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By Archer E., Wheeler

Producing copper companies were active during 1941 owing to the national defense program the United States and the requirements of the friendly belligerent nation. This activity extended to the American owned properties in South America, the major part of whose product was taken by the Metals Reserve Co., a U.S. Government agency. Anaconda Copper Mining Co. is building a new converter plant at Anaconda, with four units of the Peirce-Smith type. The addition is being built adjacent to the reverberatory plant so as to avoid the necessity of hauling matte and slag. It is expected to be completed this year.

Jan 1, 1942

By Albert Sauveur

THE senior author of this paper has expressed the belief that when gamma iron transforms into alpha iron on reaching the A3 point, each gamma grain does not change bodily into one or more alpha grains but that the alpha phase forms first at the boundaries and along the crystallographic planes of the austenite grains.1 To quote: On cooling through the point A, at about 900° C., iron passes from the gamma condition to a. new allotropic form generally designated as alpha. This transforma¬tion, however, is not instantaneous. There are good reasons for believing that it begins at the grain boundaries and along the octahedral planes of the crystals and that it progresses until each grain of gamma iron is changed into a grain of alpha iron, the resulting structure being similar to that of gamma iron. Were it possible to retain at room temperature pure iron but partially transformed, it seems probable that a Widmanstättian structure, or rather, as later explained, a martensitic structure would be observed (Fig. 1), consisting of alpha iron at the boundaries and at the octahedral planes, and of a groundmass of gamma iron. In order to confirm, if possible, the accuracy of that view, a number of experiments were conducted. Polished samples of electrolytic iron were heated in a vacuum, maintained in the gamma range for several hours and quenched in different media. It was hoped by such means to produce and retain on the polished surfaces a pattern which would reveal the mechanism of the allotropic transformation. The samples were placed in evacuated quartz tubes, heated to 1000° C. and held 2 hr. at that temperature. The tubes were then immersed in the quenching bath. Iced brine and liquid air were used first. The polyhedral austenitic structure existing at the time of immersion was very clearly revealed (Fig. 2). This is generally ascribed to iron volatilizing at the grain boundaries.

Jan 1, 1929

By Albert J. Phillips

FOR the most part, recent changes in nonferrous physical metallurgy have been gradual and of a transition nature rather than abrupt modifications of existing methods. Development of new alloys containing vital constituents in what formerly would have been considered minute amounts has necessitated more precise control of metallurgical procedure than ever before. Realization of the remarkable effects of minute amounts of specific foreign elements in many alloys has brought about a revolutionary control in the selection of even the most commonplace raw materials. In addition, recognition of the widespread occurrence of precipitation hardness in metal systems, and the utilization of many of these, has emphasized not only the need for precise temperature control but also the necessity for considering the time factor in its effect upon properties in the treatment of most metals. These needs have caused the development and widespread installation of equipment that permits precision of control undreamed of a generation ago.

Jan 1, 1935

By Charles H. Behre

KEYNOTE of the mining geology sessions was the preparation for an extensive war with all that this implies as to the need for strategic minerals, both metallic and nonmetallic. Nevertheless the sessions also included many papers of less urgent, more theoretical interest. The stress of wartime production had the effect of giving an unusually full and rich program, the longest in recent years, with sessions continuing virtually throughout the meeting. Some papers dealt with geophysical prospecting as applied to mining geology; others were concerned with the grand strategy of mineral reserves and thus involved the industrial minerals; such "borderline" subjects were presented before sessions held jointly with the Geophysics Committee or the Industrial Minerals Division.

Jan 1, 1942

By Larry Dale

An approach for predicting the development of mineral resources on public lands has been developed as an aid to resource agencies. It consists of new techniques for evaluating the economic potential of the resource, the uncertainty about mine cost estimates, and the uncertainty about the occurrance of mineral deposits. New mineral sup- plies from inferred and hypothetical resources can be predicted by using these techniques. A case study illustrates their application in predicting development of copper resources in western Montana.

Jan 1, 1982

By Charles H. White

METHODS in colorimetry are based on the assumption that the intensity of the color of a definite volume of solution is directly proportional to the quantity of the color-producing substance' present. In the preparation and examination of colored solutions there are three ways in which the depth of color may be varied. These variables are; (1) the quantity of the coloring matter, (2) the quantity of the solvent, or the dilution, and (3) the thickness or depth of that portion of the solution examined. With these three variables as a basis, three general methods in colorimetry have been devised, all of which have been used in the determination of carbon in steel. In the first method, the quantity of the solution is kept the same for both the unknown steel and the standard, and the sections compared are of equal thickness, while the quantity of the standard is varied until the color obtained is of the same intensity as that produced by a definite quantity of the unknown steel. In laboratories where this method is used there is pre¬pared and kept for permanent use a series of solutions with varying amounts of the standard, representing the percentages of carbon from the lowest to the highest demanded in that laboratory. These standard solutions are kept in bottles, or test-tubes of equal diameter, and are arranged in series from the faintest to the most intense color, so that it is a simple matter to find the place of the unknown steel in the series, when the solution is of a definite volume and is contained in a tube of the same diameter as the standard tubes. In the second method, equal quantities of the standard and of the unknown steel are dissolved, the standard is diluted to a definite volume, and the solution of the unknown steel is di-

Sep 1, 1906

By R. H. Card

THE geophysicist is interested greatly in the resistivities of different formations or parts of the earth's crust; sometimes he is interested in a single figure in the nature of an average, or what may be called the effec-tive resistivity, of all the formations or parts of the crust at a given location to a certain depth. Conversely, the engineer whose work is concerned with the coordination of power and communication lines in such a manner as to avoid electrical interference between these two types of facilities is concerned very much with this effective resistivity; to a lesser extent he is interested in the resistivities of individual forma-tions. In connection with his work the coordination engineer often estimates the effective resistivity; of the earth from measurements of the mutual impedances of lines; in other instances he derives front measure-ments of a different type the resistivities of different layers of the crust and synthesizes these to obtain an estimate of the effective resistivity. In connection with problems concerning the coordination of power and telephone lines, measurements from which earth resistivities can be estimated have been made at many points in the United States by the American Telephone and Telegraph Co. and its associated companies, in cooperation with various power companies and with railroad companies having electrified sections of line. Other such measurements have been made by the Joint Subcommittee on Development and Research of the Edison Electric Institute and the Bell Telephone System. A study has been made of the correlation of these data with the ages and types of materials of the strata of the earth's crust at each of the test locations. The purpose of this study has been primarily to provide information to assist in predicting the order of magnitude of effective resistivities to be used in the preliminary consideration of problems con-cerned with the coordination of power and communication lines. To make the results of such a study, most useful, it is essential that the geo-logical data to be secured when such a prediction is wanted be readily obtainable. Data on the ages of strata and the types of materials com-posing geological structures can usually be obtained, therefore particu-

Jan 1, 1937

By F. C. Canney

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

Jan 2, 1959

By Renato G. Bautista, Robert A. Hard

A comparative study has been made of the ex-tractability of several of the transitim metals from thiocyanate sohtions using methyl isobutyl ketone as the organic solvent. Extractions were made of scandium (III), chromium (III), manganese (II), iron (III), and cobalt (II), of which only the scandium , iron, and cobalt were readily extracted. All extractions were made at pH 2 and at 25.5 °C. A series of extraction experiments was made on scandium, iron, and cobalt to determine the effect of thiocyanate cmcentration added as the ammonium and also as the calcium salt. Also determined was the effect of metal ion concentratim and ionic strength on the distribution ratio. Analyses of the organic extracts indicated the scandium and iron were extracted principally in the uncharged trithiocyanate form, while the extractable form of the cobalt contained the cobaltotetrathiocyanate anion. THE chemical separation of metal complexes in solution by solvent extraction has in recent years been developed for large scale separation as well as bench scale work in the laboratory. It has also found increased application in analytical chemistry. Its extension to extractive metallurgy as a unit operation is best illustrated in the recovery of uranium and vanadium from their leached liquor.5-8 The extractability of some thiocyanate systems have been extensively investigated. The rate earths have been separated from one another using butanol as a olvent. Scandium was effectively extracted in ethyl ether using ammonium thiocyanate as the com-plexing agent.'' Similarly, the separation of hafnium from zirconium has been reported using ether""' or hexone13 as the organic solvent. The extractability of various metal thiocyanates in ethyl ether is given by 0ck.l Sharp and Wilkinson15 prepared nickel-free cobalt slats by thiocyanate extraction. The mechanism of extraction of titanium (IV) thiocyanate with methyl isobutyl ketone was determined by elafosse.''" The characterization of cobalt (TI) thiocyanate complex using methyl isobutyl ketone as the solvent at near neutral aqueous solution and the spectrometric studies of the ex-tractable cobalt complex has been made by Hard." It is noteworthy to mention at this point that the fundamental characteristic of ketone (oxonium) solvent extraction is that the extractable species are uncharged ion pairs solvated by the ketone molecules, but probably still partially hydrated.

Jan 1, 1963

By Satyajeet Ghosh, Edward F. Glynn

Many of the proposed techniques for calculating the reliability of rock slopes utilize Monte Carlo simulation. Monte Carlo approaches are attractive because they provide means of solving probabilistic problems with deterministic algorithms. One disadvantage shared by all simulation-based stability analyses is that they assume all the random variables are independently distributed. The paper focuses on the independence assumption and examines the effect of correlation on the probability of failure. In particular it considers the influence of possible (albeit hypothetical) correlation between joint attitudes [(8)] and joint friction angles [($)]. The study indicates that neglecting correlation can introduce significant errors into the reliability computations. With a modest positive correlation between [8] and [$] the error can lead to a 100% overestimate of the probability of failure.

Jan 1, 1982

By Baxter C. Madden

Stress corrosion is defined and certain examples are illustrated. A number of test methods, to determine the resistance of materials to stress corrosion, are being developed at Wright Field. The criterion as to the validity of a method is stated. Three types of specimens and the method of stressing each type are described. Five different corrosive conditions are set forth. The results of tests to date, although too few for any general conclusions to be arrived at, are presented. The program for the continuation of the investigation is indicated. A bibliography is included. The development of the method of loading a cantilever-type specimen, so that a nearly constant stress at a large deflection may be realized, is presented as Appendix I.

Jan 1, 1945

By Dever Ashmead

ANTHRACITE was first mined in the Wyoming Valley and sold as an article of commerce in 1808. As some preparation has always been necessary to make it ready to burn, the preparation of anthracite must date back over a hundred years. Two vital factors have determined to a large extent the degree and method of anthracite preparation. These are, first, the character of the beds worked and the methods by which they were mined and, second, the equipment used and practices followed in the burning of the coal. It is not the intention here to go deeply into these phases of preparation as a paper of no mean length could be prepared on the history of either. Rather, the intention here is to point out the main considerations and to show the influence they have exerted on the preparation of anthracite. MINING METHODS AND THEIR RELATION TO PREPARATION In the beginning f the nineteenth century, the coal beds were virgin with the possible exception of some outcroppings that had been worked, to a slight extent, by the Indians. It is known that the American aborigines had a knowledge of the use of this fuel, because when the Wyoming Valley was purchased from them, in 1754, they mentioned the fact that, by selling the land, they would lose their coal.

Jan 9, 1921

By E. L. Tucker

PREVIOUS investigations of E. L. Tucker and R. E. Head' related in particular to the effect of cyanogen compounds on galena, sphalerite, and pyrite, and their behavior in the presence of such com- pounds. The present contribution deals with the float- ability of the copper minerals and of pyrite. All experiments were performed on minus 200-mesh material, the method of preparation being the same as that described in the previous article. Flotation tests were made in a small mechanically agitated machine of 50-g. capacity designed in this laboratory by Gates and Jacobsen." General Naval Stores pine oil No. 5 and potassium xanthate were used in constant amount in all of the experiments; these agents were selected because of their present wide use in the flotation of copper minerals. Lime additions were varied to secure four 'final hydrogen ion concentrations, namely, neutral, 0.0005 per cent CaO, 0.003 per cent CaO, and 0.02 per cent CaO. The variation in lime addition was introduced to include the range of alkalinities employed in actual flotation practice.

Jan 1, 1926

Jan 1, 1893

AUGUST began with a reminder that warfare was still in progress in the world, in that the Greeks, still at war with Turkey, threatened Constantinople in spite of the small British and French forces of occupation and opened a vista of international dangers. There was in this, moreover, a recollection that religious questions in India had prevented the British from permitting a real settlement of the Turkish question, wherein the absence of Russia as a great power is missed. Civil warfare also continued in Ireland and China. Even in the military way the world is far from being at peace. Bloody fighting between the Fascisti (anti-communists) and the com-munists in Italy led to several provinces being put under martial law, but after a while these disturbances were quelled. In the latter part of the month the assassination of Michael Collins in Ireland produced a great shock. From China, however, there came rather favorable news, indicating the early reintegration of that country, which sooner or later ought to play an important part in world affairs. Economically, instead of there being any improve-ment, there grew a clearer perception of how bad things really are. Italian politics were in chaos through inability to find anyone to accept the premiership and assume the all but hopeless task of averting national bankruptcy. The French, too, began to see their own position in true colors, and especially to see how in reconstruction they had incurred debts on the strength of what they hoped to collect from Germany, with at last the understanding that the Germans could not pay and that France herself was thereby put in a desper-ate economic position. As for the Germans, them-selves their Reichsbank defaulted on ordinary bills early in August. As the month wore on the value of marks fell more and more and at the end it appeared that Germany was on the verge of economic collapse. On Aug. 7, another international conference (with America again absent) was assembled in London, this being the thirteenth since the war, and there was a general feeling that it would be the last. The prime question was what is to be done about the German reparations, but underlying was the whole subject of the entente. The conference opened like a meeting of importunate creditors, doubtful whether the debtor had any tangible assets. On Aug. 14 this conference broke up without coming to any useful conclusion. Indeed, the acrimony among the premiers was so sharp that even the entente between Great Britain and France was endangered. The international situation was saved only by the Reparations Commission excusing the Germans from the payment of 50,000,000 gold marks that come due on Aug. 15 But although the entente was nominally preserved there is no longer any illusion about the interests of Great Britain and France being different and that henceforth they will play their own hands. The London conference was really an effort to maintain the partnership, and it failed.

Jan 9, 1922

By D. J. McAdam, G. W. Geil, R. W. Mebs

As shown in previous papers by the authorsg-17t the resistance of a metal to fracture, like its resistance to plastic deformation, is a function of all three principal stresses. A technical cohesion limit is the technically determined resistance to fracture under a specific stress system. The technical cohesive strength thus comprises an infinite number of technical cohesion limits corresponding to the infinite number Of possible com-binations of the principal stresses. The technical cohesive strength, therefore, may be represented by a surface in a diagram with the three principal stresses as coordinates. As shown in previous papers,'-'' the technical cohesive strength increases with plastic deformation, with decrease in temperature, and with increase in the strain rate. The technical cohesion limit, therefore, is affected by the same factors that affect the flow stress,$ namely, the stress system, plastic deformation, temperature, and the strain rate. More- over, the influence of each of these factors on the technical cohesion limit has been shown to be qualitatively similar to the three of the same factor on the flow stress.9-18 Advance of knowledge about the conditions determining fracture of metals has been retarded by the reluctance of some investigators to abandon preconceived ideas about how metals ''should,, behave. The incorrect idea is still prevalent that the conditions determining resistance to fracture are very different from those determining the flow stress. Attempts are still being made to express resistance to fracture in terms of a single parameter, such as a limiting value of the greatest principal stress, the maximum shearing stress, or the volume stress (one-third the algebraic sum of the three principal stresses). In these attempts, a sharp distinction is sometimes made between a ductile or ,plasticdeformation, fracture,, and a "brittle frat-ture.H In making this distinction, however, the meaning attached to the term "brittle fracture" sometimes is not made clear. The term could mean fracture after very little plastic deformation or it could be used to designate the type of fracture. An ordinary fracture of a ductile metal at room temperature starts as a cleavage, generally transverse to the direction of the greatest principal stress, but eventually changes to a shearing fracture' In some conditions, however, the metal fractures entirely by cleavage, with no evidence of plastic deformation during the fracture; the surface of such a fracture may have a

Jan 1, 1948

By R. W. Mebs, G. W. Geil, D. J. McAdam

As shown in previous papers by the authorsg-17t the resistance of a metal to fracture, like its resistance to plastic deformation, is a function of all three principal stresses. A technical cohesion limit is the technically determined resistance to fracture under a specific stress system. The technical cohesive strength thus comprises an infinite number of technical cohesion limits corresponding to the infinite number Of possible com-binations of the principal stresses. The technical cohesive strength, therefore, may be represented by a surface in a diagram with the three principal stresses as coordinates. As shown in previous papers,'-'' the technical cohesive strength increases with plastic deformation, with decrease in temperature, and with increase in the strain rate. The technical cohesion limit, therefore, is affected by the same factors that affect the flow stress,$ namely, the stress system, plastic deformation, temperature, and the strain rate. More- over, the influence of each of these factors on the technical cohesion limit has been shown to be qualitatively similar to the three of the same factor on the flow stress.9-18 Advance of knowledge about the conditions determining fracture of metals has been retarded by the reluctance of some investigators to abandon preconceived ideas about how metals ''should,, behave. The incorrect idea is still prevalent that the conditions determining resistance to fracture are very different from those determining the flow stress. Attempts are still being made to express resistance to fracture in terms of a single parameter, such as a limiting value of the greatest principal stress, the maximum shearing stress, or the volume stress (one-third the algebraic sum of the three principal stresses). In these attempts, a sharp distinction is sometimes made between a ductile or ,plasticdeformation, fracture,, and a "brittle frat-ture.H In making this distinction, however, the meaning attached to the term "brittle fracture" sometimes is not made clear. The term could mean fracture after very little plastic deformation or it could be used to designate the type of fracture. An ordinary fracture of a ductile metal at room temperature starts as a cleavage, generally transverse to the direction of the greatest principal stress, but eventually changes to a shearing fracture' In some conditions, however, the metal fractures entirely by cleavage, with no evidence of plastic deformation during the fracture; the surface of such a fracture may have a

Jan 1, 1948

By Joseph Hartshorne

EVER since the introduction of the Siemens regenerative furnace, it has been recognized that a certain amount of gas is lost each time the furnace-action is reversed. This loss comes, first, from the gas which passes through the free connection between the stack and the gas-main, which exists during the time the reversing-valve is turning over; and, secondly, from the gas which is in the chamber and flues, from the hearth to the reversing-valve, and which flows back into the stack when the valve is reversed. The amount lost each time is, of course, small; but, since a furnace is reversed from four to six times every hour for, say, 300 days in the year, the aggregate is large enough to be well worth attention. The estimate has been published' that, in Germany alone, the loss from the first of the above causes amounts to 500,000 marks per year. The loss from the second cause is, of course, much greater. A process for saving this entire loss, in use at Teplitz, Bohemia, for about three years, would seem to be of value even in this country, where coal is cheaper and where the small economies are not watched as closely as they are in Europe. This process, invented by Mr. Adelbert Kurzwernhart, late Technical Director of the Teplitzer Walzwerk, has been worked out with thoroughness. The loss from each cause can be calculated very closely for any particular furnace, and the results to be expected from the process ascertained. It can easily be decided, therefore, whether it is worthwhile to adopt it under any given conditions. The amount of chamber-gas lost depends upon,-(1), the cubic contents of the chamber, ports and flues upon one side of the furnace, up to the reversing-valve (deducting, of course,

Mar 1, 1906

By DOOLITTLE E. M.

(Canal Zone Meeting, November, 1910.) The following notes are contributed, not with the idea of offering a complete history of the development of this very important process as applied to the Leadville district, but with the hope that the practice, as it was carried out in the Bi-Metallic Smelter, may still be of interest to others. The Bi-Metallic Smelter, at Leadville, Colo., was the first to make a commercial success of the process described in the following pages, which consisted essentially of smelting raw silver- and copper-bearing sulphide ores in blast-furnaces to form a matte, which was generally re-smelted with siliceous ores for a second high-grade matte. Robert Sticht, in his excellent monograph on the subject,1 has outlined briefly the plant belonging to this company. He says, in effect: The Boulder, Mont., enterprise set the example for others with whom the patentee (w. L. Austin) had been negotiating, and the old La Plata lead-smelter at Leadville, Colo., was fashioned into a crude, temporary experimental plant for the Bi-Metallic Company, and the feasibility of the process demonstrated by a trial run. Subsequently, the plant was considerably improved, and, being supplied with very favorable ores, continued to run without interruption from August, 1892, to June, 1893. In September, 1893, smelting was resumed in a completely remodeled establishment, some $200,000 having been expended upon the new work. Thus, aside from the Montana experiments, which, under the direction of W. L. Austin, began as early as- 1885 2 at Toston, and continued with indifferent success for six or seven years, the Bi-Metallic plant at Leadville may be said to have * Formerly Superintendent, Bi-Metallic Smelting Co. + Formerly Chief Assayer and Chemist, Bi-Metallic Smelting Co. 1 Modern Copper Smelting, E. D. Peters, 7th ed., p. 415 (1895). 2 Trans., xvi., p. 259 (1887-88).

Dec 1, 1910

By C. H. P. Lupis, Henri Gaye

There are m -I conditions for the stability of a solution of m components with respect to infinitesinzal flucturations. However, in most cases, only one of these conditions has to be considered to determine the domain of instability and the existence of this more restrictive condition greatly simplifies the calculations. It may be used advantageously for the prediction of miscibility gaps and the method is illustrated in details for the case of the Ag-Pb-Zn system. THE thermodynamic conditions for the formation of a miscibility gap may be viewed as a necessary consequence of the conditions for spinodal decomposition. A previous article1 has examined in detail the form of these conditions for multicomponent systems. There is only one condition for the stability of a binary system (with respect to infinitesimal fluctuations), but there are two conditions for a ternary system, and m — 1 conditions for an m-component system. The probability of violating a stability condition, and thus forming a miscibility gap, obviously increases with the number of components, a result which is rather intuitive since the atoms of the solution have now many more ways of redistributing themselves and introducing complexities in the form of the free energy hy-persurface. It is of interest to take advantage of this possibility of precipitating new phases and to examine which stability condition is the likeliest to be violated, that is, which stability condition is thermodynamically the most restrictive. The finding of such a condition would greatly simplify the application of the stability criteria since only one condition could then be considered, instead of m - 1. In Ref. 1, coherency strain energy terms were neglected, thus restricting the applications of the treatment to solutions where they are negligible, such as liquid alloys. In the following study the same assumption will be made. To generalize the treatment to systems where the strain energy terms are sizable, the reader is referred to Cahn's classical article on spinodal decomposition.2 Let us designate by Gij the second derivative of the Gibbs free energy with respect to the number of moles ni and n j. There are several equivalent sets of m — 1 stability conditions.' The one considered here expresses that the successive diagonal determinants of order 1, 2, ... m — 1, associated with the symmetric Gij matrix (for 2 5 i, j 5 m) are positive.' For a binary solution 1-2, the condition for stability is: O(u=G22^0 [1] For a ternary system 1-2-3, the condition [I.] is re- tained (the value of G22 will differ, of course, according to the concentration of 3) and another condition is introduced: £>(21 = G22G33 - Gl3 ^ 0 [2] In a composition diagram, these two conditions define two domains of instability. Starting at a point where the solution is stable (for instance at a point where the solution is very dilute) we gradually change the composition until the condition [I] or [2] is violated. As already noted in the literature, e.g., in the work of Prigogine and Defay,3 it is the boundary of the domain (2) which is first crossed. For if we assume that the boundary of the domain (1) is reached first, at this point G22 = 0 and the second condition is necessarily violated (D(2) = -& 5 0), in contradiction with our original assumption. An exhaustive study of the ternary regular solution case may be found in the work of Meijering.4 Moreover if the boundaries of the two domains have a common point, they also have a common tangent. For if the two lines were to cross each other as is illustrated in Fig. 1(a) any point M in the line QP would be such that £> = 0 and 0"' > 0 which, as shown above, are incompatible results. Thus, the two lines must be tangent at their common point Q as illustrated in the example of Fig. l(b). The reasoning of Fig. l(a) implies that the point Q is not a "singular" point for either boundary line. This singularity may be of two types. First, the lines meet without crossing each other and without being tangent. Second, the tangent at Q for D"' or 0"' is not single-valued. Other types of singularity are unlikely because of the usual analytical forms of D"' and 0"'. The exception to the common tangent requirement due to the first type of singularity was pointed out by John Morral;5 it occurs when the common point, Q' or (3" in Fig. l(b), is located at a boundary of the composition diagram, e.g., at the line X3 = 0. It may also be noted that at the common nonsingular point Q of D(1) and D(2), Fig. 1(b), G23 is necessarily equal to zero, whereas at a point such as Q' or Q", this conclusion is no longer valid because the product G22G33 is now indefinite (of the form 0. a). The exception to the common tangent requirement due to the second type of singularity occurs when two branches of the same boundary line intersect, for example when D(1)or D(2) decomposes into a product of functions, at a point which belongs to the boundary of the other condition. It is possible to show by a simple analytical calculation that, in this case, if Q is a singular point of D(1), then it is necessarily a singular point of D(2), and that the reciprocal is true except if G33 = 0 at Q. For the present article, however, more elaboration on these singularities appears to be unwarranted. To generalize the previous results to an m -component system, we use the mathematical theorem stating that if the diagonal determinant D(r) = 0, then

Jan 1, 1970