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The first part of the thesis compares Roth's method with other methods, in particular the method of separation of variables and the finite cosine transform method, for solving certain elliptic partial differential equations arising in practice. In particular we consider the solution of steady state problems associated with insulated conductors in rectangular slots. Roth's method has two main disadvantages namely the slow rate of convergence of the double Fourier series and the restrictive form of the allowable boundary conditions. A combined Roth-separation of variables method is derived to remove the restrictions on the form of the boundary conditions and various Chebyshev approximations are used to try to improve the rate of convergence of the series. All the techniques are then applied to the Neumann problem arising from balanced rectangular windings in a transformer window. Roth's method is then extended to deal with problems other than those resulting from static fields. First we consider a rectangular insulated conductor in a rectangular slot when the current is varying sinusoidally with time. An approximate method is also developed and compared with the exact method. The approximation is then used to consider the problem of an insulated conductor in a slot facing an air gap. We also consider the exact method applied to the determination of the eddy-current loss produced in an isolated rectangular conductor by a transverse magnetic field varying sinusoidally with time. The results obtained using Roth's method are critically compared with those obtained by other authors using different methods. The final part of the thesis investigates further the application of Chebyshdev methods to the solution of elliptic partial differential equations; an area where Chebyshev approximations have rarely been used. A poisson equation with a polynomial term is treated first followed by a slot problem in cylindrical geometry.

Benzyl methyl ether rearranges to l-phenyl-ethanol, under the influence of Lewis bases, such as phenyl lithium. This is one of the saturated electrophilic rearrangements which are named after Wittig. An investigation of the ether rearrangement was undertaken using optically active compounds, because the mechanism of the closely allied Stevens rearrangement has been successfully studied in this way. The thesis describes the preparation of benzyl, l-phenyl-propan-l-yl, ether and its successful rearrangement to 1,2-diphenylbutan-l-ol. The preparation and rearrangement of the optically active ether then follows. The results support the propositions of earlier workers, but some aspects of the mechanism remain obscure - especially the movement of the metal atom. An alternative approach to the problem is therefore suggested. In the previous reaction, an optically active group moved on to an inactive centre and its route was deduced. In the proposed experiment an optically inactive group would move onto an asymmetric carbon atom, thus showing the route of the metal atom. The complete mechanism of the reaction would then be demonstrated. Several attempts were made to allow ethers of this type to rearrange, but none were successful. In the second section, the resolution of l-phenyl-propan-l-ol (the ether required for the preparation of benzyl-,l-phenyl-propyl ether) is described. l-phenyl-propyl hydrogen phthalate was first prepared and then resolved by means of its quinidine and strychnine salts. On decomposition of the salts, and hydrolysis of the resulting (+)- and (-)-hydrogen phthalates, samples of ( + )- and (-)-l-phenylpropan-l-ol were obtained with the same rotatory power as those obtained by Pickard and Kenyon, by another method, nearly 50 years ago. In. the third section, the Williamson synthesis of ethers is discussed. Previous experiments by other workers had suggested that ethers prepared by this synthesis from optically pure alcohols should be optically pure, and some experiments described in the section support this supposition. However, other experiments have yielded anomalous results, and these are discussed.