MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 179 



The general value (37) 5 30, of v, the Cotesian of L, now reduces to 



v = _ (03* + ty 4. ? + 3 Cl zx + 3cjZ 8 y + Gexyz) ( 8 ft 8 u) 

 - n 8 z 8 { - a 8 ft*n* (c^ + c*y + cz) + 2afte 8 n* (ez)\ 

 + 2iiz { aWn* (cz 2 + 2^zx + 2c$z)} 



+ 2 (e*n*z s + aft/i%) 2 ( - aftn 8 ) em*, 

 i.e., 



v = aW (no* + ft?/ 3 Zexyz) ...... . . (70) 



a cubic having a node at the pole of L (now 2 = 0) with respect to *, the tangents to 

 s being also the nodal tangents of the Cotesian v. 



51. The comparison of the equations (69) (70) of m and v verifies the relation, 



(48) 37, 



m = v + 8PL. 



since, in the present form (65) of u, the Cayleyan P, for the values of the line 

 coordinates / = 0, m = 0, is simply 



P = often 3 ........... (71) 



52. Eliminating y and x successively between 







v = oar 5 -|- by 3 2exyz = 0, 



.s = c 2 2 2 + abxy = 0, 

 there result 



(a*ba* - e 3 ? 8 ) 8 = 0, 



showing that v has triple contact with * ; viz., when the loop of v is real it touches s 



at the point 



x:y:z = (&)- : (aft 8 )-* : e' 1 ; 



or at the point determined by the real lines, or any pair of them, 



(a z ft)a; = (aft*)'y = ez. 



Now, making 2 = in u and v, the points in which L meets these cubics are 

 determined by 



ax* + fey 3 = 0, 

 the real one being 



ox + ft'y = 0, 



Z =0, 

 2 A 2 



