in Homogeneous Solid Bodies. 509 



aJ . . . . 



Or x'—x-=- —c^^v since ^, is infinitely small, and therefore also 



a/ — X ; whence 





In a similar manner we should find 



y' 1 ^' 



y= a , and ^= 



But 



and hence we have 



] + ^ 1 + ^ 



^^V c,2 



a?^ y^ z^ _-, 



y" . ^\ _i 



8 — ^^ 



or 



^ii yli Ji 



for the equation to the isothermal surface whose temperature is 

 v-^-\-dvy, and which is therefore an ellipsoid described from the 

 same foci as the original isothermal ellipsoid. In exactly the same 

 manner it might be shown, that the isothermal surface whose 

 temperature is Vy + dv^ + dv\ is an ellipsoid having the same foci 

 as the ellipsoid whose temperature is v-^-\-dv-^, and consequently 

 as the original ellipsoid also. By continuing this process it 

 may be proved that all the isothermal surfaces are ellipsoids, 

 having the same foci as the original one. 



From the form of the equation found above for the isothermal 

 ellipsoid whose temperature is v-^^-\-dv,^, it follows that ^j ox ji^dn-^^ 

 is ^=a^da.^, where du^ is the increment of ff, corresponding to the 

 increment dn-^ of n^ Hence if a be one of the semiaxes of an 

 ellipsoid, a + da the corresponding semiaxis of another ellipsoid 

 having the same foci, dn the thickness at any point of the shell 

 bounded by the two ellipsoids, and^; the perpendicular from the 

 centre to the plane touching cither ellipsoid at the same point, 

 we have ' 



dn a -I- 



Ta=p ^*^ 



All that remains to be done is to find the temperature at the 

 surface of any given ellipsoid having the same foci as the given 



