1351 
A and B and expressed in natural measure'). This involves that the 
same rotation may be represented in many different ways by two 
vectors in the plane JV. 
For the rotation R we shall also use the symbol [A.B]. 
By the vector product [A.B.C] of three vectors A,B,C at a 
point of the field-figure and not lying in one plane we shall under- 
stand a vector D the direction of which is conjugate with each of 
the three vectors (and therefore with the three-dimensional extension 
A,B,C), the direction of D corresponding to those of A,B and C 
in a way presently to be indicated, while the magnitude of D, 
expressed in natural measure, is equal to that of the parallelepiped 
described on A, B and C and expressed in the same measure. This 
definition involves that the value O is ascribed to the vector product 
of three vectors lying in one and the same plane. 
A further statement about the direction of D is necessary because 
two opposite directions are conjugate with A,B,C. For one set of 
three directions A,,B,,C, we shall choose arbitrarily which of its 
two conjugate directions will be said to correspond to it. If this is 
the direction D,, then the direction D corresponding to A, B,C will 
be determined by the rule that D, passes into D by a gradual passage 
of the first three vectors from A,, B,,C, into A, B,C, this latter 
passage being effected in such a way that during the change the 
vectors never come to lie in one plane. 
The vector product [A.B.C] takes the opposite direction when 
one of the vectors is reversed as well as when two of them are 
interchanged. We must therefore always attend to the order of the 
symbols in [A.B.C]. 
The veetor product possesses the distributive property with respect 
to each of the three vectors, so that e.g. if A and A, are vectors, 
(Ag AASBECHS TAL BA 6] + [Ass BC]. 
From this we can infer that [A.B.C] depends only on C and 
the rotation R determined by A and B. For this reason we write 
for the vector product also [R.C]; in calculating it we are free to 
replace the rotation R by any two vectors by means of which it 
can be represented. 
If R, R, and R, are rotations in the same plane, such that the 
value and direction of R are found by adding R, and R, algebrai- 
cally, we have, in virtue of the distributive property 
[R, .C]+[R,.C]=[R.C] 
1) If, according to circumstances, different signs are given to R, the angle 
whose sine occurs in the formula for the area of a parallelogram must be 
understood to be positive in one case and negative in the other 
