528 Proceedings of tlie Royal Irish Academy. 



Through and the point p on the curve draw a line ; then the 

 two points on the curve, the tangents at which meet this line, represent 

 the quadratic covariant (0A)Aj;0^. In the same manner the covariant 

 (0v)0a;Va: is represented, while (Ay) A^Vx is represented by the points, 

 the tangents at which meet the line Oir or the line O'^j. 



"We shall now discuss the six remaining linear covariants. 



We saw that the linear covariant obtained by drawing a plane 

 through the corresponding point of a plane resulting from a binary 

 form a^^, and two points on the curve given by the equation A/ = 0, 

 was (Aa)-a^. IS'ow, if we substitute ^for a, we find {^K)~K„ which, 

 is Clebsch's form. 



To represent the covariant {^K^K^, di'aw a plane through the 

 chord through 0, and passing through K, the corresponding point of 

 the plane K, this plane will meet the curve again in the required 

 point. By a similar construction the covariant (v Q)" Qx is represented. 



XI. 



"We now show how to represent the forms (7rv)Vx aiid {pii>)^^. 

 We have shown how to construct the two points corresponding to the 

 form (©A) A;^©^., which we shall call for the moment p/, and we know 

 (VII.) that the linear covariant derived by drawing a plane through, 

 the two points given by p,/= and a third point a is {pdf a^. JSTow, 



p:^ = (©A)©.A.; 



therefore (p^)^= (©A) (©«) ( A(?) ; 



but (ffA)fi!y«^Aj,= («A)«/A^; 



therefore {pa)~ar= («A) (a©)^Aj.. 



Now, P:c = - 2{®ayax ; 



therefore (i'A) A^, = - 2(©«)- («A) A ; 



therefore {pafa,. = --t(fjA)A^. 



Hence ch-aw a plane through and the two points given by the 

 equation (QA)A;,Q^=0, which Avill meet the curve in a third point 

 representing the linear covariant (^A)A^. 



In the same way the form (7rv)Vx is represented. 



XII. 



The two remaining linear covariants may be represented as fol- 

 lows : — Construct the corresponding point of the plane containing the 

 chord through and the point 0' ; the plane meets the curve in points 

 given by the equation A^-^j?^ = 0, which we shall call Rj^ = A/^;^ ; 

 through the corresponding point of the plane and the chord through 

 0' draw a plane, meeting the curve in a third point given by the 

 equation 



{R^YR, = 0. 



