16 THEORY OF HEAT. [CHAP. I. 



form of this curve, and in thus including in a single formula 

 all the successive states of the solid. 







4. Let z be the constant temperature at a point m of the 



mean circumference, x the distance of this point from the source, 

 that is to say the length of the arc of the mean circumference, 

 included between the point m and the point o which corresponds 

 to the position of the source; z is the highest temperature 

 which the point m can attain by virtue of the constant action 

 of the source, and this permanent temperature z_ isj*^ function 

 /(#) of the distance x. The first part of theC^roblemj consists 

 in determining the function f(x) which represents the permanent 

 state of the solid. 



Consider next the variable state which succeeds to the former 

 state as soon as the source has been removed ; denote by t the 

 time which has passed since the suppression of the source, and 

 by v the value of the temperature at the point m after the 

 time t. The quantity v will be a certain function F (x, t) of 

 the distance x and the time t\ the object of the (pf oblem^is to 

 discover this function F (x, t), of which we only Imowas yet 

 that the initial value is f (x}, so that we ought to have the 

 equation f (.r) = F (x, o). 



5. If we place a solid homogeneous mass, having the form 

 of a sphere or cube, in a medium maintained at a constant tem 

 perature, and if it remains immersed for a very long time, it will 

 acquire at all its points a temperature differing very little from 

 that of the fluid. Suppose the mass to be withdrawn in order 

 to transfer it to a cooler medium, heat will begin to be dissi 

 pated at its surface ; the temperatures at different points of the 

 mass will not be sensibly the same, and if we suppose it divided 

 into an infinity of layers by surfaces parallel to its external sur 

 face, each of those layers will transmit, at each instant, a certain 

 quantity of heat to the layer which surrounds it. If it be 

 imagined that each molecule carries a separate thermometer, 

 -which indicates its temperature at every instant, the state of 

 the solid will from time to time be represented by the variable 

 system of all these thermometric heights. It is required to 

 express the successive states by analytical formulae, so that we 



