79 
further considerations z as base vector; for continuous change the 
ends form the logarithmie circle. If then still the mutually perpendicular 
vectors 2, and @, are introduced we can write for a free vector: 
cy AS? a ae 5 
Pp. = 7) == iS (o ex) X (as ey), 
corresponding to the previously mentioned equation : 
De (2) 
Pile BE NDA 
For the outer power, which is, indeed, a bivector, holds: 
j= EJs. ler (Le) 
or reversely, according to the choice of the positive sense of revolution, 
which is evident from the determination of the multiplical: 
e l 
So Come —_ — 1 
Eep Pete = [a], = 2 see) = Pele — [y= Re 
1 € 
When introducing the rational angles we arrive for the outer 
power and the inner respectively at the following equations: 
Sa aaa! u, 
lo, 0,| ET Q, Os CT =e e 
; 
1 
a3; =: Us 
(011 Qs) = Or» On ST 
N 
1 
Of this important applications can be made. 
§ 33. In the plane the mode of reckoning with complex powers 
is not inferior to the one with vectors. To determine the situation 
of a point in the field of ratio we can use: 
bd ae 1 
ay _— e(-) een 0, = (—) ==; eu —1 
from which ensues: 
e= erw ; yO, sr u. 
The multiplication of two directed areae (or vectors) mentioned 
in $8 leads to the rational cosinus formula: 
u,\* 
eo, — A |) (9.) . (e. 0, CT *) | 
the mutual power to the analogon of pr Morvre’s formula: 
== ns 
Oi Os = 913 Oa: aah, 
From this can again be deduced 
"cru. sriu) = er (u”) . sr (u), 
besides 
TE wT 
neu, VI) — ev : 
which can again serve for the deduction of rational goniometric 
relations and for the development in series of product. 
