(54) 
where sin is the symbol for the sine-product. This is the “vector 
product” of two vectors; it is itself a vector normal to the plane 
through the two vectors and equal to the area of the parallelogram 
described on the two vectors; the product is drawn in such a direction, 
that if the two vectors are viewed from the extremity of that product, 
the rotation of the first-mentioned vector to the second-mentioned one 
(by an angle smaller than 180°) is seen as the motion of the hands 
of a clock. As to 1,, it represents the unit vector in the direction 
of the positive normal to the wave-front. 
6. We see from (35) and (36) that B and D are both in the 
wave-front; that % is normal to €, so that 3 and D are elec- 
trically conjugate; further D is normal to $, so that B and D are 
also magnetically conjugate. J call B and D doubly conjugate. In the 
same way € and 5 are reciprocally electric and reciprocally magnetic 
conjugates or doubly reciprocal conjugates. If the wave-front is given, 
we find the directions of $ and D as the common conjugate diameters 
of its sections with the electric and the magnetic ellipsoid, the wave- 
front being supposed to pass through O. In general only one such 
a set is possible, of which either diameter can represent the electric 
indaction. If the direction of the electric induction Sis given, but 
not the wave front, we find the direction of D as the intersection 
of the plane which is electrically conjugate and that which is mag- 
netically conjugate to B. The direction of D and the wave-front will 
be determined in this way if the two planes do not coincide. These 
planes coincide only if 3 falls in the direction of one of the common 
set of three conjugate diameters of the clectric and the magnetic 
ellipsoid. We shall call these three directions the principal directions, 
which we indicate by 1, 2 and 3. The same holds if D is given 
and we want to determine 3. A principal direction may be defined 
as follows: if D and B fall in the same direction and at the same 
time € and 5 have the same direction, then the direction of D and 
B is called a principal direction; the direction of € and H will be 
called the corresponding reciprocal principal direction, and we shall 
indicate it by the same number, but distinguish it by an accent. 
If now & falls along a principal direction, D is indefinite in the 
plane of the two others. We shall only discuss the general case that 
the three principal directions are perfectly determinate; the more 
special cases are then easily treated. 
The same considerations apply to the vectors € and § with 
respect to the two reciprocal ellipsoids; instead of the wave-front we 
have then to do with the plane through € and $, which we shall 
