. Jqly — Vector E.rpressions for- Ciin-e&. 377 



4. Invariants of Unary qiiantics with non-commutative coefficients. 



Tlie following rule may consequently be stated : — In order to deter- 

 mine the pole OT of the chord joining the points determined by 



l^x^ \1hixy \l-Af- = ^^ 



foi-m the (12)' invariant of this scalar quadratic, and of the vector 

 quadratic 



(a^ - ttip) x^ + 2 (tti - a{cij) xy + (^Zg - a^^Ts) y-, 



and equate to zero the result. 



This suggests consideration of invariants derived from binary func- 

 tions of xy whose coefficients are not commutative. In other words, 

 the investigation is suggested of those functions of the coefficients of 

 the various powers of x and y in the expressions 



{PoPiP^ ■ ' • PnX'^l/T, and (Ml • • • 5'»')(^y)"'» 



which remain unaltered when a linear scalar transformation is effected 

 on X and y. As it is generally impossible to determine values of x and y 

 which shall make these binary functions vanish, it is most convenient 

 to treat these invariants by means of differential operators. 

 Suppose 



x = IX+ m Y, and y = I'X + ni' Y, 



and suppose that when this scalar transformation is made, 



{PoPl ■ • . Pn){xyr = (Po?l . . . PnXXYf, 



and 



{q,q,...q,,)[xyr={Q,Q,... a^XXF)"'; 



or, in other words, suppose that the binaries on the left-hand side of 

 these equations transform into the binaries on the right. 

 !Now, for this linear transformation, 



d d d ■!!., fi;:: 



^dY-'''dX=^^''-^"'^dy^ 

 and 



and, consequently. 



