DYNAMICAL THEOIIY OF HEAT. 



1C9 



E= fe +^0' + d fe» 



dx dy dz 



t? — & m„ dt . dt ,, 

 dx ' dy dz ' 



r dt dt dt 



Q = aZ* + dy* + Tz* 



(45). 



176. The body, being crystalline, probably possesses different electrical conduc- 

 tivities in different directions, and the relation between current and electro-motive 

 force cannot, without hypothesis, be expressed with less than nine coefficients. 

 These, which we shall call the coefficients of electric conductivity, we shall denote 

 by k, \, &c. ; and we have the following equations, expressing by means of them 

 the components of the intensity of electric current in terms of the efficient elec- 

 tro-motive force at any point of the solid : — 



-(■-^♦'('-^MZ) 

 '-* (*-£)^(*-S)**(°-£) 



(46). 



These equations (45) and (46), with 



dh 

 dx 



+ 



dx 

 dy 



+ 



dz 



= 



(47), 



which expresses that as much electricity flows out of any portion of the solid as 

 into it, in any time, (in all seven equations,) are sufficient to determine the seven 

 functions E, F, G, V, h, i, j, for every point of the solid, subject to whatever con- 

 ditions may be prescribed for the bounding surface, and so to complete the pro- 

 blem of finding the motion of electricity across the body in its actual circum- 

 stances ; provided the values of 7- , ~T ' 7h are known, as they will be when 



the distribution of temperature is given. We may certainly, in an electrical pro- 

 blem such as this, suppose the temperature actually given at every point of the 

 solid considered, since we may conceive thermal sources distributed through its 

 interior to make the temperature have an arbitrary value at every point. 



177. Yet practically the temperature will, in all ordinary cases, follow by con- 

 duction from given thermal circumstances at the surface. The equations of mo- 

 tion of heat, by which, along with those of thermo-electric force, such problems 

 may be solved, are as follows: — (1) Three equations, 



