230 Mr. J. W." Peck on the Steady 



giving 



•=±\/: 



£• • • • • • ■ M 



If we put this value of X in the solution, take J (\r) = l, 

 and choose the negative sign so as to make v finite when x is 

 infinite, we get 



,=Vex P (-*V£). . . . • (I5) 



as a first approximation. This, result is the same as the 

 Fourier solution. 



The condition of this approximation is that \a (and hence 



\ 2 a 2 

 \r) should be so small that — j- and higher powers may be 



neglected. By (14) we therefore must have %!. ov » : ~ sma ^- 



From this, or from a consideration of the dimensions of the 

 quantities, it is evident that the ratio kje has the dimensions 

 of a length. This length has of course no connexion with 

 the dimensions of any particular piece of the substance under 

 consideration, but is a quantity characterizing the thermal 

 properties of the material so far as conduction and radiation 

 are concerned. To this length a definite physical meaning- 

 may be given which is of importance in the problem under 

 consideration. In the first place let a large homogeneous 

 slab of the substance have its parallel plane faces maintained 

 permanently at unit temperature difference. Steady flow of 

 heat will take place along straight lines perpendicular to 

 these faces, and the temperature gradient will be uniform, 

 i. e. the decreasing temperatures form, for equal space steps, 

 an arithmetical progression. Let the quantity of heat that 

 passes per sq. cm. per sec. across any isothermal (a plane) be 

 noted. In the second place, let heat radiate from the surface 

 of the substance and let definite conditions as to the coating 

 of the surface, the pressure and nature of the surrounding 

 medium be specified. Let also the temperature difference 

 between the surface (supposed an isothermal if more than an 

 elementary part is considered) and the medium be unity, 

 i. e. the same as that between the faces of the slab in the first 

 case. If now the thickness of the slab in the first case be so 

 adjusted that the quantity of heat conducted per sq. cm. per sec. 

 in the first case is the same as that radiated in the second case, 

 then that particular thickness which gives this equality is the 

 length kje. This can be easily proved by a consideration of 



