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146] A MEMOIR ON CURVES OF THE THIRD ORDER. 
We have also, identically, 
ABC - FGH = i (-1 + l±)XYZ [r- (X 3 + Y 3 + Z 3 ) - (1 + 21 3 ) XYZ], 
which agrees with the relation ABC — FGH = 0. 
7. Before going further, it will be convenient to investigate certain relations 
which exist between the quantities (X, Y, Z), (X', Y', Z'), connected as before by 
the equations 
XX'+ l (YZ’ + Y'Z) =0, 
YY' + l (.ZX' + Z'X) = 0, 
ZZ' + l(XY' + X'Y) = 0, 
and the quantities 
£ = YZ' - Y'Z, 
V = ZX'~ Z'X, 
Ç = XY'-X'Y, 
a = XX' = -j (YZ' + Y'Z), 
/3 = YY' = — ~ (ZX' + Z'X), 
rt=ZZ' = — |(ZF + X'Y). 
We have identically, 
2XX' (YZ' - Y'Z) + (XT + X'Y) (ZX' - Z'X) + (ZX' + Z'X) (IF - X'Y) = 0; 
or expressing in terms of 77, £*, a, /3, 7 the quantities which enter into this 
equation, and forming the analogous equations, we have 
2lag - 777 - /37 = 0, (A) 
-7f+2£/3?7- «7=0, 
— /3f — <277+ 2^77=0. 
We have also 
X 3 Y'Z' - X' 2 YZ = 1 {-(IF + X'Y) (ZX' - Z'X) + (ZX' + Z'X) (XT - X'Y)}, 
and thence in like manner, 
X'rZ'-X'‘YZ=±(y V -l3Z), (B) 
Y‘Z'X'-T’ZX = ± <«r-7?), 
Z‘X’Y - X'-YZ = I (/Sf - ar, ). 
Again, we have 
(YZ' - Y'Z) 3 = (YZ' + Y'Zf - 4 YY'ZZ', 
(ZX' - Z'X) (XT - X'Y) = — (ZX' + Z’X) (XT + X'Y) + 2XX' (YZ' + Y'Z); 
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