two Ternary Cubic Forms. 513 



we have identically 



(l-2/+4/ 2 )(X + Y + Z) 3 -f24(/-l) 3 XYZ 



and the value of k consequently is 



l_ 8(/-l) 3 



9(l-2/+4/ 2 )' 



If, however, /=1 or /=— \, the transformation fails. In the 

 former case, viz. for /=1, the equations for the linear transforma- 

 tion become 



X=2#— y—z, 



Y=2y-z-x, 



Z = 2z—x—y, 



which give X + Y + Z = 0, so that X, Y, Z are no longer inde- 

 pendent ; and the formula of transformation becomes 



(X + Y + Z) 3 =0. 



It may be noticed that the invariant S of the form 



x 3 + y 3 + z 3 + 6lxyz 



is S= — l+l 4 , so that 7=1 is one of the values which make S 

 vanish. And the above transformation is not applicable to the 

 cubic form x 3 + y 3 + z 3 + 6xyz } which is a form for which S 

 ..vanishes. The transformation, however, holds good for 1=0, 

 which is another value which makes S vanish ; or it does apply 

 to the form aP + y 3 -^^, for which S vanishes. The transforma- 

 tion, in fact, is 



(X + Y + Z) 3 + 24XYZ = -8(ar J + 2 , 3 + 2 3 ), 

 with the linear equations 



X=-y-z, 

 Y=-*-#, 

 Z = —x—y. 

 The above two forms for which S vanishes, viz. 



# 3 + y 3 + z 3 + 6xyz, 

 xS + yS + z 3 , 



are, notwithstanding, equivalent to each other, as appears by the 

 identical equation 



(x + y + z) 3 + (x + coy + <w 2 *) 3 + (x + © 2 y + <oz) 3 

 = 3(x 3 + y 3 + z 3 + 6xyz), 

 where o> is an imaginary cube root or unity. In the latter of the 



