Prof. Tliomson on the Dynamical Theory of Heat. 379 



If from the six equations we eliminate ^, if, &c., we obtain 



D = 



c, b, -2/, 



= 



And the equation □ =0 is therefore the result of the elimi- 

 nation of I, rj, ^ from any three (other than the excepted com- 

 binations) of the six equations. But from what precedes, it 

 appears that the equation Q =0 must be satisfied when the qua- 

 dratic function breaks up into factors, and consequently D must 

 contain as a factor the discriminant 



K= \ a, h, g \ 

 \ h, b, f \ 



\ 9, f, c \ 

 of the quadratic function. This agrees perfectly with the results 

 obtained long ago by Prof. Sylvester in his paper, " Examples of 

 the Dialytic Method of Elimination as applied to Ternary Systems 

 of Equations," Camb. Math. Journ. vol. ii. p. 232; but accord- 

 ing to the assumption there made, the value of D would be (to 

 a numerical factor pi-es) abcK. The correct value is by actual 

 development shown to be D = — 2K^ It would be interesting 

 to show a priori that Q contains K^ as a factor. 



2 Stone Buildings, 

 March 28, 1856. 



XL VIII. On the Dynamical Theory of Heat. — Part VI. Thermo- 

 electric Currents. By William Thomson, M.A., Professor 

 of Natural Philosophy in the University of Glasgow. 

 [Continued from p. 297.] 



§§ 141-146. Elementary Explanations in Electro-cinematics and 



Electro-mechanics. 

 141. "\'^7'HEN we confined our attention to electric currents 

 ▼ ▼ flowing along linear conductors, it was only necessary 

 to consider in each case the ivhole strength of the cun-ent, and the 

 longitudinal electromotive force in any part of the circuit, without 

 taking into account any of the transverse dimensions of the con- 

 ducting channel. In what follows, it will be frequently necessary 

 to consider distributions of currents in various directions through 

 solid conductors, and it is therefore convenient at present to notice 



2C 2 



