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



therefore, by (i.), 



6=0, 

 and by (iii.) 



e=0; 

 finally, by (ii.) 



a s =0, 



since 6 3 = is excluded by the form of s. 



Thus, it appears that the cubic u and poloid s are reduced to 



, '". . . . (114) 

 '^V . . (115) 



The form of u shows that x = passes through the point in which 2=0 touches 

 it; or the poloid touches the cubic at the point of contact of the transversal L. 

 78. The coefficients of the reciprocal of s which do not vanish are 



4A = - a 2 2 6 3 2 < 4F = 2a 2 3 6 3 n* ; 



which values, with I = 0, m = 0, give (37), 30 (dividing out n c ), 



v= 2 2 { a. 2 



c 2 2 2 



b&z) ; .......... (116) 



the locus degenerating in this case into the line L and a conic having double con- 

 tact with s at the points where the s-polar of the point of intersection of L and u 

 meets it and s. 



79. The double chord through any point x'y' on L or z = (56), 



, 



^ "-" = 



in this case, wherein 



3s 3s 3s 

 : By : = 



18 



^^ m O ft nf t jj, ^^^ fi i\ /y *)/ I _L /j >y | O/~| / 'V* <j/ * JL O/Tf /l T* -a/ *I ' /l J/ i ___ O^v */> o j> \ _!.-.. /^ 



'2 \ vto"Q t *' T / \ "Q^ \ ^jLt-o'-'Q'*'' */ r^ ^(tnt/o*l/ w I ^^ C/o u I , >(/ ' 1* I ^^ ^/, 



or 



= (117) 



