SECT. I.] INTRODUCTION. 17 



may know at any given instant the temperatures indicated by 

 each thermometer, and compare the quantities of heat which 

 flow during the same instant, between two adjacent layers, or 

 into the surrounding medium. 



G. If the mass is spherical, and we denote by x the distance 

 of a point of this mass from the centre of the sphere, by t the 

 time which has elapsed since the commencement of the cooling, 

 and by v the variable temperature of the point m, it is easy to see 

 that all points situated at the same distance x from the centre 

 of the sphere have the same temperature v. This quantity v is a 

 certain function F (x, t} of the radius x and of the time t ; it must 

 be such that it becomes constant whatever be the value of x, when 

 we suppose t to be nothing ; for by hypothesis, the temperature at 

 all points is the same at the moment of emersion. The problem 

 consists in determining that function of x and t which expresses 

 the value of v. 



7. In the next place it is to be remarked, that during the 

 cooling, a certain quantity of heat escapes, at each instant, through 

 the external surface, and passes into the medium. The value of 

 this quantity is not constant ; it is greatest at the beginning of the 

 cooling. If however we consider the variable state of the internal 

 spherical surface whose radius is x, we easily see that there must 

 be at each instant a certain quantity of heat which traverses that 

 surface, and passes through that part of the mass which is more 

 distant from the centre. This continuous flow of heat is variable 

 like that through the external surface, and both are quantities 

 comparable with each other ; their ratios are numbers whose vary 

 ing values are functions of the distance x, and of the time t which 

 has elapsed. It is required to determine these functions. 



8. If the mass, which has been heated by a long immersion in 

 a medium, and whose rate of cooling we wish to calculate, is 

 of cubical form, and if we determine the position of each point mby 

 three rectangular co-ordinates x, y, z, taking for origin the centre 

 of the cube, and for axes lines perpendicular to the faces, we see 

 that the temperature v of the poiat m after the time t, is a func 

 tion of the four variables x, y, z, and t. The quantities of heat 



F. H. 2 



