ME. A. CAYLEY’S MEMOIE ON CTJEVES OF THE THIED OEDEE. 
439 
and subtracting, the coefficient of ^Mn F . U — 11 is 
~6/ivr 
-8^^(;j^r+rf+ev) 
-m^nT, 
which is equal to 
(1 + 8 Z^)f (r - 2|;j^ - 2|r - 6 leC). 
The expression last written down is therefore the value of or dividing by we have 
I', and then the values of yj, f are of course known, and we obtain the identical equation 
F.u-n= 
i^x+7iy-\-tzy< 
(1 + 8/^)(r - - 2|r “ 6 kX^}x 
+ (l + 8^^)(^"~2;7r--2;?r~6/rf)?/ ^ 
+(l+8^^)(r-2rr-2r^^-6/|V)2 
and the second factor equated to zero is the equation of the satellite line of 
i;X-\-riy-\-tz=0. 
33. The point of intersection of the line ^x-{-'^y-\-tz—() with the satellite line 
^x-\-ny-\-'Cz=^ is the satellite point of the former fine; and the coordinates of the 
satellite point are at once found to be 
x:y\ Z={ri^—V){rjl-\-2l^^) 
:(r-r)(ri+2/p?^) 
34. If the primary line ^x is a tangent to the cubic, then [x^, y^, z^) being 
the coordinates of the point of contact, we have 
^■.ri'‘t=ay-{-2ly^z^:y\-\-2lz^Xi : z\-\-2lx^yi ; 
these values give as before 
and they give also 
nWll^={l-^U^)y\zl 
and consequently we obtain 
x\y. z—xly\—zX) : y,{z\—x \) : z,(x\—y\y 
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-\-7iy-\-tz=^ be a tangent of the Pippian, then the locus of the 
satelhte point is the Hessian. 
