62 A. Mukhopadhyay — On a Curve of Aberrancy. [No. 1, 



whence 



pr - 3^3 = 18 (ac - 6 2 ) - 90 (ax + i)» 





8x 3b 

 a = 1- — 



5 5a 





f-**^'^—^; 



Therefore 







3a 3& 



X = ~8~Sa 





5 

 aa; + b = - (aa + 6) 



o 



and 



9 («a+ 6) ( 

 v = # — i (ac — b' 



< >~ 8a* [(a* -**)-§ (««+&)*;} 



Bat from the equation of the curve we have 



a%y = (ax -f 6) 3 + 3a (ac — 6 a ) # + a?d — & s . 

 Therefore, substituting for x and y in terms of a and /?, we have 

 64 aS/3 = - 125 a s a3 - 375 a%o? + (192 ac - 5676 2 ) aa 

 + (64a»<2 - 1896 s ), 

 or, writing x, y for a, /8, we see that the aberrancy curve of the plane 

 cubic 



y = ax % + 3bx % + 3cx + d 

 is another plane cubic of the same class 



y = Ax* + 3Sa; 2 + 3C# + D 

 where 



J. = —ha 

 B = - hb 



= - ko + (1 + h) a -^-¥- 

 a 



n %d _ jfi 

 D = -M-t-(l + &) — ~ 

 a* 



125 



If, therefore, 



H=ac-h\G = aU - Sale + 26 s 

 be the invariants of the given cubic, and H', G' the corresponding quan- 

 tities for the aberrancy cubic, viz., 



H' = AG- B% G' = AW - 3AB0 + 2B 3 , 

 we have by direct calculation 



H' = -JcE 

 G' = WG. 



