202 IOWA ACADEMY OF SCIENCES. 



C/jstopteris fragilis Bernh. Union, Page, Pottawattamie, and 

 Fremont counties. Rich woods; common. 



Onoclea strutMopteris HofCm. Decatur county. Rich woods; 

 infrequent. 



Ophioglossace.e. 



Botrychium virginianuni 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 i be a rotor such that i^ rotates any vector, A, through 

 n. 90° in the positive direction. Then 

 i x=y. 

 f A= —A 

 . • . f = —1. 

 Equating vectors from O : 



A=xcos a -\- y sin a. 

 = (cos a + i sin a) x. 

 =e** 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 length 

 (which, for the sake of simplicity, is here considered unity), 

 and a rotor (cos a-{- i sin a, or e^") which has rotated the vector 

 from unit position {x) to any other position (A) in the plane. 

 The product of two vectors is 



Aj Ag = (cos ttj + i sin a^) (cos a^ + * sin aj ^•^• 

 = [cos (flj + a^) + i sin {a^ + a,)] x. 



= 6 ^ ^^1 + S^ X. 



Since x is unity in every one of its capacities, x x = x, as 

 given above. 



The unit vector, x, is a factor of every term of this algebra, 

 and may be dropped, leaving an algebra of rotors only, which 



