294 Proceedings of the Royal Irish Academy. 



for our purpose to calculate the coefficient of x'^ in T. Forming the 

 equation A (>S, L), and putting y' = %' = 0, we find — 



XiX-i = - 2m fjivx', 



yiy% = v'^x', 



Si S2 — fX X , 



yi^2 + %y2 = - 2 (/xj/ + m\^) x', 



Sia?2^S2^i = 2m\iJix', 

 x^yo + y^x^ = 2m\vx'. 



The coefficient of x'^ is found to be — 



^6 + ^6 _ (^2 + 32m^) ^^v^ - 18m X^ix^v' 

 -2im^Xlxv{X^ + lj? + v^') - Sm^X'^ 

 - 16m^ X^ (/x^ 4- J^^). 

 JSTow, 



2 = A6 + / + V*'- (2 + 32m3)(X>H/x3v3 + j/3X3) 



- 24»i- Xfxv (A^ + /x^ + v^) - (24m + 48m*) A.^" v^ 



Substracting the above quantity from 2, we find — 



JTA^/V = (1 + Sm^) X^ { X* - 2A (/x^ + v^) -6mfrv-}. 



Hence, 



JT = (1 + Sm^), 



A' = A* - 2 A (/x^ + v^) - em/x^v^ 

 fji' = [x^- 2fi (A^ + v=^) - 6m v^'A^ 

 v' - V* - 2i/ (/x^ + A3) - 6m A-/x^ 



The form of these co-ordinates at once suggests the following 

 theorem : — 



The satellite of a given line, meeting a system of cubics passing 

 through the inflexions of V, passes through a fixed point. 



