446 Prof. Thomson on the Dynamical Theory of Heat. 



equations of the second order and degree for the determination 

 of the functions t and V. 



179. It may be remarked, however, that the second term of 

 the second member of (49), when the prefixed negative sign is 

 removed, expresses the frictional generation of heat by currents 

 through the solid, and will therefore, when the electromotive 

 forces in action are solely thermo-electric, be very small, even in 

 comparison with the reversible generation and absorption of heat 

 in various parts of the body, provided the difi'erences of tem- 

 perature between these different localities are small fractions of 

 the temperature, on the absolute scale from its zero. Excepting, 

 then, cases in which there are wide ranges (for instance, of 50° C. 

 or more) of temperature, the second principal term of the second 

 member of (49) may be neglected, and the partial differential 

 equations to which t and V are subject will become linear ; so 

 that one of the unknown functions may be readily eliminated, 

 and a linear equation of the fourth order obtained for the deter- 

 mination of the other. 



180. Further, it may be remarked that probably in most, if 

 not in all known cases, the reversible as well as the frictional 

 thermal action of the currents, when excited by thermo-electric 

 force alone, is very small in comparison with that of conduction, 

 perhaps quite insensible. [See above, § 106.] Hence, except 

 when more powerful electromotive forces than the thermo-elec- 

 tric forces of the solid itself, and of its relation to the matter 

 touching it round its surface, act to drive currents through it, 

 we may possibly in all, certainly in many cases, neglect the 

 entire second member of (49) without sensible loss of accuracy ; 

 and we then have a differential equation of the second order 

 for the determination of the temperature in the interior of the 

 body, simply from ordinary conduction, according to the condi- 

 tions imposed on its surface. To express these last conditions 

 generally, a superficial application of the three equations (48) 

 with their nine independent coefiicients is required. 



181. When t is either given or determined in any way, the 

 solution of the purely electrical problem is, as was remarked above, 

 to be had from the seven equations (45), (46), and (47). These 

 lead to a single partial differential equation of the second order 

 for the determination of V through the interior, subject to con- 

 ditions as to electromotive force and electrical currents across 

 the surface, for the expression of which superficial applications 

 of (45) and (46) will be required. When V is determined, the 

 solution of the problem is given by (45) and (46), expressing 

 respectively the electromotive force and the motion of electricity 

 through the solid. 



