PROCEEDINGS OP SECTION A. 



301 



To find T.A4, it is only necessary to join A^ to the intersection of 

 (T.A3) with L^. 



Here starling with any point 0,^ on Lq we have determined four 

 points on each of the conies S and 2, together with the tangents to the 

 conies at those points. 



§ 3. On tlie curve lohose equation is — 



If any point P {a ft' 7') be taken on the conic 2, its axis of 

 homology is — 



« + ^ + 7 ^ 0. 



a [3 y 



The envelope of this line, subject to the relation — 

 is easily shown to be- 



/a' //3' ly - Q 



v/i;+ v^K'^ v/^- ' 



v/ «- + v/ ^, + v/ 



/^=o, 



- + g- +^ 



Po 7c 



V - 27 "ll- = 0. K>/ ^ 



It may be at once verified that any point on the cubic may be ex- 

 pressed in any one of the following forms : — 



(«,X^ ^„,.^ y^v'), [„,(;. - ^,)\ (S^{. - \)\ y,(\ - /O^], 



("cOA ^00/^ 700^), 

 where X -\- ^l -{• v =z 0. 



The equation of the tangent at any point Q, («o^^' ^o/*^^> To"^) ^^ '^he 

 cubic is — 



+ 



P 



+ -L - X^t,, 



+ A + 



0, 



^o ^ 7o '^'"' V«oX^ "^ /4/^^ " 7o-V 

 showing that the tangent at Qj is the line joining Qi to the intersection 

 of Loand(X.Q,). 



The above equation easily reduces to — 



= 0, 



+ ^. + ^ 



showing that the axis of Pj touches its envelope at the point 



To find the point Qi we have that the equation of Oi2i, viz. : — 



~{^c - + ^S" - ^) + IS^ - /^) = 

 is satisfied by the values (X^ ^i^ 1^^). 



Hence Q, is determined as the intersection of Ofij and (X.P,). 



