Ellipsoid of rotation

The grounding resistance of an ellipsoid of rotation in infinite space is [4] ... 

Here Uq is the potential of the gravitational field at the surface of the ellipsoid of rotation and r the distance between the axis of rotation and points of this surface. 

Taking into account the shape of the outer surface of the ellipsoid of rotation, Sq, it is convenient to introduce the system of coordinates, where this surface coincides with one of coordinate surfaces. 

Now we demonstrate the system of coordinates, where the ellipsoids of rotation and hyperboloids of one sheet form two mutually orthogonal coordinate families of surfaces. First, we introduce the Cartesian system at the center of the mass and suppose that semi-axes of the ellipsoid of rotation obey the condition brelation between coordinates of the Cartesian and cylindrical... 

This function is a solution of Laplace s equation regardless of the values of constants, and our goal is to find such of them that the potential satisfies the boundary condition on the surface of the given ellipsoid of rotation and at infinity. In order to solve this problem we have to discuss some features of Legendre s functions. First of all, as was shown in Chapter 1, the Legendre s function of the first kind P (t]) has everywhere finite values and varies within the range... 

In order to solve the boundary value problem we have to satisfy the condition for the potential of the attraction field at the surface of the earth ellipsoid of rotation, which can be written as... 

Thus, the potential of the gravitational field on the ellipsoid of rotation is... 

Thus, we have demonstrated that the potential C/g is a solution of Laplace s equation and satisfies the boundary condition at the surface of the ellipsoid of rotation and at infinity. In other words, we have solved the Dirichlet s boundary value problem and, in accordance with the theorem of uniqueness, there is only one function satisfying all these conditions. 

The latter allows us to determine the potential of the gravitational field outside the ellipsoid of rotation, including points of its surface, provided that the mass, geometry, and angular frequency are given. Making use of Equation (2.148), we can represent the potential of the gravitational field in the form kM 1 1 2 2 SiC/s) - 1... 

Now we arrive at an important result, namely, with an accuracy of the square of the flattening this equation characterizes the ellipsoid of rotation. From the last equation we have... 

As ealeulations show, when the density inereases with a distance from the earth s surface the parameter I is smaller than 0.4. On the contrary, with a decrease of the density toward the earth s center we have 7 >0.4. Inasmuch as in reality 7 <0.4, we conclude that there is essential concentration of mass in the central part of the earth. In other words, the density increases with depth and this happens mainly due to compression caused by layers situated above, as well as a concentration of heavy components. In conclusion, it may be appropriate to notice the following a. In the last three sections, we demonstrated that the normal gravitational field of the earth is caused by masses of the ellipsoid of rotation and its flattening can be determined from measurements of the gravitational field. 

As was pointed out earlier, the determination of the elevation of points of the physical surface of the earth with respect to the geoid is the only the first step in studying the shape of the earth. The next step is the calculation of the distance between corresponding points of the ellipsoid of rotation and the geoid, and this subject will be described in the following section. 

The function T p) is called the disturbing potential and it is very small, T U). Now we focus our attention on the space between the geoid and the ellipsoid of rotation and assume the potential of the gravitational field of the geoid, W p), and the normal potential of the spheroid, U q), are equal ... 

For the ellipsoid of rotation the general constraint equations (23) and (46) take on the particular form... 

Calculations have been made of the surface of several shapes of container per liter of contents over a series of volumes. The shapes chosen were a sphere (the least possible surface per unit volume), a cube (counting all six faces), a cylinder with a height twice its diameter, and a barrel with typical inside measurements (2, 5) of the bilge (maximum) diameter 1.25 times the head diameter and the distance between the heads 1.3 times the bilge diameter. The barrel was calculated as two frustums base to base and also as a truncated ellipsoid of rotation. 

We may note that, for a nonzero internal viscosity, the system of equations for the moments is found to be open the equations for the second-order moments contain the fourth-order moments, etc. This situation is encountered in the theory of the relaxation of the suspension of rigid particles (Pokrovskii 1978). Incidentally, for 7 —> 00, equation (F.28) becomes identical to the relaxation equation for the orientation of infinitely extended ellipsoids of rotation (Pokrovskii 1978, p. 58). 

FegClg. 2HCI.4H2O. P is then the apex of a cone of which ppr and spg are a pair of rectangular sections. If, on the contrary, the fusion is accompanied by partial decomposition, as is the case here, the lines sp and vq are parts of a curve which has a vertical tangent at P, and the whole forms a figure like part of an ellipsoid of rotation or a sphere. In both cases the behaviour at constant tem-... 

For a body that possesses an axis of symmetry, such as an ellipsoid of rotation, we may define... 

The three decay constants appearing in the expression for r(t) for an ellipsoid of rotation have been calculated and are own in Table 13. As may be seen, the three relaxation times diverge rapidly with increasing axial ratio for a prolate ellipsoid. However, for an oblate ellipsoid the deviations are small even for hi axial ratios and experimentally it m prove difficult to resolve more than a single, nean relaxation time for r(t) in this case. Thus, three situatrens exist in which the emission anisotropy may decay exponentially and it is not po le, therefore, to distingui ... 

Of course, in general, the magnetization is not uniform, the particle is not an ellipsoid of rotation, the volume magneto-crystalline anisotropy is not uniaxial nor is it aligned with the particle shape, there are local orientational and magnitude variations of both SR and middle region magneto-crystalline anisotropies, etc. 

All the above alkali cyanides exist in less symmetrical low-temperature forms, in which the CN ion behaves as an ellipsoid of rotation, with long axis about 4-3 A... 

The Weber numbers We of the order of 1 constitute a practically important intermediate region in which the bubble is strongly deformed but still preserves its symmetry with respect to the midsection. For such We, the shape of the bubble is well approximated by an oblate ellipsoid of rotation with semiaxes a and b = xa, where the semiaxis b is perpendicular to the flow and x - 1 ... 

Relation (4.14.1), in particular, holds for spheres of equal radius arranged on the axis of a translational Stokes flow (the velocity distribution for this case is presented in [179,463], It also holds for a three-dimensional Stokes flow past two identical ellipsoids of rotation whose axes are parallel and perpendicular to the undisturbed flow. The direction of the line passing through their centers coincides with the direction of translational flow. 

Other examples are provided by an orbit perpendicular to the direction of the field in the case of the Zeeman effect and, in the case of the problem of two centres ( 39), by one which is confined to the surface of an ellipsoid of rotation, etc. For the purpose of illustration we shall continue to speak of circular orbits, eccentricities,... 

The equation can also be solved for spherical particles diffusing into a hole which has the shape of an ellipsoid of rotation but for most other shapes no exact solution is available. The differential equation for spheres of radius R diffusing into a hole of any given shape is the same as for the electrical capacity C of a closed surface enclosing the hole at a distance R. 

For a prolate ellipsoid of rotation with the axis a b = c the electrical capacity C is ... 

In these expressions and Ty-are characteristic times of the segmental movement of the pol5nneric chains and and Lfarc their form factors into concentrated and diluted solutions respectively. Let us note also, that the Eqs. (16) and (17) are self-coordinated since at P = P the Eq. (16) transforms into the Eq. (17). The form factors and Zy are determined by a fact how much strong the conformational volume of the polymeric chain is strained into the ellipsoid of rotation, flattened or elongated one as it was shown by author [27]. 

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