220 THEORY OF HEAT. [CHAP. IV. 



2nd. We should have the same result if the radius of the ring 

 were infinitely small. 



3rd. To find the mean temperature of the ring after a time t 

 we must take the integral \f(x)dx from x = to x=%7rr, and 



divide by 2?rr. Integrating between these limits the different 

 parts of the value of u, and then supposing x 2?rr, we find the 

 total values of the integrals to be nothing except for the first 

 term ; the value of the mean temperature is therefore, after the 

 time t, the quantity e~ M M. Thus the mean temperature of the 

 ring decreases in the same manner as if its conducibility were in 

 finite ; the variations occasioned by the propagation of heat in the 

 solid have no influence on the value of this temperature. 



In the three cases which we have just considered, the tem 

 perature decreases in proportion to the powers of the fraction e~ h , 

 or, which is the same thing, to the ordinate of a logarithmic 

 curve, the abscissa being equal to the time which has elapsed. 

 This law has been known for a long time, but it must be remarked 

 that it does not generally hold unless the bodies are of small 

 dimensions. The previous analysis tells us that if the diameter of 

 a ring is not very small, the cooling at a definite point would not 

 be at first subject to that law ; the same would not be the case 

 with the mean temperature, which decreases always in proportion 

 to the ordinates of a logarithmic curve. For the rest, it must not 

 be forgotten that the generating section of the ring is supposed to 

 have dimensions so small that different points of the same section 

 do not differ sensibly in temperature. 



4th. If we wished to ascertain the quantity of heat which 

 escapes in a given time through the surface of a given portion of 



the ring, the integral hi \ dt I vdx must be employed, and must 



be taken between limits relative to the time. For example, 

 if we took and ZTT to be the limits of x, and 0, oo , to be the 

 limits of t\ that is to say, if we wished to determine the whole 

 quantity of heat which escapes from the entire surface, during the 

 complete course of the cooling, we ought to find after the integra 

 tions a result equal to the whole quantity of the initial heat, or 

 QjrrM, M being the mean initial temperature. 



