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By AIME AIME

ON Sept. 24, 1875, a remarkable deposit of silver ore was discovered by James Ryan and Samuel Hawkes at the east base of Grampian Hill in central Beaver County, Utah.. A shaft was begun and had been sunk about 30 ft. in ore when the claim was sold on Feb. 17, 1876, to A. G. Campbell, Matthew Cullen, Dennis Ryan, and A. Byram for $25,000. To them is due the credit of having developed the mine and started the district on the road to its present state of prosperity. Their development resulted in proving the vein to a depth of over 280 ft. and about 2500 tons of ore was extracted and turned into bullion. In February, 1879, the mine came into the possession of the Horn Silver Mining Co., and since then over $65,000,000 worth of ore has been taken from the mine. Two smelters were built in 1878, near the mine; 400 men were employed underground, and 1000 on the surface, burning charcoal, hauling wood from the surrounding hills, and freighting and hauling hay for the hundreds of horses used in the work. Ore was hauled thirty miles to the

Jan 1, 1936

By Leo J. Vogel, E. D. Hummer, David J. Akers

INTRODUCTION An efficient refuse-disposal system is a necessary part of the modem cleaning plant. The large-scale refuse system and disposal area, engineered for the lifetime of the plant, has become more important due to the greater tonnages of refuse produced from increased quantities of mechanically mined and prepared coal. Laws and increased government regulations have affected the design and operation of disposal systems. THE EFFECT OF MINE AND PLANT OPERATION Refuse disposal starts at the mining face. The mining system, the type of roof support, the machinery used, and the degree of discipline at the face contribute to the amount and type of refuse that must be removed through the disposal system. Economies gained through modification of the mining system or mining machinery may be partially offset by increased dilution of the mined coal with impurities from associated rock formations. In some cases the change in the mining system and equipment has reduced the dilution. In all cases the ratio of impurities that must be removed from the raw-coal feed to the plant not only affects the maintenance and

Jan 1, 1979

By F. C. Canney

Geochemical prospecting seeks hidden mineral deposits by sampling for variations in the chemical composition of naturally occurring materials. Usually the samples are of soils and other products of weathering and erosion-surface materials extremely susceptible to contamination by human activities. Except when a survey is conducted well away from populated areas, contamination is an ever-present hazard. The program's success is particularly endangered when the contaminants are the elements sought; if these occur erratically throughout an area, spurious anomalies completely unrelated to mineralized areas can be formed. Evenly distributed contamination can be dangerous too, if it raises the background to such a level that true anomalies can no longer be easily detected. Suppose, for ex- ample, that in an area where the lead background is 20 parts per million the soil over a lead vein contains 1000 ppm of lead. Here the contrast would be 50 to 1, which is very satisfactory for geochemical surveying. Now if 1000 ppm of lead from some source were added evenly to the soil in this area, the contrast would be reduced to about 2 to 1, and the anomaly would no longer be readily detectable because a threshold of significance twice the value of the background is the minimum generally used for interpreting geochemical data.

Jan 2, 1959

By T. A. Rickard

The writing that is effective is woven with a fine texture into an agreeable pattern; it is free from knots, loose threads, and stray fluff. The instrument that weaves this literary fabric, whether it produce a homely sock or a lordly tapestry, a simple story or a learned 'Itreatise, is a disciplined intelligence, as sure as a steel needle, as precise as a swift loom. The simile breaks down at this point, for the product of the pen is instinct with the human spirit and therefore transcends in beauty anything made by a mere machine. If words are to be woven into eloquent meaning, they must be well knit. Upon the relation of words to each other and of groups of words to other groups, known as phrases, clauses, and sentences, depends the success of writing as a means of trans¬mitting thought from man' to man. To be understood beyond question, you should know not only what your words denote but how to build your sentences; you must not only choose your words aright but construct your sentences properly. A sentence is a combination of words that makes complete sense; it says something about something. ' A clause is a subordinate sentence or a sentence within a sentence. A phrase is a group of words without a predicate; it makes sense, but not complete sense. For example: I shall wait for a few minutes before I go home: The whole statement is a sentence; "I shall wait" and "before I go home" are clauses; "for a few minutes" is an adverbial phrase. The orderly arranging of words and of combinations of words is called `syntax'; and from the original Greek form, ouvaeis (suntaxis), is derived the Greek name for an editor,

Jan 1, 1931

By M. E. Steidlitz

FABRICATION of semiconductor devices.requires the formation of electrical contacts, often involving the use of lead, tin, and lead-tin solders. In several recent instances when germanium had been bonded to nickel with lead, a new metallic phase appeared, unattached to either germanium or nickel, but rather suspended as a continuous foil wholly contained within the lead matrix. Since such metallic structures do not form in nickel-lead or lead-germanium systems under identical conditions, a brief study was made of the characteristics and possible formation mechanisms of this material. EXPERIMENTAL Specimens were prepared by placing single crystal, transistor-grade germanium discs, 30-mils in diam by 6-mils thick, into a depression in a graphite boat. =A* nickel washers, 125 mils in diam by 5-mils thick, containing a center hole of 10-mils diam, and coined into a soup bowl shape, were loaded on top of the discs, so that the bottom of the

Jan 1, 1959

By John V. Beall

BROMO Seltzer blue has gone to war. The blue of the Bromo Seltzer bottle is a product of cobalt, the Nation's No. 1 strategic metal. When the National Production Authority, on Nov. 21, 1950, ordered a 70 pct cutback in domestic use of cobalt, a stab of pain was felt in the hearts of the glass, porcelain enamel, steel, paint, pottery, and electrical and electronic equipment industries. The reason: A full-scale wartime economy with astronomical requirements for jet engines, high velocity projectiles, armor plate, electronic equipment, and the many products needed to win a war. There is little potential comfort for cobalt users in nonessential manufacturing, and the military will be pinched for optimum supplies of the metal. However, if proper measures are taken immediately and current plans are completed, in two years there will be far more cobalt available than ever before. When normal times return, cobalt supplies should be sufficient to meet peacetime military requirements, and greatly expanded domestic requirements that will result from new technological development for wartime but adaptable to peacetime products. The pres-

Jan 1, 1951

By G. T. Engel, W. G. Gruzensky

Jan 1, 1960

By Paul R. Dingess, Edward H. Hare, Douglas C. Brockie

Mining in the Tri-State district of Missouri, Kansas, and Oklahoma has been nearly continuous from about 1848 until the present day, although the major activity was from about 1880 to 1955. The district, which has produced over $2 billion in zinc and lead concentrates, ranks as one of the greatest mining districts in the world. Unlike other Mississippi Valleytype deposits in the United States, which occur in Cambrian and Ordovician limestones and dolomites, the Tri-State district ores occur in Mississippian limestones containing abundant chert. Zinc is 5 to 6 times more abundant than lead. In the Picher Field, the principal sub-district which has accounted for 61 percent of the total district production, most of the ore has been mined from the single horizon, M bed. Most ore bodies, other than those of the "Sheet Ground" or blanket-type, are in large, essentially flat-lying breccia zones and have a definite mineral zonal pattern. An irregular, generally elongated central dolomitic core is surrounded progressively outward by the main ore run, the jasperoid zone, the muddy or shaly and bouldery zone, the sparry calcite limestone zone, and the fossiliferous, dominantly crinoidal, limestone zone. Some of the zones may be absent or nearly so and may overlap to some extent with an adjacent zone. Although the district has been studied by many authors, much controversy still exists about such items as genesis, paragenesis, and time of emplacement. These subjects are discussed in the paper.

Jan 1, 1968

By W. T. Bell, M. P. Lebourg

The perforating of multiple tubingless completions, in which two or more strings of 27/8-in. OD casing are installed in the same borehole, presents two basic problems. First, good completion practices require an eficient perforator that leaves no debris to interfere with subsequent completion operations. Second, and of more complicated nature, a system is needed for controlling the direction of fire so that adjacent strings are not damaged. Development of of a 2-in. OD, steel retrievable, shnped-charge gun has solved the first problem. The second was resolved through development of three different devices to provide directional perforating for the two types of completion methods being employed today. These include a mechanical orienting device and two self-orienting radiation devices. Conceivably, the latter methods could be adapted to the directional perforating (of .upper zones) of conventional dual and triple completions without removing production tubing and packers. INTRODUCTION The recent trend to multiple tubingless completions,' wherein two or more strings of 27/8-in. OD casing are installed in the same borehole, has presented two basic perforating problems. The first problem to be overcome was the design of a gun that would result in efficient perforations without causing debris to bridge in these small-diameter casings. The second problem was, of course, to devise a system for controlling the direction of fire such that adjacent strings would not be damaged. The problem of gun debris in slim-casing completions has been well defined in the numerous 27/8-in. single tubingless completions which have been effected during recent years with the expendable-type perforators. Failure of expendable-gun debris to settle out properly, even with good gun break-up, has resulted in bridging inside these small-diameter casings. This, in turn, has interfered with completion and re-conditioning operations. The past two years have clearly shown, in fact, that debris problems are not confined to this small range of casing with most designs of expendable-type guns. A high incidence of bridging has even been reported in 4%-in. OD casing. The perforating of multiple tubingless completions has entailed, therefore, the development of a completely debris-free gun of adequate performance, as well as the necessary devices to provide positive directional perforating in the two completion methods presently being employed. THE GUN —DESIGN CONSIDERATIONS The design of a shaped-charge gun for slim-casing perforating has presented a maze of interrelated prob-lems. Since the primary specification has been leveled at efficient charge performance with complete elimination of gun debris, it has been necessary to develop a retrievable steel-carrier gun of tubular construction to house the charges. The carrier must be retrievable through 2.29-in. API pump-seating nipples which are usually installed in 2 7/8-in. strings. To provide for recovery of the gun from the well, swelling of the carrier on firing must be carefully controlled. Control of swelling automatically means control of explosive weight in the shaped charge which, in turn, influences charge performance. Carrier swelling also depends upon the confinement pressure at the level being perforated, which imposes a hydrostatic pressure-depth limitation. Burr interference on the wall of the carrier, resulting from passage of the jet through the carrier wall, must be eliminated by suitable design. Clearly then, a satisfactory shaped-charge gun design for slim-casing operations represents a fine balance between carrier swelling, hydrostatic pressure, explosive weight and charge performance. Additionally, to lend itself to directional perforating, the gun design must provide a positive locking mechanism for individual charges within the gun carrier. This fixes charge orientation within the carrier and prevents

DURING September, the Coal Commission pre- pared for issue a series of 18 reports on varied aspects of- the bituminous coal industry; the final report, dated Sept. 22, sums up the facts and offers recommendations addressed to Congress, to the industry, and to the public. The few following paragraphs are selected: Bituminous coal is the real foundation of that great industrial and transportation structure which enables more than a hundred million people to live in America and be so well supplied with things necessary for their health, comfort; and convenience; it also furnishes domestic fuel for two-thirds of the homes of the land. The distribution of the output of the bituminous mines tells the story: 28 per cent. goes to the railroads, 25 per cent. to the boiler houses of factories and mills of all kinds, 15 per cent. to the coke ovens and gas plants, 10 per cent, to the homes, 7 per cent. to steel plants and the same percentage to power plants and street railways, 4 per cent. is exported, 2 per cent is used for power at the coal mines, and the remaining 2 per cent. goes to the seaboard for steamship fuel. Bituminous coal enters into interstate commerce on a scale approached by no other commodity; less than a fourth of the coal mined in the United States fails to enter interstate com-merce. All but one of the 45 coal producing districts in the United States ship coal outside the State in which it is mined. Three hundred miles of cars loaded with coal is the daily output of the bituminous mines of the United States. The public-welfare element in coal is seen in the dependence of public health and safety on an unfailing supply of fuel, in the close connection between the prosperity of most industries and the uninterrupted operation of the coal mines, and in the obvious fact that without coal the great network of railroads which binds together this great country would be idle. The railroads and the public utilities, themselves so clearly obligated to render what-ever public service is demanded of them that the constitutionality of their public regulation is unquestioned, are of all industries most dependent upon coal. Thus, coal makes up over 35 per cent. of the operating cost at the gas plant and over 25 per cent. at the electric power station, and in the operating expenses of. a railroad the cost of coal is the largest single item, next to labor. The practical logic that half a century ago clothed with a public interest the steam locomotive and in later years the electric-power station must recognize a large element of public service in the coal mine that furnishes the necessary energy to both locomotive and power-plant.

Jan 10, 1923

By Willard P. Ward

The numerous direct processes which have been patented and brought before the iron-masters of the world, differ materially from that now introduced by Mr. Wilson. After a careful examination of his process, I am convinced that Mr. Wilson has succeeded in producing good blooms from iron-ore, and I think that I am able to point out theoretically the chief reasons of the success of his method. Without going deeply into the history of the metal, I may mention the well-known fact, that wrought-iron was extensively used in almost all quarters of the globe, before pig or cast-iron was ever produced. Without entering into the details of the processes by which this wrought-iron was made, it suffices for my present purpose to say that they were crude, wasteful, and expensive, so that they can IE employed to-day only in a very few localities favored with good and cheap ore, fuel, and labor. The constructiou of larger furnaces and the employment of higher temperatures led to the production of a highly carbonized, fusible metal, without any special design on the part of the manufacturers in producing it. This pig-iron, however, could be used only for a few purposes for which metallie iron was needed; but it was produced cheaply and with little loss of metal, and the attempt to decarbonize this product and bring it into a state in which it could be hammered and welded was soon successfully made. This process of decarbonizatioa, or some modification of it, has successfully held the field against all, so-called, direct processes up to the present time. Why ? Because the old-fashioned bloomeries and Catalan forges could produce blooms only at a high cost, and because the new processes introduced failed to turn out good blooms. Those produced were invariably "red-short," that is, they contained unreduced oxide of iron, which prevented the contact of the metallic particles, and rendered the welding together of these particles to form a solid bloom impossible. The process of puddling cast-iron, and transforming it by decarhonization into wrought-iron has, as everybody knows, been in successful practical operation for many years, and the direct process referred to so closely resembles this, that a short description of the theory of puddling is not out of place here.

Jan 1, 1884

By Robert M. Funk, James E. Lawver

The development of a laboratory and pilot-scale high-tension electrostatic flowsheet for the production of iron-ore super concentrates having silica contents in the range of 0.1% is presented, A variety of hematite and magnetite ores were tested with this flowsheet, and the pertinent metallurgical data are given. The applications and limitations of this process are brought out in terms of both practical and theoretical considerations. There have been many new developments in the past few years that have required mineral beneficiation engineers to consider new techniques for iron-ore concentration. Developments in iron powder metallurgy, and particularly the making of sheet metal directly from high-purity iron-ore concentrates, and the concept of using metallized pellets as a substitute for scrap in electric-furnace steelmaking have created demands for what is commonly referred to as iron-ore superconcen-trates. In most cases, specifications for superconcentrates are that they contain less than 0.1% total gangue, with some applications requiring less than 0.05% gangue. To produce these concentrates, new and modified processes are being developed, and the three methods receiving attention as the primary concentrating methods are: 1) Magnetic separation in which dry rotor-type equipment is used for magnetite and the high-intensity magnets are used for hematite. 2) Amine flotation. 3) "High-tension" electrostatic separation (or, elec-trodynamic concentration). This paper examines the high-tension electrostatic process in terms of its limitations and abilities for the production of iron-ore superconcentrates. A technically effective and simple flowsheet was developed through laboratory and 5 and 100-ton pilot-plant studies for the production of concentrates containing from 0.02 to 0.20% SiO, with iron recoveries between 85 and 90%. The process is also shown to be economically feasible. Laboratory Test Program Separation Characteristics: The suitability of the electrostatic high-tension method for iron-ore supercon-centration stems from the fundamental mechanism governing the process, i.e., particles are separated on the basis of differences in electrical conductivities. This conductivity is a characteristic property of minerals, and if the difference between the conductivities of two min- erals is great enough, then their separation can be performed by a high-tension machine. As shown in Figs. 1 and 2, the critical factor is the time necessary for the particle charge density, which is established by a corona discharge field, to decay to a level such that the centrifugal force acting on the particle exceeds its electrostatic, or image, force. At this point, the particle can leave the rotor at some trajectory and fall into an appropriate product receptacle. This decay time is directly proportional to electrical conductivity, and the process variables, such as feed rate, rotor speed, electrode configuration, and rotor composition, are selected to optimize the ability of the separator to resolve different conductivities. Table 1 lists the measured conductivities for some of the iron and gangue minerals used in developing the flowsheet. These results were obtained by using the conductivity cell described by Lawver and Wright.' The discharge curve of most minerals that lend themselves to super concentration by high-tension is so fast that the

Jan 1, 1971

By T. Sekimoto, T. Kokubo, M. Yamamoto, K. Masuda, T. Moro, H. Sakamoto

The choking-free cyclone was developed primarily to remove solid particles suspended in a slurry containing fibrous materials, such as municipal solid waste. This cyclone is characterized by a screw conveyor at the underflow side of the cyclone. The screw conveyor has a hollow rotational shaft for air supply and is driven by a variable speed motor for discharge control. Even if the underflow discharge is highly concentrated, this type of cyclone can be stably operated without nozzle choking. High classification efficiency similar to that of a conventional cyclone can be obtained as a result of air introduction through the hollow rotational shaft. The recovery of organic matter and the removal efficiency of inorganic matter were actually improved by the application of this cyclone to municipal solid waste. After applying this cyclone to the desliming of fine sand, sand product of low water content was recovered.

Jan 1, 1980

By James P. Sloss

WHAT is the likelihood if any-that a real gold mining industry will be developed in the southern Appalachian region? Has the increase in the dollar value of gold from $20.67 to $35 per ounce potency to turn the trick? Suppose it be assumed that the price of gold will, at worst, not recede, and that the cost of digging ore and treating it of which the average level of commodity prices and wages is a fair index-will, at most, not exceed that prevailing in 1926. One may or may not accept these as reasonable assumptions; but one must arbitrarily remove some of the potential variables if one is to get anywhere on a problem like this. As a matter of fact, this article is not going to venture a conclusive answer to the question that has been posed; but it will endeavor to present some of the factors that have a bearing on the problem. The writer spent ten days visiting a dozen properties in Virginia and the Carolinas early in April, 1935, and in doing so got a first-hand slant on mining conditions in the domain of King Cotton.

Jan 1, 1935

Joseph W. RichaRds, South Bethlehem, Pa. (communication to the Secretary*): The hold experiment of Mr. James Gayley in drying the blast used in the Isabella furnace has attracted the attention of the whole metallurgical world, and has given rise to a wide discussion of the reasons for its great success. Mr. Gayley and the capitalists who backed him in his work deserve congratulations on the success of the venture, not only because it succeeded, but because they have been rewarded far beyond the maximum that could be expected from theory or probability. To state the case in its simplest terms, all that could have been predicted was: (1) that the removal of the moisture from the blaet would result in the saving of the heat required for its decomposition, representing from 3 to 5 per cent. of the heat-value of the coke used; hence about that quantity of fuel might be saved; (2) that the furnace would be somewhat more independent of variations in the weather; (3) that the campaign would be a little hotter; and (4) that the speed of the blowing-eugines would be decreased, at least in amount proportionate to the reduction in temperature of the air led to them. I do not think anyone anticipated a much larger output per day. The results actually attained were: (1) the furnace made an increase of 24.86 per cent. in the output of pig-iron per day; (2) it consumed 19.6 per cent. less coke per ton of pig-iron made; (3) the blowing-engines were run 15.8 per cent. slower; and (4) the saving in power made up for the power required to run the refrigerating-plant. In addition, the blast was hotter; the waste-gases cooler, because of their smaller volume; and there was a greater proportion of carbon dioxide to carbon

Jan 1, 1906

By Frances H. Clark

IN wartime, the fabrication and use of metals assumes increased importance, for a modern war of sizable proportions cannot be undertaken with- out a vast supply of this material. Light alloys of aluminum and magnesium, which were comparatively unknown in the last war, will play an important role in this one. Due to research on a vast scale in the interim of years, industry is today well equipped to fabricate metals on an enormous scale. Research for the past year has been most productive and a comprehensive survey in the space of this article would be impossible. It is hoped that the stressing of a few items here and there out of a great quantity of able work will indicate in a small way the advances made within recent times. Specific references to the literature will be omitted at the Editor's re- quest but some authors of papers that I have found of special interest are mentioned.

Jan 1, 1940

By H. C. Fowler, G. B. Shea

MODERN industry's incessant demands for increased operating efficiency and lower costs require that hazards attending all occupations be reduced to a minimum. Reduction of the inevitable losses that result from accidents and injury to the worker is one of the important economic problems confronting industry today and is the basis for the widespread safety movement. Dr. Charles E. Munroe, chief explosive chemist of the U. S. Bureau of Mines, is father of the statement, "The existence of a nuisance is evidence of an economic waste." This pointed expression of the internationally recognized scholar and explosives engineer has a direct relation to the prevention of accidents. Every injury and death in industry is a waste in human endeavor and a lessening of productive gain. Dr. Munroe has aptly paraphrased his previous summation with reference to wastes as follows: "A lost-time accident is evidence of an economic waste." That the economic waste entailed by occupational accidents is chargeable to industry is well substantiated by the fact that all but four states (Mississippi, South Carolina, Florida and Arkansas) now have workingmen's liability and compensation laws.

Jan 1, 1933

By R. E. Tate, L. J. Herman

We have had occasion to produce wire of Np-237 in small diameters for use in some chemical experiments. Since the mechanical metallurgy of neptunium has not been investigated extensively, it seems desirable to report our successful extrusion technique. Because of the neptunium activity and its attendant health hazard the experimental work was done in glove boxes and related equipment also used for Pu-239. At room temperature a phase neptunium' is a hard (Dph -355) orthorhombic metal of high density (p = 20.45 g per cu cm). a neptunium transforms2 to tetragonal ß neptunium at about 280°C and this in turn transforms to bcc y neptunium at about 577°C. The y phase melts at about 637°C. A temperature of 340° to 345°C was selected for the extrusion, this temperature being a compromise be- tween a possible advantage of working with the presumably more plastic cubic y phase above 577°C and the risk of having the neptunium react with the alloy steel die to form the low-melting Fe-Np eutectic. A 0.75-in.-long, 0.26-in.-diameter ingot of good-purity* Np-237 was extruded by the direct *C = 230 ppm, Pu = 200 ppm, thirty-five metallic elements below limit of detection by spectrographic methods. process through flat-face orifices to form several sizes of wire ranging from 0.010 to 0.030 in. in diameter. Molybdenum disulfide was used as the die lubricant. The extrusion was done in a vacuum of about 10-4 torr to reduce reaction of the neptunium with atmospheric gases. British workers3 at Harwell have extruded neptunium under similar conditions. The force required to extrude 0.010-in.-diameter wire at the rate of 4 in. per min was 6000 lb, as measured with an electrical load cell. Wire 0.020 in. in diameter was extruded with a force of 4700 lb at a rate of 2 in. per min, and wire 0.030 in. in diameter was extruded with a force of 4000 lb at a rate of 1.3 in. per min. An extrusion constant, c, can be computed from the measured forces, P, for the equation4 P = c log-A/a where A/a is the extrusion ratio or the ratio of the cross-sectional areas of the container and the

Jan 1, 1965

By AIME AIME

The first session, held Tuesday evening, February 23, in North Sheffield Hall, was called to order by Louis V. Pirsson, Chairman of the Local Committee, who introduced Prof. Russell H. Chittenden, Dean of the Sheffield Scientific School of Yale University. Professor Chittenden, in behalf of the Governing Board, felicitated the Institute on its visit to New Haven, and cordially welcomed the members and guests to the University and the city. Dr. James Douglas, who presided at the meeting, responded in behalf of the Institute. The Secretary announced that, upon the proposal of many members, the unanimous report of the Committee on Membership, and .the unanimous recommendation of the Council, the following members had been unanimously elected by the Board of Directors Honorary Members in the American Institute of Mining Engineers Prof. Richard Beck, of Freiberg, Saxony, in recognition of his distinguished service to technical literature, and the cordial

Apr 1, 1909

By H. H. Rachford, W. P. Schultz

The use of computing machines to solve physical problems has made it imperative to represent physical data in a form computing machines can use. Although curve-fitting is an old and well-practiced art, there still seems to be some interest in questions regarding what methods can be used and why. Even though there are many good references on this subject, it may be of some value to discuss a particular method which has been found to be useful, and to make available the associated tables for its use in the hope that there can be saved a few of the tedious hours necessary to reduce many of the suggested methods to practice. Although a very brief discussion of the mathematical background of the tables is included immediately below, those interested only in their use can skip directly to the section on Use of the Tables. Direct Fit of a Set of Points A frequently encountered problem is: Given data which can be represented by a smooth curve +(x), to find an easy-to-evaluate function F(x), with only a few coefficients, which is a good approximation of +(x) on the interval 0 5 x 5 x,,. Consider F(x) as a linear combination of m + 1 arbitrary, single-valued, continuous, linearly independent functions, f,(x), Ff.r) = a,f,,(x) + a,f,(x) + . . . -+ aifi(x) + If the value of Q(x) is known or can be read from the curve at the points xj: +(x,), +(x,), . . . +(xj) . . . +(x.), and if n = m, then all the coefficients of F(x) can be determined, and F(x,) — +(x,) = 0 for j =. 0, 1, . . . n. This follows from the solution of the set of linear simultaneous equations. +(x<.) = aofu(x,) + aJI(x0) + . . . + a,,f,,(x,) =a, = f,,(x,) + a,f,(x,) + . . . + a,f,,(x,) +(x,,) = a,,f,(x,,) + a,f,(x,) + ... + a,,f.,(x,,) ...........(2) for the coefficients ai. Although the set a, from the solutions of the set of Equations 2 yields an fix) which gives an exact value of $(x,) for the n + 1 points, there is no restraint on ~(xj in the interval xi < x < xj+,. Indeed, if the points +(x,) are not sufficiently smooth, very serious differences may arise between F(x) in the interval and the smooth curve +(x). Least-Squares Fit It has been suggested&apos; that if n is made to exceed m, and the set a, is found to minimize the function the function F(x) is much less sensitive to small errors in +(xi), and usually provides a much better fit to the smooth function +(x). This is known as the best fit in the sense of the least-squares error. To minimize $, set £ = 0,i = 0,1,... iTi,.....(3) The m + 1 equations from Equation 3 are k—n k=n a.^p /,,(a) f,(xt) + a, X A: = 0 k=O k = n k = n k=0 k=0 +(x.) f,(x,),............(4) one equation being obtained for each i = 0, 1, . . . m. Inasmuch as the arbitrary functions f(x) may be evaluated for x,, k = 0, 1, . . . n, and the +(x,) are known values of the curve to be fitted, the simultaneous Equations 4 may be solved for the set a,. Particular Functions Given the functions fi(x) and the set +(xj), the solutions for require evaluating a number of sums of products as well as the solution of an m + 1 order set

Jan 1, 1956