409 
146] A MEMOIR ON CURVES OE THE THIRD ORDER, 
these values give as before 
^-r=_( 1 + 8 * 8 )®i 8 (y 1 3 -*i , ) ; 
and they give also 
vt+Zl? = (l + W)y№> 
and consequently we obtain 
x : y : z = x, (y 1 3 - z 3 ) : y 1 (z, 3 - x 3 ) : z t {x 3 - y 3 ), 
that is, the satellite point of a tangent of the cubic is the point in which this 
tangent again meets the cubic. 
Article Nos. 35 and 36.— Theorems relating to the satellite point. 
35. If the line %x + yy + %z = 0 be a tangent of the Pippian, then the locus of 
the satellite point is the Hessian. 
Take (x, y, z) as the coordinates of the satellite point, then we have 
x : y : z = (y 3 -£ 3 )(y£+21%>) 
: (£■-?)(£+2Irf) 
: (P-if)(fr+2V?)\ 
where the parameters 77, £ are connected by the equation 
-1 (I s + y 3 + £ 3 ) + (- 1 + U 3 ) &S= 0. 
We have 
y 3 + z 3 = (£ 3 - f) s (+ 2ly 2 ) 3 
+ (I 3 — V s ) 3 (19? + 2^ 2 ) 3 , 
and it is easy to see that the function on the right-hand side must divide by rf — £ 3 : 
hence x? + y 3 + z 3 will also divide by y 3 — ¿f 3 , and consequently by (77 3 - £ 3 ) (£ 3 — f 3 ) (£ 3 — y 3 ). 
We have 
and 
(y 3 + Z 3 ) -r- (?7 3 - £ 3 ) = 
f - £ 9 - £y - - ?? 9 
+ 3p (£ 6 + £V + *? 6 ) 
- 3p (£ 3 + 77 3 ) 
+ r 
+ 6l£ 2 y 2 % 2 {— — £ 3 r) 3 — 77 s + 3 ~ 3 (£ 3 + 17 3 ) — 3p} 
+ 12l 2 ^ { - y 3 £ 3 (y 3 + ?) + 3- | 9 } 
+ 8Z 3 { - y% 6 + Sy 3 ^ 6 - (y 3 + £ 3 ) f 9 ] 
C. II. 
a*+ (y 3 - £ 3 ) = Of - 2t ? 3 ^ 3 + £ 6 ) Of £ 3 + 6££ 2 >f£ 2 + 12l^yt + 8l 3 £ s ). 
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