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Jan 1, 1881

Jan 1, 1961

By F. Milton Lewis, Roshan B. Bhappu

In Part I of this article, published in the September issue of MINING ENGINEERING, Edward S. Frohling and Robert M. McGeorge of Mountain States Mineral Enterprises, Inc., reviewed the general overall financial aspects and methods of raising the necessary capital for developing the mine. In Part 2 of this two-article series, we will present several case histories to show how realistic feasibility studies are used for evaluating the choices and options entailed in mining ventures. Basically, after a potential ore body has been discovered, and the necessary funds obtained for the development of the deposit, the next major problem is one of ascertaining what specifically is to be produced, and the costs of production. The profit potential of the mining venture can then be accurately assessed by the mine owner, operator, or loan officer.

Jan 10, 1975

Jan 1, 1928

Jan 1, 1951

Jan 1, 1949

By O. C. Wheeler

The reports for the years 1942 and 1943 were written in 1943 and 1944, respectively, and were summarized at the meetings of the Institute in February of those two years, but the manuscripts were held until released for publication in 1945. 1942 Lack of transportation facilities during 1942 resulted in a sharp curtailment of producing operations in Colombia at the middle of the year, and production of crude for export was thus greatly affected. Total production in 1942 amounted to only 10,617,008 bbl. Of this amount the Tropical Oil Co. produced 9,453,995 bbl., which includes 126,607 bbl. of petroleum condensate, and the Colombian Petroleum Co. produced 1,163,013 bbl, from its Barco property. The following summary, showing a comparison of the number of barrels of crude produced during the years 1941 . and 1942, reflects the extent to which producing operations were reduced during 1942: These above figures do not include any production recovered from Condor's (Shell's) Casabe wells during testing operations. An active exploratory drilling Program was carried out by several of the major companies, and the year 1942 established a record for wildcatting activity in Colombia. Excluding development operations on the Shell's Casabe wells on the Yondo concession, 13 wildcats were completed. Eight of these were in the Magdalena Valley, and all were abandoned; five were on the Barco Concession, and of these four were completed as producers. Particularly notable among the petroleum developments during 1942 was the continued success of drilling operations on the Shell's Yondo Concession of Compania Colombiana de Petroleos El Condor opposite Barranca-Bermeja. In 1941 the discovery well, Casabe No. I, was drilled and completed with an estimated initial production of approximately 480 bbl. of 19 1/2 A.P.I. gr. oil per day. In 1942, seven stepouts from the discovery well were successfully completed with initial production ratings ranging from 150 to 1230 bbl. per day. An extensive wildcat program on the Barco Concession of the Colombian Petroleum Co. yielded favorable results with the completion of four producers in the Socuavo and Tres Bocas areas; a fifth test, Sardinata Norte No. I, was abandoned at 500;" feet. Wildcatting operations in the lower Magdalena Valley area were initiated in 1942 With the drilling of a deep test, El Dote No. I, by Compania de petroleo Shell de Colombia. The well was a failure and was abandoned at a depth of 10,046 (?) feet. Wildcats on which drilling operations were being conducted at the end of the

Jan 1, 1945

By R. Priestner, R. F. Krause

Solid, cylindrical crystals of a iron have been fractured at 10°K by the simultaneous application of tension and torsion stresses. Thus, a complex distribution of normal (sN) and shear (ss) stresses existed in a cleavage plane at fracture. In every crystal the clearage-plane normal was inclined to the specimen axis, so that a unique point, M*, in the operative cleavage plane was defined where the quantities sn and s2 N + s2 S) were maximized. It is shown that the maxirrzum resolved normal stress on the cleavage plane at fracture decreased as the resolved shear stress in the cleavage plane at M* increased. Thus, a normal stress la?*! for fracture was not satisfied. A fracture law which takes shear stress into account. and which predicts that (s2 N +s2S) should be constant, is to be preferred. However, the data is best represented by a linear dependence of sN on sS, sS which 110 physical significance can be attached. It is suggested that the fracture-nucleating -mechanism is a short-range one. A critical stress law has not yet been unequivocally established for the propagation of cracks in metals. Measurements of the tensile fracture stresses of variously oriented zinc single crystals1 failed to establish the normal stress law expressed by In Eq. [1], sN is the fracture stress, P, resolved normal to the fracture plane, and 0 is the angle between the fracture -plane normal and the stress axis. Similarly, a comprehensive investigation failed to verify the normal stress law for the tensile fracture of a-iron single crystals.' However, a subsequent investigation,3 using iron from the same ingot, presented fracture data which obeyed the normal stress law. It has been shown previously4,5 that the Griffith criterion for fracture6 does not predict such a normal stress law. This result is of particular significance in considering the fracture of metals, such as zinc and iron, which may fail by cleavage. In these metals fracture propagates in a well-defined "cleavage" plane, whether or not an applied stress is normal to it. When the cleavage-plane normal deviates from the direction of application of a uniaxial stress, for example, the crack propagates under the combined action of normal (sN) and shear (ss) stresses. When the presence of a resolved shear stress in the crack plane is taken into account, the Griffith criterion is expressed by4

Jan 1, 1967

By Frank L. Nason

The complete statistics of zinc-ore production in the United States for 1916 are not yet available. The following figures are, therefore, only approximate. The total production of concentrates for this year, including New Jersey and Tennessee, amounts to 1,280,000 tons. Of this tonnage, five mines produced approximately 536,000 tons, or about 42 per cent. The remaining concentrates came mainly from the "Valley States," Missouri, Wisconsin, Kansas, Oklahoma, and Arkansas. The 1,280,000 tons of concentrates represent about 25,000,000 tons of crude or run-of-mine ore. Of the five mines producing the 42 per cent., the dimensions of three are approximately 25 by 1500 by 2000 ft. (7.6 by 457 by 610 m.) or more: Four of the five mines are undoubtedly veins in the common acceptance of the term. They are located in great vein systems. The mines of "Valley States" have not, so far, been classed as veins; simply as "secondary deposits." The origin of these secondary deposits has been a warmly debated subject. Many geologists contend that they are the results of leaching from overlying and adjacent rocks carrying but traces of zinc, these traces leached out and deposited in favorable places. If this theory is correct, deposition is strictly limited to shallow depths. No volume of ore comparable to that of "true fissure veins" is to be expeeted. Just what this means to the future of zinc mining in these States is here indicated. The total zinc-bearing areas of these States amount to about 9124 sq. miles (23,631 sq. km.). Assuming that 10 per cent. of this total will be more or less productive, there are 912 sq. miles (2,362 sq. km.) of "mineral" ground. The average depth of the operated mines is something like 100 ft. or less. In Bulletin 606, U. S. Geological Survey (1915), p. 202, Siebenthal makes the following statement: "The writer is informed by William Waugh, one of the drillers of the Stone City well, that cuttings from other wells which he drilled in the vicinity of Pittsburgh, Kan., have shown galena and sphalerite taken from a depth of 591 ft. (180.14 m.) and that the showing of ore at some places would be considered good enough to justify the sinking of a shaft had it been found in the Joplin district and at shallower depths." Leaving out of the question the prohibitive cost of such a shaft, there is

Jan 1, 1918

By D. W. G. White

The surface tension of zinc has been determined by the sessile-drop method in a progvam emphaszzing detailed experimental care. The surface tension oj. the pure metal was measured over a range of ternperature from the melting point (419.6"C) to 540°C. At the melting point, the suvface tension of zinc was found to be: 1) 99.99+ pct Zn - 757.0 5 dynes per cm 2) 99.999+ pct Zn - 761.0 5 dynes per CI 3) 99.9999 pct Zn - 76'7.5 5 dynes pev cna From experiments carried out in a closed, isothermal cell, the temperatuve coejyicient of the surface tension (6y/6T)p, , of zinc xas found not to be negative but to be postiue. A positive tenzperatre coefficient indicates a greater depee of order in the surface than in the body of the lquid. Houlezer, by delibevately inducing continuous zaporization, a temperature coefficient of su)face tension with a negative slope can be produced. The work reported herein was undertaken as part of a larger program of fundamental research on the properties of molten zinc and zinc alloys. This program grew out of a continuing interest at the Mines Branch in the behavior of liquid metals in the foundry and of a desire on the part of the zinc industry for reliable basic data. The work was originally begun in cooperation with the Consolidated Mining and Smelting Co. Ltd., Trail, B.C., and was subsequently continued from 1960 to date in association with the Canadian Zinc and Lead Research Committee and the International Lead and Zinc Research Organization. Although the surface tension of liquid zinc has of course, been measured before, as Table I and Fig. 1 demonstrate, there is not good agreement among the existing values. Such disparities arise because sources of experimental error are insufficiently controlled. This situation places the reader in doubt. One result may be correct within narrow limits—even fortuitously—yet, because there is not agreement between several values, none can be accepted with conviction and all become suspect. It seemed, therefore, in the circumstances, that there was room for further work embodying considerable emphasis on experimental care in order that, within a small margin of error, an accurate value might be obtained. It was in this spirit that the work was undertaken. METHOD The sessile-drop method -as adopted for this work. In this method, when applied to liquid metals at elevated temperatures, the outline of the meridional section through a drop resting freely on a horizontal surface is recorded photographically. Certain parameters

Jan 1, 1967

By R. K. Waring, E. C. Truesdale

The Research Division of The New Jersey Zinc Company (of Pa.) has conducted, over a period of years, numerous tests of the reducibility of various zinc ores and the reactivity of various kinds of coal, using a number of different experimental methods. Most of these tests have involved the reduction of mixtures of ore and coal. In such cases the following simultaneous reactions are believed to be mainly responsible for the production of zinc, as was shown by Bodenstein1 some years ago: ZnO+ CO = Zn + CO2 [1] C + CO2 = 2 CO [2] Each of these reactions is dependent upon the other for the supply of CO and CO2 , respectively, and the over-all rate of formation of zinc vapor will be determined by the slower of these two reactions. An analogous pair of reactions sometimes of importance is: ZnO + H2 = Zn + H2 O [31 C + H2O = CO + Ha [4] While it is possible to obtain an indication of the relative reducibility of various ores, or of a fine versus a coarse ore, by testing them with a given coal, and while coal reactivities may be measured in a corresponding manner, a more direct determination of relative reaction rates is possible by carrying out the desired reactions, such as I and 2, separately and under reproducible conditions. It is one purpose of this paper to present some of the results that have been obtained by the latter method. C. G. Maier2 discusses the relative rates of reactions I and 2 and concludes that while at relatively low temperatures reaction I is faster than reaction 2, this difference becomes less with increasing temperatures, although at smelting temperatures reaction I is still the faster. He also says (p. 48), "At temperatures . . . between IIOO° and I3oo°C., . . . the zinc oxide reduction reaction is intrinsically more rapid than its physical limitations; that is, it is limited by gas diffusion rates." In the course of numerous reactivity tests involving the separate use of reactions and I and 2, we have obtained data that show by direct experiment, we believe for the first time, the correctness of Maier's statements concerning these reactions. We also include data on the relative rate of reaction 3, involving reduction by hydrogen. Finally, some interesting results obtained by carrying out the reduction of a zinc ore-coal briquet in streaming atmospheres of nitrogen, carbon monoxide and hydrogen, respectively, are presented. *

Jan 1, 1943

By E. C. Truesdale, R. K. Waring

The Research Division of The New Jersey Zinc Company (of Pa.) has conducted, over a period of years, numerous tests of the reducibility of various zinc ores and the reactivity of various kinds of coal, using a number of different experimental methods. Most of these tests have involved the reduction of mixtures of ore and coal. In such cases the following simultaneous reactions are believed to be mainly responsible for the production of zinc, as was shown by Bodenstein1 some years ago: ZnO+ CO = Zn + CO2 [1] C + CO2 = 2 CO [2] Each of these reactions is dependent upon the other for the supply of CO and CO2 , respectively, and the over-all rate of formation of zinc vapor will be determined by the slower of these two reactions. An analogous pair of reactions sometimes of importance is: ZnO + H2 = Zn + H2 O [31 C + H2O = CO + Ha [4] While it is possible to obtain an indication of the relative reducibility of various ores, or of a fine versus a coarse ore, by testing them with a given coal, and while coal reactivities may be measured in a corresponding manner, a more direct determination of relative reaction rates is possible by carrying out the desired reactions, such as I and 2, separately and under reproducible conditions. It is one purpose of this paper to present some of the results that have been obtained by the latter method. C. G. Maier2 discusses the relative rates of reactions I and 2 and concludes that while at relatively low temperatures reaction I is faster than reaction 2, this difference becomes less with increasing temperatures, although at smelting temperatures reaction I is still the faster. He also says (p. 48), "At temperatures . . . between IIOO° and I3oo°C., . . . the zinc oxide reduction reaction is intrinsically more rapid than its physical limitations; that is, it is limited by gas diffusion rates." In the course of numerous reactivity tests involving the separate use of reactions and I and 2, we have obtained data that show by direct experiment, we believe for the first time, the correctness of Maier's statements concerning these reactions. We also include data on the relative rate of reaction 3, involving reduction by hydrogen. Finally, some interesting results obtained by carrying out the reduction of a zinc ore-coal briquet in streaming atmospheres of nitrogen, carbon monoxide and hydrogen, respectively, are presented. *

Jan 1, 1943

By Ralph Hultgren, Bernard York, David W. Mitchell

It is a well-known fact that magnesium-alloy castings are apt to be coarse grained if the melt is not superheated several hundred degrees above the melting point before casting. (The casting temperature varies according to the shape of the mold. A typical casting temperature is perhaps 1400°F.) British practice is to heat to 900°C. (1650°F.) for 15 min., cool rapidly to casting temperature, and cast.' In Germany superheating is carried out at 850' to 900°C. (1560° to I650°F.) depending on the amount of scrap used. It is stated that further refinement will be obtained by prolonging the period of superheating or by increasing the superheat temperature to 950°C. (I740°F.).2 In the United States superheating is carried out at 1600' to 1700°F. (871' to 927°C.), perhaps even higher in some places. It is stated that superheating effects are found as low as 1500°F. (816°C.), but several hours are required.3 These commercial treatments lead to grain sizes in sand castings that may vary from 4 to 8 grains per sq. mm.! in heavy sections and from 60 to IOO in 1/2-in. sections to perhaps 380 in thin sections. It is also generally known that the refining effect of superheating is lost if the melt is held for some time at lower temperatures before casting. Aside from these very qualitative statements, there is little to be found in the literature as to the effect of temperature and time of superheat on the final grain size. Because of the uncertainty of these factors, the Office of Production Research and Development of the War Production Board placed a research contract with the University of California, supervised by the War Metallurgy Committee. This research, which was done under the "Restricted" Project NRC-550, is the basis of this paper, which has been released for publication by the O.P.R.D. The material studied was an alloy prepared from carbothermic magnesium. The temperatures required for grain refinement proved to be widely different from those used in practice, as described in the statements in the first paragraph. This difference is attributed to the fact that the magnesium in the specimens was of carbothermic origin instead of the electrolytic magnesium in common use. Materials and Methods The magnesium alloy studied was one of the most popular sand-casting alloys, with a nominal composition of 9 Per cent aluminum, 2 per cent zinc, and 0.1 per cent (minimum) manganese and the balance magnesium. This is designated

Jan 1, 1945

By D. L. Murphy

As the world's demand for iron ore increases and available sources are depleted, increasing attention will be centered on deposits in the Quebec and Labrador portions of the Labrador Trough. Improvements in the beneficiation of low grade iron ore stimulate exploration for deposits amenable to concentration. This study is centered on the Mt. Wright-Lake Carheil area, lying between Wabush Lake and Mt. Reed in Quebec. Described in detail are the general lithology, structural and economic geology, and mineralogy of the iron-formation. The iron ore province of Quebec and Baffin Island extends from the southern shore of Baffin Island southeasterly through the Labrador Trough to Matonipi Lake, approximately 185 miles northwest of Seven Islands. (See Fig. 1.) Future geologic field studies will very likely extend the limits of the province to include the ferruginous and associated rocks of the Lake Mistassini area. At present, only the deposits near Knob Lake and Lake Jeannine (Mt. Reed) have been brought into production although in the near future several other properties including those at Wabush Lake will commence operations. Within the next decade or so, as the world's demand for iron ore increases and available sources are depleted, this region will undoubtedly become a major mining district. Initial reports of iron-rich rocks in Labrador date back to the investigations undertaken by A. P. Low near the turn of the century. During the ensuing years the country was prospected chiefly for precious and nonferrous base metals because of its inaccessibility and the low tenor of the iron ore. In 1945 an evaluation of the direct shipping ores near Knob Lake indicated that sufficient reserves were present to justify the enormous capital expenditure required to bring the deposits into production. Initial open-pit mining began in 1954 following the construction of a townsite, a 360-mile long railway to Seven Islands, and an elaborate loading dock. Recent improvements in the beneficiation of low-grade iron ore have stimulated widespread exploration for deposits amenable to concentration, particularly along the southwest extension of the Trough. The Mt. Wright-Lake Carheil area, bounded by longitudes 66O45' and 67 30' west and latitudes 52 30' and 52 45' north, is a nearly rectangular tract of slightly less than 500 sq miles. It is in Saguenay County, 174 miles north-northwest of Seven Islands (see Fig. 1). Except for the periods of the spring breakup and the fall freeze up, the area is readily accessible by air.

Jan 1, 1962

By D. Van Winkle, C. Zener, H. Nielsen

THe internal friction of metals has been studied frequently at elevated temperatures.1-4 In most cases it rises rapidly with increasing temperature. The notable exceptions are ferromagnetic materials, and lead.2 This paper describes an investigation of the general features of the high-temperature internal friction of 70-30 alpha brass. Experimental Method The internal friction of a metal is a measure of the rate at which it dissipates energy of vibration. The measure adopted here, I/Q, is (I/2p) times the specific damping capacity, a measure commonly used by engineers. In the measurements of this investigation, the stress level was kept so low that the observations were independent of the stress, The general method of making such measurements, and of observing temperature of a specimen during a measurement, has been described.5.6 The driver and detector coils were wound with wire insulated with glass thread, in order to withstand the high temperatures used. In previous work electrical eddy currents in the specimen served as the coupling between the mechanical vibration of the specimen and the electrical circuits of the driver and detector. The high values of the internal friction in the present investigation rendered this coupling insufficient. In order to increase this coupling, pole pieces cut from ordinary telephone diaphragms were attached to the specimen. A mixture of French chalk and water glass proved a satisfactory cement. All measurements were made at the fundamental frequency for transverse vibration, which lay between 600 and goo cycles per second. Specimens The measurements described herein were made upon a single cylindrical rod of brass, kindly furnished by Dr. Cyril Stanley Smith, of the American Brass Co. This rod was the specimen E of the set of alpha-brass rods described in an earlier paper.6 Successively larger grain sizes were obtained by annealing at .successively higher temperatures. A slice was cut off one end after each set of measurements. Grain counts were made on each slice by W. E. Lindlief, of the American Brass Co. Nine fields were viewed on each piece, located as shown in Fig. I. Results are given in Table I. The grain size is defined in the usual way; namely, Grain size = I/n1/2 where n is the mean number of grains per square millimeter. Experimental Results This paper is concerned exclusively with the part of the internal friction that be. comes dominant at high temperatures. It is thus desirable to separate this part of the internal friction from the part that is

Jan 1, 1942

By C. Zener, H. Nielsen, D. Van Winkle

THe internal friction of metals has been studied frequently at elevated temperatures.1-4 In most cases it rises rapidly with increasing temperature. The notable exceptions are ferromagnetic materials, and lead.2 This paper describes an investigation of the general features of the high-temperature internal friction of 70-30 alpha brass. Experimental Method The internal friction of a metal is a measure of the rate at which it dissipates energy of vibration. The measure adopted here, I/Q, is (I/2p) times the specific damping capacity, a measure commonly used by engineers. In the measurements of this investigation, the stress level was kept so low that the observations were independent of the stress, The general method of making such measurements, and of observing temperature of a specimen during a measurement, has been described.5.6 The driver and detector coils were wound with wire insulated with glass thread, in order to withstand the high temperatures used. In previous work electrical eddy currents in the specimen served as the coupling between the mechanical vibration of the specimen and the electrical circuits of the driver and detector. The high values of the internal friction in the present investigation rendered this coupling insufficient. In order to increase this coupling, pole pieces cut from ordinary telephone diaphragms were attached to the specimen. A mixture of French chalk and water glass proved a satisfactory cement. All measurements were made at the fundamental frequency for transverse vibration, which lay between 600 and goo cycles per second. Specimens The measurements described herein were made upon a single cylindrical rod of brass, kindly furnished by Dr. Cyril Stanley Smith, of the American Brass Co. This rod was the specimen E of the set of alpha-brass rods described in an earlier paper.6 Successively larger grain sizes were obtained by annealing at .successively higher temperatures. A slice was cut off one end after each set of measurements. Grain counts were made on each slice by W. E. Lindlief, of the American Brass Co. Nine fields were viewed on each piece, located as shown in Fig. I. Results are given in Table I. The grain size is defined in the usual way; namely, Grain size = I/n1/2 where n is the mean number of grains per square millimeter. Experimental Results This paper is concerned exclusively with the part of the internal friction that be. comes dominant at high temperatures. It is thus desirable to separate this part of the internal friction from the part that is

Jan 1, 1942

ONE of the features of the annual meeting was a round table conference on carbon in pig iron, on Feb. 16. This was presided over by R. H. Sweetser, and the discussion, which was so interesting as to require holding an additional session in the after-noon, ranged over such a wide field that only a brief outline of it will be attempted here. The full discus-sion together with the papers will be printed later. The first paper was one on the need for research in foundry pig iron (1648-C) by Richard Moldenke, who pointed out the trend in present practice of attaining low cost in pig iron production by hotter blast, addi-tion of. scrap to the blast furnace, and. the enormous increase in the output per furnace day. He pointed out that remelting processes are oxidizing processes, to some degree at least, and the metal cannot be given the deoxidizing finish which is characteristic of the steel-making process. The trend today is toward specifying total carbon content in foundry pig irons, and since the carbon content depends on a variety of factors, he suggested that the blast-furnace men and the foundry men study the question and agree on a minimum of total carbon for the given silicon ranges. The discussion of this paper was participated in by E. J. Lowry, Robert Jones, T. L. Joseph,. R. H. Sweet-ser, J. G. Ferguson, W. B. Coleman, E. P. Ross and D. J. Demorest and ranged from a comparison of the total carbon used in making a ton of pig iron in charcoal practice versus coke practice to a discussion of the bad effects observable from the use of scrap This paper was followed by one on carbon character-istics of copper-bearing pig iron by W.. B. Coleman. In view of the belief in many foundries that copper-bear-ing pig iron made trouble in the foundry, experiments were made by adding copper in various amounts up to 1 per cent and with the cooperation of the Robesonia Iron Co. a large number of tests were made over a run of foundry iron. The final conclusion was that the ad-dition of less than 1 per cent metallic copper did not seem to alter the physical properties and that a blast furnace using copper-bearing ore can produce pig iron suitable for making small castings, showing low-Brinell hardness test, good transverse and deflection tests, and high-total carbon.

Jan 3, 1927

By T. Koulomzine

The theory of the Askania mag-netometer, as well as a complete discussion of all factors influencing magnetometer readings, is very ably described by J. Wallace Joyce.1 We will assume that the reader is thoroughly familiar with this theory and we will use whenever possible the same notations for the physical values involved in the theory. On the other hand, we believe that the principal formulas of the magnetometer can be deduced much more easily by the use of differential calculus than by the methods employed by Joyce. Consequently, we are presenting herewith a resumé of the theory of the vertical magnetometer in order to determine the correct formula of the temperature coefficient which we propose to compensate. Short Theory of the Vertical Magnetometers We shall accept the following notations: m = total mass of the magnet system, in grams. g = acceleration of gravity, in dynes. a = projection on the axis of the magnet system of the distance between the axis of support and the center of gravity of the magnet system, in centimeters. h = projection on the perpendicular to the axis of the magnet system of the distance between the axis of support and the center of gravity of the magnet system, in centimeters. M = 2pL = M, magnetic moment of the magnet system. Z = vertical component of the earth's magnetic field. ? = deflection angle of the magnet system. = focal length of objective lens in the optical system S = scale reading. In a vertical magnetometer the axis of the magnet system is kept horizontal within 2". Therefore, referring to Fig 1, we see that the magnetic torque of the magnet system influenced by Z must be counterbalanced by the gravity torque. MZ cos ? = mga cos ? + mgh sin ? or dividing by cos ? MZ = mga + mgh tan ? If we now move to another magnetic station and the temperature conditions change, the values of all the variables will change also: M, a, and h under the influence of temperature. 2, both with time and space. ? and S, as a result of all the above changes. Taking increments in Eq 2 we obtain: (Mo + ?M)(Z + ?Z) = mgao + mg?a + mg(h + ?h) tan (? +??) but tan (? +??), because ? is small, can be replaced by tan ? + tan ??; in fact tan (?+ ??) = sin ?.cos ?? + cos ?.sin ??/ cos ?.cos ?? — sin ?sin ?? but sin ? sin ???0 tan (8 + ??) ? tan ? + tan ??. Fur-thermore, neglecting all terms of the equation containing two differentials becomes: or a differential and a tan ??, Eq 3

Jan 1, 1950