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



whence 



pr ~ 3^8 = 18 (ac - h^) - 90 (ax + 5)» 



5 5a 



ax -^-h 

 P =2/ -t- 



Tberefore 



^ = 2/ + ^f^ I 18 (ac - 62) _ 90 (a^ + 6)2 J 



_ 3a_3_6 



*" ~ "8 Sa 



5 

 aj? + 6 = - (aa + 6) 



o 



and 



Bat from the eqaation of the curve we have 



a^y = {ax + 6)^ + 3a {ac - 62) ^ + a^^Z - h^. 

 Therefore, sabstituting for x and y in terms of a and ^, we have 

 64 a2/? = - 125 a^aS - 375 a^SaS + (192 ao - 56762) aa 

 + (64^2^ - 1896^), 

 or, writing x^ y for a, /3, we see that the aberrancy curve of the plane 

 cubic 



y = ax^-\- 36;c2 j^ ^cx ^ d 

 is another plane cubic of the same class 



y= Ax^^- 3Bx^ + 3Cx-{- D 

 where 



A = — ka 

 B= -hh 



= - /&C + (1 + A;) ""' " ^' 

 D = - A;d5 -t (1 + A;) 



If, therefore, 



H= ac-h\ G = aU - 3a5c + 26^ 

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

 tities for the aberrancy cubic, viz., 



R' = AG- B\ & = A^B - ^ABG + 2B\ 

 we have by direct calculation 



W ^ -hR 

 G' = h^G, 



a 



aU - h^ 



