422 JVIE. A. CAYLEY’S I^LEMOIE OX CUEYE3 OE THE THIED OEDEE. 
10. To find the coordinates of the point G in which the line EF joining two conjugate 
poles again meets the Hessian. 
We may take for the coordinates of G, 
mX+vX', mY+wY', mZ+vZ'; 
- and substituting in the equation of the Hessian the terms containing v^. disappear, 
and the ratio w : -w is determined by a simple equation. It thus appears that we ma}‘ 
write 
^^=_3^2^XX'^+YY'^-1-ZZ'^)+(1+2Z^)(Y'Z'X+Z'XY+XY^'Z) 
' -yzi: 3^^(X^X'+YW'+Z^Z')-(1+2?^)(YZX'+ZXY'+XYZ'); 
hence introducing, as before, the quantities |, >], t, a, (B, 7, we find 
uX+vX'=Bl%r'^-(Bt)+{l+2l%X^Y'Z'-X'^YZ); 
but from the first of the equations (B), 
X“Y'Z'-X«YZ=i(y,-/3J), 
and therefore the preceding value of wX+'wX' becomes 
which is equal to 
~2T- 
Hence throwing out the constant factor, we find, for the coordinates of the point G. the 
values 
11. To find the coordinates of the point O. 
Consider O as the point of intersection of the tangents to the Hessian at the points 
E, F, then the coordinates of O are proportional to the terms of 
I 3/^X^-l+^WZ, 3ZW^-1+^®ZX, 3r^Z^-r+^^XY | 
I 3/-^X'^-rd^W'Z', 3m-T4^^ZX', 3ra''-*-r+^^XY'. | 
Hence the ^-coordinate is proportional to 
(3/W^- r+^^ZX)(3/^Z'^-- r+^^XY') - (3Z^Z^- T4^^XY)3rW'^’- T+^V.X'). 
which is equal to 
9Z^(Y^Z'^ - Y'^Z^} + 3Z^(1 + 2 Z^) YY'(XY' - X' Y) + 3Z^(1 -f- 2 P)ZZ'{ZX' - Z'X) 
-{1+21JXX'(YZ'-Y'Z ) ; 
or introducing, as before, the quantities t, a, (3, 7, to 
-9Z^«e+3Z^(l+2Z^^)(^z:+7^)-(l+2^>? 
= (_l_13/3_4^fl)«._p2Z^(l+2Z^)(Z3r+7^)- 
