The London, Edinburgh and Dublin philosophical magazine and journal of science . -j = a parameter, 236 Mr. J. W. Peck on the Steady or r2=â -x + c, A-2 (26) where c is a parameter depending upon the temperature ofthe isothermal considered. We have, therefore, a family ofsimilar coaxial paraboloids of revolution. Their concavitiesare towards the hot end, their common axis is the axis of thecylindrical rod, and the distance from focus to vertex (AS) is for each â-. A2 ^ This constant of the family, AS or â, has by (20) the A-2 value . aL</2 or vn i+ 8LJ (27) In the diagram these paraboloidal
The London, Edinburgh and Dublin philosophical magazine and journal of science . -j = a parameter, 236 Mr. J. W. Peck on the Steady or r2=â -x + c, A-2 (26) where c is a parameter depending upon the temperature ofthe isothermal considered. We have, therefore, a family ofsimilar coaxial paraboloids of revolution. Their concavitiesare towards the hot end, their common axis is the axis of thecylindrical rod, and the distance from focus to vertex (AS) is for each â-. A2 ^ This constant of the family, AS or â, has by (20) the A-2 value . aL</2 or vn i+ 8LJ (27) In the diagram these paraboloidal surfaces are shown toscale for the case of a bismuth rod of 4 cms. radius. Closeto the origin this form is deviated from because thereA226_Alx becomes comparable with \i2e~K*x, and at the originwe have the isothermal surface becoming, of course, a greater L is for a series of bars of substances undersimilar thermal conditions the greater is the value of theconstant AS, and therefore the paraboloids approach morenearly to the planes of the first =2, L=60, AS Isothermals y2= â 44r+ of Flow y = Aexf22. 11. Z=50. In this second approximation the four characteristic lengthsare the semi-radius (a/2), the thermal length modulus (L),the distance from focus to vertex of the paraboloidal surface Temperatures of a Thin Rod, 237 (AS or â), and the length o£ the bar /. Similar state-ments as to their ratios must hold as in the first approxi-mation. The lines of flow (shown in the diagram) are given by theconjugate family of logarithmic curves y = A exp (\2x/2). (28) where A is a parameter. The following table gives numerical values for the tem-peratures at the axis and at the surface for bismuth rods ofradii 4 cms. and 1 cm. respectively. For calculation theformula 9 *> is employed, and the results of the preceding table may beused. It will be seen that even for the rod of smaller radiusthe difference between the surface and axial values is quitea
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