mmmmam 
[146 
(C) 
(D) 
may be 
a conic 
In fact, 
arbitrary, 
146] A MEMOIR ON CURVES OF THE THIRD ORDER. 389 
and the function on the left-hand side is 
(l - j) («® + &y + 7z), 
which proves the theorem. 
9. The equations (A) by the elimination of (£, y, £), give 
— I (a 3 4- /3 3 + y 3 ) + (— 1 + 4Z 3 ) a/3y = 0, 
which shows that the line IJ is a tangent of the Pippian: the proof of the theorem 
is given in this place because the relation just obtained between a, /3, 7 is required 
for the proof of some of the other theorems. 
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, 
uX + vX', uY + vY\ uZ + vZ'; 
and, substituting in the equation of the Hessian, the terms containing u 3 , v s disappear, 
and the ratio u : v is determined by a simple equation. It thus appears that we 
may write 
u = - 3Z 2 (XX' 3 + FF 2 + ZZ' 2 ) + (1 + 2Z 3 ) (Y'Z'X + Z'X' Y + X' Y'Z), 
v = 31 2 (X 2 X' + Y 2 Y' + Z 2 Z') - (1 + 21 3 ) (YZX' + ZXY + XYZ'); 
hence introducing, as before, the quantities £, 77, f, a, /3, 7, we find 
uX + vX' = SI 2 (717 - /30 + (1 + 21 3 ) (X 2 Y'Z' - X' 2 YZ); 
but from the first of the equations (B), 
X'TZ’ - X'*YZ = I ( 7V - PS), 
and therefore the preceding value of uX + vX' becomes 
1 + 2Z 3 
31 2 
21 
(m - 
which is equal to 
— 1 + 4ZO 
Hence throwing out the constant factor, we find, for the coordinates of the point G, 
the values 
vv-PS, a£-y£i 
11. To find the coordinates of the point 0. 
Consider 0 as the point of intersection of the tangents to the Hessian at the 
points E, F, then the coordinates of 0 are proportional to the terms of 
3№ - 1 + 2l 3 YZ , 3№ - 1 + 2FZX , 3l 2 Z 2 - 1+ 2l 3 XY 
3 l 2 X' 2 -1 + 21 3 Y'Z', 31 2 Y' 2 - 1 + 2 l s ZX', 3 l 2 Z' 2 - 1 + 2 l 3 X' Y