scholarly journals On the determination of the terms in the disturbing function of the fourth order, as regards the eccentricities and inclinations which give rise to secular inequali­ties

1837 ◽  
Vol 3 ◽  
pp. 301-301
Keyword(s):  
Fourth Order ◽  
A Priori ◽  
Disturbing Function

The author observes, that the magnitude of the terms of the fourth order in the disturbing function, relating to the inclinations, in the theory of the secular inequalities of the planets, does not admit of being estimated à priori ; and consequently the amount of error which may arise from neglecting them cannot be appreciated. The object of the present investigation is to ascertain the analytical ex­pressions of these terms; and the method adopted for this purpose is derived from principles already explained by the author in a for­mer paper. He has bestowed great pains in putting these expressions into the simplest form of which they are susceptible; and has finally succeeded, after much labour of reduction, in obtaining ex­pressions of remarkable simplicity. He exemplifies their application by the calculation, on this method, of one of the terms given by Pro­fessor Airy as requisite for the determination of the inequality of Venus; and arrives, by this shorter process, at the same result. The same method, he remarks, is, with certain modifications, applicable to the developement of the disturbing function in terms of the true longitude.

scholarly journals IV. On the determination of the terms in the disturbing function of the fourth order as regards the eccentricities and inclinations which give rise to secular inequali­ties

1835 ◽  
Vol 125 ◽  
pp. 57-81
Keyword(s):  
Fourth Order ◽  
A Priori ◽  
Disturbing Function ◽  
Left Hand ◽  
The Third ◽  
Hand Column ◽  
The Subject ◽  
Analytical Expressions

Hitherto in the theory of the secular inequalities the terms in the disturbing function of the fourth order as regards the inclinations have been neglected. As the magnitude of these terms depends, in great measure, upon certain numerical co­efficients, it is impossible to form any precise notion à priori with respect to their amount, and as to the error which may arise from neglecting them. I have therefore thought it desirable to ascertain their analytical expressions; and the details of this calculation form the subject of this paper. Some of the secular inequalities which result from these terms are far within the limits of accuracy which Laplace appears to have contemplated in the third volume of the Mécanique Céleste. The method which I have here adopted for developing the disturbing function rests upon principles which I have already explained. Very little trouble is requisite to obtain certain analytical expressions for the terms upon which the secular inequalities depend, or for any others, in the development of the disturbing function; but it is not so easy to put these expressions in the simplest form of which they are susceptible; and this is a point to which I think hitherto sufficient attention has not been paid. It will be found that I have obtained, finally, expressions of very remarkable sim­plicity; to accomplish this, however, I have been obliged to go through tedious pro­cesses of reduction, the details of which are here subjoined, in order that my results may be verified or corrected without difficulty. In order to give an additional example of the great facility with which terms in the disturbing function are arrived at by my method, I have calculated one of those given by Professor Airy, and which is required in the determination of his inequality of Venus; and I have arrived at the result which he has given. The same method, with certain modifications, is applicable to the de­velopment of the disturbing function in terms of the true longitudes. The terms in the disturbing function which give rise to the secular inequalities of the elliptic constants, when the terms of the order of the fourth powers of the eccentricities and inclinations are retained, and higher powers of those quantities are neglected, are as follows: and I propose, as they form, in fact, a system apart, to distinguish them by the indices given in the left-hand column.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

Fourth Order Derivative Spectrophotometric Determination of Lead with 1,10-Phenanthroline and Rose Bengal

1993 ◽  
Vol 26 (3) ◽  
pp. 523-539 ◽  
Author(s):  
D. Sreevalsan Nair ◽  
T. Prasada Rao ◽  
C. S. P. Iyer ◽  
A. D. Damodaran
Keyword(s):  
Spectrophotometric Determination ◽  
Fourth Order ◽  
Rose Bengal ◽  
Order Derivative

scholarly journals Estimation of the mean normal measure from flat sections

2008 ◽  
Vol 40 (01) ◽  
pp. 31-48
Author(s):  
Markus Kiderlen
Keyword(s):  
A Priori ◽  
The Other ◽  
Consistent Estimator ◽  
Random Set ◽  
Normal Measure ◽  
Mild Assumption ◽  
The Mean ◽  
Vertical Sections ◽  
Almost All

We discuss the determination of the mean normal measure of a stationary random set Z ⊂ ℝ d by taking measurements at the intersections of Z with k-dimensional planes. We show that mean normal measures of sections with vertical planes determine the mean normal measure of Z if k ≥ 3 or if k = 2 and an additional mild assumption holds. The mean normal measures of finitely many flat sections are not sufficient for this purpose. On the other hand, a discrete mean normal measure can be verified (i.e. an a priori guess can be confirmed or discarded) using mean normal measures of intersections with m suitably chosen planes when m ≥ ⌊d / k⌋ + 1. This even holds for almost all m-tuples of k-dimensional planes are viable for verification. A consistent estimator for the mean normal measure of Z, based on stereological measurements in vertical sections, is also presented.

scholarly journals DETERMINATION OF ALKALINITY AND DISSOCIATION CONSTANTS OF HIGH SALINITY WATERS: USE OF F5BC TITRATION FUNCTION

2018 ◽  
Vol 35 (2) ◽  
pp. 79
Author(s):  
Bernadete F. Cavalcanti ◽  
Lourdes Cristina Lucena Agostinho ◽  
Luciano Nascimento
Keyword(s):  
Natural Waters ◽  
Dissociation Constants ◽  
A Priori ◽  
Linear Regression Analysis ◽  
Liquid Junction ◽  
Carbonate System ◽  
Ph Measurements ◽  
Apparent Dissociation Constant ◽  
Junction Potential

Measurements of parameters expressed in terms of carbonic species such as Alkalinity and Acidity of saline waters do not analyze the influence of external parameters to the titration such as Total free and associated Carbonic Species Concentration, activity coefficient, ion pairing formation and Residual Liquid Junction Potential in pH measurements. This paper shows the development of F5BC titration function based on the titrations developed by Gran (1952) for the carbonate system of natural waters. For practical use, samples of saline watersfrom Pocinhos reservoir in Paraiba were submitted to titration and linear regression analysis. Results showed that F5BC involves F1x and F2x Gran functions determination, respectively, for Alkalinity and Acidity calculations without knowing “a priori” the endpoint of the titration. F5BC also allows the determination of the First and Second Apparent Dissociation Constant of the carbonate system of saline and high ionic strength waters.

scholarly journals Effect of material properties on optimal radial strain gage locations in sharp V-notched configurations

2018 ◽  
Vol 172 ◽  
pp. 03002 ◽  
Author(s):  
Pranjol Paul ◽  
K.S.R. Krishna Murthy ◽  
Debabrata Chakraborty
Keyword(s):  
Strain Gage ◽  
Material Properties ◽  
A Priori ◽  
Radial Strain ◽  
Numerical Approach ◽  
Conference Paper ◽  
Notch Angle ◽  
Notch Stress ◽  
Notch Length

A simple yet reliable and powerful methodology using only one strain gage has been recommended for appropriate determination of notch stress intensity factor (NSIF) for sharp V-notched configurations subjected to mode I condition. The methodology is supported by strong theoretical postulates, and it permits the gage to be pasted prominently apart from tip of the notch thus avoiding various problems associated with singularities. Unlike the conventional methodologies, the recommended strain gage methodology also proposes optimal radial strain gage locations which are beneficial in appropriate determination of NSIFs. A FEM based numerical approach is adopted for obtaining optimal radial gage locations a priori for the aforesaid configuration. The optimal radial gage locations are observed to be influenced by parameters viz. the notch angle, the ratio of notch length to width of the plate and also material properties. Results were already published by the authors to establish that the optimal radial gage locations are influenced by the notch angle and the ratio of notch length to width of the plate. In this conference paper, a case is studied with a completely different material to check whether material properties influence the graphical trends of results or not.

scholarly journals A new lidar inversion method using a surface reference target applied to the backscattering coefficient and lidar ratio retrievals of a fog-oil plume at short range

2020 ◽  
Vol 13 (4) ◽  
pp. 1921-1935
Author(s):  
Florian Gaudfrin ◽  
Olivier Pujol ◽  
Romain Ceolato ◽  
Guillaume Huss ◽  
Nicolas Riviere
Keyword(s):  
Short Range ◽  
A Priori ◽  
Inversion Method ◽  
Backscattering Coefficient ◽  
Reference Target ◽  
Lidar Measurements ◽  
Lidar Ratio ◽  
Aerosol Plume ◽  
Good Agreement

Abstract. In this paper, a new elastic lidar inversion equation is presented. It is based on the backscattering signal from a surface reference target (SRT) rather than that from a volumetric layer of reference (Rayleigh molecular scatterer) as is usually done. The method presented can be used when the optical properties of such a layer are not available, e.g., in the case of airborne elastic lidar measurements or when the lidar–target line is horizontal Also, a new algorithm is described to retrieve the lidar ratio and the backscattering coefficient of an aerosol plume without any a priori assumptions about the plume. In addition, our algorithm allows a determination of the instrumental constant. This algorithm is theoretically tested, viz. by means of simulated lidar profiles and then using real measurements. Good agreement with available data in the literature has been found.

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