ON FARADAY'S LINES OF FORCE. 171 



On this supposition we can easily calculate the kind of alteration -which 

 the introduction of the internal medium will produce ; for wherever a unit tube 



k' k 

 enters the surface we must conceive a source producing fluid at a rate j , 



K 



and wherever a tube leaves it we must place a sink annihilating fluid at the 



k' k 

 rate j , then calculating pressures on the supposition that the resistance in 



1C 



both media is k, the same as in the external medium, we shall obtain the true 

 distribution of pressures very approximately, and we may get a better result 

 by repeating the process on the system of pressures thus obtained. 



(27) If instead of an abrupt change from one coefficient of resistance to 

 another we take a case in which the resistance varies continuously from point 

 to point, we may treat the medium as if it were composed of thin shells each 

 of which has uniform resistance. By properly assuming a distribution of sources 

 over the surfaces of separation of the shells, we may treat the case as if the 

 resistance were equal to unity throughout, as in (23). The sources will then 

 be distributed continuously throughout the whole medium, and will be positive 

 whenever the motion is from places of less to places of greater resistance, and 

 negative when in the contrary direction. 



(28) Hitherto we have supposed the resistance at a given point of the 

 medium to be the same in whatever direction the motion of the fluid takes 

 place ; but we may conceive a case in which the resistance is different in 

 different directions. In such cases the lines of motion will not in general be 

 perpendicular to the surfaces of equal pressure. If a, b, c be the components 

 of the velocity at any point, and a, ft, y the components of the resistance at 

 the same point, these quantities will be connected by the following system of 

 linear equations, which may be called "equations of conduction," and will be 

 referred to by that name. 



In these equations there are nine independent coefficients of conductivity. In 

 order to simplify the equations, let us put 



............ &c ............. &c. 



222 



