THEORY OP ELECTROMAGNETISM. 
759 
It is the line integral {including the terms contributed by SSEJJr,,) that thus appears 
on the right, which is ordinarily called the total electromotive force round the curve. 
Let us examine whether this statement is consistent with the one above about 
whole current and whole resistance. Suppose the motion steady so that K obeys the 
laws of incompressibility. Consider an infinitely small tube of flow, and let this be 
the line along which the integral is taken. Let c be the whole current flowing along 
the tube. Consider an elementary right section of the tube of length T dp, and cross- 
section T d%. Thus 
K = cXJK/Td2, dp = UKTc/p, 
and, therefore, the part contributed to the integral — JScZpRK by the element in 
question is 
- cSUKRUK. Tdp/Tdt. 
If then we choose to define as follows : (1) — SUKRUK = the specific resistance 
of the body at the point, (2) specific resistance X Tc/p/Tc/2 = the resistance of the 
element, (3) the sum of the resistances of all the elements = the whole resistance of 
the tube, we shall have 
— JScZpRK = current flowing along tube X whole resistance of tube, 
or 
conductivity of tube X (— JSc/pRK) = current flowing along tube, 
which defines the conductivity as the reciprocal of the whole resistance. Now split 
any finite tube of flow into an infinite number of such elementary tubes, call the sum 
of the conductivities of the elementary tubes the whole conductivity of the finite tube, 
and call the reciprocal of this last the whole resistance of the finite tube. We shall 
then have that the mean of the values of — JScZpPtK for the elementary tubes = the 
whole resistance of the finite tube X the whole current along it. All this may, I 
think, be said to be in complete agreement with the ordinary theory, but it serves to 
call attention to the fact—important in connection with the longitudinal effect 
mentioned in § 79 above—that anything which interferes with the ordinary lines of 
flow will alter the apparent resistance. 
89. To return to our immediate purpose, we are now at liberty to say that the line 
integral of any term contributed to the right of equation (29), § 50, round a closed 
circuit implies an equal electro-motive force round the circuit in the ordinary sense. 
Equations (22) and (22 a) are easily seen to follow. 
Comparing, now, equations (20), (21), (22), (20 a), (21a), and (22 a), with equations 
(4), (5), (6), and (7) on p. 97, vol. 8, ‘ Encyc. Brit.,’ 9th ed., we see the results of the 
theory of reversibility only differ from the ordinary theory in having cr — 6P g in place 
of o', while those of the theory of irreversibility differ widely. 
