454 THEORY OF HEAT. [CHAP. IX. 



throughout the whole extent of its course, and serves to determine 

 with facility all its results numerically. 



The functions which are obtained by these solutions are then 

 composed of a multitude of terms, either finite or infinitely small : 

 but the form of these expressions is in no degree arbitrary; it is 

 determined by the physical character of the phenomenon. For 

 this reason, when the value of the function is expressed by a series 

 into which exponentials relative to the time enter, it is of 

 necessity that this should be so, since the natural effect whose 

 laws we seek, is really decomposed into distinct parts, corre 

 sponding to the different terms of the series. The parts express 

 so many simple movements compatible with the special conditions ; 

 for each one of these movements, all the temperatures decrease, 

 preserving their primitive ratios. In this composition we ought 

 not to see a result of analysis due to the linear form of the 

 differential equations, but an actual effect which becomes sensible 

 in experiments. It appears also in dynamical problems in which 

 we consider the causes which destroy motion ; but it belongs 

 necessarily to all problems of the theory of heat, and determines 

 the nature of the method which we have followed for the solution 

 of them. 



8th. The mathematical theory of heat includes : first, the exact 

 definition of all the elements of the analysis ; next, the differential 

 equations; lastly, the integrals appropriate to the fundamental 

 problems. The equations can be arrived at in several ways ; the 

 same integrals can also be obtained, or other problems solved, by 

 introducing certain changes in the course of the investigation. 

 We consider that these researches do not constitute a method 

 different from our own ; but confirm and multiply its results. 



9th. It has been objected, to the subject of our analysis, that 

 the transcendental equations which determine the exponents having 

 imaginary roots, it would be necessary to employ the terms which 

 proceed from them, and which would indicate a periodic character 

 in part of the phenomenon; but this objection has no foundation, 

 since the equations in question have in fact all their roots real, and 

 no part of the phenomenon can be periodic. 



10th. It has been alleged that in order to solve with certainty 

 problems of this kind, it is necessary to resort in all cases to a 



