The Potentiostat and the Voltage Clamp by Jackson E

Total Page:16

File Type:pdf, Size:1020Kb

The Potentiostat and the Voltage Clamp by Jackson E The Potentiostat and the Voltage Clamp by Jackson E. Harrar n the history of science and technology, and biophysics. In electrochemistry, and this signal is transmitted to the negative there have been many instances when potentiostats are used for fundamental studies input of the differential amplifier, where it is Itwo or more persons have independently of electrode processes, analytical chemistry, compared to the desired control voltage (E) created an invention or concept at almost battery research, the synthesis of chemicals, at the positive input. The amplifier is a DC- the same time, but for various reasons, and corrosion research. Variations of the coupled, differential input amplifier. It has a one inventor takes precedence or credit. voltage clamp are employed in research high open-loop gain (>105), fast response, Examples are the telephone, the integrated on the properties of living biological cells. and its inputs have high input impedances circuit, calculus in mathematics, and the Adaptations of these instruments have also so very little current (nanoamperes) flows in theory of evolution. Once the invention or been made to control the electric current this part of the circuit. The amplifier, by the concept is introduced, further development rather than voltage. During most of this time, action of negative feedback, continuously soon proceeds along a single path. A rare however, research in these disparate fields of adjusts its output voltage and current to instance is an innovation that was developed electrochemistry and electrophysiology, and keep the potential measured by the reference by two different scientists in two different their investigators, has remained virtually electrode equal to the control voltage. fields at almost the same time, and then independent. The electrolysis current flowing between widely used for many years in these two electrodes CE and WE is measured by fields without the investigators being aware The Potentiostat additional instrumentation (M) in either of the other application. This happened the counter electrode part (as shown) or the in the case of the potentiostat and the Figure 1 shows a simplified version of a working electrode part of the circuit. voltage clamp, which are basically similar potentiostat connected to a three-electrode In electrochemistry, potentiostats are instruments, but whose actual applications electrolytic cell. The electrolysis current used with working electrodes of inert are quite dissimilar. in the electrolytic cell is provided by the elements (e.g., platinum, gold, mercury, and Both the potentiostat and the voltage output of the amplifier (Amp) to the cell via carbon), semiconductors, and for corrosion clamp operate on the principle of negative the counter electrode (CE). The desired or studies, the metal of interest. Sizes range in feedback control. Both instruments employ studied reaction takes place at the working area from >100 cm2 for controlled-potential an amplifier in a feedback arrangement to electrode (WE). The reference electrode (RE) coulometry and electrosynthesis, to very control the voltage (or electrode potential) senses the potential at the working electrode small (radius <103 cm) microelectrodes for in, respectively, an electrolytic cell or a biological specimen. In their simplest forms, they are essentially the same circuit. In the early 1940s, Archie Hickling at the University of Leicester, England, who was working in the field of electrochemistry, invented the potentiostat and coined the apt name for the device.1 He used the potentiostat to control the voltage (i.e., the potential) of an electrode to perform electrolysis in an M electrolytic cell. - In the late 1940s, at the University of Amp Chicago, Kenneth Cole, with the help of George Marmont, invented an electronic + circuit called a voltage clamp,2 which was used to investigate ionic conduction in CE nerves. Concurrently, these voltage clamp Cell RE techniques were adopted by Alan Hodgkin, Andrew Huxley, and Bernard Katz at E WE Cambridge University in England for their research in this field. In 1963, Hodgkin and Huxley were awarded the Nobel Prize in Physiology or Medicine for this work. Further development and elaboration of these circuits has been carried on for many years, including the marketing of many commercial instruments—some for electrochemistry and some for biochemistry FIG. 1. Simplified potentiostat and three-electrode electrolytic cell. 42 The Electrochemical Society Interface • Winter 2013 chemical analysis and fundamental studies.3 ions are transported through ion channels clamp arrangements that can influence the Counter electrodes are usually made of in the membrane.2,4 Later studies have dealt measurements of high-speed signals.6 platinum, while reference electrodes are with calcium and chloride ion flow. The stability of the cell/feedback loop most often made of silver, coated with silver Advances in technique and instrumenta- in controlling the potential may also be chloride, or mercury-mercurous chloride tion in the field of electrophysiology have an issue. In the case of potentiostatic (saturated calomel). led to an apparatus for clamping with a sin- measurements, this was examined in gle intracellular electrode, and patch clamp- terms of classical control-system theory in The Voltage Clamp ing, which enables measuring the properties the 1960s and 1970s,7 and has also been of single ion channels in a membrane.5,6 For addressed for voltage clamp systems.6,8 Figure 2 shows a simplified version of the invention of the patch clamp technique, Data interpretation in electrophysiology a “two-electrode” voltage clamp connected Bert Sakmann and Erwin Neher received the may also be complicated because the to an apparatus in which the properties of 1991 Nobel Prize in Physiology or Medi- microelectrode contacts a point while the the membrane of a biological specimen are cine. cell is obviously three dimensional. The examined. The specimen in this arrangement patch clamp technique is advantageous in is contained in a bathing medium such as Common Problems this respect because the microelectrode saline solution. The electrodes (CE and RE) in the Experiments in this configuration is attached directly penetrate the membrane of the specimen. to an ion channel or small group of ion Electrode CE is the current-carrying Although the electronic circuits of channels. In both electrochemistry and electrode within the specimen, while the potentiostatic and voltage clamps are similar, electrophysiology, quite sophisticated second electrode (RE) senses the potential the laboratory apparatus and experiments instrumentation has been designed and (with reference to ground) across the using them are quite different. Nevertheless, techniques of data interpretation have membrane. The current flows from electrode been developed to deal with all of these some problems that complicate the 5-9 CE through the membrane of the specimen experiments are present in both fields. problems. to ground. Measurements of the current, in Electrophysiologists are always dealing either the amplifier output circuit (M) or the with extremely small microelectrodes History ground circuit, yields information on the which have high resistances that may properties of the membrane. The electrodes introduce errors in potential control.6 Further development of potentiostatic typically are constructed of fine-tipped glass Certain investigations using potentiostats and voltage clamp instrumentation from the pipets containing a chlorided silver wire also employ very small working electrodes, simple circuits described here has proceeded or platinum, and the experiments are often and when more exact potential control is in parallel. The first circuits used amplifiers performed on a microscope stage. required, solution resistances exist that may that were assembled from the individual The similarity of the voltage-clamp have to be compensated, particularly at the electronic components (resisters, capacitors, configuration to that of the potentiostat is working electrode.7 and vacuum tubes). Some early potentiostats readily apparent. The amplifier/feedback There are also many electrical also used mechanical servomechanisms. The voltage clamp circuit functions the same capacitance effects that complicate the advent during the 1950s of commercially way it does as a potentiostat to impose a measurements. First of all, a cell membrane available, plug-in, modular amplifiers, potential equal to the command voltage at itself constitutes an electrical capacitance called operational amplifiers,10 or “op amps,” the membrane. As in potentiostatic circuits, that must be charged before the desired made possible many extensions of the basic the current passing through the membrane in potential is established. This is analogous potentiostat and voltage clamp and more measured with auxiliary circuitry not shown to potentiostatic work in which the working elaborate circuits.6,9 At first, using these in the figure. In the study of nerve cells, electrode/solution double-layer capacitance modular op amps, experimenters who were early work in voltage clamping revealed must first be charged. There are also “stray” not trained engineers, but who were versed information on how sodium and potassium capacitances in both electrolytic and voltage in electronics, could assemble their own functioning instruments. Quite advanced instruments incorporating integrated-circuit operational amplifiers are now commercially available for many specialized
Recommended publications
  • 07 Chapter2.Pdf
    22 METHODOLOGY 2.1 INTRODUCTION TO ELECTROCHEMICAL TECHNIQUES Electrochemical techniques of analysis involve the measurement of voltage or current. Such methods are concerned with the interplay between solution/electrode interfaces. The methods involve the changes of current, potential and charge as a function of chemical reactions. One or more of the four parameters i.e. potential, current, charge and time can be measured in these techniques and by plotting the graphs of these different parameters in various ways, one can get the desired information. Sensitivity, short analysis time, wide range of temperature, simplicity, use of many solvents are some of the advantages of these methods over the others which makes them useful in kinetic and thermodynamic studies1-3. In general, three electrodes viz., working electrode, the reference electrode, and the counter or auxiliary electrode are used for the measurement in electrochemical techniques. Depending on the combinations of parameters and types of electrodes there are various electrochemical techniques. These include potentiometry, polarography, voltammetry, cyclic voltammetry, chronopotentiometry, linear sweep techniques, amperometry, pulsed techniques etc. These techniques are mainly classified into static and dynamic methods. Static methods are those in which no current passes through the electrode-solution interface and the concentration of analyte species remains constant as in potentiometry. In dynamic methods, a current flows across the electrode-solution interface and the concentration of species changes such as in voltammetry and coulometry4. 2.2 VOLTAMMETRY The field of voltammetry was developed from polarography, which was invented by the Czechoslovakian Chemist Jaroslav Heyrovsky in the early 1920s5. Voltammetry is an electrochemical technique of analysis which includes the measurement of current as a function of applied potential under the conditions that promote polarization of working electrode6.
    [Show full text]
  • Hydrodynamic Electrodes and Microelectrodes
    CHEM465/865, 2004-3, Lecture 20, 27 th Sep., 2004 Hydrodynamic Electrodes and Microelectrodes So far we have been considering processes at planar electrodes. We have focused on the interplay of diffusion and kinetics (i.e. charge transfer as described for instance by the different formulations of the Butler-Volmer equation). In most cases, diffusion is the most significant transport limitation. Diffusion limitations arise inevitably, since any reaction consumes reactant molecules. This consumption depletes reactant (the so-called electroactive species) in the vicinity of the electrode, which leads to a non-uniform distribution (see the previous notes). ______________________________________________________________________ Note: In principle, we would have to consider the accumulation of product species in the vicinity of the electrode as well. This would not change the basic phenomenology, i.e. the interplay between kinetics and transport would remain the same. But it would make the mathematical formalism considerably more complicated. In order to simplify things, we, thus, focus entirely on the reactant distribution, as the species being consumed. ______________________________________________________________________ In this part, we are considering a semiinfinite system: The planar electrode is assumed to have a huge surface area and the solution is considered to be an infinite reservoir of reactant. This simple system has only one characteristic length scale: the thickness of the diffusion layer (or mean free path) δδδ. Sometimes the diffusion layer is referred to as the “Nernst layer” . Now: let’s consider again the interplay of kinetics and diffusion limitations. Kinetic limitations are represented by the rate constant k 0 (or equivalently by the 0=== 0bα b 1 −−− α exchange current density j nFkcred c ox ).
    [Show full text]
  • Μstat 4000P Multi Potentiostat
    µStat 4000P Multi Potentiostat 01 Ref. STAT4000P Following the format of our multipotentiostats with a size of only 22x20x7 cm, includes 4 channels that can act at the same time as 4 independent potentiostats; it also includes one multichannel that can act as a poten- tiostat where up to 4 working electrodes share an auxiliary and a reference electrode. With µStat 4000P users can perform up to 4 different electrochemical techniques at the same time; or carry out the study of one technique’s parameter in just one step by applying the same electrochemical technique in several channels but selecting different values for the parameter under study. These are just exam- ples of the enormous capabilities that our new instrument offers. µStat 4000P can be applied for Voltammetric or Amperometric measurements, including 12 electroanalytical techniques. In addition, µStat 4000P owners can later upgrade their instrument to a µStat 4000P by just purchasing an extension. This self-upgrade does not require any hardware modification, but it is implemented by means of a Galvanostat software update kit. This Multi Potentiostat is Li-ion Battery powered (DC charger adaptor also compatible), and can be easily connected to a PC via USB or through Wireless connection. µStat 4000P is controlled by the powerful software “DropView 8400” which is included and that allows plotting of the measurements and performing the analysis of results. DropView software provides powerful functions such as experimental control, graphs or file handling, among others. Available
    [Show full text]
  • An Improved Voltage Clamp Circuit Suitable for Accurate Measurement of the Conduction Loss of Power Electronic Devices
    sensors Article An Improved Voltage Clamp Circuit Suitable for Accurate Measurement of the Conduction Loss of Power Electronic Devices Qiuping Yu, Zhibin Zhao *, Peng Sun, Bin Zhao and Yumeng Cai State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China; [email protected] (Q.Y.); [email protected] (P.S.); [email protected] (B.Z.); [email protected] (Y.C.) * Correspondence: [email protected] Abstract: Power electronic devices are essential components of high-capacity industrial converters. Accurate assessment of their power loss, including switching loss and conduction loss, is essential to improving electrothermal stability. To accurately calculate the conduction loss, a drain–source voltage clamp circuit is required to measure the on-state voltage. In this paper, the conventional drain–source voltage clamp circuit based on a transistor is comprehensively investigated by theoretical analysis, simulations, and experiments. It is demonstrated that the anti-parallel diodes and the gate-shunt capacitance of the conventional drain–source voltage clamp circuit have adverse impacts on the accuracy and security of the conduction loss measurement. Based on the above analysis, an improved drain–source voltage clamp circuit, derived from the conventional drain–source voltage clamp circuit, is proposed to solve the above problems. The operational advantages, physical structure, and design guidelines of the improved circuit are fully presented. In addition, to evaluate the influence of component parameters on circuit performance, this article comprehensively extracts three electrical Citation: Yu, Q.; Zhao, Z.; Sun, P.; quantities as judgment indicators. Based on the working mechanism of the improved circuit and Zhao, B.; Cai, Y.
    [Show full text]
  • Lecture 1: an Introduction to Plasticity and Cellular Electrophysiology
    Lecture 1: An Introduction To Plasticity and Cellular Electrophysiology MCP 2003 Charge separation across a ATPase + semipermeable Ca++ 3Na ATPase membrane is the 2K+ (mM) basis of excitability. K+ = 140 + Na = 7 ++ + Ca Cl- = 7 3Na (mM) Ca++ = 1 x 10-4 co-transporter K+ =3 Na+ = 140 membrane pores Cl- = 140 with variable Ca++ = 1.5 permeability - + R - + m Cm gK+ gCl- gNa++ Convention: Current direction is defined by the direction of increasing positive charge. Na++ flux into a cell is an inward current. K+ flux out of a cell is an outward current. Cl- flux into a cell is an outward current. A depolarizing current is a net influx of + ions or a net efflux of negative ions. A hyperpolarizing current is a net efflux of + ions or a net influx of negative ions. I Outward rectification: when a membrane allows outward current (net + charge out) V to flow more easily than and inward current. Outward rectification I Inward rectification: when a membrane allows inward current to flow more easily V than an outward current. Inward rectification Methods of Measuring Function In Excitable Membranes Field potentials: measure current sources _ low pass filter and sinks from populations of neurons + across the electrode resistance. _ 0 Brain mV-V - t (sec) + Microelectrode extracellular recording: measures action potentials from a small _ number of neurons. 0 mV pulse amplitude window - Intracellular recording: can measure voltage, t (msec) band pass filter 1KHz-10KHz +40 0 Voltage clamp recording: resting membrane Can pass current to compensate -60 potential for voltage change. In this way voltage is held ~ constant and current applied to _ compensate is a measure of the current flowing.
    [Show full text]
  • Basics and Applications of a Quartz Crystal Microbalance Monitoring Surface Interactions Via Small-Scale Mass Changes
    Basics and Applications of a Quartz Crystal Microbalance CORROSION BATTERY TESTING Monitoring Surface Interactions via Small-scale Mass Changes COATINGS PHOTOVOLTAICS gamry.com Contents Basics of QCM ........................................................................................................................3 Calibration of a QCM ................................................................................................... 13 Investigation of a Thin Polymer Film ..........................................................................21 The eQCM 10M System ..................................................................................................... 26 The QCM-I System .............................................................................................................. 27 References .......................................................................................................................29 Additional Resources .................................................................................................... 30 2 gamry.com Basics of a Quartz Crystal Microbalance This section provides an introduction to the quartz crystal microbalance (QCM) which is an instrument that allows a user to monitor small mass changes on an electrode. The reader is directed to the numerous reviews 1 and book chapters1 & 2 for a more in-depth description concerning the theory and application of the QCM. A basic understanding of electrical components and concepts is assumed. The two major points of this section are:
    [Show full text]
  • A Practical Organic-Mediated Hybrid Electrolyser That Decouples
    Electronic Supplementary Material (ESI) for Chemical Science. This journal is © The Royal Society of Chemistry 2018 Supplementary Information for: A Practical Organic-Mediated Hybrid Electrolyser that Decouples Hydrogen Production at High Current Densities Niall Kirkaldy,a Greig Chisholm,a Jia-Jia Chena and Leroy Cronin*a a WestCHEM, School of Chemistry, University of Glasgow, University Avenue, Glasgow, G12 8QQ, UK * Corresponding author, [email protected] 1 Contents SI-1. General Experimental Remarks .................................................................................................. 3 SI-2. Electrochemical Characterisation ............................................................................................... 4 SI-3. Gas Headspace Measurements................................................................................................... 6 SI-4. Hybrid PEME Construction and Operation ................................................................................. 7 SI-5. PEME Characterisation Methods ................................................................................................ 8 SI-6. PEME Efficiency Calculations .................................................................................................... 10 SI-7. Cost Calculations ....................................................................................................................... 11 2 SI-1. General Experimental Remarks 9,10-anthraquinone-2,7-disulfonic acid disodium salt was purchased from Santa Cruz Biotechnology
    [Show full text]
  • Pulse Voltammetry Software Brochure
    Data Analysis density. This feature is particularly useful for comparing data from electrodes of different areas. The analysis of the software data is performed in the Echem Analyst. Specific analysis routines have been created to Baseline Add: Baselines can be added to the data graph by either drawing a Freehand Line or by extrapolating a handle this software data files. The general features of the Echem Analyst are described in a separate brochure entitled part of the baseline with the Linear Fit feature. Redefining Electrochemical Measurement “Overview of Gamry Software.” Integrate: Integration of the current in Differential Pulse These specific routines include: Voltammetry and Square Wave Voltammetry is possible by defining a baseline and then selecting the portion of the Pulse Voltammetry Software Peak Find: Use the Region Selector button to select a curve you want to integrate. Then select Integrate from the portion of the curve that includes the region where the drop-down menu and the result is reported on the curve The Pulse Voltammetry Software adds Differential Pulse peak is located. Click on the Peak Find button to find the and also on a new tab. This software incorporates the following pulse techniques: peak position and the peak height. A perpendicular line is Voltammetry, Square Wave Voltammetry, and other drawn on the chart from the peak to the baseline. Background Subtract: A background file can be recognized pulse voltammetry techniques to the Gamry ● Square Wave subtracted from the current active data file by selecting software product family. For qualitative and mechanistic ● Square Wave Stripping Subtract from the menu and choosing the file.
    [Show full text]
  • Emstat-Go-Description.Pdf
    z Rev. 1-2019 EmStat Go potentiostat ...............................................................................................................2 Sensor Extension module .........................................................................................................2 Sleeves in any color .................................................................................................................3 Modular design ........................................................................................................................3 Optional battery for connecting via Bluetooth ...........................................................................3 Reduce your time-to-market ....................................................................................................4 Supported techniques ..............................................................................................................4 Voltammetric techniques ......................................................................................................4 Techniques as a function of time ..........................................................................................4 Custom software options .............................................................................................................5 Specifications of general parameters ...........................................................................................6 General pretreatment............................................................................................................6
    [Show full text]
  • Development and Evaluation of a Calibration Free Exhaustive Coulometric Detection System for Remote Sensing
    University of Louisville ThinkIR: The University of Louisville's Institutional Repository Electronic Theses and Dissertations 5-2014 Development and evaluation of a calibration free exhaustive coulometric detection system for remote sensing. Thomas James Roussel University of Louisville Follow this and additional works at: https://ir.library.louisville.edu/etd Part of the Mechanical Engineering Commons Recommended Citation Roussel, Thomas James, "Development and evaluation of a calibration free exhaustive coulometric detection system for remote sensing." (2014). Electronic Theses and Dissertations. Paper 1238. https://doi.org/10.18297/etd/1238 This Doctoral Dissertation is brought to you for free and open access by ThinkIR: The University of Louisville's Institutional Repository. It has been accepted for inclusion in Electronic Theses and Dissertations by an authorized administrator of ThinkIR: The University of Louisville's Institutional Repository. This title appears here courtesy of the author, who has retained all other copyrights. For more information, please contact [email protected]. DEVELOPMENT AND EVALUATION OF A CALIBRATION FREE EXHAUSTIVE COULOMETRIC DETECTION SYSTEM FOR REMOTE SENSING by Thomas James Roussel, Jr. B.A., University of New Orleans, 1993 B.S., Louisiana Tech University, 1997 M.S., Louisiana Tech University, 2001 A Dissertation Submitted to the Faculty of the J. B. Speed School of Engineering of the University of Louisville in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Department of Mechanical Engineering University of Louisville Louisville, Kentucky May 2014 Copyright 2014 by Thomas James Roussel, Jr. All rights reserved DEVELOPMENT AND EVALUATION OF A CALIBRATION FREE EXHAUSTIVE COULOMETRIC DETECTION SYSTEM FOR REMOTE SENSING By Thomas James Roussel, Jr.
    [Show full text]
  • Recording Techniques
    Electrophysiological Recording Techniques Wen-Jun Gao, PH.D. Drexel University College of Medicine Goal of Physiological Recording To detect the communication signals between neurons in real time (μs to hours) • Current clamp – measure membrane potential, PSPs, action potentials, resting membrane potential • Voltage clamp – measure membrane current, PSCs, voltage-ligand activated conductances 1 Current is conserved at a branch point A Typical Electrical Circuit Example of an electrical circuit with various parts. Current always flows in a complete circuit. 2 Resistors and Conductors Summation of Conductance: Conductances in parallel summate together, whether they are resistors or channels. Ohm's Law For electrophysiology, perhaps the most important law of electricity is Ohm's law. The potential difference between two points linked by a current path with a conductance G and a current I is: 3 Representative Voltmeter with Infinite Resistance Instruments used to measure potentials must have a very high input resistance Rin. Capacitors and Their Electrical Fields A charge Q is stored in a capacitor of value C held at a potential DeltaV. Q = C* delta V capacitance 4 Capacitors in Parallel Add Their Values Currents Through Capacitors Membrane Behavior Compared with an Electrical Current A A membrane behaves electrically like a capacitance in parallel with a resistance. B Now, if we apply a pulse of current to the circuit, the current first charges up the Response of an RC parallel capacitance, then changes circuit to a step of current the voltage 5 The voltage V(t) approaches steady state along an exponential time course: The steady-state value Vinf (also called the infinite-time or equilibrium value) does not depend on the capacitance; it is simply determined by the current I and the membrane resistance R: This is just Ohm's law, of course; but when the membrane capacitance is in the circuit, the voltage is not reached immediately.
    [Show full text]
  • High Speed Controlled Potential Coulometry
    c1CYCLIC CHELONO, DIFPU- c2SOLVE GENERATED EQUA- 903 FORMAT (5HRR =, F10.5, SION CONTROLL, PLANE TION BEGIN AT 96 READ 8HFRACT =, F10.5) ELECTRODE, READ IN K IN NOSIG FOR ACCURACY GO TO 920 NOSIG RR FRACT, TWO 96 IF(M- 1)300,100,102 300 PRINT905 SOLUBLE ElPECIES 100 Z=Y 905 FORRSAT (2X,5HEItROR) READ 900,K,NOSlG, RR, M=M+l 920 STOP FRACT 102 IF (Z) 98,200,99 EXD DIMENSION X (100),T (1 00) , 98 IF (Y) 71,200,73 END R(100) 99 IF (Y) 73,200,71 C GENERATION OF EQUA- 71 T(N) = T(N) + 10.0 **(-LA) LITERATURE CITED TIONS GO TO 10 (1) Alden, J. R., Chambers, J. Q., Adams, DO200N = 1,K 73 T(N) = T(K) - 10.0 **(-LA) R. N., J. Electroanal. Chem. 5, 152 T(N) = 0.0 LA=LA+I (1963). M=l 199 IF (NOSIG - LA) 300,200,71 (2) Bard, A. J., ANAL. CHEM. 33, 11 (1961). LA = 0 200 CONTIXUE (3) Churchill, R. V., “Operational Mathe- 10 DO 80 I = 1,N c3EQUATION SOLVED PRINT matics,” p. 39, McGraw-Hill, New York, SUM = 0.0 ANSWER 1958. DO 60 J = I,N DO201 J = 1,K,2 (4) Galus, Z., Lee, H. Y., Adams, R. N., = 201 R(J) = T(J)/T(J 1) J. Electroanal. Chem. 5, 17 (1963). 60 SUM SUM -- T(J) + (5) Murra,y, R. W., Reilley, C. N., Ibid., X(1) = SQRTF(SUM) PRINT 903, RR, FRACT 3, 182 (1962). 80 CONTIXUE PRINT 901 (6) Piette, L.
    [Show full text]