SECT. VJ.] HEATING OF CLOSED SPACES. 63 



and passes into the external air, which we suppose to be main 

 tained at a lower and permanent temperature n. The inner air is 

 heated more and more : the same is the case with the solid 

 boundary : the system of temperatures steadily approaches a final 

 state which is the object of the problem, and has the property of 

 existing by itself and of being kept up unchanged, provided the 

 surface of the source a be maintained at the temperature a, and 

 the external air at the temperature n. 



In the permanent state which we wish to determine the air 

 preserves a fixed temperature m ; the temperature of the inner 

 surface s of the solid boundary has also a fixed value a ; lastly, the 

 outer surface s, which terminates the enclosure, preserves a fixed 

 temperature b less than a, but greater than n. The quantities 

 cr, a, 5, e and n are known, and the quantities m, a and b are 

 unknown. 



The degree of heating consists in the excess of the temperature 

 m over n } the temperature of the external air; this excess evi 

 dently depends on the area a of the heating surface and on its 

 temperature a ; it depends also on the thickness e of the en 

 closure, on the area s of the surface which bounds it, on the 

 facility with which heat penetrates the inner surface or that 

 which is opposite to it ; finally, on the specific conducibility of 

 the solid mass which forms the enclosure : for if any one of these 

 elements were to be changed, the others remaining the same, the 

 degree of the heating would vary also. The problem is to deter 

 mine how all these quantities enter into the value of m n. 



82. The solid boundary is terminated by two equal surfaces, 

 each of which is maintained at a fixed temperature; every 

 prismatic element of the solid enclosed between two opposite por 

 tions of these surfaces, and the normals raised round the contour 

 of the bases, is therefore in the same state as if it belonged to an 

 infinite solid enclosed between two parallel planes, maintained at 

 unequal temperatures. All the prismatic elements which com 

 pose the boundary touch along their whole length. The points 

 of the mass which are equidistant from the inner surface have 

 equal temperatures, to whatever prism they belong ; consequently 

 there cannot be any transfer of heat in the direction perpendicular 

 to the length of these prisms. The case is, therefore, the same 



