of the Concomitants of Ttuo Quadrics 197 



Notation. 



§ 2. Let Ui, Uo, Ui, u^ be plane coordinates; and let v, w be 

 cogredient with u. We may then typify line coordinates by 



Pij = (uv)ij = UiVj - vflij, (i, j = 1, 2, 3, 4) ; 



and X or point coordinates by cc^ = (uvtu)23^ and three similar ex- 

 pressions for X2, x-i, ^4, Then the symbolic system of Gordan can 

 be exhibited as follows. 



Let the point equations of the quadrics be 



/=<''^' = «/'=..., 

 and /" = h^- = bj^ = .... 



Let the line equations be 



u=iApy = {A'py = ..., 



U' = {Bpy = (B'pf=.... 

 Let the tangential equations be 



2 = uj = uj = ..., 



X' = M,3" = ?'j3'^ = . . . . 



Then the connections between the symbols are 



A = (ta', B = bb', a = aa'a", ^ = bb'b". 

 And all concomitants of the system can be expressed in terms of 



TO P'l'OT'^ 



4, {dd'p), (dd'd%), {dd'd"d"'), 



where d signifies a or b. But the irreducibles can be shewn to be 

 composed of the following types, 



(la, b^\ Ucc', b^, (Ap), (Bp), iia, u^, (abp), (Abu), {Ban), (AB), 



a^, ba, (A/3x), (Bax), (a^p), (AB)', F„ F, ; 



where (A^x) — a^a^' — a/a^ = «^(V , 



say, and («/3/>) = UaVp — u^Va = luvp , 



{AB)' = (Ahu)k', 



F, = {(ibp) a^' - (abp) a^ = (Abp^). 



Reciprocation. 



§ 3. By interchanging the symbols (a, a), (b, /S), (u, x) without 

 altering vl, jB or ja, we obtain from any given concomitant the 

 reciprocal form. Thus the bracket factors (A^x) and (Abu) are 



