146] 
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