( 264 ) 



Eliminatiiif;' with the aid of (4) the coefficients cj, r.-,, c^ we shall 

 find for the chonl cumiiion to </>2 and F^ the equation 



^ (an 632 - 2 «J3&ii3 + «22^2)^3 ^3 = ... (7) 



2. ff the coefficients b and c are submitted to the two conditions 



^2 '3 — «22 ' ^'3 '3 = «33 (^) 



only, it results from 



*2 ^^2 - A = ^2 ^3 -^4 (») 



that it is possible to bring a conic throuo-h the nodes P2 ' -^3 ^"'^ 

 the couples of points B^ , B.^ and Cj , Co common to F4 and each 

 of (Pci and W^. 



The equation of this conic is 



£2, = (b, c-i - «ii) ^2 .^3 + ■'•! ^ it', ''2 + /'2 q - 2 <'io) .^-^ = . (10) 



AVe shall call (p^ ^^^^^ ^\ conqjlenieidarij with regard to Dc^, JJ.^. 



Tf Cj , C2 are fixed and the conic 12^ varies, the variable pair 



jÖj , B.2 describes an involution l^' determined on it by the pencil 



Evidently the variable chord B^ Be, is represented by (7), where 

 in connection with (8) only b-^ is variable. Substituting ba, to find 

 ^3 cg for «22 and 033 into (7) we shall find for öj B^ the new equation 



((•3 To, + f2 rf'a) t-i^ + [ (^3 02 + ^-3 Cq — 2 f<23) .J'j — 2 ai3 .(-2 — 2 «12 a's] ^1 + 



+ a„ (^3 .-.2 + 62 ^3) = , . . (11) 



which is quadratic with regard to b^. So the curve of involution is 

 a conic S^-2 w'ith the equation 



4 a„ (62 .'3 + b., x„) (C2 x^ + .3 ,7-2) --= [ (62 ^3 4- ?,3 <■.) .'i - 2 ^ «,3 ^3]^ . .(12) 



From the symmetry of (12) w. r. to the quantities b and c it is 

 now evident that Sp^ is at the same time the envelope of the joining 

 lines of the pairs C^ C\ of the involution Ic' which is generatod 

 when By, B.2 are fixed and SI^ is variable. These two complementari/ 

 involutions are characterised moreover by the property that each 

 pair of 7^- oan be joined to each pair of A' by a conic containing 



