IOWA ACADEMY OF 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, fa, ib, 



(3) the plane complex, z = a+i b. 



Next let the a-^z-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 ;r?/- plane or not, about 

 the 3-axis. Let./ be a new rotor, such that./'" rotates any vector 

 through m. 90'^ in a direction from the plane of x, y, toward 

 the pole z. 



By means of these two rotors i°, ./'", a vector may be turned 

 from the unit position {x) to any other position (A); and the 

 order of the rotations is indifferent. 



j 1° x=i°^j x=J x=-z. 

 f A= —A 

 . • . /== -1. 

 It follows that ,/'" may be expressed in the forms cos b + 

 j sin &, and e^ ^. Any unit vector, A, is therefore of the form 



A= (cos a -\- i sin a) (cos b + ./ sin b) x 

 = Qia+Jb ^^ 



Prom either of these forms the product or the quotient of 

 two vectors is evident. 



Dropping x, as before, and introducing tensors w^e obtain a 

 tensor-rotor algebra which, when the l 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+i b) (c+,/ 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+./ c. 

 But unfortunately in the latter form it does not obey the 

 laws of common algebra, except in addition, subtraction, and 

 multiplication by reals. 



