JoLY — The Associative Algebra applicable to Hyj)ersp)ace. 101 



let $'1 = 1 -j\h ; then, by the operator, the first unit system (i) is 

 converted into the new system of ratitually perpendicular units 



qiMi'^i^Ji), qihqi'\ &(i., qii„,qf\ 

 Nest, as the vectors involved are units, 



Sihqr^ +j2 = (1 - Mihqr^) qMi'^ =hO- -jzqMr'-), 



and accordingly the new operator $'2 ( ) $'3"^ will convert q^nqr^ into /a, 

 and will leave unchanged any vector perpendicular to both ; such a 

 vector is/i. Again, for brevity, let q^=\ -j%q\Hq\^i and the units are 

 transformed by this double process to 



juh qzqihqr^qi^ ■ ■ • q-iq^inq^'q-r^- 



Eepeating this process, a function Q = q^qm-i • • - q%q\ is at last found, 

 and the operator Q ( ) Q~^ derived from this will convert the set of 

 arbitrary units ixH . . . im into the new ^etjij^ ■ . -Jm' 



37. Consider a little the formation and structure of these functions 

 q„ and Q„, if Q„ = q„q,,_^ . . . q^q^. 



The set of equations — 

 qi = 1 -jih, 



qi = 1 -Mihqi~\ 

 qz ^ I -M2qihqi~^q2~^ 



2'» = 1 -juqu-iqu-2 ■ • ■ iuqr^q^'^ - • . q^^l~\ 



lead to equations of the type Q,, = Q,(_i - juQu-iK- Hence, it is easy 

 to see that 



Qi = 1 -j\h, 



Qi = 1 -Jlh -J2h +J2jlhh, 



Q3 = 1 -Jlh -J2h -jzH -^-jijiHH +J3j\hh +J3j2hh, 

 and generally 



Q„, = 1 - '$j\i + %%JsjVti - %f%jJtjJJtis + &c., 



in which s, t, u, &c., are distinct integers comprised between 1 and m. 

 Of course, in this expression, J2J\i\i2=j\J2i2H for instance, as the double 

 interchange does not alter the sign of the term. 



