SECT. II.] .VARIED MOVEMENT IN A SPHERE. 91 



Denote by x the distance of any point whatever from the 

 centre of the sphere, and by v the temperature of the same point, 

 after a time t has elapsed ; and suppose, to make the problem 

 more general, that the initial temperature, common to all points 

 situated at the distance x from the centre, is different for different 

 values of x ; which is what would have been the case if the im 

 mersion had not lasted for an infinite time. 



Points of the solid, equally distant from the centre, will not 

 cease to have a common temperature ; v is thus a function of x 

 and t. When we suppose t = 0, it is essential that the value of 

 this function should agree with the initial state which is given, 

 and which is entirely arbitrary. 



112. We shall consider the instantaneous movement of heat 

 in an infinitely thin shell, bounded by two spherical surfaces whose 

 radii are x and x + dx: the quantity of heat which, during an 

 infinitely small instant dt, crosses the lesser surface whose radius 

 is x, and so passes from that part of the solid which is nearest to 

 the centre into the spherical shell, is equal to the product of four 

 factors which are the conducibility K, the duration dt, the extent 



^Trx 2 of surface, and the ratio -j- , taken with the negative sign ; 



it is expressed by AKirx* -j- dt. 



To determine the quantity of heat which flows during the 

 same instant through the second surface of the same shell, and 

 passes from this shell into the part of the solid which envelops it, 

 x must be changed into x + dx, in the preceding expression : that 



ci i) 



is to say, to the term KTTX* -T- dt must be added the differen 

 tial of this term taken with respect to x. We thus find 



- tKvx* ^dt- IKtrd (x* ^} . dt 

 dx \ dxj 



as the expression of the quantity of heat which leaves the spheri 

 cal shell across its second surface; and if we subtract this quantity 

 from that which enters through the first surface, we shall have 



x z --} dt. This difference is evidently the quantity of 



