THEORY OF ELECTROMAGNETISM. 
725 
may be expressed by saying that they are fluxes.* Similarly, T Vl is an intensity. 
This particular result can, of course, be proved by a simpler process than the above. 
We now see that the meaning of B', obtained by defining B as a flux, = 47 r H Vl, and 
likewise the meaning of b' is independent of the particular position of matter we take 
as the standard. We also see similarly that the various terms in E', e', resulting 
from regarding these vectors as intensities, and utilising equations (28), (29), § 50, 
will be independent of the particular standard position chosen. And again, by 
Prop. II., § 8, we now see that equations (3) of last section must be true. 
54. Putting now Scr = 0, St = 0, the equation SI = l" Sm + m SI" gives 
- S S¥£(K£ = - mS 8¥£<ir£ + (l" + m'^S^V'T') Sm 
+ (»;2Sv/; -1 Sipxjj~ l cr ir v"r — xs S\fjr T v"i'). 
Now, by former paper, eq. (18), 
6m = SCiUWti'l’M- 
Hence 
2 Sm = S 
or, by eq. (10) of former paper, 
Sm = — mS Sxjjt,xjj .(8). 
Similarly, since m 3 is the same function of df as m is of xfj, 
Sm= -^SSnv-'C .(9). 
Also, for future use, note that since Sm = — S S = — S Sd^Ctm^ these 
equations give (former paper, p. 105), 
0 Qm = mxjj 1 , (1m — 
( 10 ). 
* It is interesting to notice a particular result of this. Since 9 is an intensity, 0 VZ is a flux. Hence 
[Prop. IV., § 8] SV 0 VZ — mSV'eVT. Dismissing the particular notation of this paper for the moment, 
and putting x, y, z for the coordinates of p and X, p, v for those of p, this may be written 
0\ 
_0_ / Sv \ 
dp i ~ Se ) 
W 
If we add [equation (5), §52, above] — m -1 0Z/09 to the left of this equation and —dl'/dd to the 
right, we get a well-known theorem of Jacobi’s. Comparing with the form of this theorem given in 
Todhunter’s ‘History of the Calculus of Variations,’ §323, equation (2), his G, r, v, 0, II are our 
l, ml', 0 (regarded as a function of p), 0 (regarded as a function of p), and ra~ x respectively. See also 
Todhunter’s ‘Functions of Laplace, Lame, and Bessel,’ §298, equation (17), and the supplementary 
volume of Boole’s ‘Differential Equations,’ p. 216. 
