378 THEORY OF HEAT. [CHAP. IX. 



much influence on the variable temperatures of the prism ; but 

 this effect becomes weaker and weaker, and ends with being quite 

 insensible. 



379. It is necessary to remark that the reduced equation (i/) 

 does not apply to that part of the line which lies beyond the point 

 m whose distance has been denoted by X. 



In fact, however great the value of the time may be, we might 



2CLJ 



choose a value of x such that the term e 4 * would differ sensibly 

 from unity, so that this factor could not then be suppressed. We 

 must therefore imagine that we have marked on either side of the 

 origin two points, m and m , situated at a certain distance X or 

 X, and that we increase more and more the value of the time, 

 observing the successive states of the part of the line which is 

 included between m and m. These variable states converge more 

 and more towards that which is expressed by the equation 



Whatever be the value assigned to X, we shall always be able to 

 find a value of the time so great that the state of the line mom 

 does not differ sensibly from that which the preceding equation (y) 

 expresses. 



If we require that the same equation should apply to other 

 parts more distant from the origin, it will be necessary to suppose 

 a value of the time greater than the preceding. 



The equation (?/) which expresses in all cases the final state of 

 any line, shews that after an exceedingly long time, the different 

 points acquire temperatures almost equal, and that the temperatures 

 of the same point end by varying in inverse ratio of the square 

 root of the times elapsed since the commencement of the diffusion. 

 The decrements of the temperature of any point whatever always 

 become proportional to the increments of the time. 



380. If we made use of the interal 



