ANALOGIES OF THE PROBLEM OF ELECTRICAL EQUILIBRIUM. 57 



the distance of the elements, may be expressed by the following 

 formula : 



</Q = -MS. 



an 



The coefficient k is the coefficient of conductivity of the medium. 

 It represents the flow of heat for unit surface between two parallel 

 planes at unit distance, and whose temperatures differ by i. 



In the present case, the value of the flow for unit surface at each 

 point is 



Q--4' 



dn 



it is proportional to the differential of the potential in reference to 

 the perpendicular to the corresponding equipotential surface. 



The flow of heat is the same across an element dK of any surface 

 A bounded by the same tube. If 6 is the angle which this element 

 makes with the equipotential surface, and da the portion of the 

 perpendicular to the element */A, comprised within the surfaces S 

 and S', we have 



,/Q- -A/A cos B-~ -kdt*-T- -***%-- 



an an da oa 



The flow of heat across an element of any given surface is 

 therefore proportional to the partial differential of the temperature 

 in the medium with respect to the perpendicular to this surface. 



As equilibrium is established, the total flow of heat corresponding 

 to any closed surface, not containing sources of heat, must be zero. 

 Let u, v and w be the components of the flow at a point P, and 

 dxdydz an element of volume at the same point ; the flow which 

 enters by one of the surfaces dydz is udydz, that which emerges by 



the opposite surface is equal to ( u H -- dx\ dydz, the difference 



du \ dx / 



is dxdydz ; the total flow corresponding to the entire surface of 



the element is then 



/"bu cte; <)7>\ 

 dxdydz / + - + \- 



\C ty ^Z / 



as this flow must be zero, it follows that 



+ - + = -M/=0. 

 02 cy oz 



