PEIRCE AND WILLSON. — THERMAL CONDUCTIVITIES. 11 



(4) The radius of the hase of a riglit cylinder of revolution of length 

 Ms a. A function, V, harmonic within this cylinder, has the constant 

 value Fq on one (the lower) base, the constant value F, on the upper 

 base, and the conslant value Fon the convex surface. If, then, the axis 

 of the cylinder be used as axis of z with origin at the centre of the lower 

 base, Fis given by the equation 



• x^ . /i {Xp) . sinh ( '^-~ \ 



(24) 



where J^ and J^ represent Bessel's P'unctiou of the zeroth and first 

 order, respectively, and x^, is the jt»th root in order of magnitude of the 

 ecjuation J^, (./) = 0. The first ten values of x for which the Bessel's 

 Fuiictiuu of the zeroth order vanishes have been given by Meissel.* 

 We have computed the next thirty values of the xj?, by the aid of 

 Stokes's Formula,! and the values of the Bessel's Function of the first 

 order corresponding to these forty a;^'s either from the series which usu- 

 ally defines J^ (x) or from the semi-convergent series. This computation 

 was done by means of Vega's ten place table of logarithms,^ except in 

 the few cases where a greater number of places was necessary, and for 

 these we had recourse to Thoman's tables.§ All the values have been 

 checked by duplicate computation, and the first four values of Ji (x) by 

 comparison with Meissel's tables. The results of this work appear in 

 Table I. Table II. gives to seven places of decimals the values of the 

 Xj's from p = 4\ to /) = 65. The values of Kon the axis of the cylin- 

 der depend upon the corresponding values of the function 



S 





* Meissel, Matli. Abliandhiiigen der k. Akad. der Wissenscliaften zu Berlin, 

 1888. 



t Stokes, Camb. Phil. Trans., IX. Lommel, Studien iiber die Bessel'schen 

 Functionen, Leipzig, 1868. Hayleigb, The Tiieory of Sound, London, 1878. 

 Byerly, Treatise on Fourier's Series, etc., Boston, 1893. Gray and Matiiews, Be»- 

 sel Functions and their Applications to IMiysics. 



X Vega, Thesaurus Logarithniorum Completus, Lipsiae, 1794. 



§ Thoman, Tables de Logarithmes a 27 De'cimales pourles Calculs de Precision, 

 Paris, 1867. 



