756 
MR. A. McAULAY ON THE MATHEMATICAL 
the principal term in B' should be p 0 H'. It is needless to say that this will not in 
general be the case if B involves 3). In fact, in a steady field, the only conditions we 
should be able to assert that B imposed upon the field would be those which resulted 
from equations (17), (18), which do not contain p, 0 Hat all. 
85. The conclusion we are bound to come to is that on the present theory there 
must be no term in l which will result in H VZ containing D. Thus the thermo- 
magnetic effects must be left to the theory of irreversibility. 
It might be thought that all difficulty w r ould vanish if, in g, instead of D we substi¬ 
tuted d, since this would involve terms in e similar to those given above for E and 
since E = — e. This, however, is not the case. No term in e can affect steady 
currents. The equation E + e = 0 determines the value of d but does not affect the 
value of C. Again, since in equations (13) (14) we should have c instead of C, we 
see that no thermal effects would, owing to g , occur in a steady field. 
86. None of these difficulties and restrictions are met with in the theory of 
irreversibility. We shall find, however, that the thermoelectric consequences of 
that theory are inconsistent with known facts. Both theories, therefore, are assumed 
as true in jiart. 
In connection with thermoelectricity it is necessary first to establish equations (19) 
to (22) and (19a) to (22a). In these equations the notation adopted is that of 
Professor Chrystal’s article in the ‘ Encyc. Brit.,’ 9th ed., vol. viii., p. 97. He 
there considers a circuit of two unstressed isotropic metals a and b, one of the 
junctions being at temperature 6, and the other at temperature 9 0 . The positive 
direction round the circuit is taken as that from the metal a to the metal b at the 
junction 9. n, c t* are the Peltier and Thomson effects at the temperature 6, and n 0 
the Peltier effect at the temperature 9 0 . E is the electromotive force round the 
circuit due to thermoelectric effects. 
To consider such a case as this, B and C are, of course, ignored, and A is supposed 
a scalar. For the future, in order to distinguish more clearly between the theories, 
P will be substituted for A on the theory of reversibility. It is necessary now to 
distinguish between d/dO and d/d9. The former denotes differentiation when 9, T' 
are taken as independent variables. Now 7 in all the commoner experiments on 
thermoelectricity, different parts of the circuit are not similarly strained, but 
similarly stressed. This may be taken account of by regarding 'P as a function of 9. 
Regarding it as such d/d9 denotes total differentiation with regard to 9. 
87. cr, the Thomson effect, is defined by saying that the heat “absorbed” by the 
metal between two sections at temperature 9 and 9 + d9, while a unit quantity of 
electricity passes in the direction from the first to the second section, is add, or the 
rate at which heat is absorbed in this part equals the rate at which electricity is flowing 
* There seems no danger in using a here for this scalar, though in the rest of the present paper it is 
taken as the type of an intensity {vector'), nor in using E here for the electromotive force round the 
circuit, though in the rest of the paper it stands for intrinsic energy. 
