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By Garth Howlett

In 1998 the Matanuska-Susitna Borough formed the new port district at Point MacKenzie, known as Port MacKenzie, located directly across the Knik Arm from Anchorage. In 1991 PND Engineers Inc. (PND), working with the Matanuska-Susitna Borough manager and port director, envisioned a fill structure at Port MacKenzie with a 500-foot-long Open Cell? sheet-pile bulkhead face extending to EL -20 feet Mean Lower Low Water (MLLW) at the face. Now under operation the fill created 5.5 acres of tideland property using local materials for fill. The fill is protected from 2- to-3 foot-thick ice pans and 9 feet per second (fps) tidal currents by an envelope of armor rock. Construction required working in water with 30 to 40 feet of tidal variation. Construction required placement of frozen fill in water depths possibly exceeding 50 feet, creating real concerns about soil consolidation and long-term maintenance issues. A plan was prepared to consolidate the soils using Vibracompaction. Soil Density and consolidation were monitored during the work. In 2003 the port district and NPI, a wood chip export company, formalized an agreement to construct a deep water port facility at Port MacKenzie for cape-class bulk carriers. PND provided planning, studies, geotechnical investigation, design documents and construction inspection for a 500-foot-long trestle with a platform dockhead and dolphins providing 1,100 feet of dock face in EL -70 MLLW as shown in Figure 1. Design included installation of large-diameter spin-fin piles driven on a 2V:1H batter. Also included was a 300-ton tension test of smaller spin-fin piles.

Jan 1, 2006

As the chief field of the gold dredging industry in Otago is the district traversed by the Molyneux River (called above Cromwell the Clutha) and its tributaries, a brief reference to the geology of that region and the origin of its alluvial gold will not be out of place. The prevailing rocks are metamorphic schists, chiefly mica and quartz schists, pr.obably of Silurian age. Though these rocks do not contain many payable quartz reefs, gold appears to be distributed in small quantities all through them,chiefly associated with the quartz lamime and lenses. The river drift, however, contains numerous pebbles of quartzite, quartzoze conglomerate, and fme-grained slate. Seeing that these rocks are hardly known to occur in situ within the watershedof the Molyneux River, though Devonian and carboniferous rocks of similar character occur to the north-east and south-west, it seems probable that the schists were at one time overlaid by carboniferous and Devonian rocks, and that these have been removed by denuding agencies. Be this as it may, it is certain that the schists themselves have been greatly eroded by the action both of water and of ice, and that the resulting drifts have themselves, in many cases, been sorted and re-assorted by the same agents.Consequently the drifts are very varied in character and often developed on a grand scale. In geological age they vary from Early Tertiary (though some may be older) to Recent.

Jan 1, 1903

The Sandvik DE710 diamond core drill has become one of the most popular exploration drills available. It is operated on exploration and mining projects in locations around the world. The compact nature of the DE710 allows a small footprint on site. This has proven to be a great benefit in difficult access areas where site preparation can be limited. The DE710 drill is available in a range of con-figurations to suit global operational and environ-mental demands. It can be truck-mounted for greater maneuverability and ease of relocating the drill between sites. The drill can also be fitted to crawler tracks, powered from the rig hydraulic system, making it a small, self-propelled unit for areas with difficult site access. And the DE710 is available in a shack-mounted configuration for cold weather applications, ensuring a protected working environment for the drill crew. The DE710 is powered by a 6.7-liter engine that produces 141 kW (190 hp) at 2,200 rpm to power the rig hydraulic system. The engine complies with the Tier 3 emissions standards that are being introduced in many countries. The hydraulic system design for the Sandvik DE710 is based on the company?s philosophy of selecting the best available products to suit the application. The main hydraulic pumps and motors are the variable displacement design. This allows for greater fl exibility and precise control of critical drilling functions. The cooling system has also been designed for operation in ambient temperatures up to 50° C (122° F) to cope with varying global operational demands.

Jan 1, 2009

By L. Lehasa

The purpose of the project was to investigate the challenges faced by Khutala Colliery pertaining to underground road construction and maintenance. Furthermore the influence of these challenges on productivity and safety requirements were taken into consideration. The operating standards required for the construction and maintenance of the 2 seam road transport infrastructure was investigated, together with reasons why the standards of operating procedure are not met. The Code of Practice put in place by Khutala Colliery was used to view the standard operating procedures. Leading practices on underground road construction were researched. The project aims to discover the gaps that Khutala Colliery must bridge to implement enhanced practices with regards to underground road construction. During the period of study, observations were carried out to determine whether the road construction standards of operating procedure were being adhered to at all times for both road construction and road maintenance. Khutala Colliery has put in place a traffic management plan, which includes safety practices and work policies that must be adhered to while travelling on underground transport roads.

Jan 1, 2012

By Richard Samson

"RésuméComme l'exploitation d'un gisement à grande profondeur représente un défi de taille et que la profondeur à laquelle se situe celui d'Ansil varie de 1200 m à 1460 m, nous avons dû mettre en place une infrastructure de surface d'envergure incluant un treuil de production et un chevalement d'une très grande capacité. De même, l'infrastructure sous terre a été proposée en tenant compte de ce fait important; d'ailleurs, l'excavation de deux puits circulaires dont l'un est en beton et acier et l'autre exclusivement en béton en témoigne. De plus, la plupart des cheminées principales qui furent excavées sous terre sont de forme circulaire.L 'exploitation du gisement s'effectue par une méthode chantier ouvert (abattage par longs trous), remblaye immédiatement âpres avec un mélange de roche stérile et ciment. Un taux de minage de 1750 tonnes par jour est requis pour assurer la production.L 'utilisation d'une technologie de pointe tels: les ventilateurs et les portes de ventilation sous terre contrôlées de la surface, un système de relèves macroséismiques, de chargeuses-navettes contrôles à distance, en passant par le dessin et la planification assistes par ordinateur nous permet d'optimiser l'exploitation du gisement Ansil.IntroductionEtant donne la grande profondeur a laquelle le gisement Ansil se situe, no us avons à mettre en place une infrastructure souterraine et de surface d'envergure pour nous permettre d'atteindre nos objectifs d'exploitation.Infrastructure de surfaceLa mine Ansil dispose d'une infrastructure de surface d'importance possédant une grande capacité (Fig. I). Les deux facteurs qui furent déterminant pour le dimensionnement des infrastructures furent le taux d'extraction de 1990 t/jour et la profondeur a laquelle I 'exploitation s'effectue, puis que cette dernière affecte le temps de cycle et la capacité de hissage.La première composante principale de I 'infrastructure de surface est un treuil de production a 2 tambours de 3.65 m de diamètre et de 2.0 m de largeur chacun. Ce treuil de marque Fullerton, Hodgart & Barclay construit en 1970, opère avec 2 moteurs de 930 kW a une vitesse de 700 m/min. et il a une capacité maximale de 32000 kg. Deux câbles d'acier de 45 mm de diarnètre et de 160 000 kg de résistance a la rupture servant au hissage des bennes d'une capacité de 12 tonnes. De plus, un moteur/générateur de 3000 kW (M.G. Set) raccorde au treuil de production sert a stabliser son alimentation électrique."

Jan 1, 1991

By Parry. V. F., W. S. Landers

"INTRODUCTION Approximately 1,250,000 tons of 2-1/2-inch and 1-1/2-inch slack, representing over half of the output from 40 to 50 mines, is produced in the Denver region each year. Part of this slack is made by crushing lump, and some of it is rescreened to make pea and modified pea sizes measuring 1-1/2 inches by 2-1/2 inches and 3/4-inch by 1-1/2 inches, round-hole. Modified pea is popular as domestic stoker fuel. Approximately 900,000 tons of slack 0 by 2-1/2 inches, 0 by 1-1/ 2 inches, and 0 by 3/4-inch in size are consumed by industrial plants, the principal consumers being a large power plant and the beet-sugar industry. The power plant uses annually 250,000 to 275,000 tons of pulverized fuel and can pulverize slack of any size, while the sugar factories consume about the same quantity of coal in stokers. The sugar factories store about 110,000 tons of 2-1/2-inch slack under water from January to October, and burn the remainder of their consumption directly during the campaign season from October to December. For this purpose, sugar-factory power plants can use any size of slack, which is recrushed to pass a 3/4- or 1/2-inch screen before burning on chain-grate stokers. About one-third of the 2-1/ 2- inch slack produced in the field is used in smaller industrial plants that employ stokers.Certain factors affect the utilization of 2-1/2-inch slack in the Denver region. The increasing production of hydroelectric power has reduced the consumption of coal at power plants slightly. If more hydropower is generated, the trend of consumption of slack might be downward, unless industrial activity offsets this greater output. Rapid increase in number of domestic stokers during recent years has necessitated preparation of greater quantities of stoker coal, with the result that more lump is crushed, and preparation facilities are changed. At present about 19 percent of the coal produced is crushed to meet the constant demand for industrial fuel as well as the increasing but variable demand for the small domestic stoker sizes. The quantity crushed has increased 12 percent during the last 6 years. These factors are shifting constantly. The trend toward the use of clean and smaller sizes for domestic stokers gives a slack having a larger proportion of fines. In future, even greater quantities of small stoker coal probably will be prepared, and the resulting greater quantities of small slack will have to be used for industrial purposes."

Aug 1, 1942

By Matthias Baeßler, Markus Herten, Ernst Niederleithinger, Steven Georgi

"Abstract: BAW and BAM have performed a large scale comparison and calibration test on static and dynamic load capacity evaluation of bored piles in glacial sandy soil. The test was performed using eight piles at the BAM test site for technical safety at Horstwalde 50 km south of Berlin. The test area has been prepared and investigated in great detail using boreholes, cone penetration tests, pore pressure sensors and geophysical methods to assure controlled conditions for all piles and tests. The piles (10 m length, 0.9 m diameter) are mainly friction piles (low toe resistance) and have been checked by integrity testing. Five piles have been tested by five contractors using the dynamic method in a blind experiment, the other ones piles by static load and/or later on by the dynamic method. Some piles have been equipped with additional fibre optic instrumentation which proved to be robust and helpful in interpreting the results of static, dynamic and integrity tests. We have experienced a deviation of the dynamic load test results gathered in the blind experiment from the static values of up to 20% in most cases, sometimes even up to 30%. This can be related to the known soil inhomogeneities, interpretation and modelling in CAPWAP and method inherent uncertainties. In cases where the static values were known by the testers for calibration, the deviations were significantly smaller. It has to be taken into account, that the two static load tests showed different results as well. Due to the low toe resistance, use of a big drop weight (11 tons) and large drop heights most piles suffered from cracking, which was clearly seen in follow up integrity tests and confirmed by excavation. The piles are available for further research.IntroductionBored piles are used in many construction projects of the German waterways authorities, e. g. bridges, piers or locks. Their load carrying capacity varies with pile type and geometry as well as soil type, parameters and structure and is thus not easy to calculate in advance. Dynamic pile testing as a cost effective addition or replacement for expensive load tests is accepted in many places (Beim and Likins 2008) but still doubted in others. Inspired by the international prediction event for piles in residual soil carried out in the frame of ISC’2 (Viana de Fonseca and Santos 2008), governmental research institutes have designed an experiment in a geological setting typical for glacial and post-glacial areas in northern Germany. Some results have already been presented by Niederleithinger et al. (2012), Herten et al. (2013) and Baeßler et al. (2013). It was intended to perform a blind test for load capacity estimation using dynamic pile testing on six drilled shafts involving five test crews. Two other piles would have to be tested thereafter using static and dynamic methods."

Jan 1, 2014

By M. C. Bleiweiss

Various light sources were evaluated for use in a fiber optic lighting system for mining machines. It was determined that a metal halide arc lamp was best because of its resistance to damage from shock and vibration and the high luminance of the arc. Measurements were made of the percent transmission of the ½ inch thick glass window and the fiber bundles, the total lumen output of the diffuser assembly with a known input and of the average surface luminance of the diffuser assembly. These data were then used to calculate the theoretical output of the system given an estimated input from the source. A prototype 55 watt, single-ended metal halide source was then developed. In order to provide greater efficacy and higher throughput through the fiber optic cables, a variation of the lamp with thallium iodide added to the fill was developed. The lamp was modified and a ballast was developed to allow it to hot restrike.

Jan 1, 1986

In line with DML Resources continual forward management policies, the Roscommon Quarry operation has recently installed and commissioned a Nordberg 7 ft Symons cone crusher and a HP 300 cone crusher. Both machines have enhanced the production capabilities and quality control of the Roscommon comminuition circuit and they will provide efficient and reliable crushing well into the next century.

Jan 1, 1994

By Axel Preusse

For the last 150 years, subsidence damage to the surface has been an unavoidable phenomenon associated with underground coal mining. In particular, the mechanization of mining methods in multiple seam mining and the concentration of mining activities in densely populated European metropolitan areas (e.g., Ruhr, Upper Silesian industrial region) have led to mining having a very strong influence on the earth?s surface. In both these regions, surface subsidence of more than 20m has been reported up to now in comparison to the original state. Underground coal mining in these regions strongly impairs all kinds of surface use, such as urban and industrial development, infrastructure (roads, waterways, and railways) and pipelines?without restricting the functionality of those objects to a greater extent. Demanding and, in some cases, globally unique engineering solutions have been developed and implemented for this purpose. The previously mentioned mining and industrial regions are still very strongly characterized by the (increasingly declining) coal industry and will remain dependent to a large extent and for an indefinite period on artificial drainage measures because subsidence that has already occurred is irreversible. This paper will primarily discuss the prediction of subsidence that is caused by intensive multiple seam mining and its influence on surface objects (e.g., underground pipelines). However, an approach for predicting ground uplifts due to rising mine water as a consequence of mine closures will also be described.

Jan 1, 2012

By T. Patten

Inter-Mountain Labs, IML, performed a study comparing mining waste rock sample analysis by Net Carbonate Value (NCV) and Modified Sobek acid-base accounting procedures. The study looked at 468 waste rock samples and compared the data for Neutralization Potential (NP), Acid Generation Potential (AGP), Net Neutralization Potential (NNP), and the ratio of NP/AGP. The study found that a majority of the samples were below 50 Tons CaCO3/kTons. This subset of ?low-level? samples was also examined separately for Acid Base Accounting parameters. The analysis compared the NCV procedures to the Modified Sobek procedures and concludes that the NCV procedures are as protective of the environment as Modified Sobek procedures.

Feb 27, 2013

By M R. Stevens

The Kauri Block (Hauraki Hill) andMatawai Valley epithermal vein systems, near Coromandel township, are hosted by hydrothermally altered Miocene andesites and Mesozoic greywackes of the Coromandel VolcanicZone. The hydrological setting of both geothermal systems was probably that of an andesitic volcanic arc. Two types of veins are recognised: 1. Simple veins composed of fine comb quartz, and 2. Complex (breccia) veins composed of vein fragments of sulphide-bearing, chalcedonic silica cemented by saccharoidal and fine comb quartz.   Pyrite, arsenopyrite, marcasite, and gold form the ore minerals. Both deposits historically exploited hypogene and supergene bonanza gold.   At Hauraki Hill, breccia veins up to 7m wide contain abundant fragments of recrystallised colloform banded-crustiform textured quartz veins ('Golden Cross' style) and were probably major channels for ascending deep mineralising fluids. Alteration assemblages are zoned with the higher elevations of Hauraki Hill-Bunkers Hill characterised by strong kaolinite alteration whereas deeper levels are dominated by illitic clays.   At Matawai, complex veins and hydrothermal breccias pinch and swell, and inflect, within a total width of veining and alteration that exceeds 50 m. The greywacke host rocks are altered to an assemblage of illitesmectite, quartz, chlorite and pyrite although there are pockets of kaolinite.   The proposed genesis of veins and ores in both deposits invokes episodic rapid mixing (quenching) of hot mineralising reservoir fluids with cool acid condensate waters, whilst acidification and increased fO2 triggered gold deposition. These hydrothermal systems heated up during their later stages and at Matawai visible gold was deposited from a late mesothermal stage. The main conclusion is that the northwest CVZ has excellent potential for large gold deposits (= 30 tonnes contained gold).

Jan 1, 1993

By W. T. Wertman

A research program to aid development of better and more efficient methods for producing petroleum and increasing its ultimate recovery from underground formations is an integral part of the work of the Bureau of Mines on conservation of the Nation's natural resources. Economic, efficient production of recoverable oil and gas from a reservoir depends to a great extent upon the best and most accurate information that can be obtained from a study of the reservoir rock and its fluid contents. Information on oil and water saturation is pertinent and essential in wildcat, extension, and pool wells for good reservoir operations and prediction of performance. This report presents results of an investigation by the Bureau of Mines of core recovery and degree of drilling-fluid contamination occurring during cable-tool coring operations using oil-base mud. Representative core samples were obtained from 6 major oil-producing sands in 8 fields of West Virginia, Pennsylvania, and New York. Field-mixed, oil-base drilling mud was used for the first coring test, and lightweight, premixed oil-base mud, with an organic chloride tracer added, was used in coring eight other wells. Tracer analyses were made to determine contamination of core by drilling-fluid filtrate. Core analyses, including oil and water content and minimum interstitial water-saturation tests, were made on selected core samples to compare and evaluate water saturations determined by the two methods.

Jan 1, 1958

By Clarence Karr

AN EXTENSIVE characterization was conducted on the neutral oil components in a low-temperature bituminous coal tar. A total of tai individual compounds identified with respect to individual isomers, and at least 55 other compounds were indicated to be present. The amounts were determined in all instances. A comparison of isomeric distribution in the tar with thermo-dynamic equilibrium distribution, and kinetic distributions indicated that the composition of the tar largely reflected the structure of the coal from which it was formed. Detailed descriptions are presented for the separatory and qualitative and quantitative procedures for the characterization of the neutral oil component. These include micro fractional vacuum distillation, displacement liquid chromatography, gas-liquid chromatography, countercurrent distribution, and infrared and ultraviolet spectrophotometry. Descriptions are given for the synthesis of authentic specimens of alkylindenes, The ultraviolet and infrared spectra of 134 polycyclic compounds, ranging from alkylindenes to alkylphenanthrenes, either found in the distillable neutral oil, or thought likely to be present, are also given.

Jan 1, 1967

By B. J. Moore

This publication contains analyses and related source data for 269 natural gas samples from wells and pipelines in 18 States. All of the samples were obtained and analyzed during, calendar year 1979 as a part of Bureau of Mines investigations of the occurrences of helium in natural gases of countries with free market economies. This survey has been conducted in the United States by the Bureau of Mines since 1917. In late 1965, the survey was expanded to include foreign gasfields. The analyses published herein were made by mass spectrometer and a special helium analysis apparatus, which are described in Bureau of Mines Bulletins 4862 and 576.3 The results of the investigation of the occurrences of helium are published annually to make the information available as soon as possible to the gas industry and the general public. Twenty-one previous publications have presented tine results of 11,883 gas analyses made prior to 1979. The first three, Bulletins 486,4 576,5 and 617,6 contain analyses and related source

Jan 1, 1980

By Clarence O. Babcock

Model pillars of limestone, marble, sandstone, and granite with length-to-diameter (L/D) ratios of 3, 2, 1, 0.5, and 0.25 (0.286 for granite) were tested to failure in axial compression to determine to what extent an increase in end constraint increased compressive strength. Adjustable steel rings, mechanically tightened about the ends of test specimens to create a radial prestress of 3,000 or 5,000 psi prior to the application of axial load, increased end constraint more than was possible in previous tests using solid steel rings. (Solid rings, epoxy-bonded to the ends of dogbone specimens, increased compressive strength compared with that of straight specimens without ring constraint.) During the loading cycle, additional end constraint was produced by lateral expansion of the specimens against the rings. The effect of the end constraint was to increase the compressive strength 55 to 58 percent for the shorter pillars of dogbone shape compared with the compressive strength of straight pillars with the same L/D ratios without ring constraint. Most pillars with L/D ratios of less than 2 were made stronger by end constraint. Pillars with an L/D ratio of 3 did not benefit significantly from end constraint. When the results from compression testing of straight specimens of rock in the laboratory are used to design mine pillars in situ, the pillars will be larger than necessary if the horizontal compressive stress is several thousand pounds per square inch greater than that resulting from the weight of the overlying rock alone. The L/D ratio can be increased by a factor of 1.76 for ratios in the range 0.25 to 3 and still provide the same compressive strength. The safety factor applied with straight specimen test results should also be acceptable for use with constrained pillar test results.

Jan 1, 1969

By J. W. Town

The Bureau of Mines determined the physical properties of southeastern Idaho phosphate fines as related to mineral composition, particle size and liberation, beneficiation, and potential use in structural clay products. The phosphate mineral in the fines was identified by unit-cell dimensions to be essentially the same as the phosphate mineral fluorapatite identified in the coarser sizes. The mean particle frequency as determined by electron-microscopy and pipet-analyses was between 0.5 and 0.2 micron. Liberation of the phosphate could not be definitely determined because of the weathered condition of the minerals and their extreme fineness. No beneficiation scheme was developed that would concentrate the phosphate into a marketable fertilizer product; therefore, forming and firing tests were made to demonstrate that the fines can be used to manufacture structural clay products.

Jan 1, 1968

By J Mercuri, S Dunford, P Rothman, E Bartsch, J Snyder, D Newton, R Maron, D Cirulis

The purpose of a mineral comminution circuit with a ball mill running in closed-loop with a hydrocyclone classification cluster is to feed the downstream process with the target particle size. Maintaining the optimal particle size in the flotation feed can significantly improve plant cash flow. In particular, reducing the amount of coarse material is often a major challenge. Existing instrumentation to provide particle size measurement is considered standard equipment in most modern concentrator plants. However, availability is often very low, mainly owing to high maintenance requirements of the analyser and associated sampling system.CiDRA Minerals Processing has developed two novel and robust technologies that address this challenge by providing high reliability and very low maintenance systems for on-line measurement of coarse material in the overflow of individual hydrocyclones. The first system is non-invasive and measures pebbles and rocks greater than 6 mm. This paper focuses on the second system, which is a ruggedized wetted sensor design with no moving parts and provides a real-time trend of the desired target grind size parameter, such as percentage above 150 micron.Both systems have simple robust designs and measure in the primary overflow stream of the individual hydrocyclone. Individual hydrocyclone measurement enables identification of poor performing hydrocyclones in addition to providing a representative measurement of the whole hydrocyclone cluster. The systems do not require sampling and associated sample transfer piping circuits that are prone to plugging, thus avoiding high maintenance requirements.The wetted sensor will be described in detail along with its implementation in a robust control scheme in a major copper concentrator in North America. Plant data shows very high availability, achieved with very low maintenance, has enabled improved process control, thereby maintaining throughput and ball mill target power while reducing grind size.CITATION:Cirulis, D, Dunford, S, Snyder, J, Bartsch, E, Rothman, P, Maron, R, Newton, D, O’Keefe, C and Mercuri, J, 2015. Process optimisation using real-time tracking of coarse material in individual cyclone overflow streams, in Proceedings MetPlant 2015, pp 93–109 (The Australasian Institute of Mining and Metallurgy: Melbourne).

Sep 7, 2015

By Svend [Bue] Rondum

The final report contains a summary of the work performed under this contract, In addition to the report itself, other products were derived from this effort. Several hardware items were prototyped for use with underground belt conveyors, and a manual for installation and maintenance procedures was developed and published under separate cover. Recommendations have been included for implementation of the tools and the manual or, an industry-wide basis.

Jan 1, 1982

By Walter J. Karplus, Otto J. M. Smith

The transient response of earth formations may be tlcter.171ined by means of four electrodes mounted coli-rienrly on n sonde. Exponentially decaying current pules are applied to one pair of electrodes and the re.rrrlting voltage transient at the other pair is recorded. It is shown by theoreticol analysis and verified by laboratory experiments that this arrangement readily yields the direct resistivity of the formation and provides valrrable additional information as well. INTRODUCTION Although electrical transients have been employed with moderate success in geophysical prospecting, their application to well logging has received virtually no attention in technical literature. In the conventional direct resistivity logging methods only the zero frequency impedance of the formation under study is measured, and no information is obtained regarding its reactive characteristics. Since the use of certain transients appears to offer some advantages in this respect, experiments were performed on a laboratory scale to investigate this possibility. This investigation was limited to a study of a four-electrode array where both the receiving and transmitting electrode pairs are mounted on a logging sonde, and decaying exponential pulses are applied to the transmitting electrodes. THEORY Given an electrode configuration as shown in Fig. I, located in a homogenous medium of resistivity p, if each electrode has an effective radius, r, which is small compared to the spacing, s and a, it is easy to show by integrating the volume resistivity that the input resistance, Rrw, of the transmitting electrodes is Rxw=p/2pr If a capacitor of capacity C, charged to a voltage Em, is suddenly connected to these electrodes, the resulting current, I(t), will then be The voltage transient, E(t), at the receiving electrodes may then be found by substracting the resulting potential at electrode Y from that appearing at electrode Z. and by inserting Eqs. 1 and 2 Integrating Eq. 4 with respect to tim

Jan 1, 1957