SECT. II.] TRANSFER BETWEEN SEPARATE MASSES. 225 



with that towards which it has a natural tendency, and which con 

 sists in this, that the temperatures of the different points become 

 proportional to the sine of the same multiple of the arc which 

 measures the distance from the origin. The initial distribution 

 makes no change in these results. 



SECTION II. 



Of the communication of heat between separate masses. 



247. We have now to direct attention to the conformity of 

 the foregoing analysis with that which must be employed to de 

 termine the laws of propagation of heat between separate masses ; 

 we shall thus arrive at a second solution of the problem of the 

 movement of heat in a ring. Comparison of the two results will 

 indicate the true foundations of the method which we have fol 

 lowed, in integrating the equations of the propagation of heat in 

 continuous bodies. We shall examine, in the first place, an ex 

 tremely simple case, which is that of the communication of heat 

 between two equal masses. 



Suppose two cubical masses m and n of equal dimensions and 

 of the same material to be unequally heated; let their respective 

 temperatures be a and b, and let them be of infinite conducibility. 

 If we placed these two bodies in contact, the temperature in each 

 would suddenly become equal to the mean temperature \ (a + 6). 

 Suppose the two masses to be separated by a very small interval, 

 that an infinitely thin layer of the first is detached so as to be 

 joined to the second, and that it returns to the first immediately 

 after the contact. Continuing thus to be transferred alternately, 

 and at equal infinitely small intervals, the interchanged layer 

 causes the heat of the hotter body to pass gradually into that 

 which is less heated; the problem is to determine what would be, 

 after a given time, the heat of each body, if they lost at their sur 

 face no part of the heat which they contained. We do not suppose 

 the transfer of heat in solid continuous bodies to be effected in a 

 manner similar to that which we have just described: we wish 

 only to determine by analysis the result of such an hypothesis. 



Each of the two masses possessing infinite conducibility, the 

 quantity of heat contained in an infinitely thin layer, is sud- 

 F. H. - 15 



