IOWA ACADEMY OP SCIENCES, 
203 
has the laws of operation of ordinary algebra, and which com- 
bines with it to form an algebra of tensors and rotors — the 
ordinary complex algebra. 
The kinds of number involved in this algebra are: 
(1) reals, a, b, c, 
(2) plane imaginaries, m, ib, 
(3) the plane complex, z = a+^ b. 
Next let the cc|/-plane be the equatorial plane of a sphere of 
which Z is the pole. Let the power of the rotor % be extended 
so as to rotate any vector, whether in the cc?/- plane or not, about 
the 2 -axis. Let J be a new rotor, such thati“ rotates any vector 
through m. 90® in a direction from the plane of cc, toward 
the pole 2 . 
By means of these two rotors a vector may be turned 
from the unit position {x) to any other position (A); and the 
order of the rotations is indifferent. 
J x=f'j x=j x=%. 
f A== —A 
.•./--I. 
It follows that may be expressed in the forms cos h + 
j sin &, and Any unit vector. A, is therefore of the form 
A= (cos a -\-i sin a) (cos & -|- i sin h) x 
From either of these forms the product or the quotient of 
two vectors is evident. 
Dropping x^ as before, and introducing tensors we obtain a 
tensor-rotor algebra which, tohen the i and j binary factors are 
kept separate, has the laws of operation of common algebra, and 
has many of the advantages of a vector algebra without its 
limitations. 
The most general quantity in this algebra is the double 
complex 
(a-j-'i b) (c-j-J d), 
in which a, b, c, d, are connected by one relation. The double 
complex may be expressed in the form 
(‘+^) 
which is identical with 
a+i b+i c. 
But unfortunately in the latter form it does not obey the 
laws of common algebra, except in addition, subtraction, and 
multiplication by reals. 
