THEORY OF ELECTROMAGNETISM. 
753 
It will now be seen that to obtain m' or ur" from ur it is only necessary to change 
A, B, C, and H into A', B', C', and H', or into A”, B", C", and II" respectively. The 
statement is obvious so far as A, C, and H are concerned. With regard to B, we 
have, by §§ 7, 9, 
B'VwH' = 
=■ x^BVx'ojB. [Tait’s ‘Quaternions/ 3rd ed., § 157, eq. (2)], 
which, with equation (5), proves the statement. 
82. For the theory of reversibility, it is assumed that l contains a term g given by 
g = — SDct©. For the theory of irreversibility, it is assumed that x contains a 
term g given by g = — SKut 0. Denote the various parts of E, f &c., depending 
upon g by the suffix g. It conduces to clearness to arrange the general results of 
these two assumptions in parallel columns thus :— 
Theory of Reversibility . 
I contains a term g given by 
g = - SD ^0 .... ( 8 ). 
This contributes terms Ey, "E^ to the right of 
equations (29), (31), § 50, given by 
Ey = -wQ , E*, = 0 . . . (9). 
In this is not included the part of A due to g, 
but this is practically given by equations (15), 
(16), below. By equation (11), § 46, 
Xy = SD (1 + SH h V.) . . (10). 
By equations (11), (12), § 34 (putting vr g for 
0 sr/06>) 
fy = - SD (l + sh h v.) 
+ SD (1 + SH h Y.) -srA . (11). 
ft = - [SD (1 + SH h V.) wUQ a + i . . . (12). 
Hence to the left of equations (35), (36), § 40, 
are contributed for a steady field 
6ft = 6 {— SC (1 + SH h V.) w e 0 
+ SC X (1 + SH 1 h V.) ^Vj} . (13). 
Ofsy = - e [SC (1 + SE h V.) wUQ fl + J . . (14). 
By equations (34), § 50 
By = — 4 ~ h VSDv0 
= 4t r {VBD 0 - 2CHSD0} . . (15). 
MDCCCXCIL— A. 
Theory of Irreversibility. 
x contains a term g given by 
g = — SKwO .... (8a). 
This contributes terms Ey, E., y/ to tbe right of 
equations (29), (31), § 50, given by 
Ey = — w©, Ejy = 0 . . (9 a). 
In this is not included the part of a, due to g, 
but this is practically given by equation (16 a) 
below. 
Contributed to the right of equations (35), (36), 
§ 40, are terms (for any field, steady or not) 
given by 
efg = - SK (2 [ 0 ] + 3 [l] 
+ 4 [2]) 0 - . . (13a). 
efts = 0 [SKwUr]„ + } .(14a). 
5 D 
