CONSTITUTION OF MATTER AND ANALYTICAL THEOBIES OF HEAT. 27 



where c is a constant called the thermal capacity of the solid and 0^ (t + is 



the volume occapied by at time ^ + "2 ' {yd 



28. Let Y(x, t) be a function of x and t such that T^{t) equals T(x^^, t), 

 I will call Y {x, t) the auxiliarij function. 



Let R,, ^, stand for the value of „. when x,^, x^ are substituted for x„ „ 

 ^Q',ti being the values about which x^^^, fluctuate. 



Then, noting that oc^^^ — x^ and x^^, — x^, are each namerically less than /l, 

 it follows from the end of (aj that 



B^^^ = E„,„(1 + 26,AJ, |e.|<l. 



Also 



{t') - {t') = (x,,^ , - x,^ ,) { Y' {x^, n + 6, (,r,, t')], 

 where x,^ lies on the left of a;„_ j, and on the right of ^, , 



and Sl.^ (x^, f) is the greatest value of | Y" (x, t') \ within the interval 

 Now 



2- 



Therefore it follows from symmetry and from (/SJ that the quantity of 

 heat which flows across a unit area L, placed parallel to the faces of the slab 

 and distant x^ from the origin, in the positive direction of the axis of x in any 

 interval (t, t + t) is equal to 



I - Z"*"^ V' (x^, f) dt' + f^] Y' (x^, t') I df ] 2R,^, „„ (xo^ - x,J 

 t 



2\( 



./ 

 t 



\%\, IÖ5!, i9el<l, 



.(l+20,Aj(l + 6,. ^^\/^]Y'ix,,t')\dt'+%,^,P,(x„t,t + T)] ^iV'>(%.-^vJ> (1) 



J + r 



where P^(Xj^, t, t + r) stands for / £l^{x^, t')dt' and the summation extends to 



every pair of assemblages Ä^^, J.,,,,, situated on the right and on the left of x^, 

 respectively, for which 



^o,„, i' — 00.,,,, t < 2Ai 



and the line joining whose centres crosses the area perpendicularly. 

 Now, suppose it to follow from the homogeneity of the slab that 



4 * 



