378 Proceedings of the Royal Irish Academy. 



Hence 



( d d\" 

 {Im' - l'm)\{p,p, . . .p„) f — , - — j . (^0^1 . . . q,,){xyf' 



= {P,P, . . . P,.) {J^ - ^y. ( Q„Qi . . . Q,. XXYf ; 



and this is a definite covariant quantic of the order n' -nva. x and y, 

 provided the order of multiplication of the non-commutative coefficients 

 j9 and q, and P and Q is preserved. In this case the operator is written 

 to the left of the operand. The new covariant 



{Im' - rm)"{q,q, . . . q,„){zy)"'. {p,pi . . .p„) (^— , - -- j 

 = {Q,Q,...Qn){XYnP^,...P,.) 



[dY' dX 



is found when the operator is written to the right of the operand. 

 This covariant differs from the former only in the oider of multiplica- 

 tion of the p and q, and of the P and Q. Here p is always to the right 

 of q, and P to the right of Q ; in the other case, p is always to the 

 left of q, and P to the left of Q. 



When n = «', these are invariants for the linear transformation. 

 It is easy to extend this theoiy to any number of binary quanties, 

 but the order in multiplying the coefficients must be carefully 

 attended to. 



5. 2^e vanuhing of a vector invariant with respect to the parameter 

 determines the pole of a given chord of a conic. 



Forming the (12)^ invariant of the vector quadratic (in which zj is 

 an arbitrary, but given vector), 



(a^ - aQi3)x'^ + 2 (tti - flit7)a:y + (a2 - «2w)yS 



and of the scalar quadratic 



h^ + 2lxxy + l^y'^, 

 the result is 



(ttp - AqTO) ho -2 (tti - a^zy) 5i + (oj - ^2^) ^o* 



The invariant vanishes if, as in the third article, zs is the vector to the 

 pole of the chord joining the points determined by equating to zero 

 the quadratic 



h^x- + 2hixy + Jay*. 



