MR. A. CAYLEY’S MEMOIE ON CUEVES OF THE THIED OEDEE. 
419 
and by eliminating from these equations, first (X', Y', Z'), and then (X, Y, Z), we find 
+ + Y* + Z^) - (1 -f 2 /^)X Y Z = 0, 
+Z^(X'^+Y'^+Z'*)-(1+2Z^)X'Y'Z'=0, 
which shows that the points E, E are each of them points of the Hessian. 
5. I may notice, in passing, that the preceding equations give rise to a somewhat sin- 
gular unsymmetfical quadratic transformation of a cubic form. In fact, the second and 
third equations give X' : Y' : Z' = YZ — : Z^XY — IT ? : Z^ZX— ZY^. And substituting 
these values for X', Y', 71 in the form 
+ Z^(X'^ + Y'^ + Z'^) - (1 + 2 Z^)X'Y'Z', 
the result must contain as a factor 
+ Z^(X^+Y^-|-Z^)-(1-1-2Z=’)XYZ ; 
the other factor is easily found to be 
-Z^(Z^(X^+Y^-f-Z^)-f3ZXYZ). 
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'^Y'^:Z'^:Y'Z';ZX':X'Y' 
=YZ-Z^X^ : ZX-Z^Y^ : XY-Z^Z^ : Z^YZ-ZX* : Z^ZX-ZY^ : Z^XY-ZZ^ ; 
and it is, I think, worth showing how, by means of these relations, we pass from the 
equation between X', Y', 71 to that between X, Y, Z. In fact, representing, for short- 
ness, the foregoing relations by 
X'^ : Y'^ : 71 ^ : Y'Z' : Z'X' : X'Y'= A ; B : C : F : G : H, 
we may write 
X'=AF=GH, Y'=BG=HF, Z'=CH=FG, ABC=FGH; 
and thence 
X'^=AF.G^H^ Y'^=BG.HT^ Z'^=CH.F^G^ X'Y'Z'=F^G^H^; 
hence 
+ Z^(X'^-|- Y'^ -f 71^) - (I + 2Z^)X'Y'Z'. 
=FGH{-fZ\AGH+BHF-fCFG)-(lH-2Z^)FGH}. 
But we have 
-fZ^(AGHH-BHF+CFG) = -(2Z^+Z«)(X^+Y^-fZ^)XYZ+(Z^-i-2Z7)(Y^Z^+Z^X^+X^Y^) 
-(I+2Z-^)FGH = (Z^-f2Z«)(X^-fY^+Z^)XYZ+(Z^-l-2Z7)(Y^ZHZ^X^-}-X^Y^) 
-p Z3(l _ Z=>)(I +2Z^)X^Y^Z^ 
and thence 
-f Z^( AGH + BHF -f CFG) - (I + 2 Z^)FGH 
= _ Z3(l _ z^) { Z-^(X^+ Y"+Z^}XYZ - (I +2Z")X^Y=^Z*} ; 
and finally, 
-1-Z^(X'^+Y'^+Z'®)-(I+2Z^)X'Y'Z' 
= Z^( - Z+ Z^)(ZYZ - X^)(ZZX - Y^)(ZXY - Z^)XYZ 
X {Z^(X^+Y*-pZ=’)-(I-}-2Z^)XYZ}. 
3 I 2 
