84 THEORY OF HEAT. [CH. I. SECT. VIII. 



100. A. We may determine in another manner the quantity 

 of heat which the solid loses during a given time, and this will 

 serve in some degree to verify the preceding calculation. In 

 fact, the mass of the rectangular molecule whose dimensions are 

 dx, dy, dz, is D dx dy dz, consequently the quantity of heat 

 which must be given to it to bring it from the temperature to 

 that of boiling water is CD dx dy dz, and if it were required to 

 raise this molecule to the temperature v, the expenditure of heat 

 would be v CD dx dy dz. 



It follows from this, that in order to find the quantity by 

 which the heat of the solid, after time t, exceeds that which 

 it contained at the temperature 0, we must take the mul 



tiple integral 1 1 1 v CD dx dy dz, between the limits x = = ir y 



We thus find, on substituting for v its value, that is to say 



~ 9t 



e cos x cos y cos z, 



that the excess of actual heat over that which belongs to the 

 temperature is 8 CD (1 e~ gt ) ; or, after an infinite time, 

 8 CD, as we found before. 



We have described, in this introduction, all the elements which 

 it is necessary to know in order to solve different problems 

 relating to the movement of heat in solid bodies, and we have 

 given some applications of these principles, in order to shew 

 the mode of employing them in analysis ; the most important 

 use which we have been able to make of them, is to deduce 

 from them the general equations of the propagation of heat, 

 which is the subject of the next chapter. 



Note on Art. 76. The researches of J. D. Forbes on the temperatures of a long 

 iron bar heated at one end shew conclusively that the conducting power K is not con 

 stant, but diminishes as the temperature increases. Transactions of the Eoyal 

 Society of Edinburgh, Vol. xxiu. pp. 133 146 and Vol. xxiv. pp. 73 110. 



Note on Art. 98. General expressions for the flow of heat within a mass in 

 which the conductibility varies with the direction of the flow are investigated by 

 Lame in his Theorie Analytique de la Chaleur, pp. 1 8. [A. F.] 



