CONSTITUTION OF MATTER AND ANALTTICAL THEORIES DP HEAT. 



51 



The probJem of the linear conduction of heat may now be thus formulated : 

 Given the initial cliar acter istic function C(^, 0), find the characteristic fiwdion 

 at any subsequent iinie. 



Analytical representation of the conditions of the phenomenon. 



46. The following three liypotheses suffice for an analytical representation of 

 the conditions of the phenomenon: 



i. The quantity of heat which flows across unit area, placed at | at right 

 angles to the axis of along this axis in any interval {t, t-{-t) is equal to 



-Kj C'(lt')ät', 

 't 



d 



where C'{^,t') Stands for -^C{i„t') and K is a. constant, dependent on G and 



called the thermal conductivity of the slab, — tc <. ^ <z Tt. («J 



ü. The quantity of heat which is absorbed by a cylinder, with the axis of 

 X as its axis and its faces, of unit area, | = | = in any interval (t, t + x) 

 is equal to 



cf'\C(^',t + t)-C{^',t)\ä^', 



where c is a constant , dependent on 6r and called the thermal capacity of 

 the slab. (^,) 



iii. Let s and U represent cylinders , with the axis of x as their common 

 axis and their faces passing through x^, x^ and 1^, respectively , x^, x^ and 

 Ij, ^2 being any pairs of values of x and |. Thon, in Order that the energy in 

 any s be conserved, it is sufficient that the energy in any E be conserved. {y^ 



It would be convenient to consider the units so chosen that = 1. 



c 



47. Thus the principle of the conservation of energy together with the 

 supposition of the impermeability of the face , i, — —n , of the slab finds ex- 

 pression in the equation 



f\C(X',t)-C{l',0)\dl' =fc\lt')df, -%<l<^. (AJ 



— jr 0 



This is the first and the fundamental condition of the phenomenon. 

 The second condition is the following : 



T (x, t) is continuous in ^ , or , if it has any discontinuities they are of the 



second kind; further, if t^ is a point of discontinuity , T{x, tj) is contained in 



the aggregate of values assumed by T(x, t) as t approaches t^. (BJ 



This is merely the analytical representation of the supposition that when, 



7 * 



