408 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
Now the coefficient of x 3 in F. U is simply F, which is equal to 
f + v e _]_ £6 _ 2i? y _ 2£ 3 f - 2£V 
- 242 fy £* 
-24Z 2 (£ 3 + v 3 +Ç s ) ÇyÇ 
- 32Z 3 (rf? + £ 3 £ 3 + fy) 
- 48Z 4 £y£ 2 ; 
and subtracting, the coefficient of x 3 in F. 27—11 is 
£6 _ 2£y - 2f £ 3 
- 62 £y£ 2 
- 82 3 (y 3 £ 3 + £ 3 £ 3 + fy) 
- 82 3 (| 3 - y 3 ) (£ 3 - | 3 ) 
- 482 4 £V£ 2 , 
which is equal to 
(1 + 8Z 3 ) p (| 4 - 2fy - 2f Ç» - 6l V T)- 
The expression last written down is therefore the value of f 2 £', or dividing by £ 2 we 
have f', and then the values of y, £" are of course known, and we obtain the 
identical equation 
F.U-n = 
(1 + 8Z 3 ) (£a> + yy + Çzf 
(£ 4 _ f - 2ÇÇ 3 - 6ly 2 Ç 2 ) x 
+ (T; 4 — 277^ — 2 
+ (£ 4 - 2£f 3 - 2y - 62|y) 2 
and the second factor equated to zero is the equation of the satellite line of 
%x + yy + £z = 0. 
33. The point of intersection of the line %x + yy+ £z = Q with the satellite line 
g'x + yy + = 0 is the satellite point of the former line; and the coordinates of the 
satellite point are at once found to be 
x : y : z = (y 3 -£ 3 )(y£+2Zf 2 ) 
: ($*-*■)(£+ 22^) 
: (P-*?)(&> +%?)• 
34. If the primary line %x + yy +%z = 0 is a tangent to the cubic, then (x l , y 1 , Zi) 
being the coordinates of the point of contact, we have 
£ : y : f = y + 2ly x z x : y, 2 + 22^ : y + 2lx l y 1 ;