SECT. II.] CONTINUOUS MASSES IN A KING. 259 



up, and the system of variable temperatures ends soon by coin 

 ciding sensibly with that which depends on the ordinates of a 

 circle and those of a logarithmic curve. 



Since the later differences between the excess of the tempera 

 ture of a body over the mean temperature are proportional to 

 the sine of the arc at the end of which the body is placed, it 

 follows that if we regard two bodies situated at the ends of the 

 same diameter, the temperature of the first will surpass the mean 

 and constant temperature as much as that constant temperature 

 surpasses the temperature of the second body. For this reason, if 

 we take at each instant the sum of the temperatures of two 

 masses whose situation is opposite, we find a constant sum, and 

 this sum has the same value for any two masses situated at the 

 ends of the same diameter. 



277. The formulae which represent the variable temperatures 

 of separate masses are easily applied to the propagation of heat 

 in continuous bodies. To give a remarkable example, we will 

 determine the movement of heat in a ring, by means of the 

 general equation which has been already set down. 



Let it be supposed that n the number of masses increases suc 

 cessively, and that at the same time the length of each mass 

 decreases in the same ratio, so that the length of the system has 

 a constant value equal to 2?r. Thus if n the number of masses 

 be successively 2, 4, 8, 16, to infinity, each of the masses will 



be TT, -^, -r, - &c. It must also be assumed that the 



t 4 O 



facility with which heat is transmitted increases in the same 

 ratio as the number of masses in\ thus the quantity which k 

 represents when there are only two masses becomes double when 

 there are four, quadruple when there are eight, and so on. 

 Denoting this quantity by g, we see that the number k must be 

 successively replaced by g, 2g, 4&amp;lt;g, &c. K we pass now to the 

 hypothesis of a continuous body, we must write instead of m, the 



value of each infinitely small mass, the element dx ; instead of n, 



2_ 

 the number of masses, we must write ^ ; instead of k write 





n 



172 



