202 
IOWA ACADEMY OP SCIENCES. 
Gystopteris fragilis Bernh. Union, Page, Pottawattamie, and 
Fremont counties. Rich woods; common. 
Onoclea struthiopteris Hoffm. Decatur county. Rich woods; 
infrequent. 
Ophioglossace^. 
Botrychium virginianum Swartz. Pottawattamie county. 
Rich woods; frequent. 
EXTENSION OF COMPLEX ALGEBRA TO THREE- 
FOLD SPACE. 
BY T. PROCTOR HALL. 
Taking rectangular coordinates in a plane, let x, y, be unit 
vectors along the axes of X, Y, respectively; and let A be any 
unit vector from O in the plane, making an angle a with X. 
Let ^ be a rotor such that rotates any vector, A, through 
n. 90° in the positive direction. Then 
i x=y. 
f A= — A 
. • . = — 1 . 
Equating vectors from O: 
A=a:^cos a y sin a. 
= (cos a i sin a) x. 
=6^® X, (by expansion in series). 
A vector (A) is here considered to be composed of three dis- 
tinct components or factors; a unit direction (a?), a length 
(which, for the sake of simplicity, is here considered unity), 
and a rotor (cos a -f i sin a, or e^®) which has rotated the vector 
from unit position (^c) to any other position (A) in the plane. 
The product of two vectors is 
Aj Ag = (cos a^-\- i sin (cos + i sin x.x. 
== [cos (ttj + CTg) -p i sin (a^ + a^)] x. 
= e ^ X. 
Since x is unity in every one of its capacities, xx = x, as 
given above. 
The unit vector, a?, is a factor of every term of this algebra, 
and may be dropped, leaving an algebra of rotors only, which 
