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In taking the limits from t= -o to t = +00, or from M = - 1 to u = + 1, the part under the sign of integration vanishes, in consequence of the factor 1 -?; hence on integrating with respect to w from 0 to 20 we find that the first term of the right hand side of the equation is equal to zero. In the same way, on effecting the integration with respect to w of the second term of the right hand side of the equation, we find it to become

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which vanishes on taking it between the limits w = 0 and w = 27, because Ym and 2, are supposed to be rational and integral functions of sin w and cos w. Hence on integrating with respect to t and taking it between the limits t = -00 and t = + co, or = - 1 and M = + 1, the second term of the right hand side also vanishes ; therefore

{m (m + 1) - n(n + 1)} $1' du f.** dw Y, Z, = 0.

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The functions Y. and 2, are known by the name of Laplace's Functions, that mathematician baving been the first who studied their properties and pointed out their utility in the calculation of attractions. For the investigation of other remarkable theorems relating to these functions the reader is referred to the Mécanique Céleste, Liv. u., or to O'Brien's Mathematical Tracts. Mr Murphy has applied to the treatment of these functions a new and very remarkable analysis, wbich will be found in the introduction to his Elementary Principles of the Theory of Electricity.

THE END.

Works published by J. & J. J. Deighton.

Gregory (D. F.) Treatise on the Application of Analysis

to Solid Geometry. Commenced by D. F. Gregory, M.A., late Fellow and Assistant Tutor of Trinity College, Cambridge; concluded by W. WALTON, M.A., Trinity College Cambridge. 8vo. cloth, 10s. 6d.

Walton (William). Treatise on the Differential Cal

culus. 8vo. cloth, 10s. 6d.

Walton (William). A Collection of Problems in illus

tration of the Principles of Theoretical Mechanics. 8vo. cloth, 16s.

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