78 



Pi-ocpprfings of the Roi/a.I Irish Academy. 



a h - \ g 



^ -f A h f = ; 



g / c- X- 



ii. aG ^ ftF^[X-y) C = 0; 



iii. 1 - a - ^ I 



a a h - X = 0. 



ft h^X h \ 



i. is a quartic for X, and to eacti value of X we have a linear rela- 

 tion connecting a, (3, y, given by ii., and a quadratic relation bet-ween 

 a, ft, given by iii. 



a;y -f s {ax ^ fti/ + y^) = 0, 



aG- ftF^ 'X-y) 6' = 0; 



.-. alCzx- G%^~\ + ft I C%y - Fz'~\ - Cxy -f XCe^ = 



is an equation of enveloping conic ; and its form shows that it belongs 

 to a system vrbich has a Jacobian cubic curve, viz. : 



2 y 



s a; = 0, 



Cx-2Gz Cy-2Fz -2XCz 



or - 2s [AGs'- + Cxy -Fzx- Gzy'] = 0. 



Hence the system which, since it passes through two fixed points, has 

 a common Jacobian conic 



xy-^.lFx-^ Gy-] - As- = 0, 



a conic which also passes through the same two points. 

 To each value of A, we have a corresponding conic 



a{Czx- Gz-) - ft ( Czy - Fz-) ^ Cxy + XCz"" = 0. 



If this conic becomes a pair of lines, then 



C aC i 



C ftC \ = 0, 



aC ftC 2XC-2aG-2ftF 



= C^ \2aftC - 2AC r 2aG ^ 2ftF'] = ; 



