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 =e Se A , where g, denoting the rate of 
q 
variation of temperature in the direction of that line, is given by the equation 
g=(wt+u+w)-. . ; HEL 5 (86). 
Taking these values for /, m, 7, in the preceding general expression for the electro- 
motive force in any direction, we find 
P= = { Ove pers dyut G+ H uw t (Vr O)wut O +H 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 substance. This 
formula suggests the following changes in the notation expressing the general 
thermo-electric coefficients :— 
Pian sia i ates | 81) 
—P+W=22,W+%=2y, -8 +h" =23 ee ee ? 
which reduce the general equations, and the formula itself which suggests them, 
to— 
—-E=O0u+ 0+ o,w+ qG@w—Sv) 
—-F=4,u+ gut 6.wt+Qu-lw) - _. (88), 
—G=9,u+O,0+ pwt Gu—7w) 
P= 7 (G0? + pur + pur +200 +2, wu +24, uv) a9). 
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,, ,, ,, in these expressions vanish, and for which, conse- 
quently, the equations become 
—E=6u+(qw-—Sv) | 
—-F=¢v7+(Qu—Zw) (40), 
—-G= w+ (Cuv—74) 
P == (Ou + ov + yu ) BT La Pic 
166. The law of transformation of the binomial terms (y w— 3), &c., in these ex- 
pressions is clearly, that if 9 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, yu, y denote the inclinations of a cer- 
