202 PROPAGATION OF ELECTRICITY. 



215. ANISOTROPIC CONDUCTORS. We have hitherto only con- 

 sidered the case of isotropic bodies that is to say, bodies which 

 have the same properties in all directions. If the medium is 

 anisotropic, but homogeneous like crystallized bodies, the physical 

 phenomena depend on the direction in which they are regarded. 



The expansion, for instance, may be very unequal. There are 

 then three principal directions, rectangular to each other, and such 

 that the expansion of an infinitely thin cylinder considered in the 

 medium parallel to one of the principal directions takes place along 

 the axis of the cylinder. Each of these directions is denned by a 

 particular coefficient, which gives for the medium three principal 

 coefficients of expansion, /, /' and /". If we suppose in the medium 

 an infinitely thin cylinder in any given direction, making angles with 

 the principal directions, the cosines of which are a, a! and a", this 

 cylinder turns at the same time that it dilates, but remains rectilinear 

 if the medium is homogeneous. The expansion parallel to the axis 

 of the cylinder is equal to the sum of the projections on this axis 

 of the three principal expansions, and the value of the coefficient L 

 relative to this direction is 



The same considerations apply to the propagation of heat, to the 

 propagation of electricity, and to electrostatic induction. 



In an anisotropic medium, the flow of heat at a point is no longer 

 perpendicular to the corresponding isothermal surface ; but, just as in 

 the case of the expansion, and generally for all properties which are 

 linear functions of the causes on which they depend, there are again 

 three rectangular directions along which the flow of heat is perpen- 

 dicular to the isothermal surfaces, and to which correspond the three 

 principal coefficients of conductivity , k' and k". 



Across an element of surface dS, the perpendicular to which 

 makes angles with the axes the cosines of which are a, a' and a", the 

 flow of heat is equal to the sum of the flows which correspond to 

 the projections a</S, aWS, a'WS of the element perpendicular to the 

 three principal axes ; taking, then, these three directions for axes 

 of the co-ordinates, 



In like manner again, if <:, c', c" are the coefficients of electrical 



