22 THEORY OF HEAT. [CHAP. I. 



have flowed through, when the values of the time and of the 

 variable coordinates are known. Thus will be given not only the 

 differential equations which the functions that express the values 

 of the temperatures must satisfy; but the functions themselves 

 will be given under a form which facilitates the numerical 

 applications. 



14. In order that these solutions might be general, and have 

 an extent equal to that of the problem, it was requisite that they 

 should accord with the initial state of the temperatures, which is 

 arbitrary. The examination of this condition shews that we may 

 develop in convergent series, or express by definite integrals, 

 functions which are not subject to a constant law, and which 

 represent the ordinates of irregular or discontinuous lines. This 

 property throws a new light on the theory of partial differen 

 tial equations, and extends the employment of arbitrary functions 

 by submitting them to the ordinary processes of analysis. 



15. It still remained to compare the facts with theory. With 

 this view, varied and exact experiments were undertaken, whose 

 results were in conformity with those of analysis, and gave them 

 an authority which one would have been disposed to refuse to 

 them in a new matter which seemed subject to so much uncer 

 tainty. These experiments confirm the principle from which we 

 started, and which is adopted by all physicists in spite of the 

 diversity of their hypotheses on the nature of heat. 



16. Equilibrium of temperature is effected not only by way 

 of contact, it is established also between bodies separated from 

 each other, which are situated for a long time in the same region. 

 This effect is independent of contact with a medium; we have 

 observed it in spaces wholly void of air. To complete our theory 

 it was necessary to examine the laws which radiant heat follows, 

 on leaving the surface of a body. It results from the observations 

 of many physicists and from our own experiments, that the inten 

 sities of the different rays, which escape in all directions from any 

 point in the surface of a heated body, depend on the angles which 

 their directions make with the surface at the same point. We 

 have proved that the intensity of a ray diminishes as the ray 



