4a Mr maxwell, ON FARADAY'S LINES OF FORCE. 



ar.p^a+ (,n'(i-m'y)T, 



h = P//3 + {I'y - n'a)T, 



C = P^y+ima- t(i)T, 

 where ^, m, n' are the direction cosines of the fixed line with reference to the new axes. 

 If we make 



the equation of continuity 



becomes 



dp dp , dp 



= — , V = -T- 1 and ^ = -— , 

 dw' ^ dy' dz' 



da db dc 



— + 1 = 0, 



dx dy d% 



^' d.v^ ^ ^' df ^ ^' dz^ ' 



and if we make a? = v P/^, y = v -^2 »/' ^ ~ V i's^t 



d^p d'p d'p 



the ordinary equation of conduction. 



It appears therefore that the distribution of pressures is not altered by the existence 

 of the coefficient T, Professor Thomson has shewn how to conceive a substance in which 

 this coefficient determines a property having reference to an axis, which unlike the axes 

 of Pj, Pj, P3 is dipolar. 



For further information on the equations of conduction, see Professor Stokes On the 

 Conduction of Heat in Crystals (Cambridge and Dublin Math. Journ.), and Professor 

 Thomson on the Dynamical Theory of Heat, Part V. (Transactions of Royal Society of 

 Edinburgh, Vol. XXI. Part I.) 



It is evident that all that has been proved in (14), (15), (16), (17), with respect to the 

 superposition of diffiarent distributions of pressure, and there being only one distribution of 

 pressures corresponding to a given distribution of sources, will be true also in the case in 

 which the resistance varies from point to point, and the resistance at the same point is 

 different in different directions. For if we examine the proof we shall find it applicable 

 to such cases as well as to that of a uniform medium. 



(29) We now are prepared to prove certain general propositions which are true in the 

 most general case of a medium whose resistance is different in different directions and varies 

 from point to point. 



We may by the method of (28), when the distribution of pressures is known, construct the 

 surfaces of equal pressure, the tubes of fluid motion, and the sources and sinks. It is evident 

 that since in each cell into which a unit tube is divided by the surfaces of equal pressure 

 unity of fluid passes from pressure p to pressure (^ — l) in unit of time, unity of work is 

 done by the fluid in each cell in overcoming resistance. 



The number of cells in each unit tube is determined by the number of surfaces of equal 

 pressure through which it passes. If the pressure at the beginning of the tube be p and at 

 the end p', then the number of cells in it will he p - p. Now if the tube had extended from the 



