Conductors by Electric Currents. 265 



potential f^O . <£) and temperature f 2 (6 . <£), then, as Poisson 

 has shown, 



1 ( V-f»)/ 1 (fl / .Q*B 

 V ~ 4™ J ~ K 3 



where the integration extends over the surface of the sphere, 

 and 



W = r 2 — 2ar{cos 6 cos 0' + sin 0sin 6' cos ((/>- <£')} + d 2 . 



Therefore, further, as the surface-value of U will be ijcfi 2 + kf 2 , 

 we shall have 



TT __L C (* 2 -r 2 )(hVi 2 + ¥2)d& 

 U ~47raJ R 3 



and finally 



U = U-i/cV 2 , 



where U and V have the values just written. 



When, however, there is radiation from any part of the 

 bounding surface, the ordinary results cannot be so directly 

 generalized ; for the boundary equation in U is not then of 

 quite the same form as that in 3, being, when there is no flux 

 of electricity across the surface, 



-k d £-e(U-iJeY*), 



where e is the emissivity, and v the normal reckoned out- 

 wards. 



The equation (E) may of course be reduced to the form of 

 (F) by other methods or be discussed without such reduction, 

 but the transformation chosen has the recommendation of 

 being furnished by the electrical conditions and is specially 

 related, as we shall see immediately, to an important class of 

 cases of the steady state. 



4. The Steady State : a Special Solution. 



For the case of the ultimate steady distribution, (F) re- 

 duces to y2TJ = o 



and since (D) is an equation of the same form, it follows that 

 a solution is given by TT . ^^ T 



hence, determining the constants suitably, we obtain the 

 following: theorem : — 



If any two parts of the surface of a conductor, through- 

 out which the conductivities k and c are constant, are main- 



