436 
ME. A. CAYLEY’S IkLEMOIE OX CUEYES OE THE THIRD OEDEE. 
at a point (a?i, z^) of the cubic meet the cubic in the point (.T3, y,^ z^), then 
^3 : ya : z^=x,{y\—zX) : ylz\~a ^,) : zlx\—y\). 
For if the equation of the tangent i&-lx-\-ny-\-lz=^, then 
^^3 : y\yz : ^ T — f : f — 
and 
I : ?? : l=C(^,-\-2ly,z, : y\-{-2lz,x ^ : z\-\-2lx^,. 
These values give 
f]^—C= {y\ —zX){y\+A+^ lx{y^z^ — 8Zlr?) 
=(yi— ^i)x — (1 + 8^")^, 
since (^1, yi, is a point of the cubic, and forming in like manner the values of — 
and we obtain the theorem. 
31 . The preceding values of (^3, ^3, z^) ought to satisfy 
{x\-\-2ly,z,)x^-^{y\-\-2 lz,x, )y^-\-{z\+2 lx{y, )z ^ = 0 
^ 3 +y 3 +2I+ 6 lx^y^z^= 0, 
in fact the first equation is satisfied identically, and for the second equation we obtain 
Xl -\-yl-\-zl=x\{y\ — zXf +y 1(2? — + z!(x! —lylf 
= — x\{y\ — z\) —y\{z\—x\)— z^(xf -y\) 
= {x\-\ry\+z]){y\—z\){z\—x\){x\— 7 f,) 
x,y^z^= x^,z,{y\—z\){z\—x\){x\—y\\ 
and consequently 
xl+7jl-\-zl-\-Ux^y^z^={x\^y\-^z\+Uxyj,z,){;y\—z'^^{z\—x\){x\—f,)={), 
which verifies the theorem. It is proper to add (the remark was made to me by Pro- 
fessor Sylvestee) that the foregoing values 
^3 :y3 : ^3=^i(yi— 2?) : ylz\—x \) : z,{x\—y\) 
satisfy identically the relation 
oi:l-^yl + zl _ ix:\ + y\ + z\ 
^ 32 / 3^3 
Article Nos. 32 to 34 . — Fornmlmfor the Satellite line and Point. 
32 . The line ^x-\-riy-\-Z,z=^ meets the cubic 
+y ^ + 2:® -|- 6 Ixyz = 0 
in three points, and the tangents to the cubic at these points meet the cubic in three 
points lying in a hne, which has been called the Satellite line of the given line. 
To find the equation of the satellite line ; suppose that (a’l, ^1, 2:1), (.To, yo, 2o), (^3, y^. z^) 
are the coordinates of the point in which the given line meets the cubic ; then we have, 
as before, 
( 1 , 1 , 1 , IJfil—ryyj^ 7^ — c<l, arj— ={aX^-{- ,-\-7Z^){ax^-]r ^y2-]- 
