OF ARTS AND SCIENCES. 145 



since the error thus introduced into their calculations is necessarily 

 less than those arising from eri-ors in observing the phenomena. 



Dulong and Petit's experiments showed that Fourier's assumption 

 with regard to the flux of heat at the surface of a body due to ladia- 

 tion was wrong, and Principal Forbes's experiments upon heated 

 metallic bars showed that, in order to write the ^ux of heat in the 

 interior of a body 



do 



dx 



X must be regarded as a function of the temperature. Forbes's ex- 

 periments evidently offer no objection to Kelland's hypothesis, for 



— q,(v)$ and — c ^M 

 dx dx 



are equivalent expressions, if 



(p{v) = cf'(v). 



The first step in determining the form of the function y is made by 

 showing that it must satisfy a differential equation which when the 

 heated body is at its final state, reduces to Laplace's Equation. 



Consider the element of volume dxdydz, which has one of its 

 angles at the point (x, y, z) and its diagonally opposite angle at 

 {x -\- dx, y -\- dy, z -\- dz). During the instant dt, the flux of heat 

 across that face of the element which contains the point (x, y, z) and 

 is parallel to the coordinate plane xy, is 



F{v, z)= — c ^ dxdydt. 



The amount of heat which flows out at the opposite face of the 

 element is obtained by developing -F(t', z) by Taylor's Theorem : 



F(v J^dv,z-\-dz)= — c ^ dxdydt — c ^^ dxdydzdt. 



The flux across the second face is less than the flux across the first 

 face by 



c I ' dxdydzdt. 



Considering each of the other pairs of opposite faces, it is evident that 

 in the instant dt a quantity of heat equal to 



VOL. XII. (n. S. IV.) 10 



