508 Prof. Thomson on the Uniform Motion of Heat 



The first step is to find p^, whicli is proportional to the quan- 

 tity of matter at any point in the surface of an ellipsoid, when 

 the matter is so distributed that the attraction on a point within 

 the ellipsoid is nothing. Now the attraction of a shell, bounded 

 by two concentric similar ellipsoids, on a point within it is no- 

 thing if the shell be infinitely thin ; and its attraction will be 

 the same as that of matter distributed over the surface of one of 

 the ellipsoids, in such a manner that the quantity on a given 

 infinitely small area, at any point, is proportional to the thick- 

 ness of the shell at the same point. Let a,, Z»,, Cj be the semi- 

 axes of one of the ellipsoids; «j-j-Sff,, b^-\-hby, c, + Scj those of 

 the other. Let also ^Jj be the perpendicular from the centre to 

 the tangent plane at any point on the first ellipsoid, and^j, +SjOj 

 the ])erpendicular from the centre to the tangent plane at a point 

 similarly situated on the second ; then 8pi is the thickness of the 

 shell, since the two ellipsoids being similai-, the tangent planes at 

 the points similarly situated on their surfaces are parallel. Also, 

 on account of their similarity, 



«i ~ *i ~ <^i ~ Pi' 

 and consequently the thickness of the shell is proportional to J9j. 

 Hence we have, by (5), 



-4^^=^' = ^>^^' (") 



where k^ is a constant to be determined by the condition v = t\ 

 at the surface of the ellipsoid. 



To find the eqviation of the isothermal surface at which the 

 temperature is i\ + dv^, let —dv^ = C in (a). Then we have 



Q 



k,p,dn,= -7—, or^i«?Mi = ^i, where ^, is an infinitely small con- 



stant quantity ; and the required equation will be the equation 

 of the surface traced by the extremity of the line dn^, drawn 

 externally perpendicular to the ellipsoid. Let z', y', z' be the 

 coordinates of any point in that surface, and <r, ij, z those of the 

 corresponding point in the ellipsoid. Then, calling «„ /Sj, 7, 

 the angles which a normal to the ellipsoid at the point whose 

 coordinates are x, y, z makes with these coordinates, and sup- 

 posing the axes of oc, y, z to coincide with the axes of the ellip- 

 soid 2fli, 26,, 2cj respectively, we have 



—qdn^ 

 a*— a:=rfwiC08ai= y-. — ^ 5 TIT" ~~2/'i^"i= ~k^v 



\/(^ + |7 + c7) 



