78 
can be seen by bringing the members of the first equation with P, 
in mutual power; we then find: 
BP, BiB Pe 
IV. The mutual power of two free vectors is called ‘bivector.” 
This is connected with the rational area of the rational triangle 
enclosed by the vectors made to coincide in a point, and the vector 
connecting the ends: 
2 Pol /(7,.2)x (2.2) x(a), 
oP. eR 2 MIE sie 
V. A bivector is represented by the product of 3 bound vectors. 
Simultaneously we find again for a rational triangle 
zE X Ee 1. 
EEE 
VI. A bivector is equal to the product of two equal, equidistant 
hound vectors with reciprocal values (fig. 10). 
mr?) : (3 r,) = (> = xX (Pv P)=( PIKE EDT 
VII. The mutual power of point and bivector is a bound triangle; 
at the same time the mutual power of a free vector with a bound one 
Dee De 
En (3. = =P os Pp Sa gen pee 5 
VIII. The product of a bound vector with a bivector is again a 
bound vector: 
tf 
APB) (Pr EN Aen Eet and Pp x 
PP) (GE) =P 
fea 2 es ig a. 33 a 
IX. The product of two bound vectors with the same origin is 
again a bound vector. 
X. Each point in the field of ratio can be replaced by the 
product of three points, each provided with an exponent representing 
the logarithm of the weight. This can be seen in different ways: 
PE aX Boa XE | 8, 44, 4 6, — = 
We might replace e* by g, we then find: 
dn) | (L,, L,, £,) etc. 
The weights (having the character of numbers), are logarithmically 
proportional to the rational areae of the opposite triangles. If P is 
the centre of gravity of the fundamental triangle, then the weights 
are mutually equal to Be. 
§ 32. We must then still mention the difference which must be 
made between the outer and the inner power of two rational vectors, 
of which the latter is always a scalar. It is natural to take in the 
