window, feet first, and hanging the length of arms and body from the lower sill, dropped to the ground. Instantly he faced the direction taken by Roger Audemard. It was his turn now, and he felt a savage thrill in his blood. For an instant he hesitated, held by the impulse to rush to Carmen Fanché and with his fingers at her throat, demand what she and her paramour had done with Marie Anne. But the mighty determination to settle it all with Black Roger himself overwhelmed that impulse like an inundation. Black Roger had gone into the forest. He was separated from his people, and the opportunity was at hand. Positive that Marie Anne had been left with the raft, the thought that the Chateau Boulin might be devoured by the onrushing conflagration did not appall David. The Chateau held little interest for him now. It was Black Roger he wanted. As he ran toward the old spruce he picked up a club that lay in the path. This path was a faintly worn trail where it entered the forest beyond the spruce, very narrow, and with brush hanging close to the sides of it, so that David knew it was not in general use and the but few feet had ever used it. He followed swiftly and in five minutes came suddenly out into a great open, thick with smoke, and here he saw why Chateau Boulin would not burn. The break in the forest was a clearing, a rifle shot in width, free of brush and grass, and partly tilled. And it ran in a semi-circle as far as he could see through the smoke in both directions. Thus had Black Roger safeguarded his wilderness castle while providing tillable fields for his people, and as David followed the faintly beaten path he saw green stuffs growing on both sides of him and through the center of the clearing a long strip of wheat, green and very thick. Up and down through the fog of smoke he could hear voices, and he knew it was this great, circular fire-clearing the people of Chateau Boulin were watching and guarding. But he saw no one as he trailed across the open. In soft patches of the earth he found footprints deeply made and wide apart, the footprints of hurrying men, telling him Black Roger and the broken man were both ahead of him, and that Black Roger was running when he crossed the clearing. The footprints led him to a still more indistinct trail in the farther forest, a trail which went straight into the face of the fire ahead. He followed it. The distant murmur had grown into a low moaning over the treetops, and with it the wind was coming stronger and the smoke thicker. For a mile he continued along the path, and then he stopped, knowing he had come to the deadline. Over him was a swirling chaos. The fire wind had grown into a roar before which the treetops bent as if struck by a gale, and in the air he breathed he could feel a swiftly growing heat. For a space he stood there, breathing quickly in the face of a mighty peril. Where had Black Roger and the broken man gone? What mad impulse could it be that dragged them still farther into the path of death? Or had they struck aside from the trail? Was he alone in danger? As if in answer to the questions there came from far ahead of him a loud cry. It was Black Roger's voice, and as he listened it called over and over again the broken man's name. Andrei! Andrei! Andrei! Something in the cry held Kerrigan. There was a note of terror in it, a wild entreaty that was almost drowned in the trembling wind and the moaning that was in the air. David was ready to turn back. He had already approached too near to the red line of death, yet that cry of Black Roger urged him on like the lash of a whip. He plunged ahead into the chaos of smoke, no longer able to distinguish a trail under his feet. Twice again in as many minutes he heard Black Roger's voice and ran straight toward it. The blood of the hunter rushed over all other things in his veins. The man he wanted was ahead of him, and the moment had passed when danger or fear of death could drive him back. Where Black Roger lived he could live, and he gripped his club and ran through the low brush that whipped in stinging lashes against his face and hands. He came to the foot of a ridge, and from the top of this he knew Black Roger had called. It was a huge hog's back, rising a hundred feet up out of the forest, and when he reached the top of it he was panting for breath. It was as if he had come suddenly within the blast of a hot furnace. North and east the forest lay under him, and only the smoke obstructed his vision. But through this smoke he could make out a thing that made him rub his eyes in a fierce desire to see more clearly. A mile away, perhaps too, the conflagration seemed to be splitting itself against the tip of a mighty wedge. He could hear the roar of it to the right of him and to the left, but dead ahead there was only a moaning whirlpool of fire-heated wind and smoke. And out of this, as he looked, came again the cry, André, André, André! Again he stared north and south through the smoke-gloom. As of resinous clouds, black as ink, were swirling skyward along the two sides of the giant wedge. Under that death-paul the flames were sweeping through the spruce and cedar-tops like race-horses, hidden from his eyes. If they closed in there could be no escape. In fifteen minutes they would inundate him, and it would take him half an hour to reach the safety of the clearing. His heart thumped against his ribs as he hurried down the ridge in the direction of Black Roger's voice. The giant wedge of the forest was not burning, yet, and Odomard was hurrying like mad toward the tip of that wedge, crying out now and then the name of the broken man. And always he kept ahead, until at last, a mile from the ridge, David came to the edge of a wide stream and saw what it was that made the wedge of forest. For under his eyes the stream split, and two arms of it widened out, and along each shore of the two streams was a wide fire-clearing, made by the axes of Black Roger's people, who had foreseen this day when fire might sweep their world. Karagin dashed water into his eyes, and it was warm. Then he looked across. The fire had passed, the pall of smoke was clearing away, and what he saw was the black corpse of a world that had been green. It was smoldering, the deep mold was a fire. Little tongues of flame still licked at ten thousand stubs charred by the fire-death, and there was no wind here, and only the whisper of a distant moaning sweeping farther and farther away.  It was Black Roger still calling, even in that place of hopeless death, for Andrei the broken man. CHAPTER XXVI Into the stream Karagin plunged and found it only waist-deep in crossing. He saw where Black Roger had come out of the water and where his feet had plowed deep in the ash and char and smoldering debris ahead. This trail he followed. The air he breathed was hot and filled with stifling clouds of ash and char-dust and smoke. His feet struck red-hot embers under the ash, and he smelled burning leather. A forest of spruce and cedar skeletons still crackled and snapped and burst out into sudden tongues of flame about him. And the air he breathed grew hotter, and his face burned, and into his eyes came a smarting pain when, ahead of him, he saw Black Roger. He was no longer calling out the broken man's name, but was crashing through the smoking chaos like a great beast that had gone both blind and mad. Twice David turned aside where Black Roger had rushed through burning debris, and a third time, following where Audemard had gone, his feet felt the sudden stab of living coals. In another moment he would have shouted Black Roger's name, but even as the words were on his lips, mingled with a gasp of pain, the giant river-man stopped where the forest seemed suddenly to end in a ghostly, smoke-filled space, and when David came up behind him he was standing at the black edge of a cliff which leaped off into a smoldering valley below. Out of this narrow valley between two ridges, an hour ago choked with living spruce and cedar rose up a swirling, terrifying heat. Down into this pit of death Black Roger stood looking, and David heard a strange moaning coming in his breath. His great bare arms were black and scarred with heat. His hair was burned, his shirt was torn from his shoulders.  And when he saw it was David who had spoken, his great body seemed to sag, and with an unintelligible cry he pointed down. David, staring, saw nothing with his half-blind eyes. But under his feet he felt a sudden giving way, and the fire-eaten tangle of earth and roots broke off like a rotten ledge, and with it both he and Black Roger went crashing into the depths below, smothered in an avalanche of ash and sizzling earth. At the bottom David lay for a moment partly stunned. Then his fingers clutched a bit of living fire, and with a savage cry he staggered to his feet and looked to see Black Roger. For a space his eyes were blinded, and when at last he could see he made out Black Roger fifty feet away, dragging himself on his hands and knees through the blistering muck of the fire. And then, as he stared, the stricken giant came to the charred remnant of a stump and crumpled over it with a great cry, moaning again that name. Andrei! Andrei! David hurried to him, and as he put his hands under Black Roger's arms to help him to his feet, he saw that the charred stump was not a stump, but the fire-scribbled corpse of Andrei, the broken man. Horror choked back speech on his own lips. Black Roger looked up at him, and a great breath came in a sob out of his body. Then suddenly he seemed to get grip of himself, and his burned and bleeding fingers closed about David's hand at his shoulder. "'I knew he was coming here,' he said, the words forcing themselves with an effort through his swollen lips. He came home to die.' "'Home?' "'Yes. His mother and father were buried here nearly thirty years ago, and he worshipped them.  When he saw the fire, he came to shield their graves from the flames. And now he is dead.' A moan came to his lips, and the weight of his body grew so heavy that David had to exert his strength to keep him from falling. "'And you?' he cried, "'For God's sake, Audemard, tell me!' "'I, Monsieur?' "'Why, I am only St. Pierre Audemard, his brother.'  Now at last David came to the edge of the stream again, with the weight of St. Pierre Audemard on the shoulders, was a torturing nightmare which would never be quite clear in his brain. The details were obliterated in the vast agony of the thing.  But he held to St. Pierre, with his twisted, broken leg, knowing that he would die if he dropped him into the flesh-devouring heat of the smoldering debris under his feet. Toward the end he was conscious of St. Pierre's moaning, and then of his voice speaking to him. After that he came to the water and fell down in the edge of it with St. Pierre, and inside his head everything went as black as the world over which the fire had swept. He did not know how terribly he was hurt. He did not feel pain after the darkness came. Yet he sensed certain things. He knew that over him St. Pierre was shouting. For days it seemed he could hear nothing but that great voice bellowing away in the interminable distance. And then came other voices, now near and now far, and after that he seemed to rise up and float among the clouds, and for a long time he heard no other sound and felt no movement but was like one dead. Something soft and gentle and comforting roused him out of darkness. He did not move. He did not open his eyes for a time while reason came to him. He heard a voice, and it was a woman's voice speaking softly, and another voice replied to it. Then he heard gentle movement, and someone went away from him, and he heard the almost noiseless opening and closing of a door. A very little he began to see. He was in a room with a patch of sunlight on the wall. Also he was in a bed, and that gentle, comforting hand was still stroking his forehead and hair, light as thistle down. He opened his eyes wider and looked up. His heart gave a great throb. Over him was a glorious, tender face smiling like an angel into his widening eyes. And it was the face of Carmen Fauchet. He made an effort as if to speak. Hush, she whispered, and he saw something shining in her eyes, and something wet fell upon his face. She is returning, and I will go. For three days and nights she has not slept, and she must be the first to see you open your eyes. She bent over him. Her soft lips touched his forehead, and he heard her sobbing breath.  Then she was going to the door, and his eyes dropped shut again. He began to experience pain now, a hot consuming pain all over him, and he remembered the fight through the path of the fire. Then the door opened very softly once more, and someone came in, and knelt down at his side, and was so quiet that she scarcely seemed to breathe. He wanted to open his eyes, to cry out a name, but he waited, and lips, soft as velvet, touched his own. They lay there for a moment, then moved to his closed eyes, his forehead, his hair, and after that something rested gently against him. His eyes shot open. It was Marie Anne with her head nestled in the crook of his arm as she knelt there beside him on the floor. He could see only a bit of her face, but her hair was very near, crumpled gloriously on his breast, and he could see the tips of her long lashes as she remained very still, seeming not to breathe. She did not know he had roused from his sleep, the first sleep of those three days of torture which he could not remember now. And he, looking at her, made no movement to tell her he was awake. One of his hands lay over the edge of the bed, and so lightly he could scarce feel the weight of her fingers she laid one of her own upon it, and a little at a time drew it to her until the bandaged thing was against her lips. It was strange she did not hear his heart which seemed all it wants to beat like a drum inside him. Suddenly he sensed the fact that his other hand was not bandaged. He was lying on his side, with his right arm partly under him, and against that hand he felt the softness of Marie Anne's cheek, the velvety crush of her hair. And then he whispered, Marie Anne! She still lay for a moment utterly motionless. Then slowly, as if believing he had spoken her name in his sleep, she raised her head and looked into his wide open eyes. There was no word between them in that breath or two. His bandaged hand and his well hand went to her face and hair, and then a sobbing cry came from Marie Anne, and swiftly she crushed her face down to his, holding him close with both her arms for a moment. And after that, as on that other day when she kissed him after the fight, she was up and gone so quickly that her name had scarcely left his lips when the door closed behind her and he heard her running down the hall. He called after her. Marie Anne! Marie Anne! He heard another door, and voices, and quick footsteps again coming his way, and he was waiting eagerly, half on his elbow, and into his room came Neppa Penis and Carmen Fauché. And again he saw the glory of something in the woman's face. His eyes must have burned strangely as he stared at her, but it did not change that light in her own, and her hands were wonderfully gentle as she helped Neppa Penis raise him so that he was sitting up straight with pillows at his back. It doesn't hurt so much now, does it?  He shook his head. No! What is the matter? You were burned, terribly. For two days and nights you were in great pain, but for many hours you have been sleeping, and Neppa Penis says the burns will not hurt any more. If it had not been for you, she bent over him, her hand touched his face, and now he began to understand the meaning of that glory shining in her eyes. If it hadn't been for you, he would have died. She drew back, turning to the door. He is coming to see you, alone, she said, a little broken note in her throat. But I pray, God, you will see with clear understanding, David Kerrigan, and forgive me, as I have forgiven you, for a thing that happened long ago. He waited, his head was in a jumble, and his thoughts were tumbling over one another in an effort to evolve some sort of coherence out of things amazing and unexpected. One thing was impressed upon him. He had saved St. Pierre's life, and because he had done this, Carmen Fanché was very tender to him. She had kissed him, and Marianne had kissed him, and a strange dawning was coming to him, thrilling him to his fingertips. He listened. A new sound was approaching from the hall. His door was opened, and a wheelchair was rolled in by old Neppa Penis. In the chair was St. Pierre Audemard. Feet and hands and arms were wrapped in bandages, but his face was uncovered and wreathed in smiling happiness when he saw David propped up against his pillows. Neppa Penis rolled him close to the bed, and then shuffled out. And as he closed the door, David was sure he heard the subdued whispering of feminine voices down the hall. How are you, David? asked St. Pierre. Fine, nodded Caragin. And you? A bit scorched, and a broken leg. He held up his padded hand. Would be dead if you hadn't carried me to the river. Carmen says she owes you her life for having saved mine. And Marianne? That's what I've come to tell you about, said St. Pierre. The instant they knew you were able to listen, both Carmen and Marianne insisted that I come and tell you things. But if you don't feel well enough to hear me now, go on, almost threatened David. The look of cheer which had illumined St. Pierre's face faded away, and David saw in its place the lines of sorrow which had settled there. He turned his gaze toward a window through which the afternoon sun was coming, and nodded slowly.  They buried him in a casket made of sweet cedar. He loved the smell of that. He was like a little child. And once, a long time ago, he was a splendid man, a greater and better man than St. Pierre, his brother, will ever be. What he did was right and just, Monsieur David. He was the oldest, sixteen, when the thing happened. I was only nine, and didn't fully understand.  And after that, he knew how and why our mother died, but not a word of it did he tell us until years later, after the day of vengeance was passed. You understand, David? He didn't want me in that. He did it alone, with good friends from the upper north. He killed the murderers of our mother and father, and then he buried himself deeper into the forests with us. And we took our mother's family names, which was Boulin, and settled here on the yellow knife. Roger, Black Roger, as you know him, brought the bones of our father and mother and buried them over in the edge of that plane where he died and where our first cabin stood. Five years ago a falling tree crushed him out of shape, and his mind went at the same time, so that he has been like a little child and was always seeking for Roger Odomard, the man he once was. That was the man your law wanted, Roger Odomard, our brother. Our brother, cried David, who is the other? My sister. Yes? Marie Anne. Good God! joked David. St. Pierre, do you lie? Is this another bit of trickery? It is the truth, said St. Pierre. Marie Anne is my sister, and Carmen, whom you saw in my arms through the cabin window, he paused, smiling into David's staring eyes, taking full measure of recompense in the other's heartbreaking attitude as he waited. Is my wife, Monsieur David? A great gasp of breath came out of Caragin. Yes, my wife and the greatest-hearted woman that ever lived without one exception in all the world, cried St. Pierre, a fierce pride in his voice. It was she, and not Marie Anne, who shot you on that strip of sand, David Caragin. Montieux, I tell you, not one woman in a million would have done what she did, let you live. Why? Listen, Monsieur, and you will understand at last. She had a brother, years younger than she, and to that brother she was mother, sister, everything, because they had no parents almost from babyhood. She worshipped him, and he was bad, yet the worse he became, the more she loved him and prayed for him. Years ago she became my wife, and I fought with her to save the brother. But he belonged to the devil, hand and foot, and at last he left us and went south, and became what he was when you were sent out to get him, Sergeant Caragin. It was then that my wife went down to make a last fight to save him, to bring him back, and you know how she made that fight, Monsieur, until the day you hanged him. Saint Pierre was leaning from his chair, his face ablaze. Tell me, did she not fight? He cried. And you, until the last, did you not fight to have her put behind prison bars with her brother? Yes, it is so, Mermin Caragin. She hated you, went on Saint Pierre. You hanged her brother, who was almost a part of her flesh and body. He was bad, but he had been hers from babyhood, and a mother will love her son if he is a devil. And then I won't take long to tell you the rest of it. Through friends she learned that you, who had hanged her brother, were on your way to run down Roger Odomard. And Roger Odomard, mind you, was the same as myself, for I had sworn to take my brother's place if it became necessary. She was on the bateau with Marie Anne when the messenger came.  If it had been some other man, but it was you who had hanged her brother, she disappeared from the bateau that day with a rifle. You know, Mr. David, what happened? Marie Anne heard the shooting and came, alone, just as you rolled out in the sand as if dead. It was she who ran out to you first, while my Carmen crouched there with her rifle, ready to send another bullet into you if you moved. It was Marie Anne you saw standing over you. It was she who knelt down at your side. And then, St. Pierre paused, and he smiled, and then grimaced as he tried to rub his two bandaged hands together. David, fate mixes things up in a funny way.  Yes, it is true. And over you they fought for life or death, and love won, because it is always stronger than hate. Besides, as you lay there bleeding and helpless, you look different to my Carmen than as you did when you hanged her brother. So they dragged you up under a tree, and after that they plotted together and planned, while I was away up the river on the raft. The feminine mind worked strangely, Mr. David, and perhaps it was that thing we call intuition which made them do what they did. Marie Anne knew it would never do for you to see and recognize my Carmen, so in their scheming of things she insisted on passing herself off as my wife, while my Carmen came back in a canoe to meet me. They were frightened, and when I came the whole thing had gone too far for me to mend, and I knew the false game must be played out to the end. When I saw what was happening, that you loved Marie Anne so well that you were willing to fight for her honor, even when you thought she was my wife, I was sure it would all end well. But I could take no chances until I knew, and so there were bars at your windows, and St. Pierre shrugged his shoulders, and the lines of grief came into his face again, and in his voice was a little break as he continued, If Roger had not gone out there to fight back the flames from the graves of his dead, I had planned to tell you as much as I dared, Mr. David, and I had faith that your love for our sister would win.  And besides, that fight of yours against a love which you thought was dishonest interested me very much, for I saw in it a wonderful test of the man who might become my brother if he chose wisely between love and what he thought was duty.  And now even my Carmen loves you for bringing me out of the fire. But you are not listening. David was looking past him toward the door, and St. Pierre smiled when he saw the look that was in his face. Nepapinas, he called loudly, Nepapinas! In a moment there was shuffling of feet outside, and Nepapinas came in. St. Pierre held out his two great bandaged hands, and David met them with his own, one bandaged and one free. Not a word was spoken between them, but their eyes were the eyes of men between whom had suddenly come the faith and understanding of a brotherhood as strong as life itself. Then Nepapinas wheeled St. Pierre from the room, and David straightened himself against his pillows, and waited, and listened until it seemed two hearts were thumping inside him in the place of one. It was an interminable time, he thought, before Marie Anne stood in the doorway. For a breath she paused there, looking at him as he stretched out his bandaged arm to her, moved by every yearning impulse in her soul to come in, yet ready as a bird to fly away. And then, as he called her name, she ran to him and dropped upon her knees at his side, and his arms went about her, insensible to their hurt, and her hot face was against his neck, and his lips crushed in the smothering sweetness of her hair. He made no effort to speak beyond that first calling of her name. He could feel her heart throbbing against him, and her hands tightened at his shoulders, and at last she raised her glorious face so near that the breath of it was on his lips.   He could not remember all it was that he said, but at the end Marie-Anne had drawn a little away so that she was looking at him, her eyes shining gloriously and her cheeks beautiful as the petals of a wild rose. And he could see the throbbing in her white throat like the beating of a tiny heart. "'And you'll take me with you?' she whispered joyously. "'Yes. And when I show you to the old man,' Superintendent McVane, you know, and tell him you're my wife, he can't go back on his promise. He said, if I settled this Roger Audemard affair, I could have anything I might ask for. And I'll ask for my discharge. I ought to have it in September, and that will give us time to return before the snow flies. "'You see,' he held out his arms again.  Always. "'Are you a little glad, Marie-Anne?' In a great room at the end of the hall, with windows opening in three directions upon the wilderness, St. Pierre waited in his wheelchair, grunting uneasily now and then, at the long time it was taking Carmen to discover certain things out in the hall. Finally he heard her coming, tiptoeing very quietly from the direction of David Kerrigan's door, and St. Pierre chuckled and tried to rub his bandaged hands when she came in, her face, pink, and her eyes shining with the greatest thrill that can stir a feminine heart. "'If we had only known,' he tried to whisper, "'I would have had the keyhole made larger, Cherie. He deserves it for having spied on us at the cabin window. "'But tell me, could you see? Did you hear?' "'What?' Carmen's soft hand went over his mouth. "'In another moment you'll be shouting,' she warned. "'Maybe I didn't see, and maybe I didn't hear, Big Bear. But I know there are four very happy people in Chateau Boulin. And now, if you want to guess who is the happiest, "'I am Cher Kerrigan.' "'No.' "'Well, then, if you insist, you are.' "'Yes, and the next?' St. Pierre chuckled. "'David Kerrigan,' he said. "'No, no, no. If you mean that, I mean always that I am second, unless you will ever let me be first,' corrected St. Pierre, kissing the hand that was gently stroking his cheek. And then he leaned his great head back against her, where she stood behind him. And Carmen's fingers ran where his hair was crisp with the cinch of fire, and for a long time they said no other word, but let their eyes rest upon the dim length of the hall, at the far end of which was David Kerrigan's room.


Masayu Jiri has been the general manager of the Toronto Raptors since 2013 and that time they've been a perennial playoff power They've also had disappointing playoff exits Well that changed a season ago when they acquired Kawhi Leonard and they rode that wave to an NBA Championship now as his players were celebrating on the court in Oakland Masayu Jiri made his way down to the floor But he had an incident with a police officer Later that officer filed a lawsuit against you Jerry Claimed that Ujiri was the aggressor that he was the victim in the situation said that he suffered injuries to his head his Chin his jaw and his teeth and said that he will continue to suffer Mentally and emotionally as a result of the physical altercation in the incident now and basketball We tend to use the phrase ball don't lie well neither does body cam footage This was released this week footage of the actual incident back in June in Oakland Ujiri pulling out his credential there and he shoved immediately by the police officer who shouts obscenities at him and Then he shoved once again again Ujiri trying to make his way to the floor with his players Here's another look Ujiri there you see circled on the right Approaching the officer and again Shoved he's just trying to make his way to the floor and his face Shoved once again. Here's a statement now that the video has been released This is a statement from from Ujiri quote the video sadly demonstrates How horribly I was treated by a law enforcement officer last year and the midst of my team the Toronto Raptors winning its first world championship Yet unfortunately, I was reminded in that moment that there are some people who will always and only see me as something That is unworthy of respectful engagement and there's only one indisputable reason why that is the case because I am black The statement continues what saddens me most about this ordeal is that the only reason why I am getting the justice I deserve in this moment is because of my success So many of my brothers and sisters haven't had don't have and won't have the same access to resources That assured my justice and that's why black lives matter Nets governor Joe sigh tweeted this out yesterday on the court.  You have our respect Here's more reaction. A lot of just flat out Unkind behavior towards people when they don't really know the truth, right? And I I find it all I find it all very Disheartening and disappointing to be honest those heavy stuff man. It's heavy stuff Obviously, we're all privileged and my side is pretty privileged in his world and You just you just stop and think about, you know, how good we got it because there's There's people who are going to be in that same situation walking down the street who don't have money to fight the case shows while we're You know, we're supportive of the social injustices that are going on right now It shows all we're supporting of the black lives matter It showed why we need to get out there and vote. It's the worst part is that it wasn't unexpected. I'm not surprised at what I saw It's become norm and people in my community man This is what is expected. We expect this and that's that's one of the things that we're trying to get across to The white community in America, you have a black man in a suit is running an MBA franchise With security around him and he gets treated that way I walked on the court without anybody asking me for my credentials. What's the difference between me and the side? I'm white. He's black. This is why we kneel. That's why that phrase and that statement is so important Important and strong words there. That's why we say this is more than a moment. This is a movement This is why the conversation continues and the conversation about social justice and racial equality There on the MBA campus around the country and around the world


There's no denying this is a team led by rookie quarterback that was well coached and well put together last night in Detroit And when you have that play to start the game As ill-fated as it ended up being it shows you the confidence that they have In him and I'm worried anxious to talk to him. That'll be the first question out of box You know pick six to start a career But when you put that play together, you're saying we believe in this guy We don't think it's all gonna be about one play that if he throws a pick You know recover from it if he ends up hitting it You'll go for a score and we're gonna show everybody how aggressive we are in a year Michael where maybe you're right He's gonna have to win games to save his job Todd bowls that he's not gonna let a rookie quarterback Dumb his team down and be conservative. So I the first call didn't work But when you look at the way things went it tells me this is the quarterback They've wanted from the get-go going into last season. This is the quarterback. They want they got the quarterback and They're gonna play with them.  I'm sorry Michael one thing I just want to say early props to because we're gonna talk about Sam Darnell all day long And and Todd bowls. I gotta give some props to Quincy a noon. We're not only played a great game But if he doesn't hold on to that catch on the second drive on that big third down that led to them getting into the end zone Instead, maybe they settle for a field goal there. That was a moment where right away It was a great play by Darnell to he got himself out of trouble scrambles finds a noon Almost drops it holds on to it. They review it. They decide it was a catch. By the way, separate conversation everything's a catch in 2018 but That early you know early on that's something that gives Darnell a bit of confidence to get things going and also the pick six I kind of put that on the coaching staff because if you watched halftime yesterday Steve Young said in pee we I was taught you don't roll right and throw left You just don't and that's exactly what I called for because there's too much time for the defense to correct Itself and they almost they almost like said all right throw it throw it throw it.  I don't blame that on Darnell. I think that was a bad play call I was supposed to go he wasn't supposed to go there with it Was that that was where the place if you looked at the overhead if he if he's able to hit the receiver It goes for big yards. It's just that he's a young kid He panicked he should have just thrown her away But did he scramble too far right or something like do you roll too far right? He just threw a duck It was just a bad play But what I like about it I maybe I feel this way because they ended up winning the game 48 to 17 But he just tells me that they're not going to treat this kid with with kid gloves that they believe in their guy And all right what one play doesn't work We got 59 minutes to get it right and then you know what with a young quarterback again Youngest quarterback in the history of the Super Bowl here in an opening game The play I thought was to hand the ball off just to let him get his feet wet But they they called this funky play where he rolls out and throws across the field So they do a belief in this kid and then after that pick six he was 16 of 20. He did everything right He's he's he's solid He's that sort of guy that has a great head on his shoulders that doesn't get nervous that was almost like it seems like he's been Bread for this.  Of course everyone talks about Brett farf who threw a pick against the Redskins in his first appearance ever in Atlanta This was Winston and so do Winston. It's so like, you know, when does this happen? Don it must have been five minutes five minutes later. I watch him. He's made a few completions. It's gone You knew he didn't remember anymore and in fact, he settled in and was like, oh, I this is the speed of this game I'm not saying I mastered it, but I got this He seemed very comfortable and the Jets had to know that going in that we've got a guy that's gonna make his mistakes He's a rookie. There's gonna be a defense. He's not gonna be able to figure out But the one thing is he can get punched in the mouth and take it and that's the best thing because you're gonna make mistakes You're gonna have days. Hey, listen, I don't think the Lions are a very good team But Stafford's a good quarterback. He threw four interceptions. I mean, there's gonna be a defense Rathausberger looked awful on Sunday right against Cleveland, you know three turnovers But how do you recover from those turnovers? Do you put your tail between your legs and say, okay? It's not my day or do you fight through it? And what do you go three four four four on third down and he's converting and If the Jets knew that then that play makes a little bit more sense then right they had exactly two play calls I hated the third one is the other one The reverse third and one we have asked balls about that was that that play was nonsense and then the opener Well, I appreciate the confidence they had in him coming out of the gate and throwing the ball across the field in the NFL Well, you don't quite yet understand. I don't mind a pass at that point But that's kind of a fun play to call I'd rather air it out all the way down throw a bomb to Robbie Anderson rather than do what they did But besides that a well-called game by the Jets. Well, what that tells you is is that if bowls is coaching for his job Human nature is gonna be to be conservative. Let's just win the game But hey, he's he's being aggressive. He's not he's not worried about mistakes He believes he's got a guy that's gonna recover from it. So again, it's just one game But it's a hey month some about Monday night nationally televised games Just as a fan me of the Giants you the red. Oh, they're disaster. You just but you remember. Oh, yeah I mean, this is not like some Sunday in week 12. Oh, remember when we beat the Lions 4817 Andrew It's opening night on Monday the whole world shot Andrew Do you remember when the Eagles threw up a thousand spot on the Redskins in five minutes? Let me guess you remember it Well Like you don't forget those things like when it's a huge moment for this team


Okay, good morning everybody. So today I will continue speaking about Quasi crystals. And there are two things I'd like to do. The first thing is to prove this theorem about the density of a Quasi crystal. And the Quasi crystal here always means the Quasi crystal of cat and project type such a set which Jens described yesterday. So we want to prove how many points are there in such a Quasi crystal in a large region. And that proof will also allow me to illustrate the technique which is very important in applications of dynamics in number theory. So there are a number of applications where say Rattner's theorem or other equidistribution theorems in homogeneous dynamics have been applied to counting integer solutions to some Diophantine equation or solving Diophantine problems. So count how many integer points are there near some manifold or some variety in RM. And how, so it's a basic question. How can such a question be addressed using equidistribution of some orbit? Well the trick is you kind of thicken the integer points a little bit and then you are able to count such solutions. And the second thing I wanted to do is to discuss visible points in the lattice. So this is what Jens talked about at the end of the talk. So we'll come to that. But, so I will remind you first of some definitions. And I hope you have Jens and the notes from yesterday in front of you. So the picture, the first, I will try to draw some pictures as well. But, so we are going to define this catam project set in RD. And in order to do so, we extend RD by some internal dimensions. So we consider the product space, Rn equals RD times RM. And then we introduced the notations for these orthogonal projections. So we call pi the projection from Rn to the physical space RD. And we call pi interior, the projection from Rn to the interior space of the, what is it called? Model space RM. Okay, and then in Rn, we fix a lattice L. And before defining a quasi-crystal, we made some assumptions. We assumed that the, if I look at the projection of all the lattice points to the model space RM, or the interior space RM, we assume that, okay, this is always some, if I take the closure of this, it's always some close, stability and subgroup of RM. We called this closure A. But today, just for simplicity of notation, we'll assume that actually this is equal to all of RM. There are no extra difficulties. The fact that we are, this is a standard assumption to assume that we are dense in all of RM here. And the fact that in Jens and my work, we want to have a slightly more general setting, that is just for convenience. It allows us to cover a few more examples and so on. So just to refer back to Jens talk yesterday, this means that A and also the connected component of A, containing the origin, both of these are equal to RM and also the V, which was defined as the product of RD and RM. That is now equal to all of RM. So that you can refer back to this. And now we fix also a regular window set. So we'll let W be a subset of RM. In general, it's a subset of A. We'll let this be a regular window set. And recall, this means that the boundary of W has finite Lebesgue measure. And it has non-empty interior and it is a bounded set. So this means that W is bounded. It has non-empty interior and the Lebesgue measure of the boundary of W is zero. So W is a Jordan measurable set if you want. And then finally, we define, maybe I'll do it here. The quasi-crystal as the set of all points that you get, you run through all the points Y in the lattice L such that the interior project, projection of Y is in the window. So we run through all the lattice points which project into the window and we consider the projection of these two physical space. That's the quasi-crystal. And this is our fixed set P. And there wasn't one extra assumption. We assume also that any point in here, any point in this quasi-crystal only has a single representative in the lattice. So there is a certain injectivity. So we also assume that for any point in X, this was, yeah, this also Jens did this yesterday. There are many ways of writing it. There is a single point in the lattice. So one way of writing it would be like this, that if I consider all points that project down to X and which lie in the window set in the model space, there is only a single lattice point in here. So there's exactly one point that gives this. Okay. And in this general situation, we have the point that P always has an asymptotic density. So that is the theorem by Hoff and Schlotman. So this was, Jens had this yesterday also, wrote this down. They proved it independently. So no, we could allow non-connected W. So the formula, which Jens wrote out was for the number of points. I'm going to write something here, but if you take, count the number of points in some ball. So a ball of radius T and then this is, and divide by the volume of the ball, then this tends to some number. No, limit is equal to some number, theta. But I will state it slightly more generally now. So instead of taking just an expanding ball, let's take a somewhat more general expanding set. So replace this by T to the T times some fixed set D. And here, we take the volume of this. And here, this actually holds for an arbitrary. So let D be a fixed bounded subset of our D with a boundary of measures zero. So D is Jordan measurable. And for this to make sense, we have to assume also that, well, I guess that D has positive volume or equivalently D has non-empty interior. And then the statement is that this holds. And furthermore, we can tell what the asymptotic density is. We can tell what theta P is in terms of the lattice and the window set. So this holds with theta equal to the volume of the window set divided by the volume of, or the co-volume of the lattice. When I will have to do this, I will write volume for Lebesgue measure, but it depends on the context which dimension we are in. So here, it's m-dimensional Lebesgue measure. And here, it is n-dimensional Lebesgue measure. And in order to connect with the statement which Jens had yesterday, just note that in general, this would be the volume, we introduced the Lebesgue measure on the set V. And we had V divided by L intersected with V. But as Jens pointed out, if A is equal to RM as we are assuming, then we get just this. Okay, so now I wanted to outline the proof and see how it comes from Vile liquid distribution. Maybe I should have drawn a picture a bit earlier, perhaps. So here's the situation we have. It's hard to draw high-dimensional pictures, but you can imagine that the whole space, RD, lies along this. And here is the whole space, RM. And then we have some lattice. It's only, I can only draw a two-dimensional lattice. No, larger, I think. And why not? I should do as Jens did, maybe. So this is the point set L. And then we have some window. And maybe, let me put, let's say that the window is here. Let's say that, I don't know what to do. What am I doing? I don't know, let's continue the lattice a bit. Okay, and let's assume that the window is maybe here. And then we are looking at all the lattice points that project to the window. So we are actually counting lattice points. I mean, every such lattice point gives rise to a point in P by projection down to RD. So we are actually, what this is counting is the number of lattice points in a long, but in a certain region which is long in some directions, I mean, which is wide in some directions, but can be arbitrarily thin in certain other directions. So this region is nicely expanded and big in RD, but in the RM it is as small as you choose. The window set could be really small here. So this is a standard kind of lattice counting problem. I mean, it's very classical to count the number of points in a big ball and then you are interested in precise error terms also. But it's like a slight variant. Now we are counting it in a region which is long and big in certain directions, but really thin in other directions. So if I try starting the proof, what we want to prove is here I rewrite it only slightly. We want to prove that the limit as T goes to infinity of the number of lattice points inside the window times TD. So in the other way, so TD times the window. This number of lattice points is asymptotic to the volume again of TD. So it's very close to this relation. The only thing is I'm counting lattice points instead of counting points in P. And of course, when I say that those two are equivalent, I am using this in the activity assumption that there are never two lattice points that give the same point in P. And now let's look at this guy here. What is this? I want to rewrite it slightly and focus instead on the question, let's walk through the lattice points and see when it's a lattice point registered. No, no, no, no. Sorry, I want to focus on this multi-dimensional flow which we are kind of in the RD component. We are moving very widely in every direction. So I want to focus on the point lying in TD and ask when is, so point lying along the RD component and ask when does such a point happen to register as a count here? So this is equal to, we have this normalizing factor and then we are counting the number of points X in TD, such that, and if you look at the condition for such a point X, we have to, the condition is that X times some point in the window should lie in L. So in other words, if I subtract the point in the window, it implies that X comma zero, now this is a point in RN, sorry for the confusion, that this lies in minus the window set, but the window set is in the, maybe minus zero times the window set plus L. And then when I write the pair like this, I mean this is the RD component, this is the RM component. So all I did was I, instead of, yeah, so for each lattice point, we are adding the minus the window set to it. So for each, so instead of kind of looking at the window, I'm looking at the minus the window, but I added that to each lattice point. So here is lattice point plus minus the window. Here is also minus the window. And here is also lattice point plus minus the window. Now that was exactly below. So minus the window. And here lattice point plus minus the window. So we have, to each lattice point, we have now added the window, the minus the window. And the question now is, as I walk along the RD, how many times do I happen to hit? Do I go through such a, such a window? So to say, this is what I'm counting. I'm walking around in RD and I want to count how many times do I hit this M dimensional set, which is kind of transversal to the flow. And I'm going out all the way through TD, whatever TD is, it goes out maybe here. TD, it's a large big set in RD. Okay, so now I want to see this as some kind of flow in the torus that I get by modding out by the lattice L. And I want to get hold of this limit by using the fact that this D dimensional flow equidistributes this D dimensional orbit equidistributes in the torus. And then I want to see this as an ergodic average. But here I'm counting something. So it's not really an ergodic average, an ergodic for something that you can get by the, as an equidistribution. There you want to count how many times do we hit some open set, for instance. And this is, so what we do is we thicken, we thicken these windows a little bit in the RD directions. So then we will be counting some volume instead. So let's thicken it by some amount, small amount eta in the RD directions. And here we have to be, we have to be careful to take eta so small that the different thickenings are this, otherwise disjoint, otherwise the count will not be exactly equal to the volume that we want. So we take eta small, so such, so that if I now take the eta ball in RD times the window set, or it should be minus the windows, ah, it can be, that doesn't matter, times the window set, and add the lattice point, that should always be disjoint from any other such neighborhood of a lattice point. So this, these two are always disjoint when m and m prime are different lattice points. So this is empty for all m not equal to m prime in the lattice. Okay, so we have this picture. I have thickened this into an open set. Now this is an open set in RN, and I thickened it by multiplying by ball of diameter two eta, if you like. So two eta each point here. And they will be disjoint by choice of eta. Yeah, now these regions are open in RN. These regions, they are open in RN. I mean, okay, window set is, you can think of window set as an open set. The boundary smash is zero. Sorry. Okay, but of course we won't get this count on the nose because there is this, whenever I'm near, if I happen to have some such window set very near the boundary of TD, then there can be a little bit of, the part of this region could be inside and part of it could be outside of TD. So I can't, can never get it on the nose. So all I can do is estimate it from above and below. So to handle that, I will do the estimation from above. We introduce a little thickening of D. So also set D eta plus, let this be equal to the eta thickening of D. So D plus B, the ball of radius eta. This is pointwise addition. So for every point in D I consider also it's eta neighborhood and I get the slightly larger set. This is the eta thickening of D. And note that if I assume that T is larger than one as I may, then if I take T times D, and take the eta thickening of that, this is a subset of T times the eta thickening. So it's convenient somehow to have a completely fixed set. I will use this fixed set and take it times T and that will overshoot what I need, so to say. So now finally we can get a good upper bound for this thing. I'm going very slowly, I feel, but it's morning and I may not be completely awake yet. I hope you're not too, don't find it too slow. Okay, so finally we conclude that. So I'm going to replace that count by a certain volume. And so I keep this factor, volume of TD. I save some space here for an extra correction factor and then I'm now going to count the volume inside the TD or rather TD eta. I'm going to count the volume of points which fall inside these neighborhood, these lattice neighborhoods. So that's what I'm going to count and I write it as an integral. So I go through all the points in TD eta thickened and I register, so this is indicator function giving one whenever the following holds. I register this point whenever we have, whenever that point falls inside this neighborhood of the lattice point. So whenever it falls inside this plus the lattice and volume. And clearly now for every lattice point that I register in this count, since I have thickened the D a little bit, I will get the full eta ball registered here. So I will overshoot if I also divide it by the volume of the eta ball, I get something that is larger than what I needed. So okay, I'm going through this very rather carefully but as I said, this is illustrating a standard technique going from an equidistribution of some orbits of a flow to counting points into the points. And that technique has been applied over and over again to get interesting number theoretical results. Okay, and now the final thing is to view this as something taking place on the torus that I get by taking Rn modulo n, the Rn modulo l. So let pi be the projection map from Rn to that torus. And then note that this, sorry, no. This is equivalent to saying that when I project, since I have this plus l, this set is actually a set living on the torus. And what this is saying is that when I project X comma zero to the torus, it falls inside the projection of this set. I can write it as just projection of minus B eta D times W. And this is an open set in the torus. I assume that eta is, that W is open so that I am allowed to say open here. I can assume so because boundary measures zero. Okay. So now this is really an ergodic average on the torus for this D dimensional flow. So I have a D dimensional flow going in all the directions of Rd. And so now we have a general equidistribution facts on the torus. So it turns out that this D dimensional flow is in such a direction that it is ergodic and hence uniquely ergodic. So all this business, maybe I should go through it more carefully, but there is no time then. So I just say this, that the assumption we started from that the projection of the lattice is equal to all of Rm. This implies that this is more or less immediate if you think about what it means that if I take the closure of that point-wise sum, it is equal to all of Rn. And if I interpret this on the torus, it says that if I take the Rd flow, so I write it like this, and I consider it on the torus, this is dense in the torus. So the closure is equal to all of the torus. And because I have such a denseness, the flow is also uniquely ergodic. So if I take any invariant limit measure, must be the Lebesgue volume, normalized Lebesgue volume measure here. So I don't say more, I just say now that now we have equidistribution. So this ergodic average tends to the volume of this divided by the volume of the full torus. So this, as T goes to infinity, tends to, and I have to move around, maybe I removed the statement, you have it in your notes anyway. So then I can write it here. It tends to, well, this is a factor that just remains one over the d-dimensional volume of this eta ball times. And so it's not really an ergodic average. It would be if I divide by the volume of this thing. So in order to compensate for that, I have to have this compensating factor, volume of d eta plus divided by volume of d. Really, that should be a T here. But that T cancels in the numerator cancels against the denominator. So, and then now I have volume of T d eta plus. And it's really an ergodic average. And by equidistribution, it tends to the volume of this set divided by the volume of the full torus. And it was maybe minus, but that doesn't matter. And now this volume here, I can remove pi because this projection is injective. Why? It's that is equivalent to the assumption I made here. So this is equal to the Lebesgue measure in Rn of this thing. And then this is a product of the volume of this ball which cancels against that thing and the m-dimensional volume of m. So I'm left with this adjusting factor times and this cancels, so it's the volume of that thing divided by the volume of the torus. And now I'm more or less done. So of course, this factor is slightly larger than the one. But the thing is by choosing it as small, I can make it as small as I like because d is the order measurable. So what follows from this discussion? I had one inequality, so the conclusion is that the lim sup as t goes to infinity of what I'm interested in is less than or equal to the thing up here, this thing. But now I let eta go to zero in this statement and this thing goes to one because d is Jordan measurable. So now let eta go to zero and then we conclude what we want. We conclude that the lim sup of what we are interested in is less than or equal to the stated limit. This is exactly the stated theta if you look back into the statement of the theory. So lim sup is good. And then to treat the lim in, of course you do the thinning instead. In order to get an equality from the other direction then you take the eta thinning of d which would be the set of points which have a full eta neighborhood inside d. But again we will have such a fact that the volume of that thinning can be made arbitrarily close to the volume of d. And we will get again that the lim in is also okay. So I spent really a long time on this. I hope it can be helpful for at least some of you. I'm happy to discuss if there are some technicalities here that you want to. In almost the whole argument eta is fixed. And I keep eta fixed all the time until I have reached this conclusion that the lim sup at t goes to infinity. Ah, sorry. Ah, the fact that the projection is injective. There is a slight thing that I wanted to mention and it could be that the lattice is so aligned that actually there are several lattice points along zero times rm. In that case I would first, and then if the window is large I'm dead already for eta equals one. So in that case I have to first partition the window into smaller parts. Ah, but I have prejudiced this somehow. Ah, but this is a statement really in taking a fundamental domain for the tourists and looking at it. This can be done. Maybe we can discuss... Yeah, for a generic lattice there will be points arbitrarily close to rm. But this is really a statement on the tourists so it's no problem and we can discuss the technicalities. I guess I can't say anything that is really the correct thing to say but it is no problem. It's something that... Okay, this will make sense. To verify this it suffices to take m' equals the origin. If it's also for m' equals the origin and all other m's then I'm done because of periodicity. And then it's easy to... There are only a finite number of lattice points that are at all relevant somehow. You can restrict to a finite number of lattice points. I hope that makes sense. Okay, so now in the next 20 minutes I wanted to speak about visible points in p. So, remember that... I think Jens did this yesterday. We are standing at the origin and we are looking out at our points at p. And then clearly we will not see all the points because some... It may happen that we won't see all the points in p because some points may be completely shadowed. I mean hidden. They are along the same ray from the origin. And we call the points that we can see. We call them visible. Of course, if the lattice is in really generic position then I get some points at p. Then I won't have any difference between visible points and other points. It won't happen. It's a non-generic phenomenon that a point is hidden from the origin by some other points. But this happens very much for some of the classical examples of quasi-crystals. So, for instance, in the penrose tiling if you take the vertex set of the penrose tiling as Jens described yesterday then there are invisible points. So there is a real distinction between visible points and the full set. And the same holds for many other of the quasi-crystals that you get from this number field construction, which Jens did yesterday. So we define p hat to be the set of visible points. So it's a set of all points in y such that there is no point ty with t less between 0 and 1, hitting it. So it should be true that ty is never in p for all t between 0 and 1. This is the subset of visible points. I should... And I note that the origin is never in p even if the origin is in... It's never in p hat even if it is in p even if it may be in p. Okay. So now it's clearly an interesting question to ask what is the density of p hat? Can we get hold of it? So... And there's a recent paper by Bakke, Götze, Huck and Czakobi where they investigate this and they investigate also the statistics of directions the thing that Jens spoke a few... Wednesday I think. There are several such sets. p and the corresponding set of visible points. So I should have looked if this has now appeared. I'm going to refer to Archive 2014 but has it appeared, do you know? Okay. Anyway, you may look if you want to see pictures and some statistics and so on. So... Okay. Okay. So you will find it. It has appeared. Okay. But anyway, I don't have that much time but let me state a few theorems and try to give the main idea behind them. So you can actually prove that also this point set p hat always has an asymptotic density. And this is... Yeah, it's a paper by Jens and me from last year. So the statement is this. There exists... I mean p is always fixed and so also p hat is fixed. Now the statement is that there exists some kappa in the... it's strictly positive and it's less than or equal to 1 such that... again for any d... for any Jordan set d. So d is bounded subset of rd volume of the boundary is equal to zero. We have limit as t goes to infinity of the number of points in p hat intersected with td divided by the volume of td. This is equal to the original density but multiplied by kappa. So just as an example let me point out if p is zd. Okay, I should have said this earlier to illustrate just what we mean by visible points. Then p hat is the set of primitive lattice vectors. We often write z hat d for that set and it's the set of all integer vectors such that the greatest common divisor of all the coordinates is equal to 1. And in this case it is a well-known fact that the density of z hat d is equal to 1 over the Riemann zeta function. So then in this case the density of p is clearly 1 and the density of p hat or zd hat is equal to... well it's the product of all prime numbers so p runs through all prime numbers and then 1 minus 1 over p to the d. So this is 1 over zeta function of d. And the intuitive reason for this is kind of clear. For each prime you have to remove any point M which where all the coordinates is divisible by p and the density that you are then removing is 1 over p to the d. So you keep 1 minus 1 over p to the d and you do this for each prime and it's kind of independent conditions so this is what you should get. And it's easy to make this rigorous by doing counting and great Chinese remainder theory. Here we don't have really an explicit formula for kappa of p but as you will see we have somehow we have a... some... it should... there is one should be able to maybe say something about it. Maybe we won't do so but... okay. So the fun thing about this theorem is that it actually uses the Siegel-Wietz result and it uses this space of Cauchy crystals that Jens spoke about yesterday. So although this is a question only discussing the fixed points at p when we prove it we need to introduce this larger space of all Cauchy crystals associated to p. So I have to remind you of this space of Cauchy crystals associated to p. So again this refers back to Jens lecture yesterday I will be a little bit brief now but... so... we had this lattice L and we take some G in SL... in G tilde G tilde is equal to SLNR we take it so that it gives the lattice L so G is just the representative of the fixed lattice L the lattice L which is used to construct p. So such that L is equal to ZNG and in Jens lecture yesterday we also had the delta here but let me just assume that the dense that the co-volume of L is 1 we can do so after re-scaling so in like we are assuming delta equals 1 if you refer back to Jens lecture and then from this G using this G Jens defined an embedding into the homogenous space gamma tilde slash G tilde and then using Ratter Lake with this distribution or densities it follows that for any given such G there exists some closed connected subgroup HG I'm just... so it's a closed connected subgroup of G tilde and it has the property that it intersects gamma tilde in a lattice gamma tilde intersects with HG is a lattice in HG remember that gamma tilde is SLN Z so then we have a homogenous space it's HG modular this lattice so HG modular the lattice gamma tilde intersected with HG and this space can also be identified with just this subset of a certain subset of G tilde slash gamma tilde so this is a subset a nicely embedded sub manifold inside G tilde slash gamma tilde and we have with this space is also provided with a probability measure because this is a lattice there is a unique R measure on HG which gives an invariant probability measure on X and as the answer I call this mu G yeah and now having done all this we have our space of random quasi-crystals random or the space of quasi-crystals associated to the fixed quasi-crystal P so by a random quasi-crystal we mean the quasi-crystal obtained by using the window set W and then taking the lattice not Z and G as from start but we insert an H here so it's Z and HG and this is for the point H or if you like gamma tilde H random in X according to X provided with its invariant probability measure so this is our set of quasi-crystals and this is what if I take this point H at random this is what I mean by a random quasi-crystal just as yesterday we were speaking about random at random lattice now we have a natural notion of a random quasi-crystal in RD so this is also if you refer back to the more general framework with random point processes and so on this is the answer the XI so we will have so this is the XI that we get in the limit and for this XI what Jens outlined yesterday is a proof that XI T tends in distribution to XI remember this is the fixed point set P randomly rotated and then dilated dilated so just to remind you of the notation and now let me state the Siegel-Wietz formula for this so in my lecture yesterday I outlined how Siegel-Wietz general Siegel type formula should follow if you have certain conditions and if you collect various things that we have said today and yesterday you can check that really we have those conditions so we do obtain a precise Siegel-Wietz formula for this space of run of quasi-crystals so let me state it so this is the Siegel-Wietz formula for this special case for any L1 function on RD if I if I integrate over the space of quasi-crystals and really X now is a we can we have identified X with our set of quasi-crystals through this map if I take a point gamma till the H in X this corresponds to this quasi-crystal if I so I integrate over all quasi-crystals and then I add over all the points in that quasi-crystal so I M runs through all the point in the corresponding quasi-crystal this is the Siegel transform I write it directly here and I have to remove the origin as you remember yesterday and I add over all these points F and then I take the average or expected value of this sum and the answer is that this tends to just the volume integral of the Lebesgue integral of F times the appropriate number and this number is the density of P in this case okay we have this but now using exactly the same argument one can show that the same thing must hold if I instead of P here I use P hat I mean for each such for each quasi-crystal in my space of quasi-crystal I can consider the subset of visible points okay and let's do that so and there is some constant kappa P and it's something between 0 and 1 such that if I now integrate again I do the same integral I write it out again I could do it easily by just taking hat here I want to maybe have them both on the on the board to discuss so we add over all the points in same random quasi-crystal but now I take its subset of visible points and then I don't have to remove the origin because the origin is never there and I do the same thing I add over F and then average over the space and this is equal to kappa P theta P volume integral of okay so maybe to avoid so okay actually I'm not going to get time to say much more so really this is the same constant so I don't have to so in order to have the statement here I write it like this kappa P is the same constant as in theorem 2 but now I will just say in words try to give you some idea of what is happening the way we prove these two theorems is we first prove theorem 3 and theorem 3 can be proved I mean this part has already been proved if you trust everything but the second part the proof of the second part is just the same as I outlined yesterday morning because this point set can again be seen to be this random point set can be seen to be an SLDR invariant point set whenever you have a random point set if you take visible points if you apply hats to it you again get the SLDR invariant point set I mean the operation of removing all the invisible points that is invariant under the action of SLDR clearly so the argument goes through and if you remember there was some technicality that we needed an upper bound but that upper bound is already provided from here because the visible points is always a subset of the full set so there is no real problem the argument carries through and we get this so the conclusion is that if you are now trying to outline the proof of theorem 2 and theorem 3 very quickly we first prove theorem 3 and the conclusion is that there exists some constant kappa p such that this holds and then it requires quite bit more work to prove theorem 2 not that those two theorems are quite different statements theorem 3 says that if I take a fixed set I mean if I take instead of a function I take a Borel subset of rd and then I take f to be its characteristic function then this statement in theorem 3 says that for a fixed set if I run through the whole space of cosy crystals the expected number of points in the set is equal to this this kind of density whereas in theorem 2 we are instead looking just as one fixed cosy crystal we don't state anything about the family only for one fixed cosy crystal and we are looking at larger and larger subsets of that counting the number of visible points to different statements but the way we but still using the techniques of spherical averages so I would have enjoyed saying outlining the proof it's you can use the technique of spherical averages to if you go backwards so to say in what Jens showed you in his lecture this Wednesday you can still get the information from the Siegelwitz down to this individual set p but it is I think I end there


Hi, I'm Dr. Michael Fanning. I am the tactical sports medicine lead here at SCU's tactical sports medicine department. Today we'll be talking about kinesio taping. So the first tape job we're gonna do is of the knee. We're gonna do the right knee here. This is a standard kinesio taping for the knee and it's just gonna provide some proprioceptive feedback. So the patient's knee, at least, they feel some stability in there. Go ahead and strain that leg out for me.  And so we're adding just a little bit of tension on the tape as we go through the range of motion. A lot of runners and triathletes, you'll notice, will utilize that tape job. Usually they have a little patellofomore pain syndrome or what's called jumper's knee. And they like that compression and added just a smidgen of stability that can be helpful to help get them through the game and get them through that little bit of pain that they're currently with and pressure okay. I like to rub it, the glue is a little heat activated so we gotta apply some friction. Okay, last one we're gonna do is just a decompression strip right across here on the patellar tendon. We're gonna add just a little bit of tension and then apply it right across. It adds a little bit of stability to a knee. All right, good, how's that feel?