510 Prof, Thomson on the Uniform Motion of Heat 



ellipsoid. For this purpose, let us first find the value of — t- 



at any point in the surface of the isothermal ellipsoid whose 

 semiaxes are a, b, c. Now we have from (a) 



where k is constant for any point in the surface of the isothermal 

 ellipsoid under consideration, and determined by the condition 

 that the whole flux of heat across this surface must be equal to 

 the whole flux across the surface of the original ellipsoid. Now 

 the first of these quantities is equal to ^irkffpdw^ {day^ being 



an element of the surface), or to 4<7r^Jj Spdool^, since — = — . 



But ffBpdco^ is equal to the volume of a shell bounded by two 

 similar elHpsoids, whose semiaxes are a, b, c and a + ha, b-{-8b, 



c + Sc, and is therefore readily shown to be equal to 47r — abc. 



Hence ^ir-^ff^pdw^, or Afirkffpda?- is equal to ^^ir^kabc. In 



a similar manner, we have for the flux of heat across the original 

 isothermal surface ^^ir^k^a-fi^Cy, and therefore 



which gives 

 Hence we have 





k=k; 



abc 



dn -^'"^1 abc P ^""^ 



The value of v may be found by integrating this equation. 

 To effect this, since a, b, c are the semiaxes of an ellipsoid pass- 

 ing through the variable extremity of n, and having the same 

 foci as the original ellipsoid, whose axes are a^, 6„ c„ we have 



which gives 



62 = ^2 _y2 



c^=a^ —g^ 

 where f=a^^-b^% ff'^=zay^ — cf 



Hence (c) becomes 



^^' -/brrJr ^A^lP 



id) 



