SECT. III.] LAW OF THE HIGHEST TEMPERATURES. 391 



We may remark also in the first equation that the quantity 



JJH 



fe~* K 9 expresses the permanent temperatures which the 

 different points of the bar would take, if we affected the origin 

 with a fixed temperature /, as may be seen in Chapter I., 

 Article 76. 



In order to represent to ourselves the value of V, we must 

 therefore imagine that all the initial heat which the source con 

 tains is equally distributed through a portion of the bar whose 

 length is b, or the unit of measure. The temperature /, which 

 would result for each point of this portion, is in a manner the 

 mean temperature. If we supposed the layer situated at the 

 origin to be retained at a constant temperature /during an infinite 

 time, all the layers would acquire fixed temperatures whose 



_ Jw 

 general expression is fe K & , denoting by x the distance of the 



layer. These, fixed temperatures represented by the ordinates of 

 a logarithmic curve are extremely small, when the distance is 

 considerable ; they decrease, as is known, very rapidly, according 

 as we remove from the origin. 



Now the equation (8) shews that these fixed temperatures, 

 which are the highest each point can acquire, much exceed the 

 highest temperatures which follow each other during the diffusion 

 of heat. To determine the latter maximum, we must calculate 

 the value of the fixed maximum, multiply it by the constant 



/2jy\i i 

 number ( ^- ) j=- , and divide by the square root of the dis- 



W V^TT 



tance x. 



Thus the highest temperatures follow each other through the 

 whole extent of the line, as the ordinates of a logarithmic curve 

 divided by the square roots of the abscissae, and the movement of 

 the wave is uniform. According to this general law the heat 

 collected at a single point is propagated in direction of the length 

 of the solid. 



391. If we regarded the conducibility of the external surface 

 of the prism as nothing, or if the conducibility K or the thickness 

 2g were supposed infinite, we should obtain very different results. 



