DYNAMICAL THEORY OF HEAT. 165 



most general conceivable equations expressing flux of heat in terms of variations 

 of temperature along rectangular lines of reference in a solid, will show the nature 

 of the answer. 



164. The direction cosines of the line of greatest thermal variation, or the per- 



pendicular to the isothermal planes, are -,-,-, where q, denoting the rate of 



variation of temperature in the direction of that line, is given by the equation 



q = (u 2 + v 2 + w 2 ) ■ (36). 



Taking these values for I, m, n, in the preceding general expression for the electro- 

 motive force in any direction, we find 



p = - / Ou 2 + $ v 2 + -^w 2 + (<p' + 4/") v w + (-4/ + 6") wu + (& + <p") uv\ 



the negative sign being omitted on the understanding that P shall be considered 

 positive when the electro-motive force is from hot to cold in the substaDce. This 

 formula suggests the following changes in the notation expressing the general 

 thermo-electric coefficients: — 



<P' +^ = 2 6,, 4' + &' = 2 <f> lt 0' + <p- = 2 ^ 



-0' + y = 2 £ ,—4/ + 0" = 2 n , -6' +(/>" = 2 a 



which reduce the general equations, and the formula itself which suggests them, 

 to— 



-F = ^ l u + (pv+6 1 w + (§u-£w) \ . . (38), 

 — G = qj 1 u + O l v+ v|/«/ + (^- tju) 



P = - (6 u 2 + (p v 2 + ^ w 2 + 2 X v w + 2 : w u + 2 -4/j u v\ ... (39). 



165. The well-known process of the reduction of the general equation of the 

 second degree shows that three rectangular axes may be determined for which 

 the coefficients 6 X , (p x , \, in these expressions vanish, and for which, conse- 

 quently, the equations become 



— E = 6 u + (t) w — § v) 



— F = <j> v + ($ u - £ w) \ 



— G = 4w + (^v- tj u) 



V =- (Ou 2 + (pv 2 + ^w 2 \ . . . (41). 



1 66. The law of transformation of the binomial terms ( n w - 5 v), &c, in these ex- 

 pressions is clearly, that if q denote a quantity independent of the lines of reference, 

 and expressing a specific thermo-electric quality of the substance, which I shall call 

 its thermo-electric rotatory power, and if A, [x, v denote the inclinations of a cer- 



