34 president's address — section a. 



a constant over large jjarts or regions of the body, and varies 

 only from region to region. The three quantities are con- 



nected by the equation Cs = — c, -,- vi^here C. is the flow 



d\ ^^^ 



in the direction s, -5— the variation of V in this direction, 

 a s 



and c the conductivity. 



The data of the problem are — 



(1.) The value of c in all parts of the body. 



(2.) Either the value of the temperature at each point 



of the bounding surfaces of the body, or the rate 



of flow^ of heat across the boundary at this point : 



i.e., the value at all points of the boundary of 



d V 



either V or c -^— If, however, V be constant 

 du 



over any bounding surface, it will be sufficient 



to know this value of V or the surface integral 



of c -^— across the surface. 

 du 



Further, the fact that a state of steady flow has been 



attained is expressed mathematically by the equation 



d Cx d Cy dCcy ^ „ ,t ,^ • . • 1 



—j 1 — -J— H — -j:~ = or V V = : an equation obtained 



by expressing the fact that in each second as much heat leaves 



any small volume as enters it. Also at every point of a 



surface separating from one another portions of the medium 



of different conductivities c and c.,, the flow of heat towards 



the surface on one side is equal to the flow away from it on 



dY d\ 



the other : that is, c, -^— p + c., —, — = 0. 

 ^ du^ ^ du,2 



These are the data : equations can now be written down 

 whose solution would give the value of V (or c) at every 

 point of the body, and this would constitute the complete 

 solution of the problem. Diagrams showing accurately the 

 solution of various comparatively simple cases have been 

 drawn. Maxwell, for example, has drawn in his Electricity 

 and Magnetism cases of two and three circular sources in an 

 infinite and uniform solid ; and Thomson has given in his 

 papers diagrams of the flow of heat in a solid, whose con- 

 ductivity would be uniform but for the presence of a spherical 

 portion whose conductivity is different to that of the rest of 

 the body. 



Now let us turn to the theory of electrostatics. Here, in the 

 most geBeraJ case we have an insulating medium containing 



