Electromagnetic Wave-surface. 405 
therefore 
at wCVuAuB 
- .CVAB 
by the conjugate property. Now expand this quotient of two 
scalar products, and it will be found to be independent of | 
what vectors A, B, C may be. Choose them then to be i, y, k, 
three unit vectors parallel to the principal axes of w. Then 
RV pyipes) 
h= eas =F M2 Pay 
by the 77 & properties before mentioned. This proves (17). 
Transformation-Formula.—The following is very useful. 
A, B, C being any vectors, 
WAV bC= B(CA) = OCAB) in stiet is sks) 
Here CA and AB are scalar products, merely set in brackets 
to separate distinctly from the vectors Band C they multiply. 
This formula is evident on expansion. 
The Equations of Induction—HK and H being the electric 
and magnetic forces at a point in a dielectric, the two equa- 
tions of induction are ; 
femrlSseM yh FE SEES IS ELY) 
emt Baa pibleee tae ROO) 
c and yw being the capacity and permeability operators, and 
curl standing for VV as defined in equation (7). Let T' and 
G be the electric and the magnetic current, then 
T=cH /4a, G=pH/4r. . . . . (21) 
The dot, as usual, signifies differentiation to the time. The 
electric energy is HcH / 87 per unit volume, and the magnetic 
energy HuwH /87 per unit volume. If A is Maxwell’s vector 
potential of the electric current, we have also 
curl A=pH, E=—A.. . . . (21a) 
Similarly, we may make a vector Z the vector potential of the 
magnetic current, such that 
—curl Z=ch, H=—-Z .... (22) 
The complete magnetic energy, by a well-known transtor- 
mation, of any current system may be expressed in the two 
ways, 
T= >HyH /8r¥=42AT, 
the = indicating summation through all space. Similarly, 
the electric energy, if there be no electrification, may be 
