Long live the vis-viva equation. There is sometimes more than one way of telling a story with the same ending, and this is particularly true in the field of applied mathematics where there are often different ways of obtaining the same result to a given problem, so that what might be lost or not appreciated in one approach can be found and appreciated in another. It is the purpose here to illustrate this by presenting a well-known example drawn from the field of orbital dynamics, namely the development of what is called the Vis-viva equation. This equation is simply an expression relating the square of the velocity of an orbiting object, for example a planet orbiting a sun, to orbit parameters and scientific constants. It is a standard workhorse equation that is used extensively today by orbit control specialists wishing to determine and affect velocities of spacecraft orbiting significantly larger masses, and it was developed from work carried out centuries ago by Gottfried Leibniz (1646 – 1716). The first story will simply acknowledge the equation and how it arose. The second story will present an alternative approach based on calculus, trigonometry, algebra and computations set against the backdrop of an ellipse geometry. This second story leads to the vis-viva equation in disguise, so to speak, and examples of the speeds of two planets, Earth and Mercury, orbiting the Sun will be discussed. Where relevant, the nomenclature of orbit dynamics will be acknowledged. The discussion will involve primarily a planet’s orbit velocity of translation and detailed considerations will not be given to effects due to its speed of rotation.

This article sets out the results of research into effort in Dhrupad, a genre of Hindustani music. I examine gestural interactions with imaginary objects that vocalists outline when engaging with melodies. The study takes an embodied cognition stance and applies thematic analysis and inductive coding to original interviews with Dhrupad musicians in India, the UK, and the Netherlands. My findings demonstrate that from the singers’ cognitive perspectives, their hand movements are deeply connected to their voices. I also offer insights on the relationships between bodily movement, sound, and imagery that are used to inform the development of effort inference models.

Equilibrium studies on clay films exposed to NH3 and H2O vapors demonstrate the effect of exchangeable cations on surface acidity and its relationship to hydration. At a relative humidity of 98 per cent the order of acidity on the clay surface as indicated by protonation of NH3 was Al > Mg > Ca = Li > Na = K for Wyoming bentonite and Al > Mg > Li > Ca = Na = K for nontronite. At a relative humidity of 20 per cent, however, the order was Al = Mg > Ca > Li > Na > K for the bentonite and Al = Mg > Li > Ca > Na > K for nontronite. The largest change in proton donation properties due to hydration effects was in the calcium clays. For Ca-bentonite the NH4+ formation was 16 me/100 g at 98 per cent and 80 me/100 g at 20 per cent relative humidity. In Ca-nontronite, the NH4+ formation was 14 and 64 me/100 g for the wet and dry systems respectively. The differences in proton donation between the bentonite and nontronite clays are believed to be due to charge site location and its effects on ion hydration. The NH4+ formed by the protonation process seemed to exist in different environments in the bentonite and nontronite as indicated in the i.r. absorption spectra.

Interlayer K was extracted with sodium tetraphenylboron solution from a powdered sericite heated to the temperature of dehydroxylation and quenched to room temperature. By this procedure, sericite of the 2M1 type was changed to an interstratified mineral. The X-ray diffraction properties of this interstratified mineral are similar to “rectorite”.

The autoradiographic method at the electron microscope level has been applied successfully to locate the position of sorbed ions and compounds associated with kaolinite and dickite particles. For suitable isotopes, the limit of resolution was about 0·2 μm and would be conceivably less in the most favorable conditions.

Judged by the location of iodide-125 and strontium-85, the majority of positive sites and also the most reactive negative sites were situated at the edges of the clay particles. These conclusions were supported by normal electron microscope examination of mixtures of clay with charged gold and silver iodide sols.

Iron (III) ions, which have been shown to react with kaolinite by chemisorption mechanisms in 0·15 M systems and by both ion exchange and chemisorption in 0·01 M systems, were, in either case, located at edge-sites on the clay. When intentionally precipitated onto dickite, iron (III) hydroxides were distributed evenly over the clay surfaces. However, precipitation was probably initiated at the particle edges.

Minor exceptions to the above conclusions appeared to be caused by basal surface features such as fracture or stress lines, outgrowths or cavities, and if the clays had not been cleaned, by strongly associated impurities such as iron hydroxides.

Simultaneous X-ray difffraction-differential thermal analysis equipment for an X-ray diffractometer was evaluated. A brief description of the apparatus and of the method of preparing the sample was given.

Simultaneous XRD-DTA was used to study various clay minerals and inorganic hydrates. It was found in the study of clays such as hectorite, montmorillonite, kaolinite, and attapulgite that information concerning the rate of removal of water and hydroxyl units at various temperatures could be correlated to changes in the X-ray diffraction pattern. Various inorganic hydrates such as barium chloride dihydrate, nickel sulfate hexahydrate, and potassium ferrocyanide trihydrate were examined under dynamic and static conditions of analysis along with vacuum conditions. It was found that good correlation could be obtained between the X-ray diffraction pattern and the loss of water from the hydrates as indicated by the DTA curves.

The scanning electron microscope (SEM) proves to be ideally suited for studying the configuration, texture, and fabric of clay samples. Growth mechanics of crystalline units — interpenetration and interlocking of crystallites, crystal habits, twinning, helical growth, and topotaxis — also are uniquely revealed by the SEM.

Authigenic kaolins make up the bulk of the examples because their larger crystallite size, better crystallinity, and open texture make them more suited to examination by the SEM than most other clay mineral types.

X-ray mineralogical examination of the 2–20 μm fraction of oriented samples from a highly calcereous swamp soil and neighbouring Mio-Pliocene strata in Al-Hasa showed a wide range in mineralogy: attapulgite, illite, montmorillonite, chlorite and kaolinite, in decreasing order. Powder X-ray diffraction revealed an abundance of attapulgite, calcite, quartz and in a few samples, traces of dolomite, Fe-oxides, feldspars and gypsum.

The soils are believed to have been formed by deposition of highly calcareous wind-borne silt particles in an already formed swamp. The surface and subsurface horizons are underlain by an organic residue layer, 30–50 cm thick, containing 14–36% organic matter. The whole profile sits on hard, impervious, sedimentary layers that mineralogically resemble some strata of the outcrop.

Attapulgite and illite are the most abundant and common clay minerals in the soils and strata, but are most pronounced in the latter. Attapulgite not only occurs in Al-Hasa, but is widely distributed in the Arabian Peninsula; from east of the Mediterranean through the deserts of Syria and Iraq down to eastern Saudi Arabia, where it occurs only in association with limestone parent material. In this area of extreme aridity (Al-Hasa), the origin of the minerals and their relative abundance is believed to be associated with the source of sediment since minimum alteration seems to have taken place after deposition.

A preliminary survey of electronic absorption spectra of clay minerals reveals the utility of u.v.-visible spectroscopy in the elucidation of structural, physical, and chemical properties of such systems. Spectra, which were obtained in the suspension, film, and single crystal states (where applicable), are interpreted in terms of iron-associated transitions. Microcrystalline clay minerals typically show Fe(lll) in octahedral oxo-ligand geometry whereas mica-type minerals may show a range of iron species, including octahedral Fe(III), tetrahedral Fe(III), and octahedral Fe(II). Iron affects the local site geometry and in “high iron” minerals may dictate layer geometry and subsequently the crystalline form.

A series of mixed iron and titanium oxide coprecipitates ranging in composition between 0 < Ti/Ti + Fe < 1 was synthesized and aged under varying conditions of pH, temperature and time in order to establish a working model for pedogenic titanium and titano-ferric oxides. X-ray powder diffraction (XRD), selective chemical dissolution, magnetic susceptibility, charge distribution and electron optical data indicate that the freshly prepared Fe-Ti oxides consist of an Fe-rich (Ti-ferrihydrite) phase (Ti/Ti + Fe ⩽ 0.70) having pH-dependent positive charge and a Ti-rich phase (Ti/Ti + Fe ⩾ 0.7) with permanent and pH dependent negative charge.

Synthetic Ti-ferrihydrite and amorphous TiO2 were completely soluble in acid ammonium oxalate (2 hr extraction in the dark) whereas poorly crystalline anatase (width at half height, WHH > 2.0°2θ) was partly oxalate soluble. NH4-oxalate soluble Ti was particularly high in soils developed under a cool montane climate (afro-alpine) and lower in soils of warmer subtropical climate, which contain anatase and rutile.

Several mixed Fe-Ti crystalline phases were identified after aging NH3 coprecipitates of Fe and Ti nitrate at 70°C and pH 5.5 for 70 days:

1. (1) goethite and hematite in the composition range 0 < Ti/Ti + Fe ⩽ 0.20; at low Ti concentrations (<5 mole %) goethite was favored and/or hematite inhibited

2. (2) microcrystalline pseudorutile in the composition range 0.20 ⩽ Ti/Ti + Fe ⩽ 0.70

3. (3) anatase and ferriferous anatase in the range 0.70 ⩽ Ti/Ti + Fe < 1.0; with decreasing proportion of Ti the crystal-Unity of anatase decreased.

The results suggest that secondary or pedogenic Ti-Fe oxides can form by coprecipitation and crystallization in the weathering solution, and emphasize the essential role of water (as opposed to dry oxidation) in the alteration of primary titaniferous minerals.

The pH, Eh, electrical conductivity (EC), and the amounts and valency of replaceable iron were measured periodically on Fe2+- and Fe3+-saturated montmorillonite and cation-exchange resin at three temperatures. Differences in the pattern of change of pH, Eh, and EC with time appear to be related more to the histories and modes of preparation of the systems than to intrinsic differences in the hydrolysis of the iron in them. Electron transfer reactions involving crystal components of the clay can cause oxidation of adsorbed Fe2+ ions; the activation energy (Ea) for oxidation on the clay's surface was 6 kcal/mole, less than a third of the activation energy reported for Fe2+ oxidation in solution. In the Fe2+-resin, where Ea = 10.7 kcal/mole, perturbed surface-water molecules may act as electron acceptors enhancing Fe2+ oxidation.

Polymerization and precipitation of the adsorbed iron is affected by the necessity to maintain electroneutrality, the ability of the iron-hydroxy ions and small polymers to move about in the voids of the ion exchanger, and the steric hindrance posed by the matrix of the ion exchanger to the formation of large polymers. In resin, little or no iron precipitates, probably due both to steric hindrance and the inability of the resin to release ionic components to maintain electroneutrality. In clays, steric hindrance is small, and Al and Mg are released from the crystal to maintain electroneutrality, thus the precipitation of iron is abundant and is controlled by the rate of release of Al and Mg from the crystal.

Previous studies of the line profiles of the basal reflections of microcrystalline muscovites were refined by an adaptation of the method developed by Maire and Méring. In order to evaluate the variation of interlayer spacings, the method required only relative values of Fourier coefficients, without the correction for instrumental broadening, which was the source of one of the most critical problems previously. Instead of Kα radiation, Kβ radiation was used to record line profiles since difficulties associated with the separation of Kα1 and Kα2 radiations could not be overcome satisfactorily.

The data reconfirmed that the line broadening of 00l reflections was due not only to a small particle-size effect, but also to structural disorders involving the variation of the interlayer spacings. For the four specimens investigated here, the mean squares of the variation of interlayer spacings ranged from 0 to 0·0358, the square roots of which were inversely proportional to the total number of interlayer cations. It is considered that the observed distortions were mainly attributed to non-uniform interlayer spaces between silicate layers arising from an irregular distribution of interlayer cations. The data also indicated that the crystallites of all four specimens consisted of a similar number of layers. The method showed promise for the study of the nature and extent of structural disorders in micas or other silicate minerals.