512 Prof. Thomson on the Uniform Motion of Heat. 



Lame, in his memoir on Isothermal Surfaces, in Liouville's 

 Journal de Mathematique, vol. ii. p. 147, by showing that a 

 series of isothermal surfaces of the second order will satisfy the 

 equation 



d^v d'^v d-v __ 



d^^'^di^'^d? ' 



provided they are all described from the same foci. The value 

 which he finds for v agrees with (e), and he finds for the flux of 

 heat at any point the expressioii 



KA 



or, according to the notation which we have employed, 

 47r^ia,J,e, 

 \/{a^-v'']V{a^-py 



where v is the greater real semiaxis of the hyperboloid of one 

 sheet, and p the real semiaxis of the hyperboloid of two sheets, 

 described from the same foci as the original ellipsoid, and pass- 

 ing through the point considered. Hence a', v^, p^ are the 

 three roots of the equation 



or 



u-fY^^=o- 



Hence 



a'yy=fYA 

 and 



«2^'i + ay + vy=fY + (/' +ff')^^+9Y +P^'- 

 Therefore 



(a2-v2)(fl2-p2)=«^-«V-fly-vV+ ^^^ 



= a'-{fc,' + {r+f)cc^-^ffY+fz'} +2/V J 

 :=a'^-{a^-b^){a^-c'')-{2a^-b'^-c'^)a;^-{a'^-c'')f-{a'^-b^)z^ 



= a''-{a^-b^){a^-c^)-lb'^ + c^)x''-{a''-c'')i/-{a^-b'^)z'^ 



