440 Prof. Thomson on the Dynamical Theory of Heat. 



U V IV 



— , — , — , where q, denoting; the rate of variation of temperature 



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



q={u^ + v'^ + w'') (36). 



Taking these values for /, m, n, in the preceding general expres- 

 sion for the electromotive force in any direction, we find 



the negative sign being omitted on the understanding that P 

 shall be considered positive when the electromotive force is from 

 hot to cold in the substance. This formula suggests the follow- 

 ing changes in the notation expressing the general thermo-elec- 

 ti'ic coefficients : — 



which reduce the general equations, and the formula itself which 

 suggests them, to 



-F = i|r,M + <^y +e^iv + {du-^w) I . . (38), 

 — G = (f)^U + 6lV +i/r?i; + {^v —'r)u)J 



P = - {0u^ + cj)v- + y^w^ + 2d^vw + 2(\>^wu + ^^^uv) . (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 ^,, <f)^, T|rj in these 

 expressions vanish, and for which, consequently, the equations 

 become 



— E = ^M +{'r)W — dv)-\ 



-¥=(fiv +{du-^w)l . . . (40), 



— Gr=yjrw+{^v — tjujJ 



V=-{eu^ + (j)V^ + ^lrw^) .... (41). 



166. The law of transformation of the binomial terms {rjiv—^v), 

 &c. in these expressions is clearly, that if p 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 \ /j,, v denote the inclina- 

 tions of a certain axis fixed in the substance, which I shall call 

 its axis of thermo-electric rotation, to any three rectangular lines 



