Conductors by Electric Currents. 263 



diculars from a point on the three sides, then the solution may- 

 be written in the form 



3 = jq 2 ct/3y/3cka. 



It is interesting to extend two of these results — that for the 

 bar or wire of circular cross section and that for the plate or 

 approximately for any long flat strip — to suit the case where 

 the surfaces having an emissivity e are exposed in air at tem- 

 perature zero, instead of being maintained at that temperature. 

 It is only necessary to raise the temperature by such an amount 

 that the loss by emission will be equal to the rate of generation 

 of heat by the current ; and therefore for the circular wire 

 the solution will be 



$ = j q 2 {a 2 -x 2 -y 2 + 2ak/e} feck, 

 and for the flat strip, 



3 = jq 2 { a 2 - x 2 + 2ak/e } /2c*. 



In both these cases the ratio of the maximum temperature to 

 the surface-temperature is l + ae/2k. 



fi. To find the steady distribution of temperature in an 

 annular conductor in which a current circulates, the surface 

 being maintained at temperature zero. 



Taking semipolar coordinates «r, <f>, z, the equation (D) 

 reduces to ^^ = Q> 



and therefore, if q denote the current per unit area at unit 

 distance from the axis (that of z), (E) reduces to 



d'us 2 w dm dz 2 ckiz 2 



and therefore we may take 



Jf 



• -c-fcflog.fi 



where a is any solution of 



d 2 cr 1 da d?G . 

 din 2 is dvr dz 2 



and this will be the solution for the ring formed by the 

 revolution of the curve 



a =j<f\logvr\ 2 l2ck 

 about the axis of z. 

 For an example, take 



U2 



