( 370 



XXI. 



A Is^OTE OIT CEETAm CURVES CO^^^^ECTED WITH THE 

 DOUBLE ITOEMALS OF PLA2s"E BICIRCULAR QUARTICS 

 AND CYCLIDES. By J. GILBAET SMYLY, M.A., EeUow 



of Trinity College, Dublin. 



[Eead May 8, 1899.] 



Coif SIDE R a circle S, and a conic F; the bicircular quartic generated 

 from these has four centres of inversion, namely, the centre of S, and 

 the vertices of the common self-conjugate triangle of S and F. 



Let S = X- + f + 2fx + Igy + c = 0, 



a 



The polar planes of a point x'y' with regard to S and F are 



x{x' +f) + y {y' +ff)+ fx' + gy' + = 0, 



xx' yy' 

 a 



if these planes coincide, 



aix'+f) h{y' + g) 



= - (A' + ff}/'+c) = - X. 



X y 



Hence the coordinates of the vertices of the common self- conjugate 

 triangle are 



a +X' b +X' 



X being given by the equation 



ap bg- 



X-C + -^ + ^^ = 0. (i. 



a + X h + X 



The equation of the quartic is 



{■^' + y'-cy = 4 [a {x +fy+b{y + yfl 



