526 ' Proceedings of the Royal TrisJi Academy. 



iJv'oTV it is easy to shoTr that 



[aoa^-a^oQ - 3 (<ri 03 - tti c-,) } <?o 



= 6{aoAi -a-iAo} T-(r?a/f?o : 



lience 



And in tiie same Tvay 



"VTe have noTV expressed. ^ and k in terms of Clebscli's forms, and 

 can represent tliem as follQYrs : — Tiie covariant q-^ is transformed into 

 a plane -^'Hcli is the polar plane of (y Trith resard to the polar quadiic 

 of 0. 



In lite manner k^ is transformed into a plane which is the polar 

 plane of with regard to the polar quadi-ic of &. These theorems 

 ai'e immediate algebraic conseq_uences of the method of generation of 

 the covaiiants q and J:. 



Til. 



"W'e now discuss Clebseh's two linear covariants _/;;^ and tt^. 

 If through a point X, Y, Z, W. and two points on the curve, the 

 parameters of wMch are given hy the equation 



A,= = Ac^v -r 2\xix. -^ Ao^r.- - 0, 



we di'aw a plane, it will meet the cuiwe in a third point determined 

 by the equation 



Xi { Ao Y^ 2 Ai^-^ A. W] + X. { AoX -f 2Ai Y^\.Z]=0. 



And if we suppose that A is the Hessian of /, and that the point 

 X, Y, Z, TFis the corresponding point 0' of the plane ^, we find the 

 above equation becomes 



(Aa)-a^ = 0, or_/;^ = 0. 



Hence the linear covariant p^ is represented by the point in which 

 a plane thi'ough 0', and containing the chord thi'ough 0, meets the 

 curve, a similar construction representing tt^, where ~^ = {\a)-a^. 



Till. 



"We now turn to the quadratic covariants. The equation deter- 

 mining the pai'ameters of the points in which a chord through any 

 point on the line 00' is 



,VA/-2.V.0/-r/.= v.' = O. 



