60 THEORY OF HEAT. [CHAP. I. 



much more remote from the source when the bar is thicker, all 

 other conditions remaining the same. We can always raise by 

 one degree the temperature of one end of a bar of iron, by heating 

 the solid at the other end ; we need only give the radius of the 

 base a sufficient length : which is, we may say, evident, and 

 of which besides a proof will be found in the solution of the 

 problem (Art. 78). 



76. The integral of the preceding equation is 



A and B being two arbitrary constants ; now, if we suppose the 

 distance x infinite, the value of the temperature v must be 



75 



+x * 



infinitely small; hence the term Be +x * w does not exist in the in- 



/2k 



tegral : thus the equation v = Ae~* ^ u represents the permanent 

 state of the solid ; the temperature at the origin is denoted by 

 the constant A t since that is the value of v when x is zero. 



This law according to which the temperatures decrease 

 is the same as that given by experiment ; several physicists 

 have observed the fixed temperatures at different points of a 

 metal bar exposed at its extremity to the constant action of a 

 source of heat, and they have ascertained that the distances 

 from the origin represent logarithms, and the temperatures the 

 corresponding numbers. 



77. The numerical value of the constant quotient of two con 

 secutive temperatures being determined by observation, we easily 



deduce the value of the ratio -; for, denoting by v lt v a the tem 

 peratures corresponding to the distances x^ x 2 , we have 



v ~{*i-*tk/s -i /2h log v loof v 9 ,, 



~* = e v **, whence A / --=- = & 1 * Jl. 



v A/ k x x 



As for the separate values of li and k, they cannot be deter 

 mined by experiments of this kind : we must observe also the 

 varying motion of heat. 



78. Suppose two bars of the same material and different 

 dimensions to be submitted at their extremities to the same tern- 



