228 Mr. J. W. Peck on the Steady 



series from large lateral emission of heat to complete thermal 

 insulation at the surface, is more clearly seen. 



Fourier of course recognizes the approximate nature of his 

 solution when he states that as a consequence of the small 

 cross-section of the rod the temperature may be supposed the 

 same over any such section. But though this is one condition 

 of the approximation, it is also necessary to take account of 

 the relative values of the conductivity and the emissivity. It 

 will be shown below that for a rod of circular cross-section 

 of radius a the criterion of the approximation is that ea/2k 

 (a quantity of zero dimensions) should be small. Fourier 

 takes account of the a part but makes no mention of the value 

 of e/k and its relationships to the other linear dimensions of the 

 rod. In his time there were no exact measurements of emis- 

 sivity, although a rough appreciation of the smallness of e 

 relative to Jc, for metals must be evident to any observer. This 

 consideration is probably implicitly recognized when the 

 temperature is assumed the same over any cross-section. As 

 to the isothermals and tubes of flow, the difficulty would only 

 arise with the vogue of graphical methods. It seems desirable, 

 however, to recognize the complete criteria of applicability 

 of the solution, especially in view of the fact that Despretz * 

 applied the result to rods of marble, porcelain, sandstone, 

 firebrick, pine-wood; Helmersenf to quartz, granite, marble, 

 serpentine, mica, calcspar ; von Littrow J to a variety of badly 

 conducting substances ; while Wiedemann and Franz §, going 

 to the other extreme, used lengths of rods which gave for 

 good conductors a temperature drop in arithmetical progression. 

 It will be shown that two important conditions must be satisfied 

 if the result 



,=Vexp(-^/g) 

 and its well-known dependent formulae 



i; 2 h 2 ' », ' h i g(n 1+ 4/v=r)' w 



are to be valid ; or if the experimental methods of Ingenhausz, 

 Despretz, Wiedemann, and Franz are to be applicable. 



* Ann. de Chimie et de Physique (1) xxxvi. p. 422 (1827) ; Comptes 

 Pendus, xxxv. p. 540 (1852). 



t Pogg. Ann. lxxxviii. p. 461 (1853). 



X Wien. Sitzungsberiehte (2) lxxi. p. 99 (1875). 



§ Pogg. Ann. lxxxix. p. 497 (1853) ; Ami. de Chimie et de Physique, 

 xli. p. 107 (1854). 



