An Asymptotic Expansion for

1928 ◽  
Vol 24 (2) ◽  
pp. 280-289 ◽  
Author(s):  
H. P. Mulholland
Keyword(s):  
Vibrational State ◽  
Principal Axis ◽  
The Other ◽  
Band Spectrum ◽  
Molecular Gas ◽  
Moments Of Inertia ◽  
Specific Heats ◽  
Principal Moments Of Inertia ◽  
Adequate Representation ◽  
Image Position

In the theory of the specific heats of gases of diatomic molecules the functionplays a well-known and important part. The rotational specific heat Crot of a diatomic molecule, which is susceptible of representation as a rigid body with two equal principal moments of inertia, without spin about the other principal axis, is given bywhere R is the gram-molecular gas-constant andthe pair of equal moments of inertia being equal to A. Whether such a model is or is not an adequate representation is a matter for determination by a detailed study of the structure of the band spectrum, particularly of the nature of the normal electronic and vibrational state. It is known to be applicable to normal molecules of the halogen hydrides, to CO and other molecules which have a normal state of type 1S including (but for a certain special feature) H2.

Orbital and Attitude Stability of Space Shuttles in Parking Orbits

1975 ◽  
Vol 26 (1) ◽  
pp. 20-24
Author(s):  
R Arho
Keyword(s):  
Gravitational Field ◽  
Stationary State ◽  
Circular Orbit ◽  
Principal Axis ◽  
Orbital Plane ◽  
Moments Of Inertia ◽  
Attitude Stability ◽  
Principal Moments Of Inertia ◽  
Bounded Perturbation ◽  
The Stability

SummaryA unified treatment is given of the orbital and attitude stability of space shuttles in parking orbits (in vacuo) in the earth’s gravitational field. A shuttle in a circular orbit with a principal axis aligned with the horizontal in the orbital plane is found to be in stationary geostatic equilibrium. The demand for stability leads to a condition which must be satisfied by the principal moments of inertia. The stability which is achieved is not asymptotic without control. The stationary state is a stable centre about which a bounded perturbation oscillation without damping may exist.

scholarly journals Method for assessment of matching of total ankle joint endoprostheses based on CT data

2020 ◽  
Vol 6 (3) ◽  
pp. 396-397
Author(s):  
Heiner Martin ◽  
Josephine Wittmüß ◽  
Thomas Mittlmeier ◽  
Niels Grabow
Keyword(s):  
Cross Sections ◽  
Spline Functions ◽  
Moments Of Inertia ◽  
Cad System ◽  
Principal Moments Of Inertia ◽  
Principal Axes ◽  
Ct Data ◽  
The One ◽  
Relative Differences ◽  
Systemic Design

AbstractThe investigation of matching of endoprosthesis tibial components to the bone cross section is of interest for the manufacturer as well as for the surgeon. On the one hand, a systemic design of the prosthesis and the assortment is possible, on the other hand, a better matching implantation is enabled on the basis of experience of this study. CT sections were segmented manually using a CAD system and fitted by spline functions, then superseded with cross sections of the tibial component of a modified Hintermann H3 prosthesis. The principal moments of inertia, the direction of the principal axes and the area of the section were evaluated. Based on the relative differences of the principal moments of inertia, recommendations for application of the different prosthesis size and its selection with the surgery can be made.

scholarly journals The Dissociation of the Compound of Iodine and Thio-urea

1904 ◽  
Vol 24 ◽  
pp. 233-239 ◽  
Author(s):  
Hugh Marshall
Keyword(s):  
Nitric Acid ◽  
General Formula ◽  
Free Acid ◽  
The Other ◽  
Image Position

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

Extremely undecidable sentences

1982 ◽  
Vol 47 (1) ◽  
pp. 191-196 ◽  
Author(s):  
George Boolos
Keyword(s):  
Modal Logic ◽  
The Other ◽  
Propositional Modal Logic ◽  
Image Position

Let ‘ϕ’, ‘χ’, and ‘ψ’ be variables ranging over functions from the sentence letters P0, P1, … Pn, … of (propositional) modal logic to sentences of P(eano) Arithmetic), and for each sentence A of modal logic, inductively define Aϕ by[and similarly for other nonmodal propositional connectives]; andwhere Bew(x) is the standard provability predicate for PA and ⌈F⌉ is the PA numeral for the Gödel number of the formula F of PA. Then for any ϕ, (−□⊥)ϕ = −Bew(⌈⊥⌉), which is the consistency assertion for PA; a sentence S is undecidable in PA iff both and , where ϕ(p0) = S. If ψ(p0) is the undecidable sentence constructed by Gödel, then ⊬PA (−□⊥→ −□p0 & − □ − p0)ψ and ⊢PA(P0 ↔ −□⊥)ψ. However, if ψ(p0) is the undecidable sentence constructed by Rosser, then the situation is the other way around: ⊬PA(P0 ↔ −□⊥)ψ and ⊢PA (−□⊥→ −□−p0 & −□−p0)ψ. We call a sentence S of PA extremely undecidable if for all modal sentences A containing no sentence letter other than p0, if for some ψ, ⊬PAAψ, then ⊬PAAϕ, where ϕ(p0) = S. (So, roughly speaking, a sentence is extremely undecidable if it can be proved to have only those modal-logically characterizable properties that every sentence can be proved to have.) Thus extremely undecidable sentences are undecidable, but neither the Godel nor the Rosser sentence is extremely undecidable. It will follow at once from the main theorem of this paper that there are infinitely many inequivalent extremely undecidable sentences.

On functional Nörlund methods

1970 ◽  
Vol 67 (1) ◽  
pp. 47-60 ◽  
Author(s):  
B. Choudhary
Keyword(s):  
The Other ◽  
Integral Transformations ◽  
Other Hand ◽  
Nörlund Means ◽  
The One ◽  
Image Position

Integral transformations analogous to the Nörlund means have been introduced and investigated by Kuttner, Knopp and Vanderburg(6), (5), (4). It is known that with any regular Nörlund mean (N, p) there is associated a functionregular for |z| < 1, and if we have two Nörlund means (N, p) and (N, r), where (N, pr is regular, while the function is regular for |z| ≤ 1 and different) from zero at z = 1, then q(z) = r(z)p(z) belongs to a regular Nörlund mean (N, q). Concerning Nörlund means Peyerimhoff(7) and Miesner (3) have recently obtained the relation between the convergence fields of the Nörlund means (N, p) and (N, r) on the one hand and the convergence field of the Nörlund mean (N, q) on the other hand.

scholarly journals Distribution of rational points on the real line

1973 ◽  
Vol 15 (2) ◽  
pp. 243-256 ◽  
Author(s):  
T. K. Sheng
Keyword(s):  
Algebraic Number ◽  
Rational Number ◽  
Real Line ◽  
Rational Points ◽  
The Other ◽  
Liouville Numbers ◽  
The Real ◽  
Other Hand ◽  
Image Position

It is well known that no rational number is approximable to order higher than 1. Roth [3] showed that an algebraic number is not approximable to order greater than 2. On the other hand it is easy to construct numbers, the Liouville numbers, which are approximable to any order (see [2], p. 162). We are led to the question, “Let Nn(α, β) denote the number of distinct rational points with denominators ≦ n contained in an interval (α, β). What is the behaviour of Nn(α, + 1/n) as α varies on the real line?” We shall prove that and that there are “compressions” and “rarefactions” of rational points on the real line.

3. Laboratory Notes by Professor Tait (a.) Measurement of the Potential, required to produce Sparks of various lengths, in Air at different pressures, by a Holtz Machine

1878 ◽  
Vol 9 ◽  
pp. 332-333
Author(s):  
Messrs Macfarlane ◽  
Paton
Keyword(s):  
General Result ◽  
The Other ◽  
Considerable Distance ◽  
Image Position

The general result of these strictly preliminary experiments appears to show that for sparks not exceeding a decimetre in length (L), taken in air at different pressures (P), between two metal balls of 7mm·5 radius, the requisite potential (V), is expressed by the formulaThe Holtz machine employed is a double one, made by Ruhmkorff, and it was used with its small Leyden jars attached. The measurements had to be made with a divided-ring electrometer, so that two insulated balls, at a considerable distance from one another, were connected, one with the machine, the other with the electrometer.

scholarly journals Certain Mathematical Instruments for Graphically Indicating the Direction of Refracted and Reflected Light Rays

1906 ◽  
Vol 25 (2) ◽  
pp. 806-812
Author(s):  
J.R. Milne
Keyword(s):  
Optical System ◽  
The Other ◽  
Reflected Light ◽  
Light Rays ◽  
Image Position

The refraction equation sin i == μ sin r, though simple in itself, is apt to give rise, in problems connected with refraction, to formulæ too involved for arithmetical computation. In such cases it may be necessary to trace the course through the optical system in question of a certain number of arbitrarily chosen rays, and thence to find the course of the other rays by interpolation. Thelinkage about to be described affords a rapid and accurate means of determining the paths of the rays through any optical system.

Note on a Theorem regarding a Series of Convergents to the Roots of a Number

1893 ◽  
Vol 19 ◽  
pp. 15-19
Author(s):  
Thomas Muir
Keyword(s):  
The Other ◽  
Image Position

If the positive integral powers of be taken, and the expansion of each be separated into two parts, rational and irrational, thus—then the ratio of the rational portion to the coefficient of in the other portion is approximately equal to , the convergence being perfect when the power of the binomial is infinite. This is the simplest case of a theorem discovered by the late Dr Sang, and enunciated by him as the result of a process of induction in his paper “On the Extension of Brouncker's Method to the Comparison of several Magnitudes” (Proc. Roy. Soc. Edin., vol. xviii. p. 341, 1890–91).

On the Opacity of Aluminium Foil to Ions from a Flame

1906 ◽  
Vol 25 (2) ◽  
pp. 903-907
Author(s):  
George A. Carse
Keyword(s):  
High Potential ◽  
The Other ◽  
Metallic Plate ◽  
Aluminium Foil ◽  
Image Position

The apparatus used to show that aluminium was opaque to the ions from a flame consisted (see fig. 1) of a tube A, which wasfunnel-shaped at one end, the other end of which led to an enclosure B, one side of which (a b) was made of aluminium foil connected to earth, and on the other side there was an insulated metallic plate (c d), which could either be connected to earth or to a high potential battery. On the other side of the aluminium foil there was an electroscope in a closed case C. The electroscope was of the type used by C. T. R. Wilson in his experiments on the natural ionisation of air in closed vessels.

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