in Homogeneous Solid Bodies. 507 



actual sources of heat are arranged, whether over an isothermal 

 surface or not ; and the temperature produced in an external 

 point by the former sources is the same as that produced by the 

 latter. Also the total flux of heat across the isothermal surface, 

 whose temperature is v, is equal to the total flux of heat from 

 the actual sources. From this, and from what has been proved 

 above, it follows that if a surface be described round a conduct- 

 ing or non-conducting electrified body, so that the attraction on 

 points situated on this surface may be everywhere perpendicular 

 to it, and if the electricity be removed from the original body 

 and distributed in equilibrium over this surface, its intensity at 

 any point will be equal to the attraction of the original body on 

 that point, divided by 47r, and its attraction on any point with- 

 out it will be equal to the attraction of the original body on the 

 same point*. 



If we call E the total expenditm-e of heat, or the whole flux 

 across any isothermal surface, we have, obviously. 



Now this quantity should be equal to the sum of the expen- 

 diture of heat from all the sources. To verify this, we must, in 

 the first place, find the expenditure of a single source. Now the 



temperatvu-e produced by a single source is, by (1), v=—, and 



dv 

 hence the expenditui'e is obviously equal to — -7-x47jt^, or to 



47rA. If A=p^d(Oy^, this becomes 4:'irp^d(o^. Hence the total 



expenditure i8fj4nrpyd(i)^, or — // -7^ «?a),2,which agrees with 



the expression found above. 



The following is an example of the application of these prin- 

 ciples. 



Uniform Motion of Heat in an Ellipsoid. 



The principles established above afford an easy method of de- 

 termining the isothermal surfaces, and the corresponding tem- 

 peratures in the case in which the original isothermal surface is 

 an ellipsoid. 



* After having established this remarkable theorem in the manner shown 

 in the text, the author attemjrted to i)rove it by dii'ect integration, but only 

 suceeeded in doing so uinvards of a year later, when he obtained the de- 

 monstration pubbshed in a j)aper, " Propositions in the Theory of Attrac- 

 tion " (Camb. Math. Journ. Nov. 18-42), vvliieli a})peare(l almost contempo- 

 raneously with a pa])er by M. Sturm in Liouville's Journal, containing the 

 same demonstration ; exactly the same demonstration, as the author after- 

 wards (in 1845) found, had been given fourteen years eailier by Green. 



