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By W. G. Sinclair

The North Texas district incorporated in this paper corresponds with the Railroad Commission's District No. 9, and includes the counties of Archer, Baylor, Clay, Cooke, Foard, Hardeman, Jack, Knox, Montague, Wichita, Wilbarger, and Young.† Table I,Oil and Gas Production in North Texas District, has been changed to conform with the nomenclature of pools designated by the Commission in order to facilitate compilation of the data, since most of the companies operating in the area are coming more and more to recognize one central authority in the matter of naming each new oil pool. No effort is being made to discuss the structural characteristics of the region, since it is generally known by oil men that most of the oil and gas comes from Pennsyl-vanian, with some of less importance from the Mississippian and Ordovician. Development during 1944 There were 1550 wells drilled during 1944, an increase of 450. Of this number, 677 were completed as oil wells, and 11 as gassers. Production increased 1,505,025 bbl., making the year's total 43,172,005 bbl. of oil having an average gravity of 40". Development generally, however, was somewhat disappointing. The 47 new pools, extensions, or new pay horizons in old fields, had a combined initial production of only 19,146 bbl. It is estimated that these wells added approximately 10,000,000 bbl. to the reserves. The largest single addition to the reserves was in the Caddo conglomerate in the Hildreth pool, Montague County. During the year, 66 wells were completed as the result of three extensions, I to 136 miles west, northwest and north of the field. Young, Archer, and Clay were the most active counties. Eleven new potential - oil fields were developed in Young County, nine in Archer and nine in Clay. Outlook for 1945 In spite of the fact that exploration as a whole was disappointing in 1944, prospects for increased development are bright for 1945. At the close of last year, 2.1 geophysical crews were operating in the district, which, together with the Ohio Oil Company's No. I W. R. Ross, a wildcat Strawn discovery in King County, 1 1/2 miles beyond the west boundary of district 9, stimulated a lease play extending over four counties. Two of them, viz., Foard and Knox, are in the area incorporated in this report. Acknowledgments The writer is indebted to Mr. C. W. Hanley, Zone Agent of the North Texas District, and Mr. William M. Lourcey, head of the Land, Leasing and Scouting Department for the Gulf Oil Corporation, both of Fort Worth, Texas, for permission to prepare this paper.

Jan 1, 1945

By R. R. McNaughton

The Consolidated Mining and Smelting Company of Canada, Ltd. at Trail, B. C., inaugurated a comprehensive program of investigation about 15 years ago to develop the most economical process of recovering the metal in the large accumulated stock of by-products of the lead smelter and the zinc plant. Two principal methods of recovery were followed to a successful conclusion—an electrothermic process to treat zinc-plant residues and cold slag and the fuming of hot slag from the lead blast furnaces. The latter was selected because it showed a saving over the former in its ability to handle hot blast-furnace slag, thus obviating the necessity of granulating, handling and eventually remelting the same slag for the electrothermic process. The design of a fuming plant was commenced in 1928, and construction, which was started immediately, was completed in July 1030. Operating intermittently for a general tuning up, the plant was officially blown in for regular operation in August. During this month the vagaries of a slag-fuming plant were studied and an operating crew was trained. A history of slag fuming at Trail and the development of plant design is given in a paper by G. E. Murray in the Transactions of the Canadian Institute of Mining and Metallurgy for 1933. Fuming Plant The fuming plant at Trail (Fig. 1) is a separate unit although of necessity interlocked closely with the lead smelter. The plant consists essentially of a fuming furnace with its accessories, a waste-heat boiler followed by an economizer, and a suction-type baghouse. The plant was built as the first of two 400-ton units with the intention of adding the second unit within a year, so that flues, fans and buildings were built to handle double the original load. Furnace.—The furnace as originally built (Fig. 2) was of water-jacket construction, 10 ft. wide, 20 ft. long and 6 to 10 ft. high. The roof height was made variable to check experimental work, which had indicated that this dimension affected the rate of reduction. Coal and air were supplied

Jan 1, 1937

By F. J. Radd, R. L. McGlasson

The mechanisms of corrosion and corrosion fatigue damages to sucker rods are examined from a fundamental electrochemical viewpoint, and the relationships of sucker rod microstructures to these damaging effects are defined. The common corrosive agents in crude oil production and storage are discussed as to types of attacks that they cause. The variations of fatigue strength with alloy content, surface preparation, and stress ranges are analyzed. Finally, conclusions are drawn as to how microstructures largely influence and control sucker rod performance. INTRODUCTION Greater than 90 per cent of the crude oil produced by artificial lift methods is produced through sucker rod pumping systems. In the USA each year more than $14 million is spent for sucker rods in new and old pumping installations. It is important to make this investment a wise one. Essentially, the sucker rod system is a long series of steel rods, operating under alternating stress loadings and under mechanical vibrations, which connect the surface power source with the subsurface pump. These rods often operate in a variety of corrosive media, including water, brines, carbon dioxide, hydrogen sul-fide, and air. Basically, this paper is concerned with the problem of the behavior of certain steels in various corrosive media under high and dynamic stress patterns. The main objective of this paper is to show how, once the stress effects and the corrosive environments are defined, one can still realize structural control and beneficial influence upon corrosion and corrosion fatigue. FUNDAMENTAL DEFINITIONS AND ANALYSIS In the corrosion and stress behavior of this solid state mixture of iron and iron carbide called steel, one must analyze upon what major factors does its service life depend and of what, exactly, is steel composed. The former question can be referred to two external factors and two internal factors. The two major external factors are: 1. the nature of the stress patterns — size and range of stresses present 2. the nature of the corrosion — erosion environment — the corrodent and/or wear involved The two fundamental internal factors that govern the strength and useful life of a steel both in normal duty and in heavy-duty corrosion service are: 1. the nature of the steel volume properties 2. the nature of the steel surface properties For the latter question normal steel is defined as a solid mixture of iron and iron carbide particles, generally with certain accidental and intentional elements present. (The amount of iron carbide present is determined by the carbon content of the steel.) These additional elements generally dissolve into either the iron (ferrite) or the iron carbide (cementite) components.

Jan 1, 1955

By E. R. Buchanan, R. E. Reed-Hill

A transmission electron microscope study was made of magnesium single crystals deformed in torsion using the basal pole as the forsional axis. While this type of deformation is predicted to result in the formation of hexagonal networks of screw dislocations, these networks were not observed. The basic dislocation arrangement consisted of arrays of nearly parallel and approximately equally spaced dislocations. The average dislocation density was over an order of magnitude higher than that required to account for the torsional deformation. The observed high rate of work hardening in these crystals is consistent with both their high dislocation densities and the arrangement of dislocations in pile-ups. The experimental observations are also in agreement with the assumption that dislocation sources are activated not only at the surface but at interior locations as well. In considering torsional deformation, Amelinckx and Dekeyser1 postulated that when a torque is applied to a crystal oriented with a glide plane normal to the specimen axis, deformation occurs by the following steps: 1) Nucleation of dislocation sources at or near the specimen surface in areas where the shear stress, resolved along a slip Burgers vector, is a maximum; 2) the screw orientations of the dislocations accommodating the twist move inwards towards the specimen center. If two glide directions lie on the glide plane, the geometry shown in Fig. 1 should result; 3) the screw dislocations that pile up in the central part of the specimen cross-slip to form a grid. At the same time, screws of opposite sign and most edges leave the specimen. Tinder,2 using zinc crystals twisted about the basal pole, independently reached essentially the same conclusions. He suggested a mechanism involving nucleation of ?(1120) screw dislocations at the six points of highest resolved shear stress that lie on the cross-section near the surface. Under the applied shear stress these dislocations were assumed to move inward and interact so as to form a hexagonal network of screw dislocations. Today it is often accepted that rotational slip occurs by a mechanism of the above type, but to our understanding this has never been experimentally proved. This paper reports an experimental study of rotational slip, using magnesium single crystals, that indicates that in this metal rotational slip does not conform well to this mechanism. EXPERIMENTAL PROCEDURES Mechanical Testing. Specimens with a (0002) axis were prepared from magnesium crystals of 99.95 pct purity grown by Jillson's3 modified Bridgman technique. The crystals were acid machined into cylindri-

Jan 1, 1970

By H. F. Winham

The history, development, subsurface geology, production, economics and estimated reserves are discussed in this paper. The Magnolia structure is an anticline with a known maximum structural relief at 372 ft., and the areal extent of the producing acreage is 4494 acres. Production is obtained from the Smack-over oölitic lime and 116 successful wells have been drilled. The field, which was developed on 40-acre spacing, has been under strict control. It is believed to be producing with a strong water drive, To date it has produced more than eighteen million barrels and the estimate of ultimate recovery is 220 million barrels. Early History The Magnolia field is in colurnbia County, Arkansas, T. 17 s., R. 19 and 20 W. Various geologists had long been interested in this general area, and as early as 1921, John F. Magale was able to define a surface structure. The first wildcat, one of the first in Columbia County, was drilled in 1923 by Mid-States Oil Co., in sec. 10, T. 17 S., R. 20 W. This well was about 11/2 miles northwest of the Kerlyn oil Company's discovery well and was based on surface geological work done by Jewel and Dobie. Subsequently, several shallow wells that were not carried below the Upper Cretaceous beds were drilled in the immediate area of the discovery well, but none of them contained any commercial showings of oil Or gas. In the latter part of 1935, Elam and Magale (or Southwood) Garrett NO. I was located in sec. 13, T. 17 S., R. 20 W., 3/4 mile east of the discovery well. It was drilled to a total depth of 4018 ft. in the lowermost Glen Rose beds but failed to obtain any showings of oil or gas. Interest had been stimulated in this general area, however, by the discovery of several oil fields, mainly Glen Rose oil production in the Rodessa field in Louisiana in 1935, Smackover lime productioil in Snow Hill in May 1936, production from the Cotton Valley formation in Schuler in 1937, and Smackover lime production from the Buckner field in November 1937. With this background and subsequent seismic work, which was carried on partly because the old shallow wells drilled in the township had shown evidence of closure in the Upper Cretaceous beds, the discovery well, Kerlyn Oil Company's Barnett No. I, Sec. 14, T. 17 S., R. 20 W., was originally projected to a total depth of 6300 ft., to test the equivalents of the Morgan sands of the Cotton Valley series, then producing at Schuler. The well was carried to a depth of 6325 It. and tested the objective beds without finding commercial production. In the meantime, however, one well had found production in the Jones sand and another in the Smaclrover lime at Schuler. Kerlyn immediately made arrangements to deepen the Barnett well to the Smackover lime. The top of the Smackover lime was encountered at 7618 ft. and the well

Jan 1, 1943

By J. C. Swartz

New measurements of graphite and cementite solubilities in ferrite have been obtained to resolve disagreements among previous data. To measure graphite solubility, specimens of iron were equilibrated with rarious carburizing atmospheres at temperatures be-twee~z 480" and 1435°C. The carbon activity of each atmosphere relative to graphite was taken from published data. After quenching from each equilibration treatment the concentration of carbon (typically 10 pprn by wt) was determined from the height of the Snoek peak in the internal friction spectrum. A few inclividual analyses by combustion support the reszdts of Smith on the solubility of graphite near the eutectoid temperature. Combination of these combustion and internal friction analyses determines In a, (at 0.01 wt pct) = 24,000/RT - 12.72, where a, is the activtity relative to graphite in ferrite at fixed cornposition. The same specimens were used to determine the solubility of self-stressed cementite precipitates. The results indicate solubilities of 6.3, 23.2, and 68 pprn carbon by wt at respectively 400°, 500". and 600°C. The once large discrepancy between the cementite solubility determined by internal frictiori and that calculated from flee energy data has been almost completely resolved. A recent study1 (hereafter designated I) on the solubility of cementite in a iron did not fully resolve the disagreement among previous data. The remaining disagreement is illustrated by the phase diagram in Fig. L The data of I are represented by the line labeled "stress-free" cementite. They determine a heat of solution of cementite of (14.8 * 0.2) kcal per mole which contrasts with the (20 * 1) kcal per mole derived from Smith's graphite solubility2 and the formation energy of cementite.3 Toward resolving this disagreement we have obtained new data on the solubilities of graphite and cementite. METHODS As in I the Snoek peak in the internal friction spectrum was used to measure the concentration of carbon dissolved in ferrite. This method relies on the relation C=kQ-1 [1] in which C is the concentration of dissolved carbon, Q-I is the Snoek peak height, and k is a proportionality factor which is a function of the peak temperature, polycrystalline texture, and other parameters characteristic of the ferrite lattice. To determine true heats of solution via Eq. [1], it is necessary to establish that k is independent of the temperature from which the specimen is quenched. Some evidence on this point is contained in a separate report.* Additional evidence is shown by the data points in Fig. 1. These represent the Snoek peaks of two specimens carburized to about 10 ppm C, copper plated, and quenched first from 888º then from 588°C. The copper plate was removed before each Snoek peak measurement. These data show that there is no detectable dependence of k on quenching temperature. The solubility of graphite was determined by repeatedly carburizing various specimens of iron wire with gas mixtures of known carbon activity relative to graphite. To cover a wide temperature range within the a phase boundaries it is desirable to use as low a carbon concentration as is compatible with reasonable precision. With the present internal friction apparatus the best compromise of these desires is a concentration near 10 wt ppm. The dashed line in Fig. 1 was used to approximate the activity of carbon* for

Jan 1, 1970

By R. W. Powers, M. V. Doyle

ThE solution of a diatomic gas such as 0, or N2 in a metal usually follows Sieverts' law; i. e., Here C is the solute concentration at equilibrium and P, the gas pressure. The proportionality constant Ks is also the equilibrium constant for the reaction 1/2 02 (at pressure, P) - 0 (in solution at [21 concentration., C). to use oxygen as an illustrative gas. Sieverts' law is therefore interpreted to indicate that diatomic gases go into metallic solution atomically and that the solute "gas" atoms interact negligibly with each other. Of course, this interpretation is strictly valid only over the temperature interval in which the reaction indicated in [Z] can be investigated. At temperatures several hundreds of degrees below this interval, the possibility of interactions between solute atoms is conceivable. Indeed, from internal-friction measurements some evidence for the existence of groupings of oxygen atoms in tantalum solid solutions was presented a few years ago.1,2 Over a range of solute concentration, the experimental oxygen internal-friction peak in tantalum, measured as a function of temperature at constant frequency of vibration, can be described as the sum of two elemental peaks, each corresponding to a separate relaxation process. At 0.7 cps, the height of the elemental peak found at 137°C is proportional to the oxygen content, whereas the height of the one at 162°C varies as the square of the oxygen concentration. The 137°C peak was presumed to arise from the stress-induced ordering of oxygen atoms among various classes of octahedral sites in the manner proposed by Snoek.3 The 162°C peak was attributed to a somewhat similar stress-induced ordering process, in which each oxygen atom was interacting with a neighboring oxygen atom. Similarly, the experimental peak due to nitrogen in tantalum can also be described as the sum of two peaks. The height of one varies as the nitrogen concentration and the other as the square of this quantity. In fact, some evidence of solute interactions exist for all oxygen and nitrogen solid solutions of Group V transition metals. However, the interaction energy was not obtained previously for any alloy. The determination of this quantity, as a part of a more intensive study of the interaction between oxygen atoms in solid solution in tantalum, is reported in this paper. PRINCIPLE OF THE EXPERIMENT Consider an equilibrium in a solid solution of tantalum between oxygen atoms which are bound together pairwise and those solute oxygen atoms which are not so associated. 20 (Ta) - O2 (Ta) [3]

Jan 1, 1960

By H. F. Winham

The history, development, subsurface geology, production, economics and estimated reserves are discussed in this paper. The Magnolia structure is an anticline with a known maximum structural relief at 372 ft., and the areal extent of the producing acreage is 4494 acres. Production is obtained from the Smack-over oölitic lime and 116 successful wells have been drilled. The field, which was developed on 40-acre spacing, has been under strict control. It is believed to be producing with a strong water drive, To date it has produced more than eighteen million barrels and the estimate of ultimate recovery is 220 million barrels. Early History The Magnolia field is in colurnbia County, Arkansas, T. 17 s., R. 19 and 20 W. Various geologists had long been interested in this general area, and as early as 1921, John F. Magale was able to define a surface structure. The first wildcat, one of the first in Columbia County, was drilled in 1923 by Mid-States Oil Co., in sec. 10, T. 17 S., R. 20 W. This well was about 11/2 miles northwest of the Kerlyn oil Company's discovery well and was based on surface geological work done by Jewel and Dobie. Subsequently, several shallow wells that were not carried below the Upper Cretaceous beds were drilled in the immediate area of the discovery well, but none of them contained any commercial showings of oil Or gas. In the latter part of 1935, Elam and Magale (or Southwood) Garrett NO. I was located in sec. 13, T. 17 S., R. 20 W., 3/4 mile east of the discovery well. It was drilled to a total depth of 4018 ft. in the lowermost Glen Rose beds but failed to obtain any showings of oil or gas. Interest had been stimulated in this general area, however, by the discovery of several oil fields, mainly Glen Rose oil production in the Rodessa field in Louisiana in 1935, Smackover lime productioil in Snow Hill in May 1936, production from the Cotton Valley formation in Schuler in 1937, and Smackover lime production from the Buckner field in November 1937. With this background and subsequent seismic work, which was carried on partly because the old shallow wells drilled in the township had shown evidence of closure in the Upper Cretaceous beds, the discovery well, Kerlyn Oil Company's Barnett No. I, Sec. 14, T. 17 S., R. 20 W., was originally projected to a total depth of 6300 ft., to test the equivalents of the Morgan sands of the Cotton Valley series, then producing at Schuler. The well was carried to a depth of 6325 It. and tested the objective beds without finding commercial production. In the meantime, however, one well had found production in the Jones sand and another in the Smaclrover lime at Schuler. Kerlyn immediately made arrangements to deepen the Barnett well to the Smackover lime. The top of the Smackover lime was encountered at 7618 ft. and the well

Jan 1, 1943

By A. K. Snelgrove

This symposium is designed to marshal a state-of-the-art account of approaches and solutions to problems arising out of waterlogging and salinization in the world's largest irrigation system which was dismembered by legal beheading of the Eastern Rivers under the lndus Waters Treaty of 1960. In this introduction, the geological and historical backgrounds of surface waters and groundwaters in the Indus Basin are presented with a summary of remedial measures undertaken and proposed to combat waterlogging and salinization. At a time when water-resource planning is proceeding on a continental scale, a study devoted solely to the Indus River Basin may seem to require an apologia. None is offered, however, for three major reasons: (1) Although the Indus is a single river basin, it is one of great magnitude and importance. The Indus River, which comes from Himalayan sources to form part of the great Indo-Gangetic Plain drainage system, is vital to agriculture and hydroelectric power in the thickly populated northern subcontinent of Indo-Pakistan. (2) The Indus has become a plexus of extraordinary problems since the dismemberment of the world's largest irrigation system of which it was a part. The break-up of this system occurred with the legal beheading of the Eastern Waters under the Indus Waters Treaty of 1960. (3) Th River Basin has been the target of a Herculean effort to contain the twin encroachments of waterlogging and salinization that now threaten it with a shortage of fresh water. The British irrigation system of barrages and canal on the Indus was a venture admirable in conception. Unfortunately, it is now a legacy burdened with unwanted consequences due to interference with the hydrological regimen, particularly the rise of water table occasioned by leakage. To cope with this situation, there has been a possibly unprecedented concentration of domestic and international engineering and scientific talent on the Indus and its problems (about a score of organizations by recent count). To marshal a state-of-the-art account of their approaches and solutions is the purpose of this Symposium. The coordinated effort now being made on a variety of fronts under the general supervision of West Pakistan Water and Power Authority (WAPDA) is directed toward the modification and control of interactions and relationships between geological materials — soils, water and its salt balance, and air —in what has been termed a huge "underground lake" in an alluvial basin. The problem is an intriguing one in that a highly complicated case history of river-course changes visible in the present surface must be projected downwards through Recent and Pleistocene sediments in order to decipher the environmental controls of deposition in terms of vertical and lateral variations in lithology and patterns. Such studies must be extended to compaction, deformation, and fabrics of the alluvium. Geohydrology is often defined as groundwater hydrology. As employed herein, it includes the distribution, quality, and behavior of surface and subsurface waters, including their evolution, vagaries, and geological, climatological, and botanical milieux." GEOGRAPHY The Indus (Sanskrit, Sindhu = river) rises at nearly 17,000 ft above the sea in the glaciers on the northern slopes of Kailas Parbat, Tibet. It is about 1800 miles long and has a total drainage area estimated at 372,000 sq miles. The average discharge is 196,000 cfs at mouth. Leaving the Plateau of Tibet, the river flows at a level which is miles above the sea, and in places miles below Himalayan peaks, in a structurally controlled northwestward course. Joined in this course by six tributaries and swollen by the melt waters of giant glaciers, it swirls about the base of the famous

Jan 1, 1970

By J. S. Kirkaldy, D. H. Weichert, G. R. Purdy

Two-phase diffusion couples have been used to simulate the growth of proeutectoid ferrite in ternary Fe-C-Mn austenites. It has been shown, theoretically and expermentally, that the results fall into two classes: a high-super saturation class in which the major role of manganese additions is to influence the boundary conditions for carbon diffusion; and a low-supersaturation class in which the partition of manganese is required. In the latter case, a drastic inhibition of the reaction rate is to be expected. As an aid to the kinetic analysis, a portion of the iron-rich comer of the Fe-C-Mn constitution diagvam has been determined. The conclusions drawn from this work have a direct bearing on the kinetics of the grain boundary ferrite precipitation reaction and thus yield insight into the manner in which alloying elements can influence the rate of austenite-decomposition reactions. THE role of alloying elements in the austenite-decomposition reactions has provoked considerable interest, since it is apparent that an understanding of this phenomenon will yield insight into the harden-ability problem. In the present investigation, attention is focused upon the proeutectoid-ferrite transformation in the system Fe-C-Mn, and upon the manner in which manganese additions affect the growth rate of ferrite. When hypoeutectoid austenite is supersaturated, the ferrite first formed usually exhibits two distinct types of morphology. At low super saturations, grain boundary layers grow into the austenite grains with an approximately planar interface, and usually possess at least one incoherent phase boundary. At high supersaturations, Widmanstiitten plates appear, which bear an orientation relationship' to the parent austenite. In this case, there is a chance of partial lattice matching along the sides of the plates and an attendant semicoherent interface structure. Purdy and Kirkaldy2 have used a two-phase binary Fe-C diffusion couple to demonstrate that carbon volume diffusion controls the incoherent reaction, while Rouze and rube,' using thermionic-emission microscopy, have shown that the thickening of Widmanstatten ferrite plates is appreciably slower than expected for carbon-diffusion control. This suggests that the plates are bounded by partially coherent, low-mobility interfaces. The redistribution of alloying elements during transformation has been the object of studies by Aaronson4 and owmman,5 who have shown that no alloy partition occurs during ferrite precipitation in the systems Fe-C-Cr and Fe-C-Mo, respectively. More recently, Aaronson et al.6 have shown that some manganese partition occurs at low super-saturations in the system Fe-C-Mn, but that no partition occurs at higher super saturations. Pickle-simer et al.7 observed that no manganese partition occurred during the early stages of the pearlite transformation in a manganese steel, and concluded that volume diffusion of the alloying element is not the rate-deter mining factor. The diffusion coefficients pertinent to this investigation have been measured by Wells, Batz, and Mehl8 (carbon in austenite), R. P. smith9 (carbon in ferrite), Wells and Mehl 10 (manganese in austenite), and Kirkaldy and Purdy11 (diffusion of carbon on a manganese gradient in austenite). Kurdjumov 12 has shown that the diffusivity of substitutional elements in ternary austenites up-quenched from martensite may be greatly enhanced, due, apparently, to the presence of a persistent defect structure. The recent work of Krauss 19 showing high densities of tangled dislocations in reversed Fe-Ni martensite is in agreement with this conception. THEORETICAL Diffusion in ternary or higher-order systems may be described with the aid of Onsager's extenbion of Fick's first law,14 in which the flux of each of the n components is assumed to be a linear function of all concentration gradients:

Jan 1, 1964

By D. I. Bardos, A. M. Bardos, Paul A. Beck

The approximate effective silicon radii in ternary Laves phase alloys with transition elements and silicon were found to range between 1.16 and 1.21A, i.e., considerably smaller than the atomic radius calculated for elemental silicon. The atomic radii were found to increase with the size of the transition element atoms in the structure. It has been noted by Aronsson1 that silicon in solid solution in iron, cobalt, and nickel has an effective CN 12 atomic radius of approximately 1.20Å, a value very much smaller than the CN 12 radius of 1.34Å derived from the interatomic distances in elemental silicon.7 The unusually short metal-silicon interatomic distances in transition element monosilicides and disilicides have been also pointed out by various investigators.1 A similar tendency was observed by Nevitt in Cr3O type silicides.2 Recent work on Laves phases in ternary systems of silicon with transition elements4"8 suggested that extensive Laves phase fields may occur in such systems.8 It has been pointed out by Laves7 that the MgCu2 and MgZn2 type structures lend themselves particularly well for the determination of the effective radii of the component atoms. In these AB2 phases the B atoms are arranged in close packed straight lines, and their distances are related in a simple manner to the lattice parameters.8 The present work was undertaken in order to find out whether or not the B transition element in such AB2 Laves phases can be replaced to an appreciable extent by silicon. In cases where extensive substitution proves to be possible, the "average atomic radius for the B site" at various silicon contents was to be determined, and the effective atomic radius of silicon calculated by extrapolation to complete replacement of the B component by silicon. The following ternary systems were studied: Ti-Mn-Si, Ti-Fe-Si, V-Co-Si, V-Ni-Si, Nb-Fe-Si, Nb-Co-Si, Ta-Fe-Si, Ta-Co-Si, and Mo-Fe-Si. In all of these alloy systems, the Laves phase solid solutions were found to extend over a considerable range of silicon content. As previously reported,4"8 in the V-Co-Si, V-Ni-Si, and Mo-Fe-Si ternary systems Laves phases do occur, even though there are no corresponding binary Laves phases. In stabilizing the Laves phase, silicon acts as if it were decreasing the effective electron concentration of the alloy.8 A more detailed study of the boundaries of the Laves phases in the V-Co-Si and V-Ni-Si systems has been made and will be published elsewhere. In both cases, the elongated Laves phase fields extend parallel to the B-Si side of the composition triangle, clearly showing that silicon is substituting for cobalt, respectively, nickel over a wide range of compositions. The binary Laves phases NbCo2 and TaCo2 have the cubic MgCu2-type structure. It was found that, in both ternary systems with silicon, again the MgZn2-type hexagonal Laves phase is stabilized and that this phase extends to high-silicon contents. In each ternary system at least three Laves phase alloys of the composition A(B1_xSix)2 were used, x here denoting the fractional substitution of the B-transition element atoms by silicon. The alloys, listed in Table I, were prepared by arc melting in argon. Each specimen was water quenched after a homogenizing anneal of 3 days at 110(FC, except for the Mo-Fe-Si alloys, which were annealed at 1200°C. Metallographic examination showed that most of the homogenized alloys consisted of a single phase, although some had minor amounts of a second phase, as shown in Table I. X-ray diffraction patterns were prepared using an asymmetrical focusing camera of large resolution and Fe Ka radiation. The patterns were indexed, and the lattice parameters of the hexagonal Laves phases were determined from the available diffraction lines of highest indices: (213), (205), and (313). The lat-

Jan 1, 1963

By E. A. Gulbransen, K. F. Andrew

In a recent communication14 we have reported on the kinetics of the reactions of zirconium with O2, N2 and H2 as a function of the time, temperature and pressure variables. A systematic study was made and the results correlated with fundamental theories of gas-metal reactions. This paper will present a similar study for titanium. Titanium and zirconium are members of the IV group of the periodic table and possess many similar physical and chemical properties as a result of their similar electronic configuration for the outer electrons. The two metals are relatively inert to both gas and liquid phase corrosion at room temperature. However, at moderate temperatures the metals become active and react readily with the common gases including O2, N2 and H2 which are of interest in this study. A study of the kinetics of these gas-metal reactions is of interest for three reasons: (1) to understand the rate of reaction of titanium and its role in the behavior of high temperature alloys; (2) to understand the practical difficulties of the reduction, refining and working of titanium; and (3) to correlate the data with fundamental theories of gas-metal reactions and crystal structure predictions. Literature Survey Several review papers8 and books4243 exist on the preparation and properties of titanium and its alloys. THE METAL Titanium has, at room temperature, a hexagonal lattice of the zinc type. Hagg19 gives a value of 2.953A for the (a) axis, a value of 4.729 for the (c) axis and a density of 4.427 at 20°C. Burgers and Jacobs6 have observed the transformation of the hexagonal to the body-centered cubic structure at 880°C and have established a value of 3.31 for the cube edge and a density of 4.31. TITANIUM-OXYGEN Carpenter and Reavell6 using a pressure change method have studied the reaction at temperatures of 742° and 1000°C and for a pressure of one-fifth of an atmosphere. The probabilities for reaction are calculated from kinetic theory and they report a value of 10-5 for O2 at 1000°C and 10-6 at 740°C. The titanium-oxygen system has been investigated by Ehrlich.10,11 Five phases are observed. Between (TiO2 and TiO1.90) an alpha-phase, consisting ofarutile lattice, is found. A beta-phase is observed between (TiO1.80 and TiO1.70). A gamma-phase is homogeneous between (TiO1.56 and TiO1.46) and has a structure of the corundum type. The delta-phase exists between TiO1.25 and TiO0.6 and has a sodium chloride structure. From TiO0.42 to Ti the metal structure is observed. The surface oxide films have been studied by Hickman and Gulbransen.20 The rutile structure is observed in the temperature range studied, 300 to 700°C. Three crystalline modifications of TiO2 exist: rutile and anatase which are tetragonal and brookite which has an orthorhombic structure. Anatase is reported36 to exist in two forms: I and II. Anatase II changes to anatase I at 642°C. Anatase I is stable up to 915°C where rutile becomes the stable modification. At 1300°C rutile transforms to brookite which melts at 1900°C. The monoxide, TiO, may be prepared from the dioxide by high temperature reduction with carbon or magnesium. Its melting point is 1750°C. TITANIUM-NITROGEN Carpenter and Reavell6 report that at 1000°C a linear rate law is observed. The probability of reaction is given as 10-8 at 1000°C. Fast12 has studied the solubility of nitrogen and its effect on the mechanical properties of the metal. The crystal structure of TiN has been shown by several workers2'21'44 to follow the sodium chloride structure. However, the calculated density is found to differ from the pycnometric value. This is studied by Brager3,4 in detail. He has suggested that the titanium sites in the lattice are only partially filled at low temperatures. As the temperature of preparation is raised the vacant sites become occupied which expands the lattice and increases the hardness and density. An (a) value of 4.22Å is given for room temperature. TITANIUM-HYDROGEN The solubility and the crystal structures observed in this system have been reviewed in a recent book by Smith.40

Jan 1, 1950

By R. E. Reed-Hill

The lattice reorientations in (1011) and (1013) twins of pure magnesium have been investigated using polarized light. Both forms (Ire subject 20 almost complete second-order twinning on the (1012) plane of the first-order twin. The observed deviations of composition planes by 6 and 3 deg, respectively, from coherent twinning planes are caused by second-order twinning. The experimental data indicate that (1011) and (1013) are reciprocal twins with a twinning shear 0.136. In a previous paper1 it was shown that the primary fracture mechanism in single crystals of magnesium strained in tension parallel to the basal plane is parting or fracture inside twins. In the temperature interval 25oto286oC, the observed twinning form, dominant in fracture, was a very small twin which formed in bands. Because these twins were very small and appeared in extensively deformed regions they did not lend themselves to X-ray analysis. Lacking Cou-ling and Pearsall&apos;s2 recently reported polarized light technique for magnesium, the twinning plane indices (K1) were originally determined by two surface composition plane measurements. This plane fell in a major zone (axis < 1120>) at an angle of 55.75 deg to the parent crystal basal plane. Because its habit deviated by 6 deg from{l0ll), 61.9 deg, the experimental data per se were not consistent with simple (1011) twinning and, for expedience, the twin was designated with habit plane indices (30341, 54.5 deg, representing the lowest set of indices corresponding to measured angles. Couling, Pashak, and sturkey3 have also found bands of reoriented lattice with a similar habit in polycrystalline specimens of magnesium and certain dilute magnesium alloys. These bands are significant because their formation in large numbers during plastic flow permit specific magnesium alloys to undergo very extensive deformation by cold rolling. Using polarized light2 they showed that the basal plane in the reoriented material was nearly parallel to the band habit and, as a result, proposed a mechanism for band formation involving a double twinning process in which (1011) twins form and then retwin according to (1012). Because of the obvious similarity between the twin bands observed in single crystals and in polycrystalline specimens, it was decided to investigate if the two phenomena were not identical and, if possible, to verify the retwinning hypothesis. The present paper is a report of this investigation and shows that the twins with the (3034) habit correspond closely to retwinned (l011) twins. Evi- dence is also given for an analogous retwinning process in (1013) twins. EXPERIMENTAL PROCEDURE Rectangular single-crystal specimen deformed in tension (stress axis in the basal plane parallel to [1010], as previously described,4 were polished electrolytically by Jacquet&apos;s method5 and etched in acetic picral.2 All twins studied in this investigation belong to a major zone, whose axis, [1210], was perpendicular to both stress axis and two sides of each specimen [(1210) surfaces]. These latter contain the plane of shear of the twins and measurements of the basal plane trace of twins on these surfaces determines the lattice orientations of the twins. The position of the basal plane in each case was determined with a polarizing microscope using a modification of the technique of Couling and Pearsall.2 For each twin the four positions of maximum extinction were measured, the data averaged, and then the pair of extinctions corresponding to the basal plane was determined by rotating the microscope stage and observing the position at which the color changed from orange to blue with a gypsum red, first-order plate inserted between the crossed Nicol prisms. EXPERIMENTAL RESULTS (1013) Twins—In general, especially in speci-mens deformed at 150°C, (1073) twins appear in greater numbers and larger sizes than (1011) twins, and the lattice reorientation inside these twins is therefore easier to study and measure. Fig. 1 shows an area of an unetched and unpolished (1210) crystal surface containing a number of (1073) twins sloping upward to the right at about 29 deg with the basal plane of the original crystal (horizontal direction). Slip line segments may be seen in both the parent crystal (running horizontally) and in the twins (sloping downward to the right at approximately -22 deg with the horizontal). As may be seen in Fig. 2, the slip lines in the twin are not at the proper angle (64 deg from basal plane of the

Jan 1, 1961

By Roy J. Helfinstine

One of the most urgent needs in the field of coal combustion is the ability to predict the performance of a coal from knowledge gained from small-scale tests. Numerous types of analyses and tests are conducted on coal, including the proximate and ultimate analyses, heating value, varieties of sulphur, ash analysis, ash fusion temperatures, free swelling index, petrographic analysis, and Gieseler plasticity. The Illinois Geological Survey has made an intensive study of the value of these tests for judging the performance characteristics of the coal as actually utilized. To date, three reports 1 2 3 have been is- sued, and another is being prepared. The present paper briefly summarizes some of the results found in the studies.

Jan 1, 1949

By H. F. Yancey, M. R. Geer, C. L. Allyn, R. H. Eckhouse

W. M. Bertholf—This is an excellent report of a well-conducted investigation, of sufficient scope to provide generally useful information. Some years ago we had occasion to test the Humphreys spiral on the middling from our table plant. The conclusions reached in this investigation come as no surprise to us. In certain applications it is virtually impossible to improve on the spiral from the monetary returns point of view even though its "efficiency" may be low. We have noted that in all successful applications of the spiral the feed is "classified," hence we were somewhat disappointed in the showing the spiral made on the Poplar Ridge coal, the only one of the four which could really be considered the equivalent of a classified feed. As was noted in the paper, however, this coal contained a considerable amount of middle and high-gravity material, apparently more than enough to make up for the theoretical advantage of small particles in these categories. A comparison of particle size in the various specific gravity classes of the four coals is given in table XXV. Table XXV. Comparison of Particle Size in Various Specific Gravity Classes. (Unit size, 150 m to 0.105 mm) Black Eagle Clements Poplar Ridge Rosly,n 1.3 Float 7.47 10.60 8.68 9.66 1.3 x 1.4 7.89 7.96 7.58 8.20 1.4x1.6 7.24 8.36 5.58 6.85 1.6 x 1.8 7.98 8.31 3.42 7.33 Sink 1.8 7.85 7.13 5.09 7.41 Avg 7.69 10.08 7.15 8.38 This tabulation indicates that there was very little difference in the size of the various gravity fractions for three of the coals, Poplar Ridge varying the most. Examination of figs. 3, 6, 7, and 8 reveals a general similarity of behavior for all coals except Black Eagle. Would the authors care to speculate on the reasons for the difference in the distribution curve for the sink 1.60 material in the washed coal of fig. 3 as compared to the others? This difference is particularly noticeable in the 8xl4-mesh size. Was there a "shape effect"? J. D. Price and W. M. Bertholf—The performance data on three of the coals studied appear to be typical, but there are certain departures from the general pattern in the case of the Poplar Ridge coal which are large enough to deserve special consideration. For example: 1. Despite the high proportion of sink 1.60 material in the feed, it was not possible to make a corresponding proportion of high-ash reject. (See table XVI.) 2. Even though approximately 70 pct of the feed was 1.40 float with an ash content of 6.6 pct, and this

Jan 1, 1951

By C. Wachmann

The theory presented ill this paper makes possible examination of the phenomenological aspect of secondary oil recovery by alcohol flooding. Limitutions of the theory are contingent on three primary assumptions: (1) chemical equilibrium, (2) no fingering, and (5) no dispersion. The second assumption implies one-dimensional flow. In a geometric sense the theory extends the Buckley-Leverett construction to systems of two slightly miscible fluids and to systems of three components. This is made possible by considering single-phase fluids to be made up two phases, namely of an excess of one phase and a negative amount of the other phase. The procedure is discussed in the paper. The theory also permits direct determination of the relationship between the aqueous phase saturation and the concentration of water in the aqueous phase in a zone of variable composition. The velocity of planes of constant water concentration and of constant saturation can be computted likewise and the existence of saturation discontinuities inferred. Consequently the behavior of such a Zone, which may exist at some time between a displaced and a displacing fluid, May be determined. mathematical details are given in Appendices. Problems of experimental verification are pointed out on the basis of a supporting experiment. Implications of the theory and effects of violations of primary assumptions are discussed briefly. The major conclusion is that, if the primary assumptions are met, with pure alcohol and alcohol-rich .slugs piston-like displacement of oil and water is possible. INTRODUCTION The recovery of oil from porous media by alcohol displacement has been studied for several years. Extensive experimental investigations into alcohol flooding have been carried out in the laboratory although little has been done in the field. A large amount of practical information is available, the extent being indicated by the first three papers cited 1-3 and their respecrive bibliographies. This has permitted the Formulation of inferential theories. A more analytical approach in which theories are developed primarily by reasoning from known physical principles and relationships has, however, lagged behind. In fact, an analytically derived theory of alcohol displacement has not been available. The theory presented in this paper on oil and water displacement by alcohol from a porous medium is a result of the analytical approach. The theory is idealized, as certain assumptions had to be made to make the mathematics tractable and reduce complexity of argument. In particular, major assuptions concerning conditions of equilibrium, interfacial stability, longitudinal mixing, etc., led to the exclusion of fingering, for example, from all considerations. In many laboratory experiments reported in the literature fingering could not be avoided. There is, therefore, disagreement between theories based on experimental evidence available and the analytically derived theory presented here. This theory presented is based on known physical data and makes use of the Buckley-Leverett construction and its suggested extension. The mathematical derivation of the theory is given in Appendices. Three examples relevant to alcohol displacement are discussed to explain the use of this theory: (1) system of two immiscible fluids (the Buckley-Leverett case); (2) system of two slightly miscible fluids involving an extension of the Buckley-Leverett construction; and (3) system of three components, including injection of pure alcohol or alcohol-rich fluid as a special case, which involves a more general extension of the Buckley-Leverett construction. In these discussions strict assumptions were made to impose equilibrium conditions and exclude mixing and fingering. Effects of relaxing these assumptions and consequent extensions of the theory are indicated. The complexity of the alcohol displacement process makes experimental verification of the theory essential. Some relevant evidence is offered in this paper.

Jan 1, 1965

By R. H. Eckhouse, M. R. Geer, H. F. Yancey, C. L. Allyn

W. M. Bertholf—This is an excellent report of a well-conducted investigation, of sufficient scope to provide generally useful information. Some years ago we had occasion to test the Humphreys spiral on the middling from our table plant. The conclusions reached in this investigation come as no surprise to us. In certain applications it is virtually impossible to improve on the spiral from the monetary returns point of view even though its "efficiency" may be low. We have noted that in all successful applications of the spiral the feed is "classified," hence we were somewhat disappointed in the showing the spiral made on the Poplar Ridge coal, the only one of the four which could really be considered the equivalent of a classified feed. As was noted in the paper, however, this coal contained a considerable amount of middle and high-gravity material, apparently more than enough to make up for the theoretical advantage of small particles in these categories. A comparison of particle size in the various specific gravity classes of the four coals is given in table XXV. Table XXV. Comparison of Particle Size in Various Specific Gravity Classes. (Unit size, 150 m to 0.105 mm) Black Eagle Clements Poplar Ridge Rosly,n 1.3 Float 7.47 10.60 8.68 9.66 1.3 x 1.4 7.89 7.96 7.58 8.20 1.4x1.6 7.24 8.36 5.58 6.85 1.6 x 1.8 7.98 8.31 3.42 7.33 Sink 1.8 7.85 7.13 5.09 7.41 Avg 7.69 10.08 7.15 8.38 This tabulation indicates that there was very little difference in the size of the various gravity fractions for three of the coals, Poplar Ridge varying the most. Examination of figs. 3, 6, 7, and 8 reveals a general similarity of behavior for all coals except Black Eagle. Would the authors care to speculate on the reasons for the difference in the distribution curve for the sink 1.60 material in the washed coal of fig. 3 as compared to the others? This difference is particularly noticeable in the 8xl4-mesh size. Was there a "shape effect"? J. D. Price and W. M. Bertholf—The performance data on three of the coals studied appear to be typical, but there are certain departures from the general pattern in the case of the Poplar Ridge coal which are large enough to deserve special consideration. For example: 1. Despite the high proportion of sink 1.60 material in the feed, it was not possible to make a corresponding proportion of high-ash reject. (See table XVI.) 2. Even though approximately 70 pct of the feed was 1.40 float with an ash content of 6.6 pct, and this

Jan 1, 1951

F. Lynwood Garrison, Philadelphia, Pa. (communication to the Secretary): I have been specially interested in Mr. Drake&apos;s valuable contribution to our knowledge of the Chinese coalareas, as I had occasion, about a year ago, to make a reconnoissance over some of the coal-fields in the Yangtse river valley. I noticed, as he appears to have done, certain peculiarities which may eventually prove to be characteristic of the Chinese coal-measures, at least in central and northern China. The most striking feature, perhaps, is the invariable proximity of the great granite masses to those coal-deposits which have an anthracite character. In common with all other anthracite areas elsewhere, the Chinese show evidences of profound stratigraphical disturbances. The sedimentary rocks are folded, faulted and dislocated in a manner similar to those of the Pennsylvania anthracite-regions. It would seem that the intrusive granite had much to do with converting the original bituminous into present anthracite. The action of the granite was probably direct as a conveyor of heat, and indirect as dynamic force in dislocating, overturning, and compressing the coal-measures. My stay in China was unfortunately cut short, so that I was unable to collect fossils, or to give this highly interesting and important subject more than passing attention. Cholnoky* and Richthofen are practically agreed that these granite eruptions took place during the Tertiary period—probably the Eocene. This profound tectonic movement has had a most important effect upon the geological conditions in central and northern China, and will be, I am sure, a potent factor in the economic development. Evidences of disturbances and metamorphism are very abundant in the anthracite-areas of the Yangtse valley. I noticed the coal in them, as did Mr. Drake in northern China, to be often flaky and " slickensided."

Jan 1, 1902

By W. A. Backofen, D. Lee

Titanium of commercial purity (RC-70) and two all-a (hcp) alloys (4Al-1/4O2 and 5Al-2.5Sn) were tested in sheet form under conditions of combined-stress loading. Plane-strain compression and plane -strain tension together with uniaxial compression and tension provided information for constructing the tension-tension and compression-compression quadvants of the plane-stress yield locus. Coordinate values were obtained from yield-strength measurements; the slope at each point was based on a plastic-strain ratio. The materials represented a wide range of plastic anisotropy, which was reflected in correspondingly varied departures of the yield locus from that for iso-tropy. The tension-tension and compression-compression quadrants were not identical. If the uniaxial yield strength is insensitive to direction and the sign of the applied stress, the Mises or perhaps the Tresca criterion is adequate for predicting the onset of yielding under combined-stress loading. The prediction is not so straightforward in plastically anisotropic material. Depending upon details of texture, combined-stress yielding resistance may be substantially different from that indicated by the usual criteria.&apos; However. a continuum theory of yielding under such conditions is available for reasonably direct application.2,3 Whether or not it can be used successfully must, of course, depend upon how closely the conditions of the theory are satisfied in the material in question. In its present form the theory contains no provision for Bauschinger-type effects, nor does it allow for the consequences that might grow out of deformation by more than one mechanism, e.g., by slip and twinning.4 With those limitations, more importance must be attached to experimental information about the yield surface for anisotropic materials. Studies of the yield surface have generally been confined to states of plane stress (tension-tension and tension-compression) established in thin-wall tubes by externally applied forces and internal pressure. That approach would be entirely suitable for anisotropic materials if they were available in tube form or in sufficient bulk to be machined into test specimens. 6-9 It would not serve for the textured and anisotropic sheets of much current engineering interest, unless sheet could be formed into high-pressure tubing without structural change. Since there would seem to be little chance of the latter, practical alternatives are needed. One such alternative is the subject of this paper. It involves the use of a few selected but relatively simple uniaxial loading tests on sheet specimens. The underlying idea is presented here with some details relating to application and a few results bearing on how well it has worked. THE BASIS OF TESTING Conventional Measurements. Yielding for a general case of anisotropy is described by the plane-stress locus in Fig. 1. Reference directions in the sheet specimen are related to the stress coordinates as shown with the insert: x is the rolling direction, y the transverse direction, and z the through-thickness direction. The four reference-direction yield strengths (t, tension; c, compression), Xt, Xc, Yt, and Yc, are all different. Nine paths that can be followed in uniaxial loading experiments are identified. Paths 1, 2, 3, and 4 represent conventional tension and compression testing to establish Xt, Yt,.Xc, and Yc. Useful information also comes from the ratio of the transverse plastic-strain components associated with yielding and flow in conventional tension or compression. A condition to be satisfied in locus construction is that the plastic-strain vector be everywhere normal to the locus.10 The projection of the vector which would

Jan 1, 1967

By I. Javandel, P. A. Witherspoon

The finite element method was originally developed in the aircraft industry to handle problems of stress distribution in complex airframe configurations. This paper describes how the method can be extended to problems of transient flow in porous media. In this approach, the continuum is replaced by a system of finite elements. By employing the variational principle, one can obtain time dependent solutions for the potential at every point in the system by minimizing a potential energy functional. The theory of the method is reviewed. To demonstrate its validity, nonsteady-state results obtained by the finite clement method are compared with those of typical boundary value problems for which rigorous analytical solutions are available. To demonstrate the power of this approach, solutions for the more complex problem of transient flow in layered systems with crossflow are also presented. The generality of this approach with respect to arbitrary boundary conditions and changes in rock properties provides a new method of handling problems of fluid flow in complex systems. INTRODUCTION Problems of transient flow in porous media often can be handled by the methods of analytical mathematics as long as the geometry or properties of the flow system do not become too complex. When the analytical approach becomes intractable, it is customary to resort to numerical methods, and a great variety of problems have been handled in this manner. One such method relies on the finite difference approach wherein the system is divided into a network of elements, and a finite difference equation for the flow into and out of each element is developed. The solution of the resulting set of equations usually requires a high speed computer. When heterogeneous systems of arbitrary geometry must be considered, however, this approach is sometimes difficult to apply and may require large amounts of computer time. The finite element method is a new approach that avoids these difficulties. It was developed originally in the aircraft industry to provide a refined solution for stress distributions in extremely complex airframe configurations. 27 Clough has recently reviewed the application of the finite element method in the field of structural mechanics.6 The technique has been applied successfully in the stress analysis of many complex structures.l, 27, 28 Recognition that this procedure can be interpreted in terms of variational procedures involving minimizing a potential energy functional7 leads naturally to its extension to other boundary value problems. In the field of heat flow, there recently have been introduced several approximate methods of solution that are based on variational principles.2-4, 17 By employing the variational principle in conjunction with the finite element idealization, a powerful solution technique is now available for determining the potential distribution within complex bodies of arbitrary geometry. In the finite element approximation of solids, the continuum is replaced by a system of elements. An approximate solution for the potential field within each element is assumed, and flux equilibrium equations are developed at a discrete number of points within the network of finite elements. For the case of steady-state heat flow, the technique is completely described by Zienkiewicz and Cheung.33 Since the flow of fluids in porous media is analogous to the flow of heat, Zienkiewicz et al. have employed the finite element method in obtaining steady-state solutions to heterogeneous and anisotropic seepage problems. 34 Taylor and Brown have used this method to investigate steady-state flow problems involving a free surface.25 The work of Gurtin has been instrumental in laying the groundwork for the application of finite element methods to linear initial-value problems.12 As a result, Wilson and Nickell have recently

Jan 1, 1969