SECT. III.] VARIED MOVEMENT IX A CYLINDER. 95 



It might be supposed that in the initial state all the spherical 

 layers have not the same temperature : which is what would 

 necessarily happen, if the immersion were imagined not to have 

 lasted for an indefinite time. In this case, which is more general 

 than the foregoing, the given function, which expresses the 

 initial temperature of the molecules situated at distance x from 

 the centre of the sphere, will be represented by F (x) ; the third 

 equation will then be replaced by the following, &amp;lt; (x, 0) = F (x). 



Nothing more remains than a purely analytical problem, 

 whose solution w 7 ill be given in one of the following chapters. 

 It consists in finding the value of v, by means of the general 

 condition, and the two special conditions to which it is subject. 



SECTION III. 



Equations of the varied movement of heat in a solid cylinder. 



118. A solid cylinder of infinite length, whose side is per 

 pendicular -to its circular base, having been wholly immersed 

 in a liquid whose temperature is uniform, has been gradually 

 heated, in such a manner that all points equally distant from 

 the axis have acquired the same temperature ; it is then exposed 

 to a current of colder air ; it is required to determine the 

 temperatures of the different layers, after a given time. 



x denotes the radius of a cylindrical surface, all of whose 

 points are equally distant from the axis ; X is the radius of 

 the cylinder ; v is the temperature which points of the solid, 

 situated at distance x from the axis, must have after the lapse 

 of a time denoted by t, since the beginning of the cooling. 

 Thus v is a function of x and t, and if in it t be made equal to 

 0, the function of x which arises from this must necessarily satisfy 

 the initial state, which is arbitrary. 



119. Consider the movement of heat in an infinitely thin 

 portion of the cylinder, included between the surface whose 

 radius is x, and that whose radius is x + dx. The quantity of 

 heat which this portion receives during the instant dt y from the 

 part of the solid which it envelops, that is to say, the quantity 

 which during the same time crosses the cylindrical surface 



