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A MEMOIR ON CURVES OF THE THIRD ORDER.
393
17. The line through two consecutive positions of the point T is the line EF.
The coordinates of the point T are
— la 2 + /37, — 1/3 2 + ya, — Zy 2 + ot/3 ;
and it has been shown that the quantities a, ¡3, 7 satisfy the equation
-1 (a 3 4- /3 s + 7 3 ) + (- 1 + 41 3 ) a/3 7 = 0.
Hence, considering a, /3, 7 as variable parameters connected by this equation, the
equation of the line through two consecutive positions of the point T is
- 3la- + (- 1 + 4Z 3 ) ßy, - 3Iß 3 + (-1 + 4Z 3 ) ja,
x> 2,1a , 7 ,
V> 7 , - 2 Iß
— 3Zy 2 + (— 1 + 4Z 3 ) aß
ß
*, ß
a , - 2ly
= 0;
and representing this equation by
Lx + My + Nz = 0,
we find
L = (4Z 2 /3y — a 2 ) 3la 2 + (— 1 + 4l 3 )ß7)
+ (a/3+ 2Zy 2 ) (— 3Z/3 2 + (- 1 + 4Z 3 ) 7a)
+ (®7 “f 2Z/3 2 ) 3Zy 2 + (— 1 + 4Z 3 ) a/3);
or, multiplying out and collecting,
L = 3Za 4 + (— 1 — 8¥) a 2 /3y + (— 51 + 8Z 4 ) (a/3 3 + ay 3 ) + (— 16Z 2 + 16Z 5 ) /3 2 y 2 ;
but the equation
gives
and we have
- I (a 3 +/3 3 + 7 3 ) + (- 1+ 4Z 3 ) a/3y = 0
3la 4 = — 31 (aß 3 + ay 3 ) + (— 3 + 12Z 3 ) a 2 /3y,
L = (- 4 + 4Z 3 ) a 2 /3y + (- 8Z + 8Z 4 ) (a/3 3 + ay 3 ) + (- № + 16Z 5 ) /3 2 y 2
= (— 4 + 4Z 3 ) (a 2 /3y + 2Z (a/8 3 + ay 3 ) + 4Z 2 /3 2 y 2 )
= (— 4 + 4Z 3 ) (ay + 2Z/3 2 ) (a/3 + 2/y 2 );
or, in virtue of the equations (D),
L — (— 4 + 4Z 3 ) Z 2 £"£ . Z 2 ^?; = (— 4 + 4Z 3 ) l 4 ^ 2 v^ = (— 4 + 4Z 3 )
Hence, omitting the common factor, we find L : 31 : N = % : rj : and the equation
Lx -l- My + Nz = 0 becomes
C. 11.
№ + vy + & =
50