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By Ryan CG, Both RA, Mernagh TP, Large RR, Hunns SR, Gemmell JB

The Hellyer and Mt. Chalmers deposits are mound-style volcanic-hosted massive sulphide (VHMS) deposits in Australia. Textural, petrographic and microthermometric investigations of fluid inclusions in the Hellyer stringer system indicate that Type I, primary, liquid-vapour inclusions are 10-15 pm in size, and yield homogenisation temperatures of 170-220¦C in early 2A veins, 165-322¦C in main-stage 2B veins and 190-256¦C in late-stage 2C veins. These data suggest a waxing and waning thermal history. However, the average salinity remained between 8-11 NaCl equiv. wt % in all Stage 2 veins. At Mt. Chalmers, Type I inclusions up to 20 ¦m are found in quartz from the mineralised zone, and these inclusions yielded homogenisation temperatures of 160-268¦C and salinities of 5-8 NaCl equiv. wt %. Laser Raman spectroscopic (LRS) analysis indicates the presence of CO2 (

Jan 1, 1995

By Richard McCaffery

WHEN we began the study of blast furnace slags we limited our work at first to a study of those slags containing only lime, alumina and silica. In our paper1 on some of the results of this first work, the solubility of. sulfur in slags containing no magnesia was reported and the question of sulfur solubility and slag viscosity discussed. There has been, for a long time, a very great difference of opinion among metallurgists as to .the effect of magnesia in blast furnace slags, and the literature is full of expressions of the most contrary opinions on the subject. As a result, our next publication2 developed the constitution of slags considered as a four-component system, silica-alumina-lime-magnesia, and advanced an explanation of slag structure which accorded with the facts and with scientific theory. The present paper continues our study of the general subject of slags. As blast furnace slags are included in the silica-alumina-lime-magnesia system, which is a particular case of four-component solutions, we have determined that there are 22 components (Table 1), which may enter into the silica-alumina-lime-magnesia system and that 10 or 12 of these components may be present in blast furnace slags which are within the ordinary ranges of composition. To have a very definite conception of the behavior of these components in a slag, we have developed the theory of the cooling of a four-component solution from the liquid state to the solid and the conclusions that we obtained have been made use of in this paper. On account of the importance of the four oxides and 18 compounds which enter into the silica-alumina-lime-magnesia system, some method of calculating the percentages of the various compounds that are present in the slag when the percentages of the oxides are given, seemed very desirable and we develop a method later herein for doing this, and finally we give illustrations of

Jan 10, 1926

By Richard McCaffery

WHEN we began the study of blast furnace slags we limited our work at first to a study of those slags containing only lime, alumina and silica. On our paper1 on some of the results of this first work, the solubility of sulfur in slags containing no magnesia was reported and the question of sulfur solubility and slag viscosity discussed. There has been, for a long time, a very great difference of opinion among metallurgists as to he effect of magnesia in blast furnace slags, and the literature is full if expressions of the most contrary opinions on the subject. As a result, our next publication' developed the constitution of slags considered as a four-component system, silica-alumina-lime-magnesia, and advanced an explanation of slag structure which accorded with the facts and with scientific theory. The present paper continues our study of the general subject of slags. As blast furnace slags are included in the silica-alumina-lime-magnesia system, which is a particular case of four-component solutions, we have determined that there are 22 components Table 1), which may enter into the silica-alumina-lime-magnesia system and that 10 or 12 of these components may be present in blast furnace slags which are within the ordinary ranges of composition. To cave a very definite conception of the behavior of these components in a slag, we have developed the theory of the cooling of a four-component solution from the liquid state to the solid and the conclusions that we obtained have been made use of in this paper. On account of the importance of the four oxides and 18 compounds which enter into the silica-alumina-lime-magnesia system, some method of calculating the percentages of the various compounds that are present in the slag when the percentages of the oxides are given, seemed very desirable and we develop L method later herein for doing this, and finally we give illustrations of the composition of a wide series of blast furnace slags and show some applications of these composition data.

Jan 1, 1927

By Charles Mabery

So FAR as the elementary composition of petroleum is -known, it may be briefly stated. Petroleum consists principally of a few series of hydrocarbons, with admixtures of sulfur, nitrogen, and oxygen derivatives in comparatively minute proportions, which may be regarded as impurities to be removed in the preparation Of commercial products. But as each series is represented by many homologs, in the aggregate, crude petroleum is an extremely complex mixture of hydrocarbons and their derivatives. In part, these hydrocarbons individually conform in structure to the system of synthetic hydrocarbons whose structure is well defined and -represented by the typical series CnH2n+2, CnH2n, the series CnH2n-2, CnH2n-4, the members of which have not been sufficiently studied to establish their structure, and the series CnH2n-6 composed of the aromatic group, benzene and its homologs. Hydrocarbons of greater density contained in the portions of petroleum that cannot be distilled without decomposition doubtless have less hydrogen than is represented by these formulas. Since to every hydrocarbon there is a definite temperature, even in vacuum, at which its constituent carbon and hydrogen atoms fall apart, and since for the heavier bodies this temperature is not much above 360° C. in vacuum, it is evident that some other method than distillation must be devised for their separation if anything further is to be learned concerning their individual constitution. CLASSIFICATION OF PETROLEUMS There is such a wide variation in the composition of petroleum from different fields, it would seem possible to make a classification on this basis were it not that no single variety is entirely free from hydrocarbons contained in others. Such a classification has been suggested of the exceptionally pure Pennsylvania petroleum, the sulfur oil from Trenton

Jan 2, 1920

By J. C. Roy, S. Couture, C. Barbeau

The chemical composition of the asbestos tailings from different mines in the Eastern Townships of Quebec are fairly uniform in composition. The serpentine minerals antigorite and chrysotile form about 85 to 90 per cent of the total. Iron oxides add another 7 to 10 per cent; magnetite represents about 70 per cent of the oxides. Chromite is present in amounts of about 0.5 per cent. The average contents of elemental magnesium, iron and nickel are 23, 6 and 0.25 per cent respectively.

Jan 1, 1972

By G de Jong, W Sunyoto, M Cloos

"An EXTENDED ABSTRACT is available for download. A full-length paper was not prepared for this presentation. The Ertsberg mining district is a 100 km2 area in the highlands of Papua, Indonesia. The district hosts mineralised and barren, outcropping and hidden porphyry and equigranular intrusions. There are at least six major (100+ m wide) porphyry intrusions identified from surface and underground mapping and drill cores. These are the Grasberg Igneous Complex (GIC), Karume porphyry, Lembah Tembaga porphyry, Ertsberg Stockwork Zone porphyry, Kay intrusion and the Wanagon intrusion. Recent deep drilling adjacent to the GIC identified a new hidden porphyry at depth, the Gajah Tidur (Sleeping Elephant), which extends south-west beneath the Kucing Liar skarn. Other named small porphyry dykes and plugs in the district include the West Grasberg, North Grasberg, Idenberg, South East and Zaagkam intrusions. The largest equigranular intrusion is the multiphase Ertsberg diorite.Igneous activities occurred in a relatively short span (<1 million years) from about 3.4 Ma to 2.7 Ma ago, based on U-Pb zircon dating. The various intrusions in the district related to ore mineralisation are high K and appear to have been derived from the same parental chamber, based on similar whole rock rare earth element patterns. Radiogenic Nd- and Sr-isotopic variations indicate that there were multiple recharge events into the parental chamber.The larger intrusions and all economic mineralisation occur in a 4 km wide zone that parallels the N60W-trending axis of the Yellow Valley Syncline (YVS), one of the large folds generated during collisional tectonism. Intrusion pathways at ~3 Ma were localised into dilatational areas in a late and/or post-collisional NW-trending, left-lateral strike-slip zone with Riedel shear offsets between the major fault zones. An early phase of lower K magmatism produced small intrusions in the south-west limb of the YVS.CITATION:de Jong, G, Sunyoto, W and Cloos, M, 2015. Composition, lithochemistry and radiogenic isotopes of porphyritic and equigranular intrusions in the Ertsberg mining district, Papua, Indonesia, in Proceedings PACRIM 2015 Congress, pp 347–356 (The Australasian Institute of Mining and Metallurgy: Melbourne)."

Mar 18, 2015

By A. Rahmanab, A. Torpy, M. I. Pownceby, M. N. Zaman, N. Haque, W. Bruckard, J. Tardio

Ilmenite is the principal feedstock for producing TiO2 - used in paints, plastics and paper and the production of Ti metal. The Brahmaputra River system in Bangladesh represents a potentially new source of ilmenite concentrated within sandbar placer deposits and the current study presents results from beneficiation and characterisation studies conducted on Brahmaputra River Fe-Ti oxides. Characteristic of fluvial systems, the ilmenite concentration is low (~0.5 wt-%), however extensive annual dredging for waterway navigation provides the economic impetus to recover saleable ilmenite. The Fe-Ti oxide suite derived from magnetic fractionation comprises homogeneous ilmenite, titanohematite and hematite, although many grains exhibit complex exsolution within the (Ilm-Hem)ss series. The textural variability results in difficulty generating a pure ilmenite feedstock with the bulk concentrate typically being low in TiO2, containing 31.0 wt-%TiO2 and 62.7 wt-% (Fe2O3). Results are interpreted via an analysis of the implications for producing a commercial-grade ilmenite concentrate suitable for TiO2 production. Keywords: Ilmenite, titanohematite, Fe-Ti oxide exsolution, TiO2 production

Jan 1, 2020

Adaptive processing is often related to the mineral composition of the mining block currently processed. The portions of waste, ore and secondary product minerals considered are always positive and typically sum to 100% (i.e. a composition). A typical solution to predict block averages of spatially dependent quantities would be block kriging. However, kriging is based on spatial correlations and it has been repeatedly shown that correlations of compositional data are spurious and blurred by the constant sum constraint. Aitchison (1982) proposed a general solution for compositional problems, based on transforming the compositional data to a set of logarithms of ratios of components, and analysing the transformed scores. The transform is chosen to be invertible to ensure that no information is lost in the process. This methodology avoids spurious correlation problems and ensures coherence between results obtained with different subcompositions. Pawlowsky-Glahn and Olea (2004) adapted the approach for point-support geostatistics, allowing the prediction of point compositions from spatially neighbouring data, without the artefacts induced by standard multivariate cokriging. Unfortunately upscaling the results of this approach with block cokriging is not straightforward, because of the nonlinearity of logratio transforms. Furthermore, we have to consider those nonlinearities created by the dependence on the block composition of extraction efficiency and processing costs of the possible processing choices. This paper proposes a solution for predicting the conditional expectations of the benefit of processing the block with each processing choice using geostatistical simulations of the local values of the composition and the resulting block integrals. The computation for a large number of blocks can be done efficiently using Cholesky decomposition. The approach also allows the calculation of prediction errors for expected compositions and expected benefit at almost no additional computational cost. Such compositional approach is necessary, for example when processing choices depend on proportions of certain minerals after other components have been removed; or if processing alters the composition in a multiplicative way, by partially removing a portion of some components.

Aug 1, 2013

By J. R. Craig, T. N. Solberg, M. A. Linden

The central and southern Appalachian Mountains were the major sources of domestically produced gold throughout the first half of the 19th century and continue to yield gold today. Small amounts of gold were probably extracted by the Indians and the Spaniards but the first documented report of gold in the region was made by Thomas Jefferson who found, in the Rappahannock River of Virginia in 1782, a 1.8 kg (4 lb) lump of gold ore that yielded 26 g (0.85 oz). Gold was subsequently discovered in North Carolina between 1793 and 1799, in Virginia in 1806, in South Carolina in 1827, in Georgia in 1829, and in Alabama about 1830 (Becker, 1894-5; Pardee and Park, 1948). Production climbed to more than 933 kg/a (30,000 oz per year) between 1830 and 1852, but dropped sharply after the California gold rush in 1849 as experienced miners moved west in search of bigger and easier stakes. The Civil War briefly halted gold mining in the Appalachians, but production rose to an average of about 249 kg/a (8,000 oz per year) from 1865 until 1878 and to about 404 kg/a (13,000 oz per year) from 1881 until 1915 when it again fell to near nil (Pardee and Park, 1948). Minor resurgences occurred in the 1930s and 1940s. Although there has been no reported production in recent years, it is likely that "weekend panners" still remove hundreds of ounces from streams annually. The total reported production for the area through 1959 is more than 77 t (2.5 million oz). This paper reports some initial findings of a long-term project designed to investigate the mineralogy, chemistry, mode of occurrence, and petrology of gold occurrences in the Appalachians. The goal of the initial phase was to determine the compositional range of gold in the Appalachians as a step toward deciding if sufficient variations were present to allow the definition of "compositional signatures." The very few published data on gold fineness in the Appalachians are based on assays that commonly include gold and silver from several mineral species. Analyses in the present study were performed on individual grains of electrum using a spot size of 10 µm x 10 µm to 100 µm X 100 µm (394 X 394 micro in. to 3,940 X 3,940 micro in.), and thus reveal the true composition of the principal precious metal-bearing phase.

Jan 1, 1984

By Georgette Smits

The gun-sprayed refractory lining of electrothermal furnaces used for the manufacture of carbon disulphide from liquid sulphur and charcoal at between 800 and l000°C are severely attacked during the process. A mineralogical study of samples of eroded refractory material disclosed that carbon disulphide had penetrated the lining, causing the precipitation of sulphides and carbon, and the decomposition of mullite, a major constituent of the lining, to corundum and amorphous silica. Calcium, iron, titanium, and chromium are undesirable cations in a refractory that is used in furnaces for the production of carbon disulphide, since they react readily with this substance. A series of tests, in which samples were exposed to different amounts of carbon disulphide for various residence times and at different temperatures, was carried out with eight types of aluminosilicate-based castable and brick refractory materials, to compare their resistance to attack by carbon disulphide. Mineralogical investigation of the test cores showed that castable refractory materials are unsuitable for this type of process because of the presence of calcium aluminates and other impurities in the cement. Of all the refractories that were tested, a mullite brick of low porosity, containing a minimum amount of impurities, appeared to be the most suitable for use in this type of operation.

Jan 1, 1989

By D Craw, L C. Kerr, P Wopereis

Gold placers in the Nokomai and Nevis valleys are dominantly Quaternary in age. They occur mainly in western tributaries of the upper Nevis, eastern tributaries of the middle and upper Nokomai, and in the trunk valleys themselves. Placer gold in both valleys is dominantly a-phase Au-Ag-Hg alloy. Au-Ag alloy is absent or rare in most of the deposits, except in the lower Nokomai. Alpha-Au-Ag-Hg alloy is typically coarse-grained (up to 2 cm), angular, and rarely flattened or folded. Crystalline texture, quartz intergrowths, and psuedo-hexagonal crystal-pluck cavities are common. Fluvial transport distance predicted from the maximum Flatness Index ofAu-Ag-Hg alloy particles is typically less than 10-20 km. Coarse (up to 2 cm) cinnabar is commonly associated with the Au-Ag-Hg alloy, and both were probably derived from hydrothermal sources in western tributaries ofthe upper Nevis and eastern tributaries of the upper Nokomai. Minor secondary Au-Ag-Hg alloy occurs locally in the lower Nokomai, where it coats or cements detrital a-phase Au-Ag-Hg and Au-Ag alloy particles. Au-Ag alloy dominates over Au-Ag-Hg alloy in the lower half of the lower Nokomai alluvial plain and gravel of a raised channel incised into semischist basement adjacent to the plain. The Au-Ag alloy is rounded and commonly flattened and folded. Crystalline texture, quartz intergrowths, and pluck cavities are rare, and a fluvial transport distance of 25-40 km is estimated from maximum Flatness Index ofAu-Ag alloy particles. The Au-Ag alloy, three types of gamet, magnetite, clinozoisite and well foliated schist boulders in the abandoned channel are not found in the pumpellyite-actinolite facies bedrock in the Nokomai. These components were derived from greenschist facies schist and sedimentary sources in Otago, many tens of kilometers north of the Nokomai catchment. They were transported to the Nokomai either in Wakatipu Glacier till or fluvio-glacial outwash that entered the valley via Nokomai Saddle, the confluence with the Mataura, or both.

Jan 1, 2000

By V. K. Banakar

The occurrence of hydrogenous ferromanganese encrustations (Fe-Mn crusts) on the Cretaceous age Afanasiy-Nikitin Seamounts (ANS) in the eastern Equatorial Indian Ocean (EEIO) was first reported by Banakar et al. (1997, Marine Geology). The major element composition of the samples from the upper flank of the ANS (~1650 m water depth) indicated enrichment of Co up to 0.9 % (Parthiban and Banakar, 1999, Indian Mineralogist). Subsequently a project supported by the Department of Ocean Development to explore the ANS for Co-enriched Fe-Mn crusts was launched in 2002 by the National Institute of Oceanography. Sampling at regular depth intervals along two transects on the slopes of the northern elevated region of the ANS was carried out. The Fe-Mn oxide deposition is evident on almost all types of substrates viz., basalts, conglomerate limestone, fossil corals etc., and the thickness of the oxide layer varies from a patina to 7 cm. The Fe-Mn crust growth is evident only above a water depth of 3200 m, below which semi-lithified carbonate ooze is dominant. The ANS Fe-Mn crusts from all the water depths exhibit Mn/Fe < 1.5 suggesting pure hydrogenetic accumulation of metal-hydroxides. Bulk sample Ce-contents for a few selected depth-representative samples are remarkably high, up to 3700 ppm with an average of ~2500 ppm. The positive Ce-anomaly decreases from ~8 at 1600 m depth to ~1.6 at 3200 m depth, which is in contrast to the modern dissolved oxygen depth-profile in the region. This observation suggests that major time-period of the Fe-Mn crust accretion history is dominated by ambient waters with lower-oxygen at deeper depths than at the intermediate depths. This in turn probably suggests a major reorganization in the deep-intermediate hydrography of the region in the past.

Jan 1, 2005

By Garland R. Dever

Extensive deposits of high-calcium limestone occur in the Upper Mississippian Warsaw and Ste. Genevieve Limestones in the region of the lower Cumberland, Tennessee, and Ohio Rivers in western Kentucky. Analyses of foot-by-foot samples from 20 quarries and cores in 8 counties show variations between the chemical composition of high-calcium deposits in the 2 formations. Knowledge of probable variations in composition should aid in prospecting for high-calcium stone for a specific use and in evaluating potential deposits. High-calcium strata in the Warsaw consist of bryozoan- and crinoid-rich bioclastic limestones, and in the Ste. Genevieve, predominantly oolitic limestones. Values used for comparison were obtained by averaging the foot-by-foot analyses of each high-calcium deposit, 10 in the Warsaw and 15 in the Ste. Genevieve. Average MgCO3 and Fe2O3 contents are generally lower in the Warsaw than in the Ste. Genevieve. Average SO2, S, and P contents are- generally higher in the Warsaw. No characteristic differences are apparent between the average CaCO3 and A1203 contents of the two formations. The higher S and P contents of the Warsaw are believed attributable to the predominance of biogenic material in these limestones. Within the Warsaw Limestone, higher average S and A1203 contents and lower average Fe2O3 contents were found in deposits in the northern half of the region. In the Ste. Genevieve, vertical variations in composition occur within individual high-calcium deposits containing oolitic and nonoolitic limestones. Lower average CaCO3 contents and higher average contents for the other constituents were found in the nonoolitic intervals.

Jan 1, 1972

By W. E. Robinson

Oil-shale samples selected from a core of the Green River Basin (Wyoming) were studied to ascertain differences in the composition and/or chemical structure of the organic material in the samples relative to stratigraphic position within the Green River Formation. This involved a systematic study of the soluble organic material and the insoluble kerogen present in the samples. Emphasis was placed upon the distribution of the alkanes (normal and isoprenoid compounds) in the soluble extract and upon the aromaticity of the kerogen. In general, it was concluded that significant differences exist in the chemical structure of the oil-shale organic material from various strata within the Green River Formation of Wyoming. Most of the compositional and structural changes were not related to burial depth but were related to source material variations, environmental differences, or other factors.

By P Mason, J F. Huntington, K M. Scott

Short-wave infrared (SWIR) spectral reflectance of the hydrothermally altered volcanic rocks in the footwall of the Hellyer Zn-Pb massive sulfide deposit was measured with a portable PIMATM infrared spectrometer. The Al-OH absorption band was used to derive information on the cation composition of white mica. The wavelength range of the Al-OH band from 2194 nm to 2222 nm reflects variation in the number of octahedral Al (Alvi) in white mica from 3.9 to 3.2 (based on four octahedral cations per formula). This Alvi range indicates a substantial compositional variation, covering most of the compositional region between muscovite (Alvi = 4.0) and phengite (Alvi = 3.0). Moreover, the spectral reflectance data show that the compositional variations of white mica are spatially related to the hydrothermal alteration zoning, such that the white mica of more phengitic composition tends to form in the main upflow fluid channels in the centre of the footwall alteration system. The results suggest that the octahedral Al content of white mica, which is spectrally definable by the wavelength of the Al-OH band using a field instrument such as the PIMATM, can be used to map fossil hydrothermal systems and potentially as vectors to mineralisation.

Jan 1, 2004

By D. W. Rogers

Several commercial polyacrylamide flocculants most widely used in Western Canadian mineral processing plants have been analyzed chemically and compared in standardized settling tests as aids to thickening and clarification. Data are included for both dry and liquid-type flocculants. Percentages of active ingredients in these commercial products varied Widely. Performances were compared on the basis of: equal additions of active ingredients, nearly equal molecular weights and nearly equal degrees of ionic character. Based on zone settling velocities and supernatant clarities of siliceous slurries coagulated by lime additions, dry flocculants proved more economical than liquid flocculants. Using these same two criteria, the most highly anionic were more effective than either weakly anionic or non ionic polyacrylamide- type flocculants.

Jan 1, 1978

By C Ward, U Mueller

Compositional geostatistics is an approach developed to ensure that the constant sum constraint induced by geochemical iron ore analysis is respected in estimation and simulation. Jack-knifing and cross-validation were used within this framework to compare the results from ordinary cokriging of the additive log ratio (alr) variables with those from conventional ordinary cokriging. Two methods were used to back-transform the alr estimates: the crude additive generalised logistic back transformation and numerical integration via Gauss-Hermite quadrature. Previous work on a blasthole data set from a Yilgarn iron ore mine showed promising results, with the Gauss-Hermite quadrature method returning results comparable to ordinary cokriging from the point of view of bias, but superior in that the constant sum constraint imposed by geochemistry of the data was honoured. Other practitioners have noted that the linear interpolation of non-linear log ratio transformations will result in biased results, and that this bias can be masked in a (grade control) setting where the spatial density is high. To test this assertion, the methodology has been extended to successively lower spatial density data sets. The results and applicability to routine iron estimation are discussed.CITATION:Ward, C and Mueller, U, 2013. Compositions, log ratios and bias - from grade control to resource definition, in Proceedings Iron Ore 2013 , pp 313-320 (The Australasian Institute of Mining and Metallurgy: Melbourne).

Aug 12, 2013

By José Ricardo Nunes de Macedor

Com o crescente zelo ambiental, os polímeros biodegradáveis ganharam importante papel no mercado mundial. Em especial, o amido é um destaque por apresentar preço baixo e alta demanda, contudo possui grande fragilidade mecânica e os seus filmes são muitos hidrofílicos. Outro importante polímero biodegradável é o poli(ácido lático) que possui, alta resistência mecânica e transparência, quando comparado com o amido termoplástico, permitindo sua aplicação em embalagens alimentícias. Contudo, o PLA ainda possui custo elevado, quando comparado ao amido e até commodities, como o PET, PP e PE. Estudos com algodão, material em grande produção e baixo custo, retratam ganhos nas propriedades mecânicas sem perda da biodegradabilidade em compósitos. Assim, este trabalho objetivou preparar, por meio de homogeneizador termocinético KMixer e prensa térmica, placas de compósitos de PLA contendo amido (0, 3 e 5% m/m) e fibras de algodão in natura processadas (0, 10 e 20% m/m). As amostras foram caracterizados por espectroscopia na região do infravermelho com transformada de Fourier (FTIR), ângulo de contato estático e ensaios mecânicos (módulo de Young, resistência à tração e alongamento na ruptura). As análises mostraram que a adição de fibra de algodão compensa a perda de resistencia à tração ocasionada pela adição de TPS. Tanto a fibra como o amido aumentaram a molhabilidade do compósito.

Aug 8, 2018

INTRODUCTION AND SCOPE This chapter is intended as a brief review of the mathematics of compound interest and of the meaning of time equivalent money. In order to make valid economic evaluations of various investment alternatives, it is important that they be analyzed on an equivalent basis, taking into account the time value of money. Compound interest has been said to be one of the eighth wonders of the world. Certainly compound interest is a fact of life in today&apos;s economy. Whenever we deal with money over time-investments, earnings, savings, loans, etc.-we are immediately confronted with the principles and effects of compound interest. In such an environment it is necessary to be able to determine time value equivalence of various sums of money received or paid at different dates. It follows that all future payments or series of payments that would repay a present sum with interest at a stated rate can be said to be equivalent with each other. Equivalence, then, is a function of the interest rate. MATHEMATICS OF COMPOUND INTEREST FACTORS In order to develop mathematical formulas for various time value equivalence of compound interest we will use symbol designations: P is Present or principal sum of money; usually refers to value at time zero. F is a single sum of money at some designated future date. R is The amount of each payment in a uniform series of equal payments made at the end of each period. n is Number of interest compounding periods (may represent annual, semiannual, quarterly, or even daily periods). i is the period compound interest rate. Figure 20-1 illustrates the increase of princi-

Jan 1, 1980

By William E. Ford, Edward Salisbury Dana

Twin crystals are those in which one or more parts regularly arranged are in reverse position with reference to the other part or parts. They often appear externally to consist of two or more crystals symmetrically united, and sometimes have the form of a cross or star. They also exhibit the composition in the reversed arrangement of part of the feces, in the striae of the surface, and in re-entering angles; in certain cases the compound structure can only be surely detected by an examination in polarized light. The above figures (Figs. 384-386) are examples of typical kinds of twin crystals, and many others are given on the pages following.

Jan 1, 1922