TrueDoc run 97 on olmOCR-bench: 86.8, with all 1,403 page outputs and the scorer's log (part 5)

README.md

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+---
+license: apache-2.0
+datasets:
+- allenai/olmOCR-bench
+tags:
+- ocr
+- document-parsing
+- pdf
+- pipeline
+- olmocr-bench
+---
+
+# TrueDoc: a PDF-to-markdown pipeline, measured on olmOCR-bench
+
+**This repository holds a benchmark result, not model weights.** TrueDoc converts PDFs to markdown with a YAML
+front matter (the Open Knowledge Format) and is built for *meaning accuracy*: does the output say what a reader of
+the page would understand it to say? Its code is at https://github.com/islandtimer/TrueDoc (Apache-2.0). The models it uses are other
+people's: Infinity-Parser2-Flash and -Pro (infly), olmOCR-2-7B-1025 (allenai), docling-layout-heron
+(docling-project), all Apache-2.0.
+
+**olmOCR-bench, run 97: 86.8** (95% bootstrap CI 86.0-87.7), official scorer, all eight sections,
+dataset revision `54a96a6f`. [RESULTS.md](RESULTS.md) has the section scores, exactly what read which pages, and
+how to re-score the 1,403 outputs in `outputs/` yourself.
+
+The arrangement, in one line: the PDF's own text for the 1,122 pages that have it (no model), Infinity-Parser2-Flash
+for the 281 that do not, Infinity-Parser2-Pro behind it for the 102 the pipeline's own OCR cannot read, olmOCR 2
+for 92 picture regions; readings recorded on a rented GPU and replayed; no post-processing keyed on the
+benchmark's folder names.

RESULTS.md

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+# TrueDoc on olmOCR-bench: run 97
+
+**Overall 86.8** (macro-average of the eight sections; 95% bootstrap CI 86.0-87.7), scored on
+2026-09-18 with the benchmark's official scorer (`olmocr` 0.4.27's `benchmark.py`) at dataset revision
+`54a96a6fb6a2bd3b297e59869491db4d3625b711`, over all 1,403 pages and all 7,019 checks plus the 1,394 baseline checks.
+
+| section | score | checks passed |
+|---|---|---|
+| arxiv_math | 88.6 | 2594 / 2927 |
+| old_scans_math | 86.7 | 397 / 458 |
+| table_tests | 89.3 | 913 / 1022 |
+| old_scans | 58.4 | 307 / 526 |
+| headers_footers | 96.7 | 735 / 760 |
+| multi_column | 83.6 | 739 / 884 |
+| long_tiny_text | 91.4 | 404 / 442 |
+| baseline | 99.9 | 1392 / 1394 |
+
+## What produced these pages
+
+TrueDoc is a PDF-to-markdown conversion pipeline, not a model. For this run:
+
+- **1,122 pages with a digital text layer** were read from the PDF's own text on CPU, with a layout model
+  (docling-layout-heron) for structure. No vision model read those pages.
+- **281 pages with no text layer** (scans, photographs of paper) were read by **Infinity-Parser2-Flash**
+  (2.2B, Apache-2.0). The **102** of those that the pipeline's own OCR could not read (its word-likeness gate)
+  were read by **Infinity-Parser2-Pro** (35B, Apache-2.0) instead.
+- **92 picture regions on digital pages** (pictures that hold text: a scanned table, a screenshot) were read by
+  **olmOCR-2-7B-1025** (Apache-2.0).
+- The model readings were recorded on a rented GPU on 17 September 2026 (Infinity-Parser2 through vLLM 0.17.1 with
+  the settings of its authors' olmOCR-bench guide, raw markdown, none of their post-processing) and on 8 September
+  2026 (olmOCR 2), and replayed from disk during the run. The readings are in the code repository.
+- No post-processing keyed on the benchmark's category folder names. Every page was converted the same way.
+
+The pipeline's code is at https://github.com/islandtimer/TrueDoc, tag `run-97` (commit `9f9b77b`); its README says how to reproduce this run
+from the committed readings, and `docs/BENCHMARKS.md` there holds every run before this one.
+
+## What is in this repository
+
+- `outputs/<section>/<page>_pg1_repeat1.md`: the 1,403 page outputs exactly as scored, in the scorer's layout.
+  Put the folder beside the dataset's `bench_data/pdfs` and run the scorer to re-score in minutes.
+- `scoring/summary.json`, `scoring/score_output.txt`, `scoring/failed_tests.jsonl`: the scorer's own output, the
+  bootstrap interval, and every failed check.
+- `.eval_results/olmocrbench.yaml`: the entry the dataset's leaderboard reads.
+
+## How to read the number
+
+- It is a self-reported run of the official scorer, not an independent one. Everything needed to check it is
+  here.
+- A second scoring of the same pages moves a section by about three checks, so the tenth is not significant.
+- A fifth of the benchmark's pages (`bench/holdout.txt` in the code repository) was never looked at while the
+  pipeline's rules were built; that fifth scores 86.0 in this run, the rest 84.8.
+- As of 2026-09-18 this overall would rank second among the entries on the leaderboard by point estimate; the leading
+  entry's 87.6 sits inside this run's interval. A live integration of the served readers (rather than replayed
+  readings) is pending.

outputs/old_scans/73_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+Blue Anchor
+
+THE EVENING CRITIC.
+Published Daily (Sundays Excepted)
+BY THE Evening Critic Publishing Company
+511 NINTH STREET NORTHWEST,
+WASHINGTON, D.C.
+
+TERMS:
+SINGLE COPY ..... 2 CENTS
+BY CARRIER, per Month ..... 85 CENTS
+BY MAIL (Postage Paid) One Year ..... \$4.50
+BY MAIL (Postage Paid) Six Months ..... \$2.85
+Mail Subscription Invariably in Advance.
+ADDRESS EVENING CRITIC PUBLISHING COMPANY,
+WASHINGTON, D.C.
+
+Entered at the Post Office in Washington as second class matter.
+
+Friday, March 31, 1882.
+
+THE country breathes easier. We are to
+
+The Blue Anchor.
+
+The timely enterprise of the Women's Relief Association in giving a donation party and promenade concert for the benefit of the victims of the Southern floods at Willard Hall tomorrow evening is sure to be successful. Many of the most popular and well known of our resident ladies, as well as members of the association from other States, are on the reception committee. Mrs. Waite, Mrs. Miller, Mrs. Mathews, Mrs. Rodgers, Mrs. Windem, Mrs. Hunt, Mrs. Jonas, Mrs. Van Voorhees, Mrs. Pound, Mrs. J. B. Edmonds, Mrs. Tulloch, Mrs. Sayles Brown, Mrs. Dr. J. M. Brown, Mrs. Henry A. Willard, and Mrs. Metserott are among the number.
+
+A Novel Exhibition.
+
+"Our renderers never sew 's Handus a one
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/74_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+The American Association of the Red Cross.
+
+For the relief of suffering by war, pestilence, famine, flood, fires, and other calamities of sufficient magnitude to be deemed national in extent. The Association and its auxiliary societies operate under the provisions of the Geneva Treaty, promulgated at Geneva, Switzerland, in 1864, and signed since then by all the nations of the earth, including the United States, which gave in its adhesion through President Arthur, in March, 1882.
+
+CHESTER A. ARTHUR,
+PRESIDENT OF THE BOARD OF CONSULTATION.
+Executive Officers.
+CLARA BARTON, PRESIDENT.
+WALTER P. PHILLIPS, GENERAL SECRETARY.
+GEORGE KENNAN, TREASURER.
+Trustees.
+CHARLES J. FOLGER.
+ROBERT T. LINCOLN.
+GEORGE B. LORING.
+
+Copy from Demorest Monthly.
+May 1884.
+
+The Womans National Relief Association - President Mrs. Chief Justice Waite, of Washington - is doing a work worthy the active interest of all women. Its annual report says: Eighteen months ago it undertook first to supplement government work in the life saving service by providing the stations with bedding, clothes, medicine and proper diet for the shipwrecked and suffering crews and passengers rescued from death. Seventy stations have been so supplied. The Michigan sufferer next received systematic and efficient aid through this organization. Then it undertook to furnish the marine hospitals with supplies not provided by Government, such as libraries, clothing for indigent sailors. The association is now occupied with collecting and distributing relief for the sufferers by the Mississippi floods. The main object of the association is to organize the benevolent women of the nation into an intelligent, harmonious body which, in the event of any great disaster from war, famine, flood, fire, pestilence, or any other cause, shall be ready to bring at once a vast, effective machinery of relief to bear upon it. The association has its headquarters in Washington D.C. It is proposed that there shall be auxiliary societies in every State of which several are formed and in active operation. Any person can become a member by the annual payment of \$5.00. Women who desire to form State auxiliary societies should apply for information to Mrs. Hannah McLehland No. 616 Third Street N.W. Washington D.C.
+
+A POUND PARTY
+AND
+Entertainment
+WILL BE GIVEN AT
+Masonic Temple,
+Monday, April 24th, 7:30 P.M.,
+UNDER THE AUSPICES OF THE
+DISTRICT AUXILIARY.
+Blue Anchor Association,
+FOR THE RELIEF OF THE
+Sufferers by the Mississippi Flood.
+ADMISSION FREE:
+All are invited, and are requested to
+bring or send donations of Groceries
+or clothing old or new.
+R. O. POLKINHORN, PRINTER.
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/75_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+Women's National War Relief Association.
+
+The Mount Vernon branch of the Women's National War Relief Association held an executive meeting on Tuesday last.
+
+Several new members were enrolled and gratifying reports were heard from all committees. It was decided to forward to the general society in New York the sums already raised by subscription, amounting to \$55, which money is to be used in the furnishing of the hospital ship.
+
+One hundred and six abdominal bandages have also been made by this society and have been approved and accepted.
+
+It is not generally known that this Association is the only one working under Government direction. No salaries are paid and no expenses incurred. The society does the work, raises the money and the Government of the United States directs, controls and superintends in every way its outlay. The ship Relief has been fitted out as a floating hospital, complete in every detail, by the W. N. W. R., the awnings alone costing \$1,500.
+
+The Mount Vernon branch is working for this great object, and feels that if by their effort even one of the American soldiers or sailors, sick or wounded, can be spared the discomforts of the ordinary hospital transport, its work will be well done; and the public have the great satisfaction of knowing that every cent given to this Association is used for the comfort and benefit of our own brave men.
+
+Prominent members of the society will from time to time, address the meetings, which will be held (by the kindness of the W. C. T. U.) in Willard Hall every Tuesday at 4 o'clock P. M.
+
+All are most cordially invited to attend.
+
+Women's National War Relief.
+The Mount Vernon branch of the Women National War Relief Association will hold its meetings every Tuesday at 4 o'clock p. m, at Willard Hall. This society is not international in character, it acts under the direct supervision of the United States government, and cares for our soldiers and sailors. Everyone is cordially invited to these meeting, where the work accomplished and to be accomplished will be fully explained.
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/76_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+I enclose these clippings from Nat V. papers to have you see what obstacles I have fought against and overcome. Nothing dies, but error and untruth — ; Red Cross will live forever
+
+C. C. H.
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/77_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+[Apr. 19, 1861] Individual interests are absorbed in this great National calamity. I know the fact but cannot realize it. I look out upon the same beautiful landscape — the same clear blue sky — the same floating clouds — the face of nature is unchanged — nothing there that indicates that the darkest page in our countrys history is now being written in lines of blood! But I turn and one glance on the face of man reveals the terrible certainty of some dark impending war. The 19th of April! Is it not an omen of evil import to those who have dared to raise the hand of rebellion against the common country; for that day 1861 is again made memorable by opening the "Dance of Death" by the shedding the blood of the sons of Mass. as in the days of the Revolution 1775. And was this massacre at Baltimore a necessary prelude to the melancholy dance which has already
+
+[42] You must before this have learned from my letter from Mrs D. Washington D.C. Dec 16, 1861
+
+Mrs. Miller Sec. You must before this S. of Ladies Relief Com'tt have learned Worcester Mass.
+
+Dear Madam Your letter mailed to me on the 11th came duly to hand at a moment when I was more than busy, and as I had just written Mrs Dickin (of whom I read the articles) a detailed acct. of their hasting and final declination, I have ventured with much regret to allow your letter to remain unanswered for a day that I might find time to write you at greater length. You must before this have learned from my letter to Mrs. D. the occasion of the delay (viz — uncertain orders, rainy weather and Maryland muds) and decided with me that the (anxious) package has long before this [perf] accomplished its mission of charity & love. The bundles were all packed together in a stout box, securely nailed, and [delivered] given to the Suttle of the 15th Regt. who promised to deliver
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/78_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+began, and which God alone knows when, & how it will end —
+
+The above was a part of a letter written by me to a friend the morning after the news reached here of the reception of the Mass troops in the city of Baltimore the 19th, of April 1861 — M. E. Stone, P. M
+
+<table><tr><td rowspan="2">Free. No.</td><td colspan="6">PAID HERE.</td><td rowspan="2">Rater. Cents.</td></tr><tr><td colspan="3">In Money.</td><td colspan="3">By Stamps.</td></tr><tr><td></td><td>No.</td><td>Dolls.</td><td>Cents.</td><td>No.</td><td>Dolls.</td><td>Cents.</td></tr></table>
+
+them safely at Head quarters — I have no doubt but it has all been properly done, — A box for the 25th I had delivered to Capt. Atwood & Co — and heard with much satisfaction the gratification it afforded the various recipients — The men was looking splendidly, and I need not tell you that the 25th "is a live" regiment from its Col & Chaplain down. Worcester Co his just cause for pride.
+
+I come now to the expression in your excellent letter which I had all along feared — are our labors needed at all day or good? shall we "work" or shall we "failures" — for the fact I have dreaded lest a sense of vague uncertainty in regard to matters here should discourage the efforts of our patriotic [ladies women] ladies at home — it was this fear and only this which ever gave me courage to [renew] assemble address the worthy ladies of your committee (so worthy my superiors) [on every day] to confer upon a [subject] matter after which they seemed perfectly familiar with knowing so little. But even now I scarce know how to reply — It is said upon high authority that our Army is supplied — Well this may be so it is not for me to gainsay — and so far as our New England troops are concerned [this] it may be that in these days of quiet idleness they have really no pressing wants — But in the
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/79_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+event of a Battle who can tell what [our needs] their necessities might grow to in a single day They would want them faster than you could make — But only a small portion of an Army comparatively speaking are New England troops — N.Y. Penn. Ohio Ind & Missouri have sent their hundreds of thousands and I greatly fear that these States lack somewhat the intelligent active industrious organizations at home which so characterize our New England circles. I think I deserve treas of this in this Camp. I feel while passing through them that they could be better supplied without danger of encumbrance from luxuries — Still it is said that "our army is supplied" — it is [also] said also, upon the same authority that we "need no nurses," either male or female and none are admitted — I wished an hour ago that you had been with me — In compliance with a request of my sister in this city I went to her house and found there a young Englishman a brother of one of their domestics who had enlisted
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/7_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+ack 5/27.11x
+
+Hay Side. Long Island. New York, May 22, 1914.
+
+Honorable Colonel Theodor Roosevelt. Oyster Bay, Long Island.
+
+Dear Sir:-
+
+I am very glad that you arrived home in good health. It has been a long time since I want to write to you and to tell you what a great admiration and respect I have for you, that I cannot express on the paper. But there is one instant,
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/80_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+during the summer in a Regt of Pennsylvania Cavalry. They are stationed at Camp Pierpont; the sister heard that her brother was sick and with the energetic habit of a true English woman carried the county on foot nine miles out to his camp & back the same day - found him in an almost dying condition and begged that he be sent to her, she has taken shortly after in an ambulance and upon his arrival his condition was found to be most deplorable - he had been attacked with ordinary fever - and had lain unmoved until the flesh upon all points of the body which rested hard upon whatever was under him had decayed [and] grown perfectly black and was falling out. his heels had around the same appearance [his toes also and were dropping off at the joints] his stockings had never been removed during all his illness and his toes were matted and grown together and are now dropping off at the joints the cavities in his back are absolutely putrid - When intelligent medical attendance was summoned from the city the verdict [of] rendered upon examination was that his extremities were perishing for want of nourishment, he had been neglected until he was literally starving - too little nourishment had been taken into the
+
+system during his illness to preserve life on the extremities - This conclusion [would] seems all the more reliable from the famished appearance which he presents - I am accustomed to see people hungry when recovering from a fever - but I find that hunger and starvation are two distinct conditions - he can lie only on his face with his insteps propped up with hair pillows to prevent his toes from touching the bed (for with the life engendered by food & care, sensation is returning to them) and asks only for "something to eat" food is placed by him at night, and with the earliest dawn of day commences his bowls of broths & soups and a little meat and he eats and begs for "more", and sleeps & eats & begs - three of his toes are to be amputated today. The surgeon of the Regt comes to see him, but had no idea of his condition, said that their Asst Surgeon was killed and that li
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/81_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+"was true that the man had not [had the] [had] received proper care. he was very sorry" With the attention which this young man is now receiving [she] will probably recover — but how had it been otherwise? Only thus. [after] the [after] not far from this time [some] the city [new] papers under caption of Deaths of Soldiers would have continued the paragraph — Berg. Pollard private Comp Dorp Vermont — and this would have been [all] the end. Who ever could have mistrusted that this soldier had [de] starved to death through [want of] [one] lack of proper attendance? Ah me all [soldiers] of our poor boys have not a sister within nine miles of them — And still it is said by authority "We have no need of nurses" and that "our army is supplied" — How this can be so I fail to see still again it is not for me to question — we are loyal and our [government] authorities must be respected though our men perish — I only mention such facts as
+
+come under my own observation and only a fraction of those — This is not to any manner in accordance with our home style of judging. If all New England [ladies] people say men lying in camp uncared for until the toes rotted from their feet with not [people] persons enough about them to take care of the dead should think they needed more nurses, if with plenty of persons about who failed to care for them we should think they needed better I can only repeat that I fail to see it clear — I greatly fear that the few privileged elegantly dressed ladies who ride over and sit in their carriages to witness "splendid reviews" and "inspect the [soldiers of] army of the Potomac" and come away "delighted" learn very little of what lies then under covers. Since receiving your letter I have taken occasion to converse with [the] a number of the most intelligent and competent ladies who are or have been connected with the Hospitals in this city. [at with one]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/82_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+and all agree upon one point viz that our army cannot afford that our ladies lay down their needles and fold their hands if their [labors] contributions are not needed just today they may be tomorrow, - and somewhere they are needed today - and again all agree in advising that whatever be sent be gotten as nearly direct as possible from the hands of the donors to the very spot for which it is designed - not to pass through too general distribution, strengthening their advice by many reasons & circumstances which I do not feel at liberty to lay before you. no one can fail to perceive that a house of general receipt & distribution of store of all description for the whole U S must be a mammoth concern - abounding in confusion - which always involves loss & destruction of property. I am confident that this idea cannot be incorrect and therefore I will not hesitate to admit it upon my own responsibility - viz that every State should have in the vicinity of her greatest body of troops a [house] depot of her own where all her contributions should be sent - and dispersed if her own soldiers need it all to them, if
+
+Washington then let her share generously and intelligently with those whose real but know what she has and what she gives. we shall never have any other precise method of discovering the real wants of our soldiers, when the store house of any state should be found empty - it would be safe to conclude that her troops are in need then let the full givers render the required assistance - this would systematize the whole [the about] matter, and do any with all necessary confusion at once and uncertainty. - it would preclude all possibility of loss, as it would be the business of [the] each house to [see] look to its own [own goods] property. There is some truth in the old maxim that 'what is everybody's business is no ones business, I believe that as long ago as the early settlements of our country it was found that the plan
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/83_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+general labor, general storehouse, and general distribution, proved in effect, and reduced our one little colony to a state of [utter] confusion & almost ruin. There were one hundred and one, persons here [four] hundred thousand. If pecuniarily I were able, Massachusetts should have the depot in this city and I should have no fears of mobility this to one would be an experiment, for however dimly & slowly I discern other points, this has been clear to me from the first, strengthened by eight months daily observation.
+
+When I write another [thought] idea seems to me - has it been thought of to provide each of our Regiments that are to accompany the next expedition with some strong well filled boxes of of useful articles to stores, which are not to be opened until some [action] battle or other strong necessity render supplies necessary - these necessities are sure to follow, and unless anticipated and guarded against - no activity on the part of friends at home can prevent the suffering which their absence will create with regard to our 23, 25 & 27. regts I cannot speak but our 21st I know have no such provision, and will not have unless thoughts of at home and the consequence of neglect will be that by & by our army hearts will be among by acts of our best officers and desert from being their lives compromised by the loss the anchors of taller
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/84_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+candle in the mouth of a bottle, and pitchy darkness close down upon men bleeding to death, or vainly trying to staunch their wounds with bushes & straw.
+
+A note just over informs me that our 6 companies of Engineers for 31 Independent name started at the arrival of this same ten miles from any in writing for their supplies per [the Great] Boats are compelled to sleep on bare loose floors with a single blanket and are taking severe colds and coughing [frightfully] fearfully. — my inhuman points no way of relief but to buy soaking warm ticks to be filled with hay to raise them from the draft a little. They are far more exposed than they would be on the ground under a good tent. I almost say our ladies when so many of you can work together and accomplish so much — while my poor ladies are so single handed. The future often looks dark to me & and it seems sometimes that the smiles of Heaven are almost withdrawn from our poor rent & distracted country and yet
+
+there is everything to be grateful for and by no means the least is this strangely mild winter
+
+But I must desist and crave pardon for my (perhaps unpardonably) long letter, for if you have followed me thus far, and especially at comparatively as rapid a rate as I have written you must be weary — I did not intend to say so much but let my interest be my apology. And with one more final word in answer to your rational question I have done, Ladies remember that the call for your organized efforts in behalf of our army, was not from any Commission or Committee but from Abraham Lincoln & Simon Cameron And when they no longer need your labors they will tell you
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans/85_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+before you again, only that I incidentally learned that our excellent Dr. Hitchcock has taken back from Roanoke other papers relating to the same subject, which will doubtless be laid before you, and as I have an entirely different boon to crave, I find it necessary to speak.
+
+I desire your Excellency's permission to go to Roanoke. I should have preferred my request weeks earlier, but I am called home to witness the last hours of my old Soldier father, who is wearing out the remnant of an oak and iron Constitution, seasoned and tempered in the wild wars of "Mad Anthony". His last tale of the Red Man is told, — a few more suns, and the old Soldiers weary march is ended, — Honorably discharged, — he is journeying home, —
+
+With this, my highest duties close, and I would fain be allowed to go and administer comfort to our brave men, who peril life and limb in defence of the priceless boon the fathers so dearly won.
+
+If I know my own heart, I have none but right motives. I ask neither pay or praise, simply a soldier's fare and the sanction of (Your Excellency) to go and do with my might, whatever my hands find to do.
+
+In General Burnside's noble Command are upwards of forty young men who in former days were my pupils*
+
+I think I am safe in saying that I possess the entire confidence and respect of every one of them. For the Officers, their signatures are before you.
+
+If my request appear unreasonable, and must be denied, I shall submit, patiently, though sorrowfully, but, trusting, hoping better things. I beg to subscribe myself
+
+With the Highest Respect Yours &
+
+Clara H. Barton
+
+* I am glad to know that somewhere they have learned their duty to their country, and have come up neither cowards, or traitors.
+
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+John A. Andrew. Governor of the Commonwealth of Massachusetts Boston.
+
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+I do not think, at the present time, Miss Barton had better undertake to go to Burnside's division to act as a nurse. Alfred Hithcock March 25th 1862.
+
+J. W. Flotey letter & request to send Miss Barton to nurse sick & wounded soldiers is reasonably decided.
+
+Respectfully referred for the information of Miss Barton, J. A. Anderson, Mar. 15/62
+
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+
+Official Business,
+Commonwealth of Massachusetts.
+EXECUTIVE DEPARTMENT.
+
+To.
+Miss Clara C. Barton.
+North Oxford
+Mass.
+
+Copy.
+COMMONWEALTH OF MASSACHUSETTS.
+EXECUTIVE DEPARTMENT.
+
+Boston, March 24th, 1862.
+
+Miss Clara H. Barton
+North Oxford, Mass.
+I beg to assure you, Miss Barton, of my cordial sympathy with your most worthy sentiments and wishes; and that if I have any power to promote your design in aid of our soldiers I will freely use it. Whenever you may be ready to visit Gen'l. Burnsides division I will cheerfully give you a letter of introduction, with my hearty approval of your visit and my testimony to the value of the service to our sick and wounded, it will be in your power to tender.
+With high respect I am
+Your ob. servant
+
+John A. Andrews.
+
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+62 Letter & request to send Miss Barton to nurse sick & wounded soldiers in Burnside Div.
+
+Commonwealth of Massachusetts. Executive Department. Boston, March 24th 1862
+
+Miss Clara H. Barton North Oxford, Mass'ts Very to assume Gov. Miss Barton, of my cordial sympathy with yr most worthy sentiments & wishes; & that if I have any power to promote yr design in aid of our soldiers I will freely use it. Whenever you may be ready to visit Genl Burnside's division I will cheerfully give you a letter of introduction, with my hearty
+
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+
+that can prove you it and from which you can imagine and convince yourself of the truth of my words. My first son who is attending a Public School at Bay Side in Long Island is named in your honor.
+
+Inclosing this letter, I really hope and wish from the dept of my heart that some day I shall have the greatest honor of seeing you and saying to you one word in a voice of voice.
+
+I remain
+
+your humble
+servant
+O. Belmont
+
+P.S. I would highly appreciate if you would send me your picture.
+
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+
+opinion of your visit & my testimony to the value of the same, your sick & wounded, it will be in the power to render,
+
+With highest respect I am
+Yr ob servt
+
+Hub. Anderson
+
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+J. A. Andrews / 62 official mess letter offering aid to C. B.
+
+Col. Alexander of Louisville - formerly Maj. Alexander who kept the Soldiers Home in Washington had among other servants - one "Louisa" Rudd. who had been the property of his wife Sallie Rudd - and one Wesley Williams - who became the husband of said Louisa while in Louisville - and they had four children. two of which Mary & Robert they took with them to Washington and two others Rachael and Louisa were left in Louisville with capt. Rudd. or Mrs. Alexander and now with Mrs. Alexander. When Maj. Alexander went to Louisville in 1862 or 3, Louisa & Wesley Williams declined going with him, and remained in Washington. On the Boat St Nicholas plying between Louisville & Cincinnati, I met Rachael Alexander the mother of Louisa (Wesley) Williams of Washington, who desires to hear from her and wishes to tell her and her husband to send to
+
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+
+Louisville and get their two children who reside now with Mrs. Alexander as she fears they may not be well treated.
+
+This Mission is transmitted to Rosa, who will make inquiry.
+
+St Nicholas 10 1/2 elr Aug. 21. Midst of a dance
+
+They must address their letter to Mrs. Nellie Churchill (another daughter) Cincinnati, O. Water St. Bet. Elm & plum No. 178. to whom the Mother is on the way now —
+
+Letter directing Rosa how to hunt up Wesley & Louisa Williams formerly at Soldiers Home Wash
+
+P M Watson Esq Asst Sec of War
+
+Sir As I prefer to a request that I desire to make at your hands, allow me without charge of egotism to relate a few circumstances [incidents] connected with they past year life — When our first regt of troops [enters the city] came to the defense of Washington I commenced to look after the wants — and believe I carried the first basket of food which was ever taken to Mr Solon in the City. from that day I cared for the home until our reserves commanded by Go Pope at the Battle of Second Manassas. I was the first who reached the with supplies [and remained] at the Battle of 2d. Bull Run I reached Fairfax Station with a car load of
+
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+
+of food & cattle [supplies] on Sunday morning and remained without sleep or rest until Tuesday night when the last wounded man was away, the Battle of Chancellorsville had been fought and the Union fleet to be destroyed. I [then followed] next the [scene of the] army obliged for Col Buck on Army wagon and with it loaded to the utmost met the Battle of Antietam on the morning of the 17 Sept and remained until my supper and strength well with exhaustion, and returned to re-supply myself and [return from the] with [with much] meet if possible the expected battle at Harpers Ferry - as I neared the point our army crossed the bridge, and I followed, coming from the dict. and feeding them all the way down the mountains to Harpers Ferry
+
+where I came [here with] with the 1st West Virginia Genl Storges by his own special request take charge of my team, which I joined again at Harpers Ferry a few days before the 1st battle, at the landing and I lost my post at the Key House and with the exception of two days when I formed army to tender suffering in Frederick - by I remained there until the hundred of dying wretched [suffering] until they were all taken away I believe my services have been faithful I know they have been laborious and appreciated. I have charged my own position and I then mentioned it when I had to stand by the soldier better the bullet and the city hospital
+
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+
+I trust that I have over stoved
+for duty or danger - and new
+shill, while this tender new host
+I must to serve with it - and for
+carter seems I am done to go
+to Port Royal and will
+most respectfully ask of
+you honored Dept - perman
+to do so, and to take with me
+the two such supplies as I
+very fondly aming with
+confer of the suffering among
+me
+
+I am with great
+Solly me
+
+CLARA BARTON MEMORIAL ASSOCIATION
+
+MRS. JOHN A. LOGAN, PRESIDENT
+
+2523 13TH STREET, WASHINGTON, D. C.
+
+MRS. ADA H. WEISS, 1ST VICE-PRESIDENT
+
+WASHINGTON, D. C.
+
+MISS GRACE M. PIERCE, RECORDING SEC'Y
+
+WASHINGTON, D. C.
+
+MRS. MARY LOGAN TUCKER, CORRESPONDING SEC'Y
+
+WASHINGTON, D. C.
+
+MR. JOHN JOY EDSON, TREASURER
+
+WASHINGTON LOAN & TRUST CO., WASHINGTON, D. C.
+
+GEN'L NELSON A. MILES
+
+2nd Vice-President
+
+Washington, D. C.
+
+GEN'L HARRISON GRAY OTIS
+
+3rd Vice-President
+
+Los Angeles, Calif.
+
+MR. S. B. HEGE,
+
+4th Vice-President
+
+Washington, D. C.
+
+MISS JANET RICHARDS
+
+5th Vice-President
+
+Washington, D. C.
+
+HON. NEWELL B. WOODWORTH
+
+6th Vice-President
+
+Syracuse, N. Y.
+
+MR. W. B. SCOFIELD
+
+7th Vice-President
+
+Worcester, Mass.
+
+HON. LEO RASSIEUR
+
+8th Vice-President
+
+Granite Building.
+
+St. Louis, Mo.
+
+WASHINGTON, D. C., 191
+
+Washington, D. C. Sept. 4, 1862
+
+COPY.
+
+Dear Friends:-
+Yours awaited me on my return from Fairfax Tuesday evening (or night rather). I left here on Sunday morning in the rain in company with Mr. Wills, Mrs. Morell, Mr. Haskell, Mrs. Alvord, et al, and took train at Morgan Bulley's office, and soon found ourselves at Fairfax.
+
+I cannot tell you the scenes which awaited our eyes; the wounded were constantly coming; but no hospitals this time; only God's great one under the blue canopy.
+
+The men were brought down from the field and laid on the ground beside the train, and so back up the hill 'till they covered acres. The bales of hay for forage were broken open and the ground was "littered", like "bedding" for horses. They came 'till dark and then it was dark indeed: One lantern on the ground; made requisition for candles; drew a few. The wind blew just enough to put them out every few minutes, and the men lay so thick we could not take one step in the dark. By midnight the men lay so thick we could not take one---- By midnight there must have been three thousand helpless men lying in that hay. We had two water buckets--five dippers--the stores which we carried to eat besides hard crackers. My one stew pan which I remembered to take, and this made coffee for them. All night we made compresses and slings and bound up and wet wounds when we could get water, for what we could traveled miles in the dark over those poor helpless wretches, in terror lest some ones candle fall into the hay and consume them all. At length morning came and we sent up the train with 1250. Next 1000, next 1100, next 940 and so on. Still the ambulances came down and the cars went out and we worked on. Took the meat from our own sandwiches and gave it to them, and broke the bread into wine and water to feed the poor sinking wretches as they lay in the ambulances
+
+On Monday the cavalry appeared in the wood opposite and a raid was hourly expected. (Ineglected to tell you that Mrs. Fales sent to me before I started to know if she could go with me. I had the train wait and sent back an ambulance for her and her stores; and this made three ladies--Mrs. Carner is away.
+
+On Monday p.m. all the wounded then in were sent off and the danger became so iminent that Mrs. Fales thought it best to leave, altho she only "went for stroes". I begged to be excused from accompanying her as the ambulances were up to the field for more wounded and I knew I should never leave a wounded man there if I knew it, though I were taken prisoner forty times.
+
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+
+MRS. JOHN A. LOGAN, PRESIDENT
+2523 13TH STREET, WASHINGTON, D. C.
+MRS. ADA H. WEISS, 1ST VICE-PRESIDENT
+WASHINGTON, D. C.
+
+MISS GRACE M. PIERCE, RECORDING SEC'Y
+
+WASHINGTON, D. C.
+
+MRS. MARY LOGAN TUCKER, CORRESPONDING SEC'Y
+WASHINGTON, D. C.
+
+MR. JOHN JOY EDSON, TREASURER
+WASHINGTON LOAN & TRUST CO., WASHINGTON, D. C.
+
+WASHINGTON, D. C., 191
+
+GEN'L NELSON A. MILES
+2nd Vice-President
+Washington, D. C.
+
+GEN'L HARRISON GRAY OTIS
+3rd Vice-President
+Los Angeles, Calif.
+MR. S. B. HEGE,
+4th Vice-President
+Washington, D. C.
+
+MISS JANET RICHARDS
+5th Vice-President
+Washington, D. C.
+
+HON. NEWELL B. WOODWORTH
+6th Vice-President
+Syracuse, N. Y.
+
+MR. W. B. SCOFIELD
+7th Vice-President
+Worcester, Mass.
+
+HON. LEO RASSIEUR
+8th Vice-President
+Granite Building,
+St. Louis, Mo.
+
+At six o'clock it commenced to thunder and lighten, and all at once the artillery began to play joined by the musketry, about two miles distant. We sat down in our tent and waited to see them break in upon us; but Reno's forces held them back. The old 21st Mass. lay between us and the enemy and they couldn't pass.
+
+God only knows who is lost, I do not, for the next day all fell back. Poor Kearney, Stephens and Webster were brought in, and in the p.m. Kearney and Heintzleman's divisions fell back through our camp on their way to Alexandria; we knew this was the last. We put the thousand wounded we had then into the train. I took one car load of them, Mrs. Morell another, the men took to horse. We steamed off and two hours later there was no Fairfax Station. Reached Alexandria at ten o'clock at night, and oh, the repost which met those poor men at the train. The people of the Island are the most noble band I ever saw or heard of. I stood in my car and fed the men 'till they could eat no more, then the people would take us home and feed us and after this we came home.
+
+I had slept 1-1/4 hours since Saturday night; but I am well and a strong and wait to go again if I have need.
+
+Our forces are all back again in the old places around the city. McClellan's army here again and he in command of it all.
+
+I am going to reach for my friends now. I have told you nothing of the old friends who met us among the wounded and dying on that bloody field. I have no heart to tell it today. Two huge boxes from Jersey have arrived. I don't know where we shall need them next. To Archie and Lizzie Shaver, Yours,
+
+Clara Barton,
+Frankfort, N.Y.
+
+Headquarters Military District of Washington,
+Washington, D. C., October 23 1862.
+
+Miss. Clara H. Barton
+
+I have
+
+confended with Gen. Wadsworth on the subject of obtaining a house for your use, and he has directed that I place a house at your disposal as soon as a suitable one can be found.
+
+Very respectfully
+
+Yr. Obt. Secy.
+
+John P. Sherburne
+Asst. Adj. Gen.
+
+No. 488½ 7th Street
+3rd Story - Room No 9.
+
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+
+Camp near Falmouth, Va Head quarters Genl. Sturges (2nd) Division Dec. 8th 1862. —
+
+Mr J Brown & Co.
+
+Dear friends
+
+Mr Wells returns tomorrow, and I improve the opportunity to send a line by him to you not feeling quite certain if posted matter reaches directly when sent from the Army.
+
+We reached Aquia Creek safely in the time anticipated, and to my great joy learned immediately that our old friend Capt (Maj.) Hall (of the 21st) was Q. Master as soon as the Boat was unloaded he came on board and spent the remainder of the evening with me, — we had a home chat I assure you — — remained till the next day, sent a bbl of apples etc up to the Capt's quarters, and proceeded with the remainder of our luggage for which, it is needless to say, ready transportation
+
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+
+was found, and the Capt. chided me for having left any thing behind, at the depot, as I told him I had done, = on reaching Falmouth Station we found another old friend, Capt. Bailey in charge who instituted himself as watch over the goods until he sent them all up to Head Quarters — My Ambulance came through that P.M, but for fear it might not, Genl Sturgis had his taken down for me, and had supper arranged and a splendid serenade; I don't know how we could have had a warmer "welcome home" as the officers termed it.
+
+Head Quarters are in the door yard of a farm house, one room of which is occupied by Miss G, and myself. — My wagons are a little way from me, out of sight, and I am wishing for a tent and stove to pitch and live near them. — the weather is cold, and the ground covered with snow, but I could make me comfortable with a good tent, floor and stove, and should prefer it to a room in a rebel house and one so generally occupied. —
+
+The 21st are a few rods from me, — many of
+
+the officers call to see me every day. Col Clerk is very neighborly, he is looking finely now he was in this P.M, and was going in search of Col Morse whom he thought to be a mile or two distant. I learned to night that the 15th are only some three miles away. the 36th I cannot find yet. I have searched hard for them and shall get on track soon I trust. —
+
+Of Army movements nothing can be said with certainty, no two persons, not even the Genls agree in reference to the future programme the snow appears to have deranged the plans very seriously. I have recd calls from two Genls today and in the course of conversation I discovered that their views were entirely different — Genl Burnside stood a long time in front of my door today but to my astonishment, he did not express his opinion. Strange!
+
+I have not suffered for want of the boots yet, but should find them convenient I presume, and shall be glad to see them. — the sore finger is much the same, not very troublesome, although somewhat, so,
+
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+
+me at home without a question under the circumstances he was wise — Or perhaps he thought I should [come] go without an asking of the Expedition if I could be of any service all of which are erroneous — I made the request of the Gen. at that particular time because the arrival of the petitions made it necessary for me to say something — and intended when I should be at liberty to ask of such friends as I had in the Expedition if I could serve them in any way when an [long] hour of trial and suffering [should come] and disease, for me should come upon them for more terrible than armed men — And this is what I have to say to you — and my honored friend Mr. James — That if in the course of your [life and] [domination] journey in the treacherous soil and climate, there be a time or place when my strength, firm health, and willing hand [can] and somewhat of experience can benefit the just and holy cause in which it is your glorious privilege to serve — for which to [forfeit] forego comfort and peril life, you will tell me and let me come and do it — I know the thousands who are also ready and willing waiting, and I offer not because I can do better or as well perhaps as others, but I am stronger, better acclimated, have firmer health, better able to forego comforts than ladies in general — I had almost said men, — and what is more, my life if I should lose it is worth less to others, there are many few to mourn; and among the most willing, "your fellow servant I'
+
+Washington D.C. Feb 22nd /63 To the members of the Military Comtee U.S.S. Senators Nothing less than a strong conviction of duty owed to one of the brave defenders of our Nation's honor could induce me to intrude for a moment upon the already burdened, and limited term of action yet remaining to your Honorable Body. During the late Battle of Fredericksburg the 21st Massachusetts Regiment of Volunteers were ordered to charge upon a Battery across an open field; in the terrible fire which assailed them the colors were three times in quick succession bereft of their support; the third time they were seized by Sergeant Thomas Plunkett of Co. E. and borne over some three hundred yards of open space when a shell from the Enemies Battery in its murderous course killed three men of the regiment and shattered both arms
+
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+
+of the Sergeant. He could no longer support the colors upright, but planting his foot against the staff, he endeavored to hold them up, while he strove by his shouts amid the confusion to attract attention to their condition; for some minutes he sustained them against his right arm, torn and shattered just below the shoulder, while the blood poured over and among the sacred folds, literally obliterating the stripes, leaving as fit emblem of such heroic sacrifice, only the crimson and the stars. Thus drenched in blood, and rent by the fury of eight battles, the noble Standard could be no longer borne, and while its gallant defender lay suffering in quiet hospital from amputation of both arms, reverently wrapped by Col Clark and returned to the State House in Boston, with the request that others might be sent them; The 21st had never lost their colors but they had worn them out,
+
+The old flag and its brave bearer are alike past their usefulness save as examples
+
+for emulation and titles of glory for some bright page of our nation's history, and while the one is carefully treasured in the sacred archives of the State, need I more than ask of this noble body to put forth its protecting arm to shelter, cherish and sustain, the other. If guarantees were needful for the private character of so true a soldier, it would have been found in the touching address of his eloquent Colonel (Clark) delivered on Christmas beside the stretcher waiting at the train at Falmouth to convey its [burden] helpless burden to the car, whither he had been escorted not only by his regiment but his General The tears which rolled over the veteran cheeks around him were ample testimony of the love and respect he had won from then and today his hearts deepest affections twine around his gallant regiment as the defenders of their nation's country.
+
+A moments reflection will obviate the necessity of any suggestions in reference
+
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+
+May 22nd, 1914.
+
+Hon. Theodore Roosevelt,
+
+Oyster Bay, N.Y.
+
+Dear Sir:
+
+In Dec. 1903, I had passed through the Senate and House, at Salem, Oregon, a Resolution committing the Legislature of the State of Oregon, to your renomination at that time. I am still making another prediction, which you will observe from the inclosed interview. I am still with you, as I have always been, because I have felt that while you have seemed to think that I was guilty of criminal misconduct, someday I am going to convince you that you have been deceived, and I have never had any hard feeling towards you because I have felt that you had to believe what certain parties in Oregon felt in their duty, at all times to impress upon you.
+
+I was a candidate for governor this year, and polled a splendid vote although there were eight other candidates in the field.
+
+I am not writing this letter with any hopes of reward from you, or anything of that kind; I simply believe that you are the man to meet the demand of the country in the next Presidential contest, and I am for you. You will carry Oregon with a big majority. Everything indicates that the sentiment here now, is very strong for you.
+
+Very sincerely yours,
+
+Geo. C. Brownell
+
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+
+EXAMPLES ON THE INTEGRAL CALCULUS.
+
+CHAPTER I.
+
+ELEMENTARY INTEGRALS TO BE COMMITTED TO MEMORY.
+
+\[(1.) \int \frac{dx}{x} = \log x.\]
+
+\[(2.) \int \frac{dx}{a+bx^2} = \frac{1}{\sqrt{ab}} \tan^{-1} \left( x \sqrt{\frac{b}{a}} \right), \text{ or } \int \frac{dx}{a^2+x^2} = \frac{1}{a} \tan^{-1} \frac{x}{a}.\]
+
+\[(3.) \int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1} \frac{x}{a}.\]
+
+\[(4.) \int \frac{-dx}{\sqrt{a^2-x^2}} = \cos^{-1} \frac{x}{a}.\]
+
+\[(5.) \int \frac{dx}{\sqrt{x^2 \pm a^2}} = \log \frac{(x + \sqrt{x^2 \pm a^2})}{a}.\]
+
+\[(6.) \int \frac{dx}{\sqrt{2ax-x^2}} = \operatorname{vers}^{-1} \frac{x}{a}.\]
+
+\[(7.) \int \frac{dx}{\sqrt{x^2 \pm 2ax}} = \log (x \pm a + \sqrt{x^2 \pm 2ax}).\]
+
+\[(8.) \int \frac{dx}{x \sqrt{x^2-a^2}} = \frac{1}{a} \sec^{-1} \frac{x}{a}.\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/1_pg113_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+(8.) To find the volume of a parabolic spindle.
+
+Put \(CD = h\) , \(AB = 2c\) , and
+
+\(x = AG\) , and \(y = FG\) .
+
+From the property of the parabola
+
+\[AD^2 : AG \cdot GB :: CD : EG,\]
+
+\[c^2 : x(2c - x) :: h : y,\]
+
+\[\therefore y = \frac{h(2cx - x^2)}{c^2}, \therefore y^2 = \frac{h^2}{c^4} (4c^2x^2 - 4cx^3 + x^4),\]
+
+\[\therefore V = \pi \int y^2 dx\]
+
+\[= \pi \frac{h^2}{c^4} \int (4c^2x^2 - 4cx^3 + x^4) dx = \frac{\pi h^2}{c^4} \left( \frac{4c^2x^3}{3} - cx^4 + \frac{x^5}{5} \right),\]
+
+the volume of AFH, since \(C = 0\) ; when \(x = c\) , we have
+
+\[\frac{\pi h^2}{c^4} \left( \frac{4c^5}{3} - c^5 + \frac{c^5}{5} \right) = \frac{8}{15} \pi h^2 c =\]
+
+volume of the semi-spindle; and, as we have found the volume of the part AFH, if we subtract this from the whole semi-spindle, we shall have the frustum EHCF, the double of which will be volume of the whole frustum EHIL.
+
+In the same manner as in the last example, we may find the surface of the parabolic spindle.
+
+(9.) In a parabola find the area included between the curve, its evolute, and its radius of curvature.
+
+\[\text{area parabola ANP} = \int y dx\]
+
+\[= 2 \sqrt{a} \int \sqrt{x} dx = \frac{4}{3} \sqrt{a} x^{\frac{3}{2}},\]
+
+\[\text{area evolute AN'P'} = \int \beta d\alpha = \frac{2}{3\sqrt{3a}} \int (\alpha - 2a)^{\frac{3}{2}} d\alpha\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/1_pg125_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+\[\therefore \text{semi-area} = \frac{c(c^2 - a^2)}{4a} \sin^{-1} \frac{\sqrt{c^2 - a^2}}{\sqrt{c^2 - a^2}}\]
+
+\[= \frac{c(c^2 - a^2)}{4a} \frac{\pi}{2}\]
+
+\[\text{area circle} = \frac{a}{2} 2\pi b = \pi ab,\]
+
+\[\text{area between epicycloid and circle} = \frac{c(c^2 - a^2)}{4a} \pi - \pi ab\]
+
+\[= \frac{(a + 2b)4b(a + b)}{4a} \pi - \pi ab\]
+
+\[= \frac{\pi}{a} (a^2b + 3ab + 2b^3 - a^2b)\]
+
+\[= \pi b^2 \left( 3 + \frac{2b}{a} \right).\]
+
+(18.) Find the length of the curve where
+
+\[x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}},\]
+
+\[s = \int \sqrt{1 + \frac{dy^2}{dx^2}} = \int \sqrt{1 + \frac{y^{\frac{2}{3}}}{x^{\frac{2}{3}}}}\]
+
+\[= \int \sqrt{\frac{x^{\frac{2}{3}} + y^{\frac{2}{3}}}{x^{\frac{1}{3}}}} = \int \frac{a^{\frac{1}{3}}}{x^{\frac{1}{3}}} = \frac{3}{2} a^{\frac{1}{3}} x^{\frac{3}{2}}\]
+
+Taking it between the limits \(x = 0\) , \(x = a\) ,
+
+\[s = \frac{3}{2} a.\]
+
+The whole length of the curve \(4 \times \frac{3}{2} a = 6a\) .
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/1_pg131_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+\[\text{And } \int \frac{dx}{\sqrt{2ax+x^2}} = \log \{x+a+\sqrt{2ax+x^2}\},\]
+
+\[\therefore \int \frac{x dx}{\sqrt{2ax+x^2}}\]
+
+\[= \sqrt{2ax+x^2} - a \log \{a+a+\sqrt{2ax+x^2}\}.\]
+
+\[\therefore \int \frac{x^2 dx}{\sqrt{2ax+x^2}} = \frac{x\sqrt{2ax+x^2}}{2} - \frac{3a}{2} \sqrt{2ax+x^2}\]
+
+\[+ \frac{3a^2}{2} \log \{x+a+\sqrt{2ax+x^2}\}.\]
+
+\[(3.) \quad \int_a^0 \frac{x^2 dx}{(2ax-x^2)^{\frac{3}{2}}}\]
+
+\[\text{let } p=x \quad dp=dx, \quad dq = \frac{x dx}{(2ax-x^2)^{\frac{3}{2}}},\]
+
+\[\therefore q = \int \frac{x dx}{(2ax-x)^{\frac{3}{2}}} = \int (2a-x)^{-\frac{3}{2}} x^{\frac{1}{2}} dx =\]
+
+\[\int (2ax^{-1}-1)^{-\frac{3}{2}} x^{-2} dx\]
+
+\[= -\frac{1}{2a} \int (2ax^{-1}-1)^{-\frac{3}{2}} \times -2ax^{-2} dx\]
+
+\[= \frac{(2ax^{-1}-1)^{-\frac{1}{2}}}{a} = \frac{1}{a} \sqrt{\frac{x}{2a-x}}.\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/1_pg19_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+\[\text{Let } \frac{x^2}{(x^2 + 1)(x^2 + 4)} = \frac{A}{(x^2 + 1)} + \frac{B}{(x^2 + 4)}\]
+
+\[x^2 = A(x^2 + 4) + B(x^2 + 1).\]
+
+\[\text{Let } x = \sqrt{-1}, \text{ or } x^2 = -1,\]
+
+\[\therefore -1 = 3A, \quad \therefore A = -\frac{1}{3},\]
+
+\[\therefore x^2 + \frac{1}{3}(x^2 + 4) = B(x^2 + 1),\]
+
+\[\therefore \frac{4(x^2 + 1)}{3} = B(x^2 + 1), \quad \therefore B = \frac{4}{3}\]
+
+\[\int \frac{x^2 dx}{(x^2 + 1)(x^2 + 4)} = -\frac{1}{3} \int \left\{ \frac{dx}{(x^2 + 1)} - \int \frac{4 dx}{(x^2 + 4)} \right\}\]
+
+\[= \frac{1}{3} \int \left\{ \frac{dx}{(1 + x^2)} - \frac{dx}{\left(1 + \frac{x^2}{4}\right)} \right\}\]
+
+\[= \frac{1}{3} \left\{ 2 \tan^{-1} \frac{x}{2} - \tan^{-1} x \right\}.\]
+
+\[\text{Or thus, } \int \frac{x^2 dx}{(x^2 + 1)(x^2 + 4)} = \frac{1}{3} \int \frac{3x^2 dx}{(x^2 + 1)(x^2 + 4)}\]
+
+\[= \frac{1}{3} \int \frac{\{4x^2 + 4 - (x^2 + 4)\} dx}{(x^2 + 1)(x^2 + 4)}\]
+
+\[= \frac{1}{3} \int \left( \frac{4 dx}{(x^2 + 4)} - \frac{dx}{x^2 + 1} \right)\]
+
+\[= \frac{1}{3} \left( 2 \tan^{-1} \frac{x}{2} - \tan^{-1} x \right)\]
+
+\[(5.) \quad du = \frac{(3x^2 + x - 2) dx}{(x - 1)^3 (x^2 + 1)}.\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/1_pg40_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+and the integral required is therefore
+
+\[\log \sqrt{x^2 + 1} + \frac{1}{x^2 + 1} - \frac{1}{4(x^2 + 1)^2},\]
+
+\[= \frac{4x^2 + 3}{4(x^2 + 1)^2} + \log \sqrt{x^2 + 1}.\]
+
+\[(22.) \int \frac{x^2 dx}{x^4 + 1},\]
+
+\[(x^m + 1) = \left( x^2 - 2x \cos \frac{\pi}{m} + 1 \right) \left( x^2 - 2x \cos \frac{3\pi}{m} + 1 \right) \dots\]
+
+continued to the factor \(\left( x^2 - 2x \cos \frac{m-1}{m} \pi + 1 \right)\) when \(m\) is an even number.
+
+This gives
+
+\[(x^4 + 1) = \left( x^2 - 2x \cos \frac{\pi}{4} + 1 \right) \left( x^2 - 2x \cos \frac{3\pi}{4} + 1 \right),\]
+
+\[\text{or } (x^4 + 1) = (x^2 - x\sqrt{2} + 1)(x^2 + x\sqrt{2} + 1).\]
+
+\[\text{Since, } \frac{\pi}{4} = 45^\circ \cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}, \cos \frac{3\pi}{4} = -\sin \frac{\pi}{4}.\]
+
+Assume therefore
+
+\[\frac{x^2}{x^4 + 1} = \frac{Ax + B}{x^2 - x\sqrt{2} + 1} + \frac{Cx + D}{x^2 + x\sqrt{2} + 1};\]
+
+\[\therefore x^2 = (Ax + B)(x^2 + x\sqrt{2} + 1) + (Cx + D)(x^2 - x\sqrt{2} + 1).\]
+
+\[\text{If } x^2 + x\sqrt{2} + 1 = 0, \quad x = \frac{\sqrt{-1} - 1}{\sqrt{2}} \quad (1.)\]
+
+\[\text{If } x^2 - x\sqrt{2} + 1 = 0, \quad x = \frac{\sqrt{-1} + 1}{\sqrt{2}} \quad (2.)\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/1_pg61_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+\[\therefore \int \frac{dx}{(1+x)\sqrt{1-x-x^2}} = - \int \frac{dz}{z^2 \frac{1}{z} \sqrt{1 + \frac{1}{z} - \frac{1}{z^2}}}\]
+
+\[= - \int \frac{dz}{\sqrt{z^2 + z - 1}} = - \int \frac{dz}{\sqrt{\left(z + \frac{1}{2}\right)^2 - \frac{5}{4}}}\]
+
+\[= - \log \left\{ \sqrt{z^2 + z - 1} + z + \frac{1}{2} \right\}\]
+
+\[= - \log \left\{ \frac{\sqrt{1-x-x^2}}{1+x} + \frac{1}{1+x} + \frac{1}{2} \right\}\]
+
+\[= - \log \left\{ \frac{2\sqrt{1-x-x^2} + x + 3}{2(1+x)} \right\}\]
+
+\[= \log \left\{ \frac{2(1+x)}{2\sqrt{1-x-x^2} + x + 3} \right\}\]
+
+\[= \log \left\{ \frac{2(1+x)(x+3) - 2\sqrt{1-x-x^2}}{x^2 + 6x + 9 - 4 + 4x + 4x^2} \right\}\]
+
+\[= \log \frac{2(1+x)(x+3 - 2\sqrt{1-x-x^2})}{5(1+x)^2}\]
+
+\[= \log \frac{x+3 - 2\sqrt{1-x-x^2}}{1+x} + c.\]
+
+\[(36.) \quad \int \frac{dx}{\sqrt{1+2x-x^2}} = \int \frac{dx}{\sqrt{2-(1-x)^2}}\]
+
+\[= - \int \frac{d(1-x)}{\sqrt{2-(1-x)^2}} = \cos^{-1} \frac{1-x}{\sqrt{2}}.\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/1_pg63_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+\[(39.) \quad \int \frac{\sqrt{x} \, dx}{\sqrt{a^3 - x^3}}. \quad \text{Let } x^3 = z^2\]
+
+\[x = z^{\frac{2}{3}}, \quad dx = \frac{2}{3} z^{-\frac{1}{3}} dz, \quad \sqrt{x} = z^{\frac{1}{3}},\]
+
+\[\therefore \int \frac{\sqrt{x} \, dx}{\sqrt{a^3 - x^3}} = \frac{2}{3} \int \frac{dz}{\sqrt{a^3 - z^2}} = \frac{2}{3} \sin^{-1} \frac{z}{a^{\frac{3}{2}}}\]
+
+\[= \frac{2}{3} \sin^{-1} \frac{x^{\frac{2}{3}}}{a^{\frac{3}{2}}}\]
+
+\[\text{Let } \theta = \sin^{-1} \frac{x^{\frac{2}{3}}}{a^{\frac{3}{2}}}, \quad \sin \theta = \frac{x^{\frac{2}{3}}}{a^{\frac{3}{2}}},\]
+
+\[\tan \theta = \frac{\sin \theta}{\sqrt{1 - \sin^2 \theta}} = \frac{x^{\frac{2}{3}}}{\sqrt{a^3 - x^3}},\]
+
+\[\therefore \int \frac{\sqrt{x} \, dx}{\sqrt{a^3 - x^3}} = \frac{2}{3} \tan^{-1} \sqrt{\frac{x^3}{a^3 - x^3}}.\]
+
+\[(40.) \quad \int \frac{dx}{(2ax + x^2)^{\frac{3}{2}}} = \int \frac{dx}{((x+a)^2 - a^2)^{\frac{3}{2}}}.\]
+
+\[\text{Let } x + a = \frac{1}{z}, \quad dx = -\frac{dz}{z^2}\]
+
+\[= - \int \frac{dz}{z^2 \left( \frac{1}{z^2} - a^2 \right)^{\frac{3}{2}}} = - \int \frac{z \, dz}{(1 - a^2 z^2)^{\frac{3}{2}}}.\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/1_pg72_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+\[\therefore \int \frac{dx}{(a^2 + x^2)^n} = \frac{1}{a^2} \frac{x}{(2n-2)(a^2 + x^2)^{n-1}}\]
+
+\[- \frac{1}{a^2 \cdot (2n-2)} \int \frac{dx}{(a^2 + x^2)^{n-1}} + \frac{1}{a^2} \int \frac{dx}{(a^2 + x^2)^{n-1}}\]
+
+\[= \frac{1}{a^2} \frac{x}{(2n-2)(a^2 + x^2)^{n-1}} + \frac{2n-3}{a^2(2n-2)} \int \frac{dx}{(a^2 + x^2)^{n-1}}.\]
+
+Also for \(\int \frac{x^m dx}{(a^2 + x^2)^n} = \int x^{m-1} dx \frac{x}{(a^2 + x^2)^n}\) . We have,
+
+if \(p = x^{m-1}\) , \(dq = x(a^2 + x^2)^{-n} dx\) :
+
+\[\int \frac{x^m dx}{(a^2 + x^2)^n}\]
+
+\[= \frac{1}{2-2n} \frac{x^{m-1}}{(a^2 + x^2)^{n-1}} + \frac{m-1}{m-2} \int \frac{x^{m-2} dx}{(a^2 + x^2)^{n-1}}\]
+
+\[du = (a^2 - x^2)^{\frac{n}{2}} dx\]
+
+\[(a^2 - x^2)^{\frac{n}{2}} = a^2 (a^2 - x^2)^{\frac{n-2}{2}} - x^2 (a^2 - x^2)^{\frac{n-2}{2}};\]
+
+\[\therefore u = a^2 \int (a^2 - x^2)^{\frac{n-2}{2}} dx - \int x \cdot x (a^2 - x^2)^{\frac{n-2}{2}} dx\]
+
+\[= a^2 \int (a^2 - x^2)^{\frac{n-2}{2}} dx + \frac{x(a^2 - x^2)^{\frac{n}{2}}}{n} - \frac{u}{n};\]
+
+\[\therefore u = \frac{x(a^2 - x^2)^{\frac{n}{2}}}{n+1} + \frac{na^2}{n+1} \int (a^2 - x^2)^{\frac{n-2}{2}} dx.\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/2_pg189_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF's own text layer is an OCR layer, not the author's text.[^inferred]
+
+Integration gives,
+
+\[\frac{dx}{dt} = C_1, \quad \frac{dy}{dt} = -gt + C_2.\]
+
+When \(t = 0\) , \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\) are the components of \(v_0\) . Hence
+
+\[C_1 = v_0 \cos \alpha, \quad C_2 = v_0 \sin \alpha,\]
+
+and
+
+\[\frac{dx}{dt} = v_0 \cos \alpha,\]
+
+\[\frac{dy}{dt} = v_0 \sin \alpha - gt.\]
+
+Integrating again, we get
+
+\[x = v_0 t \cos \alpha,\]
+
+\[y = v_0 t \sin \alpha - \frac{1}{2} gt^2,\]
+
+the constants being zero because \(x\) and \(y\) are zero when \(t = 0\) .
+
+5. Curves with a Given Slope. —If the slope of a curve is a given function of \(x\) ,
+
+FIG. 5.
+
+FIG. 4.
+
+then
+
+\[\frac{dy}{dx} = f(x),\]
+
+and
+
+\[dy = f(x) dx\]
+
+\[y = \int f(x) dx + C\]
+
+is the equation of the curve.
+
+Since the constant can have any value, there are an infinite number of curves having the given slope. If the curve is required to pass through a given point \(P\) , the
+
+value of \(C\) can be found by substituting the coördinates of \(P\) in the equation after integration.
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/2_pg238_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF's own text layer is an OCR layer, not the author's text.[^inferred]
+
+✓ 4. Find the volume generated by revolving about \$OY\$ the area bounded by the coördinate axes and the parabola \(x^{\frac{1}{2}} + y^{\frac{1}{2}} = a^{\frac{1}{2}}\) .
+
+5. Find the volume generated by revolving about the \(x\) -axis the area bounded by the catenary \(y = \frac{a}{2} \left( e^{\frac{x}{a}} + e^{-\frac{x}{a}} \right)\) , the \(x\) -axis and the lines \(x = \pm a\) .
+
+6. Find the volume generated by revolving one arch of the sine curve \(y = \sin x\) about \$OX\$ .
+
+7. A cone has its vertex on the surface of a sphere and its axis coincides with a diameter of the sphere. Find the common volume.
+
+8. Find the volume generated by revolving about the \(y\) -axis, the part of the parabola \(y^2 = 4ax\) cut off by the line \(x = a\) .
+
+9. Find the volume generated by revolving about \(x = a\) the part of the parabola \(y^2 = 4ax\) cut off by the line \(x = a\) .
+
+10. Find the volume generated by revolving about \(y = -2a\) the part of the parabola \(y^2 = 4ax\) cut off by the line \(x = a\) .
+
+11. Find the volume generated by revolving one arch of the cycloid
+
+\[x = a(\phi - \sin \phi), \quad y = a(1 - \cos \phi)\]
+
+about the \(x\) -axis.
+
+12. Find the volume generated by revolving the curve
+
+\[x = a \cos^3 \phi, \quad y = a \sin^3 \phi\]
+
+about the \(y\) -axis.
+
+13. Find the volume generated by revolving the cardioid \(r = a(1 + \cos \theta)\) about the initial line.
+
+14. Find the volume generated by revolving the cardioid \(r = a(1 + \cos \theta)\) about the line \(x = -\frac{a}{4}\) .
+
+15. Find the volume generated by revolving the ellipse
+
+\[x^2 + xy + y^2 = 3\]
+
+about the \(x\) -axis.
+
+16. Find the volume generated by revolving about the line \(y = x\) the part of the parabola \(x^{\frac{1}{2}} + y^{\frac{1}{2}} = a^{\frac{1}{2}}\) cut off by the line \(x + y = a\) .
+
+29. Volume of a Solid with Given Area of Section. —Divide the solid into slices by parallel planes. Let \(X\) be the area of section at distance \(x\) from a fixed point. The plate \$PQRS\$ with lateral surface perpendicular to \$PQR\$ has the volume
+
+\[PQR \cdot \Delta x = X \Delta x.\]
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

outputs/old_scans_math/2_pg349_pg1_repeat1.md

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+> This page was read from its image by a model (olmocr); the PDF's own text layer is an OCR layer, not the author's text.[^inferred]
+
+215. Moment of inertia of a right circular cylinder about a line tangent to its base.
+
+216. Moment of inertia of a segment cut from a sphere by a plane, about a diameter parallel to that plane.
+
+217. Center of gravity of a right circular cone whose density varies as the distance from the center of the base.
+
+218. Volume bounded by the \$xy\$ -plane, the cylinder \(x^2 + y^2 = 2ax\) and the cone \(z^2 = x^2 + y^2\) .
+
+219. Find the attraction of a uniform wire of length \(l\) and mass \(M\) on a particle of unit mass at distance \(c\) from the wire in the perpendicular at one end.
+
+220. Find the attraction of a right circular cylinder on a particle at the middle of its base.
+
+221. Show that the attraction of a homogeneous shell bounded by two concentric spherical surfaces on a particle in the enclosed space is zero.
+
+CHAPTER X
+
+Solve the following differential equations:
+
+222. \(y \, dx + (x - xy) \, dy = 0\) .
+
+223. \(\sin x \sin y \, dx + \cos x \cos y \, dy = 0\) .
+
+224. \((2xy - y^2 + 6x^2) \, dx + (3y^2 + x^2 - 2xy) \, dy = 0\) .
+
+225. \(x \frac{dy}{dx} + y = x^3y\) .
+
+226. \(x \frac{dy}{dx} + y = \cot x\) .
+
+227. \(x \, dy - \left( y + e^{\frac{1}{x}} \right) \, dx = 0\) .
+
+228. \((1 + x^2) \, dy + (xy + x) \, dx = 0\) .
+
+229. \(x \, dx + y \, dy = x \, dy - y \, dx\) .
+
+230. \((\sin x + y) \, dy + (y \cos x - x^2) \, dx = 0\) .
+
+231. \(y (e^x + 2) \, dx + (e^x + 2x) \, dy = 0\) .
+
+232. \((xy^2 - x) \, dx + (y + xy) \, dy = 0\) .
+
+233. \((1 + x^2) \frac{dy}{dx} + xy = 2y\) .
+
+234. \(x \, dy - y \, dx = \sqrt{x^2 + y^2} \, dx\) .
+
+235. \((x - y) \, dx + x \, dy = 0\) .
+
+236. \(x \, dy - y \, dx = x \sqrt{x^2 + y^2} \, dx\) .
+
+237. \(e^{x+y} \, dy + (1 + e^y) \, dx = 0\) .
+
+238. \((2x + 3y - 1) \, dx + (4x + 6y - 5) \, dy = 0\) .
+
+239. \((3y^2 + 3xy + x^2) \, dx = (x^2 + 2xy) \, dy\) .
+
+240. \((1 + x^2) \, dy + (xy - x^2) \, dx = 0\) .
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+> This page was read from its image by a model (olmocr); the PDF's own text layer is an OCR layer, not the author's text.[^inferred]
+
+Using V, with \(u = x^2 + 1\) , \(v = x^2 - 1\) ,
+
+\[dy = \frac{(x^2 - 1) d(x^2 + 1) - (x^2 + 1) d(x^2 - 1)}{(x^2 - 1)^2}\]
+
+\[= \frac{(x^2 - 1) 2x dx - (x^2 + 1) 2x dx}{(x^2 - 1)^2}\]
+
+\[= -\frac{4x dx}{(x^2 - 1)^2}.\]
+
+Ex. 5. \(y = \sqrt{x^2 - 1}\) .
+
+Using VI, with \(u = x^2 - 1\) ,
+
+\[\frac{dy}{dx} = \frac{d}{dx} (x^2 - 1)^{\frac{1}{2}} = \frac{1}{2} (x^2 - 1)^{-\frac{1}{2}} \frac{d}{dx} (x^2 - 1)\]
+
+\[= \frac{1}{2} (x^2 - 1)^{-\frac{1}{2}} (2x) = \frac{x}{\sqrt{x^2 - 1}}.\]
+
+Ex. 6. \(x^2 + xy - y^2 = 1\) .
+
+We can consider \(y\) a function of \(x\) determined by the equation. Then
+
+\[d(x^2) + d(xy) - d(y^2) = d(1) = 0,\]
+
+that is,
+
+\[2x dx + x dy + y dx - 2y dy = 0,\]
+
+\[(2x + y) dx + (x - 2y) dy = 0.\]
+
+Consequently,
+
+\[\frac{dy}{dx} = \frac{2x + y}{2y - x}.\]
+
+Ex. 7. \(x = t + \frac{1}{t}\) , \(y = t - \frac{1}{t}\) .
+
+In this case
+
+\[dx = dt - \frac{dt}{t^2}, \quad dy = dt + \frac{dt}{t^2}.\]
+
+Consequently,
+
+\[\frac{dy}{dx} = \frac{1 + \frac{1}{t^2}}{1 - \frac{1}{t^2}} = \frac{t^2 + 1}{t^2 - 1}.\]
+
+Ex. 8. Find an approximate value of \(y = \left(\frac{1-x}{1+x}\right)^{\frac{1}{3}}\) when \(x = 0.2\) .
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+> This page was read from its image by a model (olmocr); the PDF's own text layer is an OCR layer, not the author's text.[^inferred]
+
+Using the formula just proved,
+
+\[d \cos u = d \sin \left( \frac{\pi}{2} - u \right) = \cos \left( \frac{\pi}{2} - u \right) d \left( \frac{\pi}{2} - u \right) = -\sin u \, du.\]
+
+38. Proof of IX, X, XI, and XII. — Differentiating both sides of the equation
+
+\[\tan u = \frac{\sin u}{\cos u}\]
+
+and using the formulas just proved for the differentials of \(\sin u\) and \(\cos u\) ,
+
+\[d \tan u = \frac{\cos u \, d \sin u - \sin u \, d \cos u}{\cos^2 u} = \frac{\cos^2 u \, du + \sin^2 u \, du}{\cos^2 u}\]
+
+\[= \sec^2 u \, du.\]
+
+By differentiating both sides of the equations
+
+\[\cot u = \frac{\cos u}{\sin u}, \quad \sec u = \frac{1}{\cos u}, \quad \csc u = \frac{1}{\sin u},\]
+
+and using the formulas for the differentials of \(\sin u\) and \(\cos u\) , we obtain the differentials of \(\cot u\) , \(\sec u\) and \(\csc u\) .
+
+Example 1. \(y = \sin^2 (x^2 + 3)\) .
+
+Since
+
+\[\sin^2 (x^2 + 3) = [\sin (x^2 + 3)]^2,\]
+
+we use the formula for \(u^2\) and so get
+
+\[dy = 2 \sin (x^2 + 3) \, d \sin (x^2 + 3)\]
+
+\[= 2 \sin (x^2 + 3) \cos (x^2 + 3) \, d (x^2 + 3)\]
+
+\[= 4x \sin (x^2 + 3) \cos (x^2 + 3) \, dx.\]
+
+Ex. 2. \(y = \sec 2x \tan 2x\) .
+
+\[\frac{dy}{dx} = \sec 2x \frac{d}{dx} \tan 2x + \tan 2x \frac{d}{dx} \sec 2x\]
+
+\[= \sec 2x \sec^2 2x (2) + \tan 2x \sec 2x \tan 2x (2)\]
+
+\[= 2 \sec 2x (\sec^2 2x + \tan^2 2x).\]
+
+EXERCISES
+
+In the following exercises show that the derivatives and differentials have the values given:
+
+✓1. \(y = 2 \sin 3x + 3 \cos 2x\) , \(\quad \frac{dy}{dx} = 6 (\cos 3x - \sin 2x)\) .
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+with the axes). By differentiation we see at once that these components are the derivatives of the function
+
+\[\frac{\sigma}{\sqrt{((\xi_1 - x_1)^2 + (\xi_2 - x_2)^2 + (\xi_3 - x_3)^2))}}\]
+
+with respect to the co-ordinates \(x_1, x_2, x_3\) respectively. The force vector apart from a constant factor is therefore the gradient of the function
+
+\[\frac{1}{r} = \frac{1}{\sqrt{((\xi_1 - x_1)^2 + (\xi_2 - x_2)^2 + (\xi_3 - x_3)^2))}}.\]
+
+If a field of force is obtained from a scalar function by forming the gradient, this scalar function is often called the potential function of the field. We shall consider this concept from a more general point of view in the study of work and energy (Chapter V, p 350, and Chapter VI, pp 415, 468-81).
+
+4. The Divergence and Curl of a Vector Field.
+
+By differentiation we have assigned to every function or scalar a vector field, the gradient. Similarly, by differentiation we can assign to every vector field a certain scalar, known as the divergence of the vector field. Given a specific co-ordinate system, the \(x\) -system, we define the divergence of the vector \(\mathbf{u}\) as the function
+
+\[\operatorname{div} \mathbf{u} = \frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_2} + \frac{\partial u_3}{\partial x_3},\]
+
+i.e. the sum of the partial derivatives of the three components with respect to the corresponding co-ordinates. Suppose now that we change the co-ordinate system to the \(\xi\) -system. If the divergence is really to be a scalar function associated with the vector field and independent of the particular co-ordinate system, we must have
+
+\[\operatorname{div} \mathbf{u} = \frac{\partial \omega_1}{\partial \xi_1} + \frac{\partial \omega_2}{\partial \xi_2} + \frac{\partial \omega_3}{\partial \xi_3},\]
+
+where \(\omega_1, \omega_2, \omega_3\) are the components of \(\mathbf{u}\) in the \(\xi\) -system. In fact, the truth of the equation
+
+\[\frac{\partial u_1}{\partial x_1} + \frac{\partial u_2}{\partial x_2} + \frac{\partial u_3}{\partial x_3} = \frac{\partial \omega_1}{\partial \xi_1} + \frac{\partial \omega_2}{\partial \xi_2} + \frac{\partial \omega_3}{\partial \xi_3}\]
+
+can be verified immediately by applying the chain rule and the transformation formulæ of p 84.
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+is no longer an "affine" transformation, but a transformation to general curvilinear co-ordinates
+
+We again assume that when \((x, y)\) ranges over a region \(R\) of the \$xy\$ -plane the corresponding point \((\xi, \eta)\) ranges over a region \(B\) of the \(\xi\eta\) -plane, and also that for each point of \(B\) the corresponding \((x, y)\) in \(R\) can be uniquely determined, in other words, that the transformation is one-to-one. The inverse transformation we again denote by \(x = g(\xi, \eta)\) , \(y = h(\xi, \eta)\)
+
+By the co-ordinates of a point \(P\) in a region \(R\) we can mean any number-pair which serves to specify the position of the point \(P\) in \(R\) uniquely. Rectangular co-ordinates are the simplest case of co-ordinates which extend over the whole plane. Another typical case is the system of polar co-ordinates in the \$xy\$ -plane, introduced by the equations
+
+\[\xi = r = \sqrt{(x^2 + y^2)},\]
+
+\[\eta = \theta = \text{arc tan } (y/x) \quad (0 \leq \theta < 2\pi).\]
+
+When we are given a system of functions \(\xi = \phi(x, y)\) , \(\eta = \psi(x, y)\) as above, we can in general assign to each point \(P\) \((x, y)\) the corresponding values \((\xi, \eta)\) as new co-ordinates. For each pair of values \((\xi, \eta)\) belonging to the region \(B\) uniquely determines the pair \((x, y)\) , and thus uniquely determines the position of the point \(P\) in \(R\) , this entitles us to call \(\xi, \eta\) the co-ordinates of the point \(P\) . The "co-ordinate lines" \(\xi = \text{const}\) and \(\eta = \text{const}\) are then represented in the \$xy\$ -plane by two families of curves, which are defined implicitly by the equations \(\phi(x, y) = \text{const}\) and \(\psi(x, y) = \text{const}\) respectively. These co-ordinate curves cover the region \(R\) with a co-ordinate net (usually curved), for which reason the co-ordinates \((\xi, \eta)\) are also called curvilinear co-ordinates in \(R\) .
+
+We shall once again point out how closely these two interpretations of our system of equations are interrelated. The curves in the \(\xi\eta\) -plane which in the mapping correspond to straight lines parallel to the axes in the \$xy\$ -plane can be directly regarded as the co-ordinate curves for the curvilinear co-ordinates \(x = g(\xi, \eta)\) , \(y = h(\xi, \eta)\) in the \(\xi\eta\) -plane; conversely, the co-ordinate curves of the curvilinear co-ordinate system \(\xi = \phi(x, y)\) , \(\eta = \psi(x, y)\) in the \$xy\$ -plane in the mapping are the images of the straight lines parallel to the axes in the \(\xi\eta\) -plane. Even in the interpretation of \((\xi, \eta)\) as curvilinear co-ordinates in the \$xy\$ -plane
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+complete symmetry. Lagrange's rule is usually expressed as follows.
+
+To find the extreme values of the function \(f(x, y)\) subject to the subsidiary condition \(\phi(x, y) = 0\) , we add to \(f(x, y)\) the product of \(\phi(x, y)\) and an unknown factor \(\lambda\) independent of \(x\) and \(y\) , and write down the known necessary conditions,
+
+\[f_x + \lambda\phi_x = 0, \quad f_y + \lambda\phi_y = 0,\]
+
+for an extreme value of \(F = f + \lambda\phi\) . In conjunction with the subsidiary condition \(\phi = 0\) these serve to determine the co-ordinates of the extreme value and the constant of proportionality \(\lambda\) .
+
+Before proceeding to prove the rule of undetermined multipliers rigorously we shall illustrate its use by means of a simple example. We wish to find the extreme values of the function
+
+\[u = xy\]
+
+on the circle with unit radius and centre the origin, that is, with the subsidiary condition
+
+\[x^2 + y^2 - 1 = 0\]
+
+According to our rule, by differentiating \(xy + \lambda(x^2 + y^2 - 1)\) with respect to \(x\) and to \(y\) we find that at the stationary points the two equations
+
+\[y + 2\lambda x = 0\]
+
+\[x + 2\lambda y = 0\]
+
+have to be satisfied. In addition we have the subsidiary condition
+
+\[x^2 + y^2 - 1 = 0.\]
+
+On solving we obtain the four points
+
+\[\xi = \frac{1}{2}\sqrt{2}, \eta = \frac{1}{2}\sqrt{2},\]
+
+\[\xi = -\frac{1}{2}\sqrt{2}, \eta = -\frac{1}{2}\sqrt{2},\]
+
+\[\xi = \frac{1}{2}\sqrt{2}, \eta = -\frac{1}{2}\sqrt{2},\]
+
+\[\xi = -\frac{1}{2}\sqrt{2}, \eta = \frac{1}{2}\sqrt{2}\]
+
+The first two of these give a maximum value \(u = \frac{1}{2}\) , the second two a minimum value \(u = -\frac{1}{2}\) , of the function \(u = xy\) . That the first two do really give the greatest value and the second two the least value of the function \(u\) can be seen as follows: on the circumference the function must assume a greatest and a least value (cf. p. 97), and since the circumference has no boundary point, these points of greatest and least value must be stationary points for the function.
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.

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+> This page was read from its image by a model (olmocr); the PDF holds no text for it.[^inferred]
+
+Hence by the definition of \(\lambda\) and \(\mu\)
+
+\[f_w + \lambda \phi_w + \mu \psi_w = 0.\]
+
+Similarly, if we multiply the equations
+
+\[\phi_v + \phi_z \frac{\partial z}{\partial y} + \phi_t \frac{\partial t}{\partial y} = 0\]
+
+and
+
+\[\psi_v + \psi_z \frac{\partial z}{\partial y} + \psi_t \frac{\partial t}{\partial y} = 0\]
+
+by \(\lambda\) and \(\mu\) respectively and add them to the equation
+
+\[f_v + f_z \frac{\partial z}{\partial y} + f_t \frac{\partial t}{\partial y} = 0,\]
+
+we obtain the further equation
+
+\[f_v + \lambda \phi_v + \mu \psi_v = 0\]
+
+We thus arrive at the following result.
+
+If the point \((\xi, \eta, \zeta, \tau)\) is an extreme point of \(f(x, y, z, t)\) subject to the subsidiary conditions
+
+\[\phi(x, y, z, t) = 0,\]
+
+\[\psi(x, y, z, t) = 0,\]
+
+and if at that point \(\frac{\partial(\phi, \psi)}{\partial(z, t)}\) is not zero, then two numbers \(\lambda\) and \(\mu\) exist such that at the point \((\xi, \eta, \zeta, \tau)\) the equations
+
+\[f_w + \lambda \phi_w + \mu \psi_w = 0,\]
+
+\[f_v + \lambda \phi_v + \mu \psi_v = 0,\]
+
+\[f_z + \lambda \phi_z + \mu \psi_z = 0,\]
+
+\[f_t + \lambda \phi_t + \mu \psi_t = 0,\]
+
+and also the subsidiary conditions, are satisfied
+
+These last conditions are perfectly symmetrical. Every trace of emphasis on the two variables \(x\) and \(y\) has disappeared from them, and we should equally well have obtained them if, instead of assuming that \(\frac{\partial(\phi, \psi)}{\partial(z, t)} \neq 0\) , we had merely assumed that any one of the Jacobians \(\frac{\partial(\phi, \psi)}{\partial(x, y)}\) , \(\frac{\partial(\phi, \psi)}{\partial(x, z)}\) , \(\dots\) , \(\frac{\partial(\phi, \psi)}{\partial(z, t)}\) did not
+
+[^inferred]: Marked text was read from the page image or inferred by a model, not taken from the document's own text. See truedoc.inferred in the front matter.