138 PROCEEDINGS OF THE AMERICAN ACADEMY 



For a solid 



f(y)=v = E-\--^ 



p 



When r z= (x> v = ,; U = ,; v =. — 



r 



^ ..... 



„ r = V = — := cc , wliich agaiu is impossible. 



The same thing may be proved as follows. Assume, for the sake of 



argument, that the flux is — c — tor the direction x, — c — for y, 



dx dy 



— c-f- for z. Then for all homosfeneous bodies, 

 dz ® ' 



dl— \dx^ "T" dy^ + dz-j' 



This is the only condition that a function v must satisfy, in order to 

 represent an actual possible case. Any function that is a particular 

 solution of this equation may represent an actual distribution of heat. 

 First. \i V :=! (jp(a:, y,z, t) represent the temperature throughout a 

 body, it cannot have any (rue maxima and minima for x, y, z. This 



is evident, since the conditions of a maximum or minimum are -j~ =. 0, 



■4- :=: 0, -^ = 0, which by hypothesis cannot be zero without mak- 

 ing the flux = 0. No point can be hotter or colder than the points 

 around it, and there not be a flux to or from the point. The physical 

 conditions forbid the mathematical condition of the existence of maxi- 

 ma and minima. There must be points hotter than points around 

 them, and therefore they must be shooting points, and not maxima or 

 minima. 



Secondly. If Vj and v.^ are two particular solutions of the Partial 

 Differential E(iuation, their sum is also a solution, and therefore cor- 

 responds to an actual distribution of heat, in which the temperature of 

 any point is equal to the sum of the temperatures, which it would have 

 under the conditions represented by v^ and v.^. It will now be easy to 

 show 



Thirdly. The points of hottest temperature, when the solution is v, 

 fall in exactly the same places in the body as the liottest points of the 

 two solutions v^ and v^ taken jointly. That is to say : if there are n^ 

 points of highest temperature when the solution is f j, and n„ when the 

 solution is v^, there will be n^ -\- n.^ points of highest temperature when 

 the solution is v^^ -\- v.^ = v, unless v^ and v.^ have some hot points in 



