168 ON FARADAY'S LIMES OF FORCE. 



of (14) we can determine the distribution of sources to which a given distri- 

 bution of pressures is due. 



(20) We have next to shew that if we conceive any imaginary surface 

 as fixed in space and intersecting the lines of motion of the fluid, we may 

 substitute for the fluid on one side of this surface a distribution of sources 

 upon the surface itself without altering in any way the motion of the fluid 

 on the other side of the surface. 



For if we describe the system of unit tubes which defines the motion of 

 the fluid, and wherever a tube enters through the surface place a unit source, 

 and wherever a tube goes out through the surface place a unit sink, and at the 

 same time render the surface impermeable to the fluid, the motion of the fluid 

 in the tubes will go on as before. 



(21) If the system of pressures and the distribution of sources which pro- 

 duce them be known in a medium whose resistance is measured by k, then in 

 order to produce the same system of pressures in a medium whose resistance 

 is unity, the rate of production at each source must be multiplied by k. For 

 the pressure at any point due to a given source varies as the rate of produc- 

 tion and the resistance conjointly ; therefore if the pressure be constant, the 

 rate of production must vary inversely as the resistance. 



(22) On the conditions to be fulfilled at a surface which separates two media 

 whose coefficients of resistance are k and k'. 



These are found from the consideration, that the quantity of fluid which 

 flows out of the one medium at any point flows into the other, and that the 

 pressure varies continuously from one medium to the other. The velocity normal 

 to the surface is the same in both media, and therefore the rate of diminution 

 of pressure is proportional to the resistance. The direction of the tubes of 

 motion and the surfaces of equal pressure will be altered after passing through 

 the surface, and the law of this refraction will be, that it takes place in the 

 plane passing through the direction of incidence and the normal to the surface, 

 and that the tangent of the angle of incidence is to the tangent of the angle 

 of refraction as k' is to k. 



(23) Let the space within a given closed surface be filled with a medium 

 different from that exterior to it, and let the pressures at any point of this 

 compound system due to a given distribution of sources within and without 



