134 PROFESSOR W. THOMSON ON THE 
in each metal respectively. It is easily shown (as will be seen by the treatment 
of the subject to follow immediately) that if the values of ¢,, ¢,, &c., depend 
either on the section of the conductor, or on the rate of variation of temperature 
along it, or on any other variable differing in different parts of the conductor, except 
the temperature, a current might be maintained by the application of heat to a ho- 
mogeneous metallic conductor. We may, therefore, at once assume them to be, if 
not invariable, absolute functions of the temperature. From this it follows, that 
if ¢ ¢ denote any function of ¢, the value of the sum, / ptodt, for any conducting 
arc of homogeneous metal, depends only on the temperatures of its extremities ; 
Za : : 
and therefore the parts of the sums Sa, and ia , corresponding to the successive 
metals in the principal conductor, are respectively 
Ty T, TnI Tr 
- o,dt, — OT, At, vrrereeee — o,dt, — o,dt, 
1 2 1 
He Us Ta T 
0 
i, uy LA Th 
and =f Pia, af Bed tn =f on dt, =f Sud tn 
Di. it nb Dn Toy ae 
T) Ty Tr-1 Th 
F=J { 1, +0,+ woe +0,—f o,dt-[ o,dt—-~~f aud oat}. . (10) 
. 7, Ty T, 
T a n—1 Tr 
rh Bi, Th 007, “if 1 -f Tn o, 
~ 1 f 2 4 reese ——— ee Se een pati fy Nee —_1qdi=0 . 11) 
as T, Jy, ¢ 7, # ge r, # ( 
which are the fundamental equations of thermo-electricity in non-crystalline con- 
ductors. In these, along with the equation 
P+F 
aa SulgnneSlRit gun epee aii 
which shows the strength of the current actually sustained in the conductor when 
an independent electro-motive force, P, is applied between the principal electrodes 
k, FE’, we have a full expression of the most general circumstances of thermo- 
electric currents in linear conductors of non-crystalline metals. 
114. The special qualities of the metals of a thermo-electrie circuit must be 
investigated experimentally before we can fix the values of m,, m,, &c., and 
c,, 7, &c., for any particular case. The relation between these quantities 
expressed in the general equation (11), having, as we have seen, a very high 
degree of probability, not merely as an approximate law, but as an essential 
truth, may be used as a guide, but must be held provisionally until we have suf- 
ficient experimental evidence in its favour. The first fundamental equation (10) 
admits of no doubt whatever in its universal application, and we shall see (j 123 
below) that it leads to most remarkable conclusions from known experimental 
facts. 
