THEORY OF ELECTROMAGNETISM. 
719 
transformed into a surface integral. Transforming and noticing that the part of the 
surface integral is zero, the last term in equation (13) is obtained. 
We may now obtain E. To do this, combine the first integral of equation (13) with 
d (Jlf Of ds + If Of s ds)jdt. We thus get by equations (11), (12) of § 34, and equation 
( 4 )> § 5 
[j] Ofds + [j Ofds - [J 0 S dtfJX. 
Thus, from eq. (13), § 34, 
E = [j] 9}<h + J] Bf.ds - fffjS/ [D„ (p' + V'W) - 
- Sc d V/ + SC (dcVl/dt + A - f7l)} ds > 
- [| Spy dt - f[ iSdt (YAH/477 + 0 0 VA) 
J 
(14). 
C. The Equations of Motion. 
47. The symbols E, F, <J>, &c., will now again be supposed to stand for the whole 
external forces including those due to friction. The parts contributed by all of these 
except d> to SQ Sq can be written down at once. By the former paper p. 107 the 
force per unit volume (of present position of matter) due to d> is d>'A'. d? is assumed 
to be self-conjugate* and of Class I of § 9 above. Thus d>' is also self-conjugate, and 
therefore there is, due to it, no couple per unit volume. The force per unit surface at 
a surface of discontinuity is — [d>'UV] K + Thus the part contributed to XQ Sq 
by <f> is 
jj S 3 p'& dt - f jj S Sp'dq'V/ ds. 
Hence collecting all the terms 
SQ Sq = - [j] S Bp (F + dq'V/) ds - j] S Bp' (F, ds - d>' dt) J 
<■ • ( 15 )- 
+ | j] (Se Sd + SE SD) ds + jj (Se, Sd + SE, SD) ds 
48. To obtain the equations of motion from these results, it must be remembered 
(§ 38) that while Sp' and Sd are quite arbitrary, this is not the case with SD and SH. 
We adopt the same method here as in § 38, i.e., we add to the SL for all space 
* It is clear by the work in tbe former paper (pp. 106 to 108) that there is no necessity to make 
this simplification. On the other hand nothing seems gained by not making it. 
