THEORY OF ELECTROMAGNETISM. 
721 
so that P may be called the “ power ” of the external forces and heat sources. P may 
be divided into Py the part due to the frictional forces, i.e., the expression on the 
left of eq. (38) § 42, and P 6 , the power of the really external forces and sources, i.e., 
forces and sources included neither in l nor x. From the equations of motion just 
obtained, and from eq. (14) § 46 above, 
E = P + jjjscVy* - [j^SCdS - j]«{ Sp'fdt + S dt (VAH/4tt + 0 0 VX)}. 
Now, SCVy = SV(Cy), since SVC = 0. Hence [eq. (4), § 5] 
— [LySCd£ =[[ySCdS — j'pySCdS = 
Hence 
E = P - J[ 4 {S p'fidZ + S dt {-yC + VAH/47T + 0 e VX)} . . (24). 
Putting now P = P c + Py and substituting the expression on the right of eq. (38), 
§ 42 for Py, we get 
E = P„ — j|,{Sp' (f + 4>» dt + SdS [ - (y + Y) C + V (A + a) H/4*- 
+ (0 e VX + d e Vx)]}.(25), 
where now 4?/ has been put for the 4>' of eq. (38), § 42, to distinguish it from the 4>' of 
equations (26), (27) below. 
50. In §§ 38, 39, it will be remembered that E, F, &c., stood for those parts only of 
the external forces which were due to X. Let, now, these symbols stand for those 
parts only of the external forces which are not involved in X. Thus in equation (20) 
we must change E into E -f c Vx -bad- VY [eq. (28), § 38], and similarly for the 
rest of the equations of motion. We thus get 
D/p' = - D,YW + (f + $>/ + 4> , )i v i / + F.(26). 
0 = — [(</> -f- dy + d>) JJv ] a + 1 + Pj.(27). 
e = d VZ - c Vx .(28). 
E = D V/ — ( dcVl/dt + c V£c) - (dA/dt + a) - V (y 4 Y) . . (29). 
e, = 0.(80). 
E* = \_(y + T ) TJvl + ij .(31). 
MDCCCXCII.—A. 4 Z 
