386 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
5. I may notice, in passing, that the preceding equations give rise to a somewhat 
singular unsymmetrical quadratic transformation of a cubic form. In fact, the second 
and third equations give X' : Y' : Z'=YZ—l 2 X 2 : PXY—IZ 2 : PZX — IY 2 . And sub 
stituting these values for X', Y', Z' in the form 
Z 2 (X' 3 + F 3 + X 3 ) - (1 + 21 3 ) X' YZ', 
the result must contain as a factor 
l 2 +Y 3 + Z z ) - (1 + 2Z 3 ) XYZ; 
the other factor is easily found to be 
- I 3 (Z 3 (X 3 + F 3 + Z 3 ) + 3IXYZ). 
Several of the formulae given in the sequel conduct in like manner to unsymmetrical 
transformations of a cubic form. 
6. I remark also, that the last-mentioned system of equations gives, symmetrically, 
X 2 : F 2 : X 2 : YZ' : Z'X' : X'Y 
= YZ — l 2 X 2 : ZX-PY 2 : XY-PZ 2 : PYZ-IX 2 : PZX-IY 2 : PXY-IZ 2 ; 
and it is, I think, worth showing how, by means of these relations, we pass from 
the equation between X', Y, X to that between X, F, Z. In fact, representing, for 
shortness, the foregoing relations by 
X' 2 : F 2 : X 2 : YZ' : Z'X' : X'Y = A : B : G : F : G : H, 
we may write 
X' = AF=GH, Y'=BG = HF, Z' = CH = FG, ABC = FGH; 
and thence 
X' 3 = AF . G 2 H 2 , Y' 3 = BG.H 2 F 2 , Z' 3 = GH . F*G 2 , X'Y'Z' = F 2 G 2 H 2 ; 
hence 
P (X' 3 + F 3 + Z' 3 ) - (1 + 2P) X' YZ' = FGH [P (AGH + BHF + CFG) - (1 + 2P) FGH}. 
But we have 
P (AGH + BHF + CFG) = - (21 5 + P) (X 3 + F 3 + Z 3 ) XYZ + (P + 2P) (YZ 3 + XX 3 + X 3 F 3 ), 
- (1 + 2P) FGH = (P + 2P) (X 3 + F 3 + Z 3 ) XYZ + (P + 2P) (Y 3 Z 3 + XX 3 + X 3 F 3 ) 
+ P (1-P) (1 + 2P) X 2 Y 2 Z 2 ; 
and thence 
P (AGH + BHF+ CFG) - (1 + 2Z 3 ) 
= - l 3 (1 - l 3 ) [P (X 3 + F 3 + X) XFF - (1 + 2P) X 2 F 2 X}; 
and finally, 
P(X' 3 +Y' 3 + Z' 3 )-(l + 2P)X'Y'Z' = P(-l + P)(lYZ-X 2 ) (IZX - F 2 ) (ZXF-X)XFF 
x {Z 2 (X 3 + F 3 + X) - (1 + 2Z 3 ) XYZ\.