Chapter 4 In which some more first appearances are made on the stage of these adventures. Though the offices of Dombie and Sun were within the liberties of the City of London, and within hearing of bobells, when their clashing voices were not drowned by the uproar in the streets, yet were their hints of adventurous and romantic story to be observed in some of the adjacent objects. Gog and Magog held their state within ten minutes walk. The royal exchange was close at hand. The Bank of England, with its vaults of gold and silver down among the dead men underground, was their magnificent neighbour. Just around the corner stood the rich East India house, teeming with suggestions of precious stuffs and stones, tigers, elephants, howders, hookers, umbrellas, palm trees, palanquins, and gorgeous princes of a brown complexion sitting on carpets, with their slippers very much turned up at the toes. Anywhere in the immediate vicinity there might be seen pictures of ships speeding away full sail to all parts of the world, outfitting warehouses ready to pack off anybody anywhere, fully equipped in half an hour, and little timber midshipmen in obsolete naval uniforms, eternally employed outside the shop doors of nautical instrument makers in taking observations of the hackney carriages. So Master and Proprietor are one of these effigies, of that which might be called familiarly the woodnest, of that which thrust itself out above the pavement, right leg foremost, with the suavity the least endurable, and had the shoe-buckles and flapped waistcoat the least reconcilable to human reason, and bore at its right eye the most offensively disproportionate piece of machinery. So Master and Proprietor of that midshipman, and proud of him too, an elderly gentleman in a Welsh wig, had paid house rent, taxes, rates, and dues for more years than many a full-grown midshipman of flesh and blood has numbered in his life, and midshipmen who have attained a pretty green old age have not been wanting in the English Navy. The stock-in-trade of this old gentleman comprised chronometers, barometers, telescopes, compasses, charts, maps, sextants, quadrants, and specimens of every kind of instrument used in the working of a ship's course, or the keeping of a ship's reckoning, or the prosecuting of a ship's discoveries. Objects in brass and glass were in his drawers and on his shelves, which none but the initiated could have found the top of, or guessed the use of, or having once examined could have ever got back again into their mahogany nests without assistance. Everything was jammed into the tightest cases, fitted into the narrowest corners, fenced up behind the most impertinent cushions, and screwed into the acutest angles to prevent its philosophical composure from being disturbed by the rolling of the sea. Such extraordinary precautions were taken in every instance to save room and keep the thing compact, and so much practical navigation was fitted and cushioned and screwed into every box, whether the box was a mere slab, as some were, or something between a cocked hat and a starfish, as others were, and those quite mild and modest boxes as compared with others. That the shop itself, partaking of the general infection, seemed almost to become a snug, sea-going, ship-shaped concern, wanting only good sea-room in the event of an unexpected launch to work its way securely to any desert island in the world. Many minor incidents in the household life of the ship's instrument-maker, who was proud of his little midshipman, assisted and bore out this fancy. His acquaintance lined chiefly among the ship-chandlers and so forth. He had always plenty of the veritable ship's biscuit on his table. It was familiar with dried meats and tongues, possessing an extraordinary flavour of ropian. Pickles were produced upon it, in great wholesale jars, with dealer in all kinds of ship's provisions on the label. Spirits were set forth in case-bottles with no throats. Old prints of ships, with alphabetical references to their various mysteries, hung in frames upon the walls. The tartar frigate, under way, was on the plates. Outlandish shells, sea-weeds and mosses decorated the chimney-piece. The little wainscotted back-parler was lighted by a sky-light like a cavern. Here he lived too, in skipper-like state, all alone with his nephew Walter, a boy of fourteen, who looked quite enough like a midshipman to carry out the prevailing idea. But there it ended for Solomon Gills himself, more generally called Old Sol, was far from having a maritime appearance. To say nothing of his Welsh wig, which was as plain and stubborn a Welsh wig as ever was worn, and in which he looked like anything but a rover, he was a slow, quiet-spoken, thoughtful old fellow, with eyes as red as if they had been small suns looking at you through a fog, and a newly awakened manner, such as he might have acquired by having stared for three or four days successively through every optical instrument in his shop, and suddenly came back to the world again to find it green. The only change ever known in his outward man was from a complete suit of coffee-colour cut very square, and ornamented with glaring buttons, to the same suit of coffee-colour minus the inexpressibles which were then of the pale nankine. He wore a very precise shirt-frill, and carried a pair of first-rate spectacles on his forehead, and a tremendous chronometer in his phab. Rather than doubt which precious possession, he would have believed in their conspiracy against it on part of all the clocks and watches in the city, and even of the very sun itself. Such as he was, such he had been in the shop and parlor behind the little midshipmen for years upon years, going regularly aloft to bed every night in a howling garret remote from the lodgers, where, when gentlemen of England who lived below at ease had little or no idea of the state of the weather, it often blew great guns. It is half past five o'clock, and an autumn afternoon, when the reader and Solomon Gilles become acquainted. Solomon Gilles is in the act of seeing what time it is by the unimpeachable chronometer. The usual daily clearance has been making in the city for an hour or more, and the human tide is still rolling westward. The streets have thinned, as Mr. Gilles says, very much. It threatens to be wet tonight. All the weather-glasses in the shop are in low spirits, and the rain already shines upon the cocked hat of the wooden midshipmen. Where's Walter, I wonder, said Solomon Gilles, after he had carefully put up the chronometer again. Here's dinner, been ready half an hour, and no Walter. Turning round upon his stool behind the counter, Mr. Gilles looked out among the instruments in the window, to see if his nephew might be crossing the road. No. He was not among the bobbing umbrellas, and he certainly was not the newspaper boy in the oil-skin cap who was slowly working his way along the piece of brass outside, writing his name over Mr. Gilles' name with his forefinger. If I didn't know he was too fond of me to make a run of it, and go and enter himself aboard ship against my wishes, I should begin to be fidgety, said Mr. Gilles, tapping two or three weather-glasses with his knuckles. I really should. All in the downs, eh? Lots of moisture. Well, it's wanted, I believe, said Mr. Gilles, blowing the dust off the glass-top of a compass-case, that you don't point more direct and due to the back-parler, and the boy's inclination dars after all. And the parler couldn't bear straighter either, due north, not the twentieth part of a point either way. Hello, Uncle Sol! Hello, my boy! cried the instrument-maker, turning briskly round. What! you are here, are you? A cheerful-looking merry boy, fresh with running home in the rain, fair-faced, bright-eyed, and curly-haired. Well, Uncle, how have you got on without me all day? Is dinner ready? I'm so hungry. As to getting on, said Solomon, good-naturedly, it would be odd if I couldn't get on without a young dog like you a great deal better than with you. As to dinner being ready? It's been ready this half-hour, and waiting for you. As to being hungry? I am. Come along, then, Uncle! cried the boy. Hurrah for the admiral! Confound the admiral, and turned Solomon Gills. You mean the Lord Mayor. No, I don't! cried the boy. Hurrah for the admiral! Hurrah for the admiral! Forward! At this word of command, the Welsh wig and its wearer were born without resistance into the back parler, as at the head of a boarding-party of five hundred men. And Uncle Sol and his nephew were speedily engaged on a fried soul with a prospect of stake to follow. The Lord Mayor, Wally, said Solomon, forever. No more admirals. The Lord Mayor's your admiral. Oh, is he, though? said the boy, shaking his head. Why, the swordbearer's better than him. He draws his sword sometimes. And a pretty figure he cuts with it for his pains, returned the Uncle. Listen to me, Wally. Listen to me. Look on the mental shelf. Why? Who has cocked my silver mug up there on a nail? exclaimed the boy. I have, said his Uncle, no more mugs now. We must begin to drink out of glasses to-day, Walter. We are men of business. We belong to the city. We started in life this morning. Well, Uncle, said the boy, I'll drink out of anything you like, so long as I can drink to you. Here's to you, Uncle Sol, and hurrah for the Lord Mayor, interrupter the old man, for the Lord Mayor, sheriffs, common counsel, and livery, said the boy, long life to him. The Uncle nodded his head with great satisfaction. And now, he said, let's hear something about the Firm. Oh, there's not much to be told about the Firm, Uncle, said the boy, plying his knife and fork. It's a precious dark set of offices. And the room where I sit, there's a high fender and an iron safe, and some cards about ships that are going to sail, and an almanac, and some desks and stools and an ink bottle, and some books and some boxes and a lot of cobwebs. And in one of them, just over my head, a shriveled up blue bottle that looks as if it had hung there ever so long. Nothing else, said the Uncle. No, nothing except an old birdcage. I wonder how that ever came there. And a coal scuttle. No banker's books, or checkbooks, or bills, or such tokens of wealth rolling in from day to day, said old Sol, looking wistfully at his nephew out of the fog that always seemed to hang about him, and laying an unctuous emphasis upon the words. Oh, yes, plenty of that, I suppose, returned his nephew carelessly. But all that sort of thing is in Mr. Karker's room, or Mr. Morphin's, or Mr. Donby's. Has Mr. Donby been there today? inquired the Uncle. Oh, yes, in and out all day. He didn't take any notice of you, I suppose. Yes, he did. He walked up to my seat. I wish he wasn't so solemn and stiff, Uncle, and said, Oh, you are the son of Mr. Gill's, the ship's instrument maker. Nephew, sir, I said. I said, nephew boy, said he. But I could take my oath, he said, son, Uncle. Your mistake, and I dare say. It's no matter. No, it's no matter, but he needn't have been so sharp, I thought. There was no harm in it, though he did say, son. Then he told me that you had spoken to him about me, and that he had found me employment in the house accordingly, and that I was expected to be attentive and punctual, and then he went away. I thought he didn't seem to like me much. You mean, I suppose, observed the instrument maker, that you didn't seem to like him much. Well, Uncle, returned the boy, laughing, perhaps so. I never thought of that. Solomon looked a little graver, as he finished his dinner, and glanced from time to time at the boy's bright face. When dinner was done, and the cloth was cleared away, the entertainment had been brought from a neighbouring eating-house. He lighted a candle, and went down below into a little cellar, while his nephew, standing on the mouldy staircase, dutifully held the light. After a moment's groping here and there, he presently returned with a very ancient-looking bottle, covered with dust and dirt. Why, Uncle Sol, said the boy, what are you about? That's the wonderful Madeira. There's only one more bottle. Uncle Sol nodded his head, implying that he knew very well what he was about, and having drawn the cork in solemn silence, filled two glasses, and set the bottle and a third clean glass on the table. You shall drink the other bottle, Wally, he said, when you come to good fortune, when you are a thriving, respected, happy man. When the start in life you have made today shall have brought you, as I pray heaven it may, to a smooth part of the course you have to run, my child. My love to you. Some of the fog that hung about, old Sol, seemed to have got into his throat, for he spoke huskily. His hand shook, too, as he clinked his glass against his nephews. But having once got the wine to his lips, he tossed it off like a man, and smacked them afterwards. Dear uncle, said the boy, affecting to make light of it, while the tears stood in his eyes, for the honour you have done me, et cetera, et cetera, I shall now beg to propose Mr. Solomon Gill's, with three times three, and one cheer more, hurrah! And you'll return thanks, uncle, when we drink the last bottle together, won't you? They clinked their glasses again, and Walter, who was hoarding his wine, took a sip of it, and held the glass up to his eye, with as critical an air as he could possibly assume. His uncle sat looking at him for some time in silence. When their eyes had last met, he began at once to pursue the theme that had occupied his thoughts aloud, as if he had been speaking all the time. You see, Walter, he said, in truth this business is merely a habit with me. I'm so accustomed to the habit that I could hardly live if I relinquished it. But there's nothing doing, nothing doing. When that uniform was worn, pointing out towards the little midshipman, then, indeed, fortunes were to be made and were made. But competition, competition, new invention, new invention, alteration, alteration, the world's gone past me. I hardly know where I am myself, much less where my customers are. Never mind, airman-uncle. Once you came home from weekly boarding school at Peckham, for instance, and that's ten days, said Solomon, I don't remember more than one person that has come into the shop. Do, uncle, don't you recollect? There was a man who came to ask for change for a sovereign. That's the one, said Solomon. Why, uncle, don't you call the woman anybody who came to ask the way to Myland Turnpike? Oh, it's true, said Solomon. I forgot her two persons. To be sure they didn't buy anything, cried the boy. No, they didn't buy anything, said Solomon quietly. No, want anything? cried the boy. No, if they had, they'd have gone to another shop, said Solomon, in the same tone. But there were two of them, uncle, cried the boy, as if that were a great triumph. You said only one. Well, Wally, resumed the old man after a short pause. Not being like the savages who came on Robinson Crusoe's Island, we can't live on a man who asks for change for a sovereign and a woman who inquires the way to Myland Turnpike. As I said just now, the world has gone past me. I don't blame it, but I no longer understand it. Tradesmen are not the same as they used to be. Apprentices are not the same, business, commodities are not the same. Seven-eighths of my stock is old-fashioned. I am an old-fashioned man in an old-fashioned shop, in a street that is not the same as I remember it. I have fallen behind the time, and am too old to catch it again. Even the noise it makes, a long way ahead, confuses me. Walter was going to speak, but his uncle held up his hand. Therefore, Wally, therefore it is that I am anxious you should be early in the busy world and on the world's track. I am only the ghost of this business, its substance vanished long ago, and when I die its ghost will be laid. As is clearly no inheritance for you, then, I have thought at best to use for your advantage almost the only fragment of the old connection that stands by me through long habit. Some people suppose me to be wealthy. I wish for your sake they were right, but whatever I leave behind me or whatever I can give you. You in such a house as Donby's are in the road to use well and make the most of. Be diligent. Try to like it, my dear boy. Work for a steady independence, and be happy. I'll do everything I can, uncle, to deserve your affection. Indeed, I will," said the boy earnestly. I know it," said Solomon. I am sure of it. And he applied himself to a second glass of the old Madeira with increased relish. As to the sea, he pursued, that's well enough in fiction, Wally, but it won't do in fact. It won't do at all. It's natural enough that you should think about it, associating it with all these familiar things, but it won't do. It won't do. Solomon Gills rubbed his hands with an air of stealthy enjoyment, as he talked of the sea, though, and looked on the seafaring objects about him with inexpressible complacency. Think of this wine, for instance," said old Sol, which has been to the East Indies and back. I'm not able to say how often, and has been once round the world. Think of the pitch-dark nights, the roaring winds and rolling seas, the thunder, lightning, rain, hail, storm of all kinds," said the boy, to be sure," said Solomon, that this wine has passed through. Think what a straining and creaking of timbers and masts, what a whistling and howling of the gale through ropes and rigging.  Why, when the charming Sally went down in the Baltic Sea, in the dead of night, five and twenty minutes past twelve, when the captain's watch stopped in his pocket, he lying dead against the main mast, on the fourteenth of February 1749," cried Walter, with great animation. I, to be sure," cried old Saul, quite right, then there were five hundred casks of such wine-board, and all hands except the first mate, first lieutenant, two seamen and a lady in a leaky boat, going to work to stave the casks, got drunk and died drunk, singing Rul, Redania, when she settled and went down and ending with one awful scream in chorus. But when the George II drove ashore on the coast of Cornwall in a dismal gale, two hours before daybreak on the fourth of March 71, she had near two hundred horses aboard, and the horses breaking loose down below, early in the gale, and tearing to and fro and sampling each other to death, made such noises and set up such human cries, that the crew believing the ship to be full of devils, some of the best men losing heart and head, went overboard in despair, and only two were left alive at last to tell the tale. And when, said old Saul, when the polyphemus, private west into your trader, burdened three hundred and fifty tons, Captain John Brown of Depford, owners, wigs and co," cried the same, said Saul, when she took fire, four days sail, with a fair wind out of Jamaica harbour in the night, there were two brothers on board," interposed his nephew, speaking very fast and loud, and there not being room for both of them in the only boat that wasn't swamped, neither of them would consent to go, until the Elch took the younger by the waist and flung him in, and then the younger rising in the boat cried out, Dear Edward, think of your promised wife at home, I'm only a boy, no one waits at home for me, leap down into my place, and flang himself into the sea. The kindling eye and heightened colour of the boy, who had risen from his seat in the earnestness of what he said and felt, seemed to remind old Saul of something he had forgotten, or that his encircling mist had hitherto shut out. Instead of proceeding with any more anecdotes, as he had evidently intended, but a moment before, he gave a short, dry cough, and said, Well, suppose we change the subject. The truth was, that the simple-minded uncle in his secret attraction towards the marvellous and adventurous, of which he was, in some sort, a distant relation by his trade, had greatly encouraged the same attraction in the nephew, and that everything that had ever been put before the boy to deter him from a life of adventure, had had the usual unaccountable effect of sharpening his taste for it. This is invariable. It would seem as if there never was a book written, or a story told, expressly with the object of keeping boys on shore, which did not lure and charm them to the ocean as a matter of course. But in addition to the little party now made its appearance, in the shape of a gentleman in a wide suit of blue, with a hook instead of a hand attached to his right wrist, very bushy black eyebrows, and a thick stick in his left hand covered all over, like his nose, with knobs.  He was evidently the person for whom the spare wine-glass was intended, and evidently knew it, for having taken off his rough outer coat, and hung up on a particular peg behind the door, such a hard, glazed hat as a sympathetic person's head might ache at the sight of, and which left a red rim round his own forehead as if he had been wearing a tight basin, he brought a chair to where the clean glass was, and sat himself down behind it. He was usually addressed as captain, as visitor, and had been a pilot, or a skipper, or a privateersman, or all three perhaps, and was a very salt-looking man indeed. His face, remarkable for a brown solidity, brightened as he shook hands with uncle and nephew, but he seemed to be of a laconic disposition, and merely said, Oh, goes it! Oh, well, said Mr. Gills, pushing the bottle towards him. He took it up, and having surveyed and smelt it, said with extraordinary expression, Thee! Thee! returned the instrument-maker. Upon that, he whistled as he filled his glass, and seemed to think they were making holiday indeed. Waller! he said, arranging his hair which was thin with his hook, and then pointing it at the instrument-maker. Look at him! Love, honour, and obey! Overhaul your catechism till you find that passage, and when found, turn the leaf down. Success, my boy! He was so perfectly satisfied, both with his quotation and his reference to it, that he could not help repeating the words again in a low voice, and saying he had forgotten them these forty years. But I never wanted two or three words in my life that I didn't know where to lay my hands on him, Gills. He observed, it comes of not waste and language, as some do. The reflection perhaps reminded him that he had better, like young Norval's father, increase his store. At any rate he became silent, and remained so, until old Saul went out into the shop to light it up, when he turned to Walter and said without any introductory remark, I suppose he could make a clock, if he tried. I shouldn't wonder, Captain Cattle, returned the boy. And it would go, said Captain Cattle, making a species of serpent in the air with his hook. Lord, how that clock would go! For a moment or two he seemed quite lost at contemplating the pace of this ideal timepiece, and sat looking at the boy as if his face were the dial. But he's chock full of science, he observed, waving his hook towards the stock in trade. Looky here! Here's a collection of him. Earth, air, or water, is all one. Only say where you'll have it. Up in a balloon, there you are. Down in a bell, there you are. Do you want to put the North Star in a pair of scales and weigh it? He'll do it for you. It may be gathered from these remarks that Captain Cattle's reverence for the stock of instruments was profound, and that his philosophy knew little or no distinction between trading in it and inventing it. Arr! he said with a sigh. It's a fine thing to understand him, and yet it's a fine thing not to understand him. I hardly know which is best. It's so comfortable to sit here and feel that you might be weighed, measured, magnified, electrified, polarized, played the very devil with, and never know how. Nothing short of the wonderful Madeira, combined with the occasion which rendered it desirable to improve and expand Walter's mind, could have ever loosened his tongue to the extent of giving utterance to this prodigious oration. He seemed quite amazed himself at the manner in which he had opened up to view the sources of the taciturn delight he had had in eating Sunday dinners in that parlor for ten years. Becoming a sadder and wiser man, he mused and held his peace. Come! cried the subject of this admiration returning. Before you have your glass of grog, Ned, we must finish the bottle. Stand, boy! said Ned, filling his glass. Give the boy some more. No more, thank you, Uncle. Yes, yes, said Saul, a little more. We'll finish the bottle to the house, Ned, Walter's house. Why, it may be his house one of these days in part. Who knows? Sir Richard Whittington married his master's daughter. Turn again, Whittington, Lord Mayor of London, and when you are old you will never depart from it, interposed the captain. Waller overhauled the book, my lad. And although Mr. Dumbie hasn't a daughter, Saul began, Yes, yes, he has, Uncle, said the boy reddening and laughing. Has he? cried the old man. Indeed, I think he has, too. Oh, I know he has, said the boy. Some of them were talking about it in the office today, and they do say, Uncle and Captain Cuttle, lowering his voice, that he's taken a dislike to her and that she's left unnoticed among the servants, and that his mind's so set all the while upon having his son in the house that although he's only a baby now, he is going to have balances struck oftener than formerly, and the books kept closer than they used to be, and has even been seen, when he thought he wasn't, walking in the docks, looking at his ships and property and all that as if he was exulting like over what he and his son will possess together. That's what they say. Of course, I don't know. He knows all about her already, you see, said the instrument-maker. Nonsense, Uncle! cried the boy, still reddening and laughing, boylike. How can I help hearing what they tell me? The sun's a little in our way at present, I'm afraid, Ned, said the old man, humoring the joke. Very much, said the Captain. Nevertheless, we'll drink him, pursued Saul. So here's to Dombie and son. Oh, very well, Uncle! said the boy merrily, since you've introduced the mention of her and have connected me with her, and have said that I know all about her, I shall make bold to amend the toast. So here's to Dombie and son and daughter. For more information, or to volunteer, please visit LibriVox.org. Recorded by Mill Nicholson. Dombie and son, by Charles Dickens.  Every day, too, he was more and more ardently cherished by Miss Tox, whose devotion was so far appreciated by Mr. Dombie that he began to regard her as a woman of great natural good sense, whose feelings did her credit and deserved encouragement. He was so lavish of this condescension that he not only bowed to her in a particular manner on several occasions, but even entrusted such stately recognitions of her to his sister as pray tell your friend Louisa that she is very good, or mention to Miss Tox, Louisa, that I am obliged to her. Specialties which made a deep impression on the lady thus distinguished. Whether Miss Tox conceived that having been selected by the Fates to welcome the little Dombie before he was born in Kirby, Beard and Kirby's best mixed pins, it therefore naturally devolved upon her to greet him with all other forms of welcome in all other early stages of his existence, or whether her overflowing goodness induced her to volunteer into the domestic militia as a substitute in some sort for his deceased mama, or whether she was conscious of any other motives, are questions which in this stage of the firm's history herself only could have solved. Nor have they much bearing on the fact, of which there is no doubt, that Miss Tox's constancy and zeal were a heavy discouragement to Richards, who lost flesh hourly under her patronage, and was in some danger of being superintended to death. Miss Tox was often in the habit of assuring Mrs. Chick that nothing could exceed her interest in all connected with the development of that sweet child, and an observer of Miss Tox's proceedings might have inferred so much without declaratory confirmation. She would preside over the innocent repasts of the young heir with ineffable satisfaction, almost of an heir of joint proprietorship with Richards in the entertainment. At the little ceremonies of the bath and toilet, she assisted with enthusiasm. The administration of infantile doses of physics awakened all the active sympathy of her character, and being on one occasion secreted in a cupboard, whether she had fled in modesty, when Mr. Dombie was introduced into the nursery by his sister to behold his son in the course of preparation for bed, taking a short walk uphill over Richards' gown, in a short and airy linen jacket, Miss Tox was so transported beyond the ignorant present as to be unable to refrain from crying out, Is he not beautiful, Mr. Dombie? Is he not a cupid, sir? and then almost sinking behind the closet door with confusion and blushes. Louisa said Mr. Dombie one day to his sister, I really think I must present your friend with some little token on the occasion of Paul's christening. She has exerted herself so warmly in the child's behalf from the first and seems to understand her position so thoroughly, a very rare merit in this world, I'm sorry to say, that it would really be agreeable to me to notice her. Let it be no detraction from the merits of Miss Tox to hint that in Mr. Dombie's eyes, as in some others that occasionally see the light, they only achieved that mighty piece of knowledge, the understanding of their own position, who showed a fitting reverence for his. It was not so much their merit that they knew themselves as that they knew him and bowed low before him. My dear Paul, returned his sister, you do miss Tox for justice, as a man of your penetration was sure I knew to do. I believe if there are three words in the English language for which she has a respect amounting almost to veneration, those words are Dombie and son. Well, said Mr. Dombie, I believe it, it does Miss Tox credit. And as to anything in the shape of a token, my dear Paul, pursued his sister, all I can say is that anything you give Miss Tox will be hoarded and prized, I am sure, like a relic. But there is a way, my dear Paul, of showing your sense of Miss Tox's friendliness in a still more flattering and acceptable manner, if you should be so inclined. How is that? asked Mr. Dombie. Godfathers, of course, continued Mrs. Chick, are important in point of connection and influence. I don't know why they should be to my son, said Mr. Dombie coldly. Very true, my dear Paul, here Mrs. Chick faltered Mrs. Chick with an extraordinary show of animation to cover the suddenness of her conversion. And spoken like yourself, I might have expected nothing else from you. I might have known that such would have been your opinion. Perhaps here Mrs. Chick faltered again as not quite comfortably feeling her way. Perhaps that is reason why you might have the less objection to allowing Miss Tox to be godmother to the dear thing, but only as deputy and proxy for someone else. That it would be received as a great honour and distinction, Paul, I need not say. Louisa, said Mr. Dombie after a short pause. It is not to be supposed. Certainly not! cried Mrs. Chick hastening to anticipate a refusal. I never thought it was. Mr. Dombie looked at her impatiently. Don't flurry me, my dear Paul, said his sister. For that destroys me. I am far from strong. I have not been quite myself since poor dear Fanny departed. Mr. Dombie glanced at the pocket-hanker-chief which his sister applied to her eyes and resumed. It is not to be supposed, I say. And I say, murmured Mrs. Chick, that I never thought it was. Good heaven, Louisa, said Mr. Dombie. No, my dear Paul, she remonstrated with tearful dignity. I must really be allowed to speak. I am not so clever or so reasoning or so eloquent or so anything as you are. I know that very well, so much the worse for me. But if they were the last words I had to utter, and last words should be very solemn to you and me, Paul, after poor dear Fanny, I would still say I never thought it was. And what is more, added Mrs. Chick, with increased dignity, as if she had withheld her crushing argument until now, I never did think it was. Mr. Dombie walked to the window and back again. It is not to be supposed, Louisa, he said. Mrs. Chick had nailed her colours to the mast and repeated, I know it isn't. But he took no notice of it. But that there are many persons who, supposing that I recognised any claim at all in such a case, have a claim upon me superior to Miss Toxes. But I do not. I recognise no such thing. Paul and myself will be able, when the time comes, to hold our own. The house, in other words, will be able to hold its own and maintain its own and hand down its own of itself and without any such commonplace aids. The kind of foreign help which people usually seek for the children I can afford to despise, being above it, I hope. So that Paul's infancy and childhood pass away well, and I see him becoming qualified without waste of time for the career on which he is destined to enter. I am satisfied. He will make what powerful friends he pleases in afterlife, when he is actively maintaining and extending, if that is possible, the dignity and credit of the firm. Until then I am enough for him, perhaps, and all in all. I have no wish that people should step in between us. I would much rather show my sense of the obliging conduct of a deserving person like your friend. Therefore, let it be so, and your husband and myself will do well enough for the other sponsors, I dare say. In the course of these remarks, delivered with great majesty and grandeur, Mr. Dombay had truly revealed the secret feelings of his breast, an indescribable distrust of anybody stepping in between himself and his son, a haughty dread of having any rival or partner in the boy's respect and deference. A sharp misgiving recently acquired that he was not infallible in his power of bending and binding human wills, a sharp jealousy of any second check or cross. These were, at that time, the master-keys of his soul. In all his life he had never made a friend. His cold and distant nature had neither sought one nor found one. And now, when that nature concentrated its whole force so strongly on a partial scheme of parental interest and ambition, it seemed as if its icy current, instead of being released by this influence and running clear and free, had thawed for but an instant to admit its burden and then frozen with it into one unyielding block. Elevated thus to the god-mothership of little Paul, in virtue of her insignificance, Miss Tox was from that hour chosen and appointed to office. And Mr. Dombay further signified his pleasure that the ceremony, already long delayed, should take place without further postponement. His sister, who had been far from anticipating so signal her success, withdrew as soon as she could to communicate it to her best of friends, and Mr. Dombay was left alone in his library. He had already laid his hand upon the bell-rope to convey his usual summons to Richards when his eye fell upon a writing-desk, belonging to his deceased wife, which had been taken, among other things, from a cabinet in her chamber. It was not the first time that his eye had lighted on it. He carried the key in his pocket, and he brought it to his table and opened it now, having previously locked the room door with a well-accustomed hand. From beneath a leaf of torn and canceled scraps of paper, he took one letter that remained entire, involuntarily holding his breath as he opened this document, and, baiting in the stealthy action something of his arrogant demeanour, he sat down, resting his head upon one hand, and read it through. He read it slowly and attentively, and with a nice particularity to every syllable. Otherwise, and as his great deliberation seemed unnatural and perhaps the result of an effort equally great, he allowed no sign of emotion to escape him. When he had read it through, he folded and refolded it slowly several times, and tore it carefully into fragments. Checking his hand and the act of throwing these away, he put them in his pocket, as if unwilling to trust them even to the chances of being reunited and deciphered, and instead of ringing as usual for little Paul, he sat solitary all the evening in his cheerless room. There was anything but solitude in the nursery, for there Mrs. Chick and Miss Tox were enjoying a social evening, so much to the disgust of Miss Susan Nipper, that that young lady embraced every opportunity of making rye faces behind the door. Her feelings were so much excited on the occasion that she found it indispensable to afford them this relief, even without having the comfort of any audience or sympathy whatever. As the night errands of old relieved their minds by carving their mistress' names in the deserts, and wildernesses, and other savage places where there was no probability of there ever being anybody to read them, so did Miss Susan Nipper curl her snub-nose into drawers and wardrobes, put away winks of disparagement in cupboards, shed derise of squints into stone pictures, and contradict and call names out in the passage. The two interlopers, however, blissfully unconscious of the young lady's sentiments, saw little Paul safe through all the stages of undressing. Airy exercise, supper, and bed, and then sat down to tea before the fire. The two children now lay through the good offices of Polly in one room, and it was not until the ladies were established at their tea-table that, happening to look towards the little beds, they thought of Florence. "'How sound she sleeps,' said Miss Tox. "'Why, you know, my dear, she takes a great deal of exercise in the course of the day,' returned Mrs. Chick, playing about little Paul so much. "'She is a curious child,' said Miss Tox. "'My dear,' retorted Mrs. Chick in a low voice, "'her mama all over.' "'Indeed,' said Miss Tox. "'Oh, dear me!' A tone of most extraordinary compassion Miss Tox said it in, though she had no distinct idea why, except that it was expected of her. "'Florence will never, never, never be a donby,' said Mrs. Chick. "'Not if she lives to be a thousand years old.'  "'I quite fret and worry myself about her,' said Mrs. Chick, with a sigh of modest merit. "'I really don't see what is to become of her when she grows older, or what position she is to take. "'She don't gain on her papar in the least. "'How can one expect she should, when she is so very unlike, a donby?' Miss Tox looked as if she saw no way out of such a cogent argument as that, at all. "'And the child, you see,' said Mrs. Chick, in deeper confidence, "'has poor dear Fanny's nature. "'She'll never make an effort in afterlife,' I'll venture to say. "'Never. She'll never wind and twine herself about her papar's heart like—' "'Like the ivy,' suggested Mrs. Tox. "'Like the ivy,' Mrs. Chick assented. "'Never. She'll never glide and nestling to the bosom of her papar's affections like the startled fawn,' suggested Mrs. Tox. "'Like the startled fawn,' said Mrs. Chick. "'Never. Poor Fanny. Yet how I loved her.' "'You must not distress yourself of my dear,' said Mrs. Tox in a soothing voice. "'Now, really, you have too much feeling.' "'We have all our faults,' said Mrs. Chick, weeping and shaking her head. "'I dare say we have. I never was blind to hers. "'I never said I was, far from it. Yet how I loved her!' "'What a satisfaction it was to Mrs. Chick, "'a commonplace piece of folly enough, "'compared with whom her sister-in-law had been a very angel "'of womanly intelligence and gentleness, "'to patronize and be tender to the memory of that lady, "'in exact pursuance of her conduct to her in her lifetime, "'and to thoroughly believe herself and take herself in "'and make herself uncommonly comfortable "'on the strength of her toleration. "'What a mighty-pleasant virtue toleration should be "'when we are right, to be so very pleasant when we are wrong, "'and quite unable to demonstrate how we come to be invested "'with the privilege of exercising it. "'Mrs.  "'She had risen, as the nerfs said, "'and the lashes of her eyes were wet with tears. "'But no one saw them glistening, save Polly. "'No one else slant over her and whispered soothing words to her "'or was near enough to hear the flutter of her beating heart. "'Oh, dear nurse,' said the child, looking earnestly up in her face, "'let me lie by my brother.' "'Why, my pet,' said Richards. "'Oh, I think he loves me,' cried the child wildly. "'Let me lie by him. Pre-do.' Mrs. Chick interposed with some motherly words about going to sleep like a deer, but Florence repeated her supplication with a frightened look and in a voice broken by sobs and tears.   "'Oh, pray, pray, let me lie by my brother to-night, "'for I believe he's fond of me.' Richard took her without a word and carrying her to the little bed in which the infant was sleeping, laid her down by his side. She wrapped as near him as she could without disturbing his rest and stretching out one arm so that it timidly embraced his neck and hiding her face on the other over which her damp and scattered hair fell loose, lay motionless. "'Poor little thing,' said Miss Tox. "'She has been dreaming, I dare say.' "'Dreaming, perhaps, of loving tones forever silent, "'of loving eyes forever closed, "'of loving arms again wound round her "'and relaxing in that dream within the dam "'which no tongue can relate. "'Seeking, perhaps, in dreams "'some natural comfort for a heart deeply and sorely wounded, "'though so young a child's, "'and finding it, perhaps, in dreams "'if not in waking, cold, substantial truth.' "'This trivial incident "'had so interrupted the current of conversation "'that it was difficult of resumption, "'and Mrs. Chick, moreover, "'had been so affected by the contemplation "'of her own tolerant nature "'that she was not in spirits. "'The two friends accordingly "'soon made an end of their tea, "'and a servant was dispatched "'to fetch a hackney cabriolet for Miss Tox.' "'Miss Tox had great experience in hackney cabs, "'and her starting in one was generally a work of time, "'as she was systematic in the preparatory arrangements. "'Have the goodness, if you please, Dowlinson,' "'said Miss Tox, "'first of all, to carry out a pen and ink "'and take his number legibly.' "'Yes, Miss,' said Dowlinson, "'then, if you please, Dowlinson,' "'said Miss Tox, "'have the goodness to turn the cushion, "'which,' said Miss Tox, "'apart to Mrs. Chick, "'is generally damp, my dear.' "'Yes, Miss,' said Dowlinson, "'I trouble you also, if you please, Dowlinson,' "'said Miss Tox, "'with this card and this shilling. "'He's to drive to the card "'and is to understand "'that he will not on any account "'have more than the shilling.' "'No, Miss,' said Dowlinson, "'and I am sorry to give you so much trouble, Dowlinson,' "'said Miss Tox, looking at him pensively. "'Not at all, Miss,' said Dowlinson. "'Mention to the man then, if you please, Dowlinson,' "'said Miss Tox, "'that the lady's uncle is a magistrate, "'and that if he gives her any of his impertinence "'he will be punished terribly. "'You can pretend to say that, if you please, Dowlinson, "'in a friendly way, "'and because you know it was done to another man who died.' "'Certainly, Miss,' said Dowlinson. "'And now, good night to my sweet, sweet, sweet God-son,' "'said Miss Tox, "'with a soft shower of kisses "'at each repetition of the adjective. "'And, Louisa, my dear friend, "'promise me to take a little something warm "'before you go to bed, "'and not to distress yourself.' "'It was with extreme difficulty "'that Nipper, the black-eyed, "'who looked on steadfastly, "'contained herself at this crisis, "'and until the subsequent departure of Mrs. Chick. "'But the nursery being at length free of visitors, "'she made herself some recompense for her late restraint. "'You might keep me in a straight wiscuit for six weeks,' "'said Nipper, "'and when I got it off, "'I'd only be more aggravated. "'Whoever heard the like of them two griffins, Mrs. Richards?' "'And then a talk of having been dreaming, "'poor dear,' said Polly. "'How you beauties!' cried Susan Nipper, "'affecting to salute the door by which the ladies had departed. "'Never be donby, won't she? "'It's to be hoped she won't. "'We don't want any more such. "'One's enough.'" "'Don't wake the children, Susan, dear,' said Polly. "'I'm very much beholden to you, Mrs. Richards,' said Susan, "'who was not by any means discriminating in her wrath, "'and really feel it is an honour to receive your commands, "'being a black-slave and a mulotter, Mrs. Richards. "'If there's any other orders you can give me, "'prime mention them.'" "'Nonsense orders,' said Polly. "'I'll bless your heart, Mrs. Richards,' cried Susan. "'Temporaries always order's permanencies here. "'Did you know that?  "'And whenever, and however, which is best known yourself, "'you may bear in mind, please, "'that it is one thing to give orders "'and quite another to take them. "'You may tell a person to dive off a bridgehead "'for most into five and forty feet of water, Mrs. Richards, "'but a person may be very far from diving.'" "'There, now,' said Polly.  "'And yet you turn round on me because there's nobody else.' "'It's very easy for some to keep their tempers "'and be soft-spoken, Mrs. Richards,' returned Susan, slightly mollified, "'when their child's made as much of as a prince "'and is petted and petted till it wishes its friends further. "'But when a sweet young pretty innocent "'that never ought to have a crossword spoken to or of it "'is run down, the case is very different indeed.  Here, Miss Nipper made a horrible lowing as opposed to issue from a conscientious goblin of the wool species, impatient to discharge the severe duty of his position. Having further composed her young charge by covering her head with the bed-clothes and making three or four angry dabs at the pillow, she folded her arms and screwed up her mouth and sat looking at the fire for the rest of the evening. Though little poor was said, in nursery phrase, to take a deal of notice for his age, he took as little notice of all this preparations for his christening on the next day but one, which nevertheless went on about him as to his personal apparel and that of his sister and the two nurses with great activity. Neither did he, on the arrival of the appointed morning, show any sense of its importance, being, on the contrary, unusually inclined to sleep and unusually inclined to take it ill in his attendance that they dressed him to go out. It happened to be an iron-grey autumnal day, with the shrewd east wind blowing, a day in keeping with the proceedings. Mr. Dombie represented in himself the wind, the shade and the autumn of the christening. He stood in his library to receive the company as hard and cold as the weather, and when he looked out through the glass-room, at the trees in the little garden, their brown and yellow leaves came fluttering down as if he blighted them. Ugh! They were black cold-rooms and seemed to be in mourning like the inmates of the house. The books precisely matched as to size and drawn up in line like soldiers, looked in their cold hard slippery uniforms as if they had but one idea among them and that was a freezer. The bookcase, glazed and locked, repudiated all familiarities. Mr. Pitt, in bronze on the top, with no trace of his celestial origin about him, guarded the unattainable treasure like an enchanted moor. A dusty urn at each high corner, dug up from an ancient tomb, preached desolation and decay as from two pulpits, and the chimney-glass reflecting Mr. Dombie in his portrait at one blow seemed fraught with melancholy meditations. The stiff and stark fire-iron appeared to claim a nearer relationship than anything else there to Mr. Dombie, with his buttoned coat, his white cravat, his heavy gold watch-chain and his creaking boots. But this was before the arrival of Mr. and Mrs. Chick, his lawful relatives, who soon presented themselves. My dear Paul! Mrs. Chick murmured as she embraced him, the beginning, I hope, of many joyful days. Thank you, Louisa, said Mr. Dombie grimly. How do you do, Mr. John? How do you do, sir? Said Chick. He gave Mr. Dombie his hand, as if he feared it might electrify him. Mr. Dombie took it as if it were a fish or seaweed or some such clammy substance, and immediately returned it to him with exalted politeness. Perhaps, Louisa, said Mr. Dombie, slightly turning his head in his cravat as if it were a socket, you would have preferred a fire. Oh! My dear Paul, no! said Mrs. Chick, who had much ado to keep her teeth from chattering. Not for me! Mr. John, said Mr. Dombie, you are not sensible of any chill? Mr. John, who had already got both his hands in his pockets over the wrists, hisองed same canoin chorus which had given Mr. Chick so much offence on a former occasion, protested that he was perfectly comfortable. He added in a low voice, with my tiled, did tiled l'llllllllllllllllllllllllllllllllll. hen he was providentially stopped by Talonson, who announced, Miss Torx, and entered that fair and slavor with a blue nose and indescribably frosty face, referable to her being very thinly clad answering odds and ends, to do honour to the ceremony. "'How do you do, Miss Tox?' said Mr. Dombie. Miss Tox, in the midst of her spreading gauzes, went down altogether like an opera-glass, shutting up. She curtsied so low, in a gloragement of Mr. Dombie's advancing a step or two to meet her. "'I can never forget this occasion, sir,' said Miss Tox, softly. "'Tears impossible. My dear Louisa, I can hardly believe the evidence of my senses.' If Miss Tox could believe the evidence of one of her senses, it was a very cold day. That was quite clear. She took an early opportunity of promoting the circulation in the tip of her nose by secretly chafing it with her pocket-hanker-chief, lest, by its very low temperature, it should disagreeably astonish the baby when she came to kiss it. The baby soon appeared, carried in great glory by Richards, while Florence, in custody of that active young Constable Susan Nipper, brought up the rear. Though the whole nursery-party were dressed by this time in lighter mourning than at first, there was enough in the appearance of the bereaved children to make the day no brighter. The baby, too, it might have been, as Tox's nose, began to cry, thereby, as it happened, granting Mr. Chick from the awkward fulfilment of a very honest purpose he had, which was to make much of Florence. For this gentleman, insensible to the superior claims of a perfect Dombie, perhaps an account of having the honour to be united to a Dombie himself, and being familiar with the excellence, really liked her, and showed that he liked her, and was about to show it in his own way now, when Paul cried, and his helpmate stopped him short. Now Florence's child, said her aunt, briskly, what are you doing, love? Show yourself to him, engage his attention, my dear." The atmosphere became, or might have become, colder and colder, when Mr. Dombie stood frigidly watching his little daughter, who, clapping her hands and standing on tiptoe before the throne of his son and heir, lured him to bend down from his high estate and look at her. Some honest act of Richard's may have aided the effect, but he did look down and held his peace.  Was Mr. Dombie pleased to see this? He testified no pleasure by the relaxation of a nerve, but outward tokens of any kind of feelings were unusual with him. If any sunbeam stole into the room to light the children at their play, it never reached his face. He looked on so fixedly and coldly, that the warm light vanished even from the laughing eyes of little Florence, when at last they happened to meet his. It was a dull grey autumn day indeed, and in a minute's pause and silence that took place, the leaves fell sorrowfully. Mr. John, said Mr. Dombie, referring to his watch, and assuming his hat and gloves, Take my sister, if you please. My arm today is Miss Toxas. You'd better go first with Master Paul, Richard's. Be very careful. In Mr. Dombie's carriage, Dombie and Son, Miss Tox, Mrs. Chick, Richard's and Florence. In a little carriage following it, Susan Nipper and the owner, Mr. Chick. Susan looking out of window, without intermission, as a relief from the embarrassment of confronting the large face of that gentleman, and thinking whenever anything rattled that he was putting up in paper an appropriate pecuniary compliment for herself. As upon the road to church, Mr. Dombie clapped his hands for the amusement of his son. At which instance a parental enthusiasm, Miss Tox, was enchanted. But exclusive of this incident, the chief difference between the christening party and a party in a morning-coach consisted in the colours of the carriage and horses. Arrived at the church steps, they were received by a portentous beetle. Mr. Dombie dismounting first to help the ladies out, and standing near him at the church door, looked like another beetle. A beetle less gorgeous, but more dreadful. The beetle of private life. The beetle of our business and our bosoms. Miss Tox's hand trembled as she slipped it through Mr. Dombie's arm, and felt herself escorted up the steps, preceded by a cocktail and a Babylonian collar. It seemed for a moment like that other solemn institution. Will thou have this man, Lucretia? Yes, I will. Pleased to bring the child in quick out of the air there, whispered the beetle, holding open the inner door of the church. Little Paul might have asked, with Hamlet, into my grave, so chill and earthy was the place. The tall shrouded pulpit and reading desk, the dreary perspective of empty pews stretching away under the galleries, and empty benches mounting to the roof and lost in the shadow of the great grim organ, the dusty matting and cold stone slabs, the grisly free seats in the aisles, and the damp corner by the bellrope, where the black tressels used for funerals were stowed away, along with some shovels and baskets, and a coil or two of deadly-looking rope. The strange, unusual, uncomfortable smell and the cadaverous light were all in unison. It was a cold and dismal scene. There's a wedding just on, sir, said the beetle, but it'll be over directly if you'll walk into the westerie here. Before he turned again to lead the way, he gave Mr. Dombey a bow and a half-smile of recognition, importing that he, the beetle, remember to have had the pleasure of attending on him when he buried his wife, and hoped he had enjoyed himself since. The very wedding looked dismal as they passed in front of the altar. The bride was too old, and the bridegroom too young, and a superannuated bow with one eye and an eyeglass stuck in its blank companion was giving away the lady, while the friends were shivering.  One of a long series of similar volumes, gorged with burials. Over the fireplace was a ground-plan of the vaults underneath the church, and Mr. Chick, skimming the literary portion of it aloud by way of enlivening the company, read the reference to Mrs. Dombey's tomb in full before he could stop himself. After another cold interval, a wheezy little pew-opener afflicted with an asthma, appropriate to the churchyard if not to the church, and them to the font, a rigid marble basin which seemed to have been playing a churchyard game at cup and ball with its matter-of-fact pedestal, and to have been just at that moment caught on the top of it. Here they waited some little time while the marriage-party enrolled themselves, and meanwhile the wheezy little pew-opener, partly in consequence of her infirmity, and partly that the marriage-party might not forget her, went about the building coughing like a grampus. Only the clerk, the only cheerful-looking object there, and he was an undertaker, came up with a jug of warm water, and said something as he poured it into the font about taking the chill off, which millions of gallons boiling hot could not have done for the occasion. Then the clergyman, an amiable and mild-looking young curate, but obviously afraid of the baby, appeared like the principal character in a ghost's story, a tall figure all in white, at sight of whom Paul rented the air with his cries, and never left off again till he was taken out, black in the face. Even when that event had happened, to the greater relief of everybody, he was heard under the portico during the rest of the ceremony, now fainter, now louder, now hushed, now bursting forth again with an irrepressible sense of his wrongs. This so distracted the attention of the two ladies, that Mrs. Chick was constantly deploying into the centre aisle to send out messages by the pew opener, while Miss Tox kept her prayer book open at the gunpowder plot, and occasionally read responses from that service. During the whole of these proceedings, Mr. Dombie remained as impassive and gentlemanly as ever, and perhaps assisted in making it so cold that the young curate smoked at the mouth as he read. The only time that he unbent his visage in the least was when the clergyman, in delivering very unaffectedly and simply the closing exhortation relative to the future examination of the child by the sponsors, happened to rest his eye on Mr. Chick. And then Mr. Dombie might have been seen to express by a majestic look that he would like to catch him at it. It might have been well for Mr. Dombie if he had thought of his own dignity a little less, and had thought of the great origin and purpose of the ceremony in which he took so formal and so stiff apart a little more. His arrogance contrasted strangely with its history. When it was all over, he again gave his arm to Miss Tox, and conducted her to the vestry, where he informed the clergyman how much pleasure it would have given him to have solicited the honour of his company at dinner, but for the unfortunate state of his household affairs. The register signed, and the fees paid, and the pew opener, whose cough was very bad again, remembered, and the beetle gratified, and the sexton, who was accidentally on the doorsteps, looking with great interest at the weather, not forgotten. They got into the carriage again, and drove home in the same bleak fellowship. There they found Mr. Pitt turning up his nose at a cold collation, set forth in a cold pump of glass and silver, and looking more like a dead dinner lying in state than a social refreshment. On their arrival Miss Tox produced a mug for her godson, and Mr. Chick a knife and fork and spoon in a case. Mr. Dombie also produced a bracelet for Miss Tox, and on the receipt of this token Miss Tox was tenderly affected. Mr. John, said Mr. Dombie, will you take the bottom of the table, if you please? What have you got there, Mr. John? I've got a cold fillet of veal here, sir," replied Mr. Chick, wrapping his numbed hands hard together. What have you got there, sir? This, you turn, Mr. Dombie, is some cold preparation of carves' head, I think. I see cold fowls, ham, patties, salad, lobster. Miss Tox will do me the honour of taking some wine. Champagne to Miss Tox. There was a toothache in everything. The wine was so bitter-cold that it forced a little scream from Miss Tox, which she had great difficulty in turning into a hem. The veal had come from such an airy pantry, that the first taste of it had struck a sensation as of cold lead to Mr. Chick's extremities. Mr. Dombie alone remained unmoved. He might have been hung up for sale at a Russian fair as a specimen of a frozen gentleman. The prevailing influence was too much, even for his sister. She made no effort at flattery or small talk, and directed all her efforts to looking as warm as she could. Well, sir, said Mr. Chick, making a desperate plunge after a long silence, and filling a glass of sherry. I shall drink this, if you will allow me, sir, to little Paul. Bless him, murmured Miss Tox, taking a sip of wine. Dear little Dombie, murmured Mrs. Chick. Mr. John, said Mr. Dombie with severe gravity, my son would feel and express himself obliged to you, I have no doubt, if he could appreciate the favour you have done him. He will prove in time to come, I trust, equal to any responsibility that the obliging dispositions of his relations and friends, in private, or the onerous nature of our position in public, may impose upon him. The tone in which this was said, admitting of nothing more, is to Chick relapsed into low spirits and silence. Not so Miss Tox, who, having listened to Mr. Dombie with even a more emphatic attention than usual, and with a more expressive tendency of her head to one side, now lent across the table and said to Mrs. Chick softly, Louisa, my dear, said Mrs. Chick, onerous nature of our position in public, may I have forgotten the exact term. Expose him too, said Mrs. Chick. Pardon me, my dear, returned Miss Tox. I think not. It was more rounded and flueing. Obliging dispositions of relations and friends in private, or onerous nature of position in public, may impose upon him. Impose upon him, to be sure, said Mrs. Chick. Miss Tox struck her delicate hands together lightly in triumph, and added, casting up her eyes. Eloquence indeed. Mr. Dombie in the meanwhile had issued orders for the attendance of Richards, who now entered curtsying, but without the baby, poor being asleep after the fatigues of the morning. Mr. Dombie, having delivered a glass of wine to this vassal, addressed her in the following words. Miss Tox previously settling her head on one side and making other little arrangements for engraving them on her heart. During the six months, Sir Sir Richards, which have seen you an inmate of this house, you have done your duty. Desiring to connect some little service to you with this occasion, I considered how I could best affect that object. And I also advised, with my sister, Mrs. Chick, interposed the gentleman of that name. You hush, if you please, said Miss Tox. I was about to say to you, Richards, presumed Mr. Dombie with an appalling glance at Mr. John, that I was further assisted in my decision by the recollection of a conversation I held with your husband in this room on the occasion of your being hired, when he disclosed to me the melancholy fact that your family, himself at the head, were sunk and steeped in ignorance. This quailed under the magnificence of the reproof. I am far from being friendly, pursued Mr. Dombie, to what is called by persons of levelling sentiments general education, but it is necessary that the inferior classes should continue to be taught to know their position and to conduct themselves properly. So far I approve of schools. Having the power of nominating a child on the foundation of an ancient establishment called, from a worshipful company, the Charitable Grinders, where not only is a wholesome education bestowed upon the scholars, but where a dress and badge is likewise provided for them, I have, first communicating through Mrs. Chick with your family, nominated your eldest son to an existing vacancy, and he has this day I am informed, assumed the habit. The number of her son, I believe, said Mr. Dombie, turning to his sister and speaking of the child as if he were a hackney coach, is one hundred and forty-seven. Louisa, you can tell her. One hundred and forty-seven, said Mrs. Chick. The dress, Richards, is a nice warm, blue-blazed tailed coat and cap, turned up with orange-coloured binding, red-woisted stockings, and very strong leather small clothes. One might wear the articles oneself, said Mrs. Chick with enthusiasm, and be grateful. There, Richards, said Miss Tox, now indeed you may be proud, the Charitable Grinders. I am sure I am very much obliged, sir, returned Richards faintly, and taken very kind that you should remember my little ones. At the same time a vision of Byla was a Charitable Grinder, with his very small legs encased in the serviceable clothing described by Mrs. Chick, swam before Richards' eyes and made them water. I am very glad to see you have so much feeling, Richards, said Miss Tox. It makes one almost hope it really does, said Mrs. Chick, who prided herself on taking trustful views of human nature, that there may yet be some faint spark of gratitude and bright feeling in the world. Richards deferred to these compliments by curtsying and murmuring her thanks. But finding it quite impossible to recover her spirits from the disorder into which they had been thrown by the image of her son in his precocious nether garments, she gradually approached the door, and was heartily relieved to escape by it. Such temporary indications of a partial thaw that had appeared with her, vanished with her, and the frost set in again, as cold and hard as ever. Mr. Chick was twice heard to hum a tune at the bottom of the table, but on both occasions it was a fragment of the dead march in Saul. The party seemed to get colder and colder, and to be gradually resolving itself into a congealed and solid state, like the collation round which it was assembled. At length Mrs. Chick looked at Miss Tox, and Miss Tox returned the look, and they both rose and said it was really time to go. Mr. Domby, receiving this announcement with perfect equanimity, they took leave of that gentleman, and presently departed under the protection of Mr. Chick, who, when they had turned their backs upon the house, and left its master in his usual solitary state, put his hands in his pockets, threw himself back in the carriage, and whistled. With a hey-ho chevy, all through, conveying into his face, as he did so, an expression of such gloomy and terrible defiance, that Mrs. Chick dared not protest, or in any way, molest him. Richards, though she had little Paul on her lap, could not forget her own first-born. She felt it was ungrateful, but the influence of the day fell even on the charitable grinders, and she could hardly help regarding his pewter badge, number 147, as somehow a part of its formality and sternness. She spoke, too, in the nursery of his blessed legs, and was again troubled by his spectre in uniform. "'I don't know what I wouldn't give,' said Polly, to see the poor little dear, before he gets used to them. "'Why, then, I'll tell you what, Mrs. Richards,' retorted Nipper, who had been admitted to her confidence. "'See him, and make your mind easy.'" "'Mr. Donby wouldn't like it,' said Polly. "'Uh, wouldn't he, Mrs. Richards?' retorted Nipper. "'He'd like it very much, I think, when he was asked.' "'You wouldn't ask him, I suppose, at all,' said Polly. "'No, Mrs. Richards, quite contrary,' returned Susan, and them two inspectors, Tox and Chick, not intended to be on duty to-morrow, as I heard them say, me and Miss Floyd will go along with you to-morrow morning, and welcome, Mrs. Richards, if you like, for we may as well walk there as up and down a street and better, too.' Polly rejected the idea pretty stoutly at first. But by little and little she began to entertain it, as she entertained more and more distinctly the forbidden pictures of her children and her own home. At length, arguing that there could be no great harm in calling for a moment at the door, she yielded to the Nipper proposition. The matter being settled thus, little Paul began to cry most piteously, as if he had a foreboding that no good would come of it. "'What's the matter with the child?' asked Susan. "'East cold, I think,' said Polly, walking with him to-and-fro and hushing him. It was a bleak or tumble afternoon indeed, and as she walked and hushed, and glancing through the dreary windows, the little fellow closer to her breast, the withered leaves came showering down. End of chapter 5


Hello, welcome to NPTEL NOC, an introductory course on point set topology part 2, module 3. So, today we shall begin with some preliminaries required from one variable real analysis. I can say that it is one variable calculus, but in calculus courses we do not go this deeper. So, I take this opportunity to do this one. I will explain why this is needed a little later. Start with any function defined on an open interval j to r. For any point x inside j, we will define the upper denierite and derivatives. They are attributed to Dini, he was one of the French mathematicians I believe. So, let us just call them upper right hand derivatives, ok. They are not right hand derivatives, you are already familiar with the right hand derivatives and left hand derivatives. Now, we are only taking upper right hand derivatives which are nothing but, so notation is t plus of f at x, lim sup of these divided differences, f of x plus delta minus fx divided by delta. The only thing which makes it right derivative is delta tends to 0 plus, so only from positive part or delta is always taken positive here, ok. Delta tends to 0, but positive and you are not taking the limit which may not exist, but you are taking the lim sup. The whole idea is that lim sup always exists, no matter what the function is. If you allow infinity also for example, this always exists. If it is a bounded function, then it will be finite, it will also exist, ok. So, I am just recalling the definition of lim sup here which is nothing but, infremum of all these quantities over n, for all n here, where the inside thing is supremum over all the deltas of f of x plus delta minus fx divided by delta, delta between 0 and 1 by 2, ok. So, if you make say 1 by 2, 1 by 3, 1 by 4 and so on, as the interval for delta becomes smaller and smaller, supremum also becomes smaller. So, this sequence as a sequence in n is a monotonically decreasing sequence, therefore its infremum exists, ok. It is actually limit of this one, it is monotonically decreasing sequence, ok. So, this always exists, I do not need to call this one, but I am rewriting it just to tell you what is lim sup that is all. So, d plus fx always exists, this is called upward deney right hand derivative. Similarly, I can define the lower deney right hand derivative, the only thing I change is instead of lim sup, I take lim inf, which is in the first you take infremum over in this interval over delta, then you get a strictly, you get a monotonically increasing sequence, you take supremum of that. It can be shown that, I mean if you are a clear from this one, lim sup is, lim inf is always smaller than lim sup and so on. If both of them exist and, sorry, both of them always exist, but if they are equal then it will be actually the limit, namely the right hand derivative, ok. So, these things you must be knowing, but now we have these symbols here, d plus and d minus for upward right hand derivative and lower right hand derivative, ok. So, it is easily checked that if f is differentiable at x, then d plus is equal to d minus equal to the derivative.  So, these are just linearity of d plus, which follows by the corresponding property for lim sup, ok. The same thing for lim inf also, then in d minus also. By the way, I have only talked about the right hand derivative, it is exactly same way you can fact two more d is here from the left hand also. So, there will be four such quantities, if f exists, sorry if f is differentiable then all these four quantities are equal to the derivative itself, indeed it is a field only, ok. So, that is the easiest way to see that, but I am not interested in right now the left hand derivatives here at all, ok. So, neither I am going to do a lot of things with these things, these things are very very helpful in general, you know, analysis, so I take the support just to introduce them. My main aim is to use them to prove the so called big mean value theorem in the case of Banach spaces, ok. So, let us go ahead with that. So, this is theorem which will help us to do that job, this is very simple thing, ok, depends only on d plus which always exists. So, all that I start is start with a function f defined on say some open interval and a continuous function, real value function assume f 0 equal to 0, this is a harmless assumption, you can always assume f 0 equal to something and then subtract that, that is not a problem. So, this is technically, this is just a technical assumption. The basic assumption here is that this is a continuous function, ok. Suppose for some alpha positive, d plus of f is less than or equal to alpha for all x inside 0, 1, suppose we have found out a bound, upper bound for d plus of f, ok. So, pay attention, I am taking it only 0 to 1, ok, all that I want is, this could be 0 to 1, but I want to have a function even defined at 1 also, ok. So, the conclusion is that f x is less than equal to alpha x for entire of 0, 1, ok. So, I do not want to write 1 plus epsilon and something. So, I have taken just 1 comma 0 comma 2, there is nothing lost here, no extra assumption or nothing, I mean this as general as it could be, ok. So, just a continuous function, if the dv derivative t plus is bounded, it would give you bound, a very specific bound in terms of bound for all the function f x itself, ok. So, this bound must be inside the open and true 0, 1, I mean 0 open and 1 open, 0 closed and 1 open, but the conclusion is for the entire 0 to 1 closed and true, ok. So, so how do I go ahead it? If I show that alpha x, f of x is less than equal to alpha plus epsilon x, alpha plus epsilon times x for every epsilon positive, then because it is true for every epsilon positive, it must be true for alpha itself, right. So, I consider the function f x minus alpha plus epsilon x, ok, right and show that this function is less than equal to 0 g 0, g x is this function, g x is less than equal to g 0, g 0 here is put to be x equal to, if x equal to 0 here, this f 0 which is 0 is 0, so this is 0. So, this is negative, non positive for all x. The same thing as saying that f x is less than equal to alpha plus epsilon times x, for every epsilon this is true, f x is equal to alpha x. So, how do I prove that g x is less than equal to 0 for all x in 0, 1, ok. Now, what the only hypothesis that I have is that g is continuous because f is continuous, ok, this is just a difference of two continuous functions here. So, g is continuous, continuous function on a closed interval attains its minimum on each closed interval is 0 to x I have taken, ok, for all x inside 0, 1, ok. Look at fix some x inside 0, 1 after all I have to prove for all this one here, right. I do not want to take 1 yet, but you can take for all x this is true after all, g is continuous therefore, attains its minimum on the whole of 0 x, ok, for all x. So, we claim this minimum value is at x itself, if the minimum is at x, ok, for all other things g x must be less than equal to that minimum, bigger than equal to that minimum. So, g x will be less than that, in particular g x will be less than g 0 because g 0 is also one of the values, right. So, we want to prove g x is less than g 0, we are proving a very strong thing that the function itself takes minimum value at x, ok. So, there is lot more we are proving here, but we are not using finally, we are just using that g x is non-negative, ok. So, how do we prove that? Claim is that the minimum value at x, minimum value of this g is at x, ok. So, that is where the Dini derivative D plus will help us. By the way, every time you are to do with limb soup, you have to do this epsilon business, give an epsilon, subtract or add whatever then the something happens that is the only way to catch this limb soup and limb means, not only even the limits also there is in the way. So, for limb soup also is what you have to do. So, D plus of g, remember g was difference of two functions, right. D plus of f plus D plus of this function minus of alpha plus epsilon times x, ok. But what is the derivative of this one? It is this is linear map. So, it is alpha plus epsilon. Whenever derivative exists, D plus will exist, D plus will be equal to that. So, it is just the derivative, it is alpha plus epsilon. This is D plus as a as it is, ok. So, this D plus g is nothing, but D plus f minus alpha plus epsilon which is we started with our assumption that this is negative, ok. And D plus is less than or alpha itself what we have assumed, ok. So, apply this one to a point y inside 0 to x. This is true for all points inside this open interval 0, 1, ok. Take a, take fix an x and then take y inside 0 to x. We see that there exists an n belonging to n such that now I am going to use what is Limb Soap correctly, ok. Such that supremum over this interval is less than 0. If something D plus is less than 0, if all the supremums, ok, where greater than equal to 0, then the limits Limb Soap will be also greater than equal to 0. So, for some n this must be less than 0 because D plus is infimum over all such things, ok. So, for some n this must be less than 0, ok. So, which is the same thing as saying if you clear, remember now delta is positive because 0 and 1 by n, g of y plus delta minus g of y is negative for every delta in 0 to 1 by n, ok. That is where this delta is ranging over this one. This must be negative for all the y because the supremum itself is negative. Therefore, each term must be negative. Therefore, see if g y is minimum, then g y, g y plus delta cannot be smaller than g y. Therefore, g y is not the minimum of g in the entire of 0 to x because y is taken between 0 to x. 0 to close x, it is minimum. In the open interval, it cannot be the minimum. Therefore, where is the minimum? It must be at x.  The minimum is not, may not be attained in 0 open interval, open part. But it is the minimum we are looking at 0 to x closed. But this minimum cannot be inside the 0 open x. That is what this says. Therefore, the minimum must be at x. Hence, g x must be the minimum value of g in 0 to x. And that is precisely what we wanted to say. Therefore, g x is less than or equal to g 0 in the entire of 0. So, now, we are ready to do the generalization of weak mean value theorem. Why I am calling generalization? Weak mean value theorem is true for all differentiable functions on a convex domain into R n into R m, any vector value functions. Now, we are going to do it on Banach's basis. Same thing, same statement for Banach's basis is what we are going to do. The usual proof for in the case of R n and R m becomes easier because the norm square function is differentiable on R n. What is norm square? Namely, the Euclidean norm. Euclidean norm square is just summation xi square. So, we can use that to do our job because all other norms are all equivalent to the Euclidean norm. But in the general case, we do not have any such theorem. And we do not know. In fact, it is not even true perhaps that given a Banach space with a norm, that norm in our definition may not be differentiable. Even the square of norm may not be differentiable. Putting that norm being differentiable requires too much of a restriction. Almost we are begging that it must be a till work space. So, that is why the Diris derivatives are brought in to help us to prove the weak mean value theorem. So, the proof itself is not at all difficult now. Let us go through this one. Start with V and W Banach's basis. U is a convex neighborhood of 0 belonging to V. By the way, I have already told you that this assumption that 0 belongs to V is just a technical thing. You can do it for any other point also. Suppose, g from U to W is a differentiable function on U and there exists a lambda positive such that all the derivatives are bounded by lambda. Norm of g T g at all the points U is less than or equal to lambda.  So, this is weak mean value inequality. In the case of one variable calculus, this was deduced by using the mean value theorem wherein there is equality namely g U minus g 0 equal to g prime of something in the interval. And then g prime is bounded by lambda. So, you get this one. But we do not have the mean value theorem itself in the case of vector valid functions. But what we have is inequality directly. So, this is what we are going to prove it in the case of Banach's basis directly. Of course, it will work for any R n also because R n's are also Banach's basis. So, fix one point U inside U and define this h T equal to norm of g T U minus g 0. See, if you just take g T U minus g 0, this is precisely one variable calculus. Only thing is, this would be a W valued function, not a real valid function. How to get a real valid function? We are left with only taking norm or norm square or some such thing. Okay, taking norm square denthalpy, just take the norm function. This is a continuous function. We do not know that it is differentiable. Okay, now we have got into one variable calculus. T is sum between 0 and 1 because U is convex. When you take T times U, it will be still inside U. So, h from j to R is defined on an interval containing 0, 1 because in open subset, I can extend it just a little bit. Okay, 0, 1. I can make it 0, 2 also, but then I will have to take some T by 2 or some such thing, unnecessarily complication. I just need 0, 1, something positive. Okay, and some open interval j containing that. We claim that the derivative satisfies d plus h t is less than 2 lambda times 0. So, this was the condition that we needed. So, this lambda is the same thing as this lambda in the hypothesis of this proposition. Okay, once we have proved this one, you can use this earlier theorem that this function h itself is less than 2 lambda times norm U times T. The T is there after all here. Okay, for all T inside 0, 1. Okay, but then you can put T equal to 1, then you get h t. What is h t here? Just see. When T equal to 1, h 1 is g of U minus d 0. That will be less than equal to lambda times norm U. That is what we wanted to prove. Okay, so we have to prove this equation 12, this formula. Okay, so this comes very easily now. Given T and delta positive, such that both T and T plus delta are inside j, because j is the interval on which the function is defined. So, you have to choose only delta as small as, you know, small. So, that T plus delta is also inside. So, once that is satisfied, I can look at h of T plus delta as well as h t of T and take the difference. Divide by delta, take the supremum and then a ring supremum that is D plus. Okay, so start with h of T plus delta minus h t. That is by definition g of T plus delta times U minus g 0, take the norm minus g of T U minus g 0, norm of that. That by, you know, triangle inequality, you see add and subtract g T U, this plus this is less than equal to this one. So, if you take this term on the other side, so this minus this is less than equal to g of T plus delta times U minus g of T, norm of that. Okay, this is triangle inequality. Now, once again, you add and subtract this term is as it is dg of T U operating on delta U, dg at T U operating upon delta U. Okay, add and subtract so that the norm will be less than equal to norm of this plus norm of this second factor which I have subtracted here. So, there the delta comes out, delta is positive, which is nothing but norm of dg T U into norm of U. Okay, so this is less than or equal to, norm of this is less than or equal to, that is why I have put less than equal to. This is norm not equal to. Now, that why I have pulled out this delta because now I can divide out by this delta. When you divide out by this delta, okay, on the left hand side what you have is the corresponding term for our D plus delta definition, h of T plus delta minus h T divided by delta. What is this one? This is the, this occurs in the definition of the derivative, right, g of T U plus delta U minus g of T U delta times dg at T U, okay, operating upon U. So, this is the directional derivative at the, in the direction of U because there is this word divided by delta, okay. And this last term remains as it is, this delta has gone away. When you take the lim soup here, it will be less than lim soup of this plus lim soup of this, but here there is no delta. So, it is just a constant term, okay. What is the lim soup of this one? Lim soup, this limit itself exists, the limit itself is 0. Therefore, lim soup, limit for right hand, left hand, all of them are 0. So, this is 0. So, what you get is h, this one, d plus of h will be less than or equal to this one, okay. Because when you take lim soup, what you get is the definition of d plus of h, okay. That is what we wanted to prove. d plus of h T is less than or equal to this term which is, we know is all less than or equal to lambda times. By the assumption here, dg of u is less than or equal to lambda, okay. dg for all elements inside the convex set is less than or equal to lambda. So, lambda times, norm you get, okay. So, what we have proved is this mean value inequality for Banach spaces. For all differentiable functions on a convex neighbor, convex open subset, okay. So, let us now convert this into the following theorem ready made to use for our implicit function theorem, so on. Let V and W be Banach spaces, u is a convex open subset, f is a differentiable function on u, there exists a lambda positive and T is a some linear, bounded linear function. See, this part is same thing as the previous proposition. But here, now I am bringing an arbitrary T which is a bounded linear function on V to W such that df V minus T norm of this is less than or equal to lambda. In earlier case, if you put T equal to 0, then you get the earlier case, okay. So, now this is an extension df of V minus T is less than or equal to lambda for all V inside u. Then the conclusion is also slightly different, namely f of V2 minus f of V1 minus T times V2 minus V1 is less than or equal to lambda times V2 minus V1 norm for every V1 V2 inside this convex neighborhood, okay. This u itself is a subset of the Banach space V, okay. So, this is very easy now, of course, using this one, using this one. We do not need any more DNA derivative, we directly use this ready made theorem there. So, first consider the case when V1 itself is 0, okay. This is my assumption, this may not be the case, but I am assuming this is a special case. Now, u is a convex neighborhood of 0 because V1 is after all an element of u, okay.  This is a special case we want to prove when V1 is 0 and V2 is just u, okay. So, this is what we want to prove in the special case. For this, what I do? Put g equal to f minus T because in the hypothesis here, df V minus T is bounded, the norm of this is bounded. So, I take g equal to f minus T. What is the derivative of g? It is df minus T. Therefore, this dg is now bounded. Therefore, we can apply previous proposition and go over, right. Now, one more simplification we have to remove. Namely, I assume V1 is 0, okay. So, how to do in general case? In the general case, you first take the domain itself to be u minus V1. Translate u by V1, u minus V1. So, shift the origin. That means, all points V minus V1 where V is inside u, okay. On this one, you change the function also now.  V1 is a point of u. Therefore, 0 is a point of u minus V1. u is convex. Therefore, u minus V1 is convex. So, minus V1 is a linear, is a isometric, okay. It is not nonlinear, it is affine linear. It is an isometric. It preserves the norms. So, it is convex also and so on. So, we can apply the previous conclusion which was done for when V1 equal to 0 for this step, instead of that, you do it for f quiddle. What you get is precisely a statement now, okay.  So, what I have to do, if I want to get f of V2 here, f quiddle of V2 minus V1 will be f of V2. f quiddle of 0 will be f of V1 and so on. So, this one follows, okay. So, whatever we wanted, we have proved it by just shifting the origin or just shifting the function, okay. The shift of the function is occurring by a linear map here, okay. So, essentially this 1.21 is nothing but just a little more modification of our proposition here. And this was proved by using deny derivative, okay. So, next time we shall prove implicit function theorem. Just for the sake of what we are up to, I will just begin it, just statement and then we will prove this one next time. So, implicit function theorem, the statement itself is somewhat long here, okay. The proof is not as long as that because you know, you do not have to be threatened by the big statement, okay. The first part is, I mean the preparation here, what are the hypothesis here? V and W are Banach spaces. Y is any topological space, okay. So, that is the generality that we have achieved here. Take m cross n to be any open subset of Y cross V. In other words, m is an open subset of Y and n is an open subset of V. I will what we specifically that, okay. f from m cross n to W be a continuous function such that for some point Y naught V naught belonging to m cross n, we have f of Y naught V naught is 0. So, for each Y inside m, the function f Y from n to W given by f Y of V could f of Y V, namely Y is fixed now, okay. It is a function of just V from n to W. That is differentiable and this derivative, derivative will be a function from for each Y, there is a function n to the bounded linear maps from V to W, okay. This function must be continuous. So, for each fixed Y, it is continuous. That is not enough. As a bounded function, as a function of two different variables, it must be continuous, okay, jointly continuous. Finally, third one is that the derivative at Y naught V naught, f Y naught derivative of f Y naught at V naught, that is t. Then this t is a similarity, okay. So, hypothesis is this hypothesis is very important that here we have an isomorphism, a similarity. Other things are joint continuity of the derivative. Before that, the function must be differentiable, okay. Only in terms of Y, in terms of n, okay, namely in terms of V, the Y part is arbitrary topological space, there is no differentiation there. You understand that why is an arbitrary topological space? In the other words, you should think of this as a family f Y of V of differentiable function. Only thing is the family itself is a continuous family, okay. Not only each function is continuous and differentiable, the family itself is continuous. That is the way we have to express that function is m cross n itself is contained. With these hypothesis, now the conclusions, there are two conclusions here, okay. For the second conclusion, we will need little more hypothesis, that is why it is separated out. The first conclusion is that you can take a neighborhood prime of m of Y naught, okay, a smaller neighborhood, not the whole of m, some smaller neighborhood and a neighborhood and a row positive that will be neighborhood of that will be neighborhood of zero inside n there, okay, such that for all Y inside m prime, okay, there exists a G Y belonging to B row bar of V naught, the closed ball around V naught with the property that f of Y G Y is zero, okay. So, you see f of Y naught V naught was zero. One solution was think of f Y V as equal to zero as a equation which we want to solve. One solution has been given, okay.  So, that is precisely what is achieved here. In fact, when you are trying to get a one solution, you are assuming you are already getting a unique solution. This is the only way to get a solution after all, put sufficient conditions so that it becomes unique cause. In all existence theorems, the uniqueness part helps you a lot, okay.  Only thing is with these hypotheses we have to cut down the domain properly. Some opens up set we do not know, okay. Some row we do not know, okay. On the closed ball, okay, for each point inside m prime, G Y will be unique such that f of Y G Y is zero. And this G itself is a continuous function, okay. One is not satisfied with just continuity, we would like to have differentiability also. For that we have to put little more hypotheses because m prime was a subset of just Y, right. Y was a topological space. So, it does not make sense to tell you or demand that G must be differentiable. So, in order to make that sense, we have to at least say that m prime is an open subset of some norm linear space. So, that is why we would like to assume that this is, this Y is a Banach space, okay.  You see I have used now upper index here to indicate that now I am fixing the right end slot as fixed now. v naught is fixed, Y is a variable. But the function is same a function f, okay. It is restricted to v naught and Y is varying. That must be differentiable, okay. We are not demanding anywhere that the m prime to n, this may have been function, so m cross n to w itself is differentiable from the product space. But we are demanding that this is continuous. Whereas, differentiability is by partial. Fixed Y naught, you get a differentiability. Fixed v naught, you must be differentiable. That is all we are demanding, okay. So, this part is that f v naught, where v naught is fixed, that must be differentiated at Y naught. And the derivative, let us say, is H naught of H of Y naught v naught. This is just a notation for the derivative of this function, okay. Then what happens? G will become differentiable at Y naught. What is this G? It is a unique solution given by the part a, okay. That will be differentiable to Y naught and its derivative is given by minus t inverse composite with H. See, H is a linear map, bounded linear map. T is a bounded linear map. This t is invertible, it is similarity, so I can talk about t inverse. Take t inverse of H, take the minus of that, that will be G Y naught. Which happens to be, I rewrite it because G is just, this t is just G Y naught v naught, okay. So, let us prove this statement next time. Thank you.