CONSTITUTION OP MATTER AND ANALYTICAL THEORIES OP HEAT. 



31 



For esample , when the area is a rectangle {^^'^j' ^leither of the two 



quantities of heat can be approximated to, since ä is unity. 



It should be noted that Y(x, t) need not satisfy any conditions other than 

 those indicated above and the approximate conditions of the phenomenon, ex- 

 pressed in terms of Y{x, t), which I will give in the next article. For example, 

 T(x, t) need not be difFerentiable with respect to t. 



81. The resnlt as regards the translation of the actual conditions of the 

 pJienonienon into approximate conditions, expressed in terms of Y(x, t), may be 

 stated now. 



The first of the three actual conditions of the phenomenon is that the prin- 

 ciple of the conservation of energy be satisfied. In other words , the first con- 

 dition is that the quantity of heat which flows into any closed sarface S in any 

 interval be equal to the quantity of heat absorbed by the enclosed solid in the 

 same interval. Now this condition, in all its generality, cannot be approximated 

 to. For example, when S is such that the greatest length, parallel to the axis 

 of X, taken inside it is less than k^, or, the greatest length, in even one direction 

 perpendicular to the axis of x, is less than A, then the fact of the conservation 

 of the energy inside S cannot be approximated to ; on the other band , when S 

 is not reentrant and is, further, such that the greatest length, in any direction, 

 taken inside it exceeds , say , unity , then an approximation to the fact of the 

 conservation of the energy inside S can be obtained, the degree of the approxi- 

 mation depending on the size and shape of S. 



The fact of the conservation of the energy inside any circular cylinder 0, 

 with its axis parallel to the axis of x and its faces, of unit area, x = x', 

 X = a^j-l-l, is expressed by the equation 



i t 

 Ic' K I Y' {x, + l,t') dt' - hK I Y' {x,, t') dt' 

 0 'o 



X +1 



= c/ ' I Y{x', t) - Y{x', 0) j dx' + m,, E, 1 0, J < 1, (1) 



Xi 



— jr < iCj < TT— 1, 



where JS stands for E{d), cl being the value of d for a circle, and it foUows, 

 from Art. 28, that {k, h') equals 1 or Qc^, h[) according as {x^, x^ + 1) lies or does 

 not lie in the interval (-Jt + 2A,, ä-2AJ, 0 < h[)<l. 



I r* I 



Let £ stand for the greatest value oi K\j Y'{x, t')dt'\, t having any value 



0 



and X lying outside the interval (— ä4-2Aj, ;t — 2AJ. Then, if (x^, x^ + 1) does 

 not lie in this interval, the left side of (1) equals 



xfl Y' (X, + 1, f ) - Y' {x^, f) I dt' + \, s, |9,,| < 1. 



0 



