SECT. V.] STEADY TEMPERATURE IX A BAR. 61 



perature A ; let l t be the side of a section in the first bar, and 1 2 

 iii the second, we shall have, to express the temperatures of these 

 two solids, the equations 



Vl = Ae~ 1 and v 9 =Ae~ 



i\, in the first solid, denoting the temperature of a section made 

 at distance x lf and v z , in the second solid, the temperature of a 

 section made at distance x z . 



When these two bars have arrived at a fixed state, the tem 

 perature of a section of the first, at a certain distance from the 

 source, will not be equal to the temperature of a section of the 

 second at the same distance from the focus ; in order that the 

 fixed temperatures may be equal, the distances must be different. 

 If we wish to compare with each other the distances x^ and x&amp;lt; 2 

 from the origin up to the points which in the two bars attain 

 the same temperature, we must equate the second members of 



these equations, and from them we conclude that -\ = j. Thus 



x z 2 



the distances in question are to each other as the square roots of 

 the thicknesses. 



79. If two metal bars of equal dimensions, but formed of 

 different substances, are covered with the same coating, which 

 gives them the same external conducibility 1 , and if they are 

 submitted at their extremities to the same temperature, heat will 

 be propagated most easily and to the greatest distance from the 

 origin in that which has the greatest conducibility. To compare 

 with each other the distances x l and x z from the common origin 

 up to the points which acquire the same fixed temperature, we 

 must, after denoting the respective conducibilities of the two 

 substances by k^ and & 2 , write the equation 



/** /^ -r 2 I* 



e -W^ = e -W w f w hence ^ = p . 



x * k 2 



Thus the ratio of the two conducibilities is that of the squares 

 of the distances from the common origin to the points which 

 attain the same fixed temperature. 



1 Ingenhousz (1789), Sur les mgtaux comme conducteurs de la chalenr. Journal 

 de Physique, xxxiv., 68, 380. Gren s Journal der Physik, Bd. I.&quot; [A. F.] 



