8 GANESHPRASAD, 



The following tivo Itypotheses suffice for an analytical representation of the 

 principle of energy as stated above : 



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

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



-Kf T'(x,f')dt', 



d 



where T' (x, t') Stands for T {x, t') and K \b a. constant called the thermal 

 conductivity of the solid, 



— % x < TC. (a) 



ii. The qnantity of heat which is absorbed by a cylinder , with the axis 

 of X as its axis and its faces, of unit area, x = x^, x = x^, in any interval 

 (t, t + t) is equal to 



c/""] T{x', f + r)- T(x', i) j dx', 



where c is a constant called the thermal capacity of the solid, [ß) 



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



c 



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

 supposition of the impermeability of the face, x = —n, of the solid finds ex- 

 pression in the equation 



X t 



f j T (x', t) - T(x', 0) Idx' = fl' (x, t') dt', -jkx<jc. (A) 



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 is a point of discontinuity, T(x,tJ is contained in 

 the aggregate of values assumed by T(x, t) as t approaches (B) 



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

 with a given x, T{x,t) passes from one value to another it assumes all the inter- 

 mediate values. 



The third condition, 



lim Q {x, t) = 0, lim Q {x, f) = 0, (C) 



X = TT + O X = Tt 0 



expresses the impermeability of the faces, Q{x, t) being the quantity of heat 

 which flows across unit area, at x, in the interval (0, t). 



