758 
MR. A. McAULAY OR THE MATHEMATICAL 
and 
- (2A + o dk/dO) ScZpVdSC dt. 
But the rate of absorption of heat also = cr d6 X the rate of flow through the element 
from 9 to 9 -f dO 
= O-S dpVO SC dt. 
Equating these different expressions for the same quantity, we get equations (19) and 
(19a). 
Equations (20) and (20a) are obtained in an exactly similar way. It need only 
now be said that from equation (14) the rate of absorption of heat at an element of 
the boundary of the metals a and b [fig. 3] is 
6 [PSC dt~] a+b = — 9 [P] a _ b SC dt a , 
Fig. 3. 
and also that by the definition of the Peltier, effect IT, it is = 14 X the rate of flow 
from the metal a to the metal b 
= - nsc dt a , 
from which equation (20) follows. Similarly for equation (20a). Equations (21) and 
(21a) are easily deduced from those now established. 
88. In connection with equations (22) and (22a) it is advisable to make what may be 
looked upon as a digression, to examine whether, on the present theory, we have 
a right to identify the line integral of any part of E round a circuit with what, in the 
laboratory, is known as the electromotive force of the particular kind round it. To test 
this, we must see whether the total line integral of IlK round the circuit = what is 
called the whole resistance of the circuit X the whole current. 
In equations (29), (31), § 50 occurs a scalar y -f- Y or v which, unlike the other 
terms in the equations, does not depend directly or indirectly (as is the case with 
dk/dt + a) upon the form of l or x. Consider now any closed curve which, if it anywhere 
crosses a surface of discontinuity, passes, we shall suppose, from the region a to the 
region b. Then, in the expression — J SE dp — SSE,U^ a this unknown scalar does not 
appear as can be easily seen by equations (29), (31). Before taking this line integral, 
remove PJ£ to the left of equation (29), § 50, keeping all the other terms on the right. 
