OF ARTS AND SCIENCES. 137 



The solution for one dimension "^ ,^ = , as well as that for 



two, admits of experimental verification. 



Let a long rod, covered with non-conducting material, except at the 

 ends, be heated at the points A and B. Then for A : f(i\) = Jc^r^ -\~ 

 h^. For B: f{r.,) = k.^r^ -\- K'^ ^i ^"*^^ ^2 l^eing the distances from A 

 and B measured on the rod. 



Let k^ = ^2 fv =z k^(7\ -\- r.-^) -f- ;.. 



/(v) between A and B is constant, and may be represented by a 

 straight horizontal line. Whenever /(y) has a constant value, v has a 

 constant value, and conversely. Therefore, v, as well as/(v), must be 

 represented between A and B by a straight line. If A and B are not 

 heated exactly alike, the line will still be straight, but inclined to the 

 horizontal. 



We have not yet tried the experiment, but intend to do so, that 

 there may be the greatest possible assurance in regard to the existence 

 of f(v), seeing that it exists in every case which can be experimentally 

 tested. 



The second portion of our work is the determination of f(v). For 



a rod heated to the final state, it must satisfy the condition — = 0, 



or /(f) = Ax -\- B. 



For a plate: A-r- + '~pr~ ^= 0' or/(v) = C-\- D log r (chang- 

 ing to polar co-ordinates). 



Forasolid: -^ + --^ + ___ = 0, or/(.) = ^ + -. 



This function cannot be v itself. For, in the case of a rod, heated at 

 one point, let/y = v. 



V = Ar 4- B. r ■ 



' 1 since by physical conditions 



When r = 1 v :=z A -\- B =^ co J the temperature must here 

 „ r = v = Bdp^ pe finite. 



„ r =. CO v = Acc-\-B=^0, which is impossible. 



For a plate 



f(v) =v= G^ Dlogr 



When r = I v = A =^ co 



„ r = V =: C -\- ( — 00 ), which again 



is contrary to the physical conditions. 



