本文將要介紹正交曲線座標中的長球體座標以及與其類似的扁球體座標，並推導出這兩種座標的量度係數、比例因數以及Laplace方程式，然後用這兩種座標係統以及邊界條件，來求得在溫度場中的基質與內含物的溫度場的一般解，然後逐漸增加內含物的覆蓋層，求其溫度場一般解的遞迴關係，接著再將兩座標所求得的溫度場一般解相對照，並找出其異同。

In this thesis we first make a description of two curvilinear orthogonal coordinates: prolate and oblate spheroidal coordinates. We review the detailed derivation procedures for the metric coefficients, scale factors and the governing equation for the Laplace equation. By using these coordinates and boundary conditions, the general solution of steady state temperature field for a spheroidal inclusion in an infinite matrix is examined. The same boundary value problem is also considered for a multicoated spheroidal inclusion in an unbounded matrix. We find the recursion relation for the general solution by incorporating the effects of multiple coatings with different constituent properties and volume fraction. The general solutions for these two different coordinates are compared. Finally, we propose a concept of neutral inclusion for modeling the imperfect interface between the inclusion and the matrix.

Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972.

Arfken, G., Mathematical Methods for Physicists, Harcourt/Academic Press, San Diego, 2001.

Benveniste Y. and Miloh T., The Effective Conductivity of Composites with Imperfect Thermal Contact at Constituent Interfaces, Int. J. Engng Sci. 24, pp.1537-1552, 1986.

Dunn M. L. and Taya M., The Effective Conductivity of Composites with Coated Reinforcement and The Application to Imperfect Interface, J. Appl. Phys. 73 pp.1711-1722, 1992.

Duschlbauer D., Pettermann H. E. and Bohm H. J., Heat Conduction of a Spheroidal in Homogeneity with Imperfectly Bonded Interface, J. Appl. Phys. 94 pp.1539-1549, 2003.

Hatta H. and Taya M., Thermal Conductivity of Coated Filler Composites, J. Appl. Phys. 59, pp.1851-1860, 1986.

Hildebrand F. B., Advanced Calculus for Applications, Prentice-Hall, Englewood Cliffs, N.J., 1976.

Hobson E. W., The Theory of Spherical and Ellipsoidal Harmonics, Chelsea Pub. Co., New York, 1931.

J. C. Maxwell, Treatise on Electricity and Magnetism. 3 rd Edition, Oxford University Press, Oxford, 1904.

Lamb H., Hydrodynamics, The University press, Cambridge [Eng.], 1932.

L. M. Jiji, Heat Transfer Essentials: A Textbook, Begell House, Inc. New York, 1998.

Moon, P. and Spencer, D. E., Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions, Springer-Verlag, New York, 1988.

Morse, P. M. and Feshbach H., Methods of Theoretical Physics, McGraw-Hill, New York, 1953.

Ozisik, M., Boundary Value Problems of Heat Conduction, International Textbook Co., Scranton, 1968.

Saada, Adel S., Elasticity theory and applications, Malabar, Fla, Krieger, 1993.

Williams M M R, Thermophretic Forces Acting on a Spheroid, J. Phys. D: Appl. Phys.19 pp.1631-1642, 1986

邱益成，複合橢圓柱之扭轉與傳導問題，國立成功大學土木工程研究所論文，2003。