ME. A. CAYLEY’S MEMOIE ON CURVES OE THE THIRD ORDER. 
437 
The equation of the three tangents is 
n = [(^ h,x,)y-^{z \ + 2 lx,y,)z'\ ^ = 0 
X [_{xl + 2 ly^z^)x-\- {yl-\-2 lz^^)y + ( 2 :^+ 2 lx.,y^)z'\ 
X [(^+2^y323)^+(3/3+2fe3^3)3/+(2'3+2^^3y3)2:] _ 
and if we put 
F=(f+^^+r)"-24z^(r+;!^+r}i^r+(-24^-48^^}ivr+(-4+32z^X^T+rrH-iv) 
(F is the reciprocant FU of my Third Memoir), then we have identically 
¥ .V—Il={^x-\-ny-{-lzf{^x-\-r!y+'Cz\ 
and the equation of the satellite line is ^x-\-ny-¥‘Cz=^. In fact the geometrical theory 
shows that we must have 
F . U — NIT ={^x-{-ny-\- lz)\^x-¥n'y + ^z), 
and it is then clear that N is a mere number. To determine its value in the most simple 
manner, write ?=0, y=0, x=t, 2 =— |, we have then F.U — Nn=0, where 
F=f+;j®+^®-2;j'r-2rf-2|Y, U=:r-f. 
The value of 11 is n=F . U, and we thus obtain N=l. For substituting the above values, 
Il={x\l— zl^)(xl^ — zll)(xlt — zll) 
_ V3 /yi2 ^2 ^2 
—Vl{x\xlzl+ &c.) 
-\-ll\oc,zlzl-\- &c.) 
and we have 
and thence 
and consequently 
x^x^^=yi^—V 
^1^22^3+ &c.= 3n 
^I^2^3 + &C. = — 3^1^ 
> 
— ^ 5 
xw,zi^ &c.=9rr+6rr(p?^-r)=3rr+62:iv 
xy,zi + &c. = 9rr- 6ri(r-p;=’)= 3rr + 
-ri.3rr(rH-^*) 
+2:r.3ri(r+;,®) 
= _j_ ;j6 q_ ^6 _ _ 21%^ - 21Y)- 
Now considering the equation 
¥ .'[5 — U.={lx-\-r)y-]r'C;z)\^x-\-fly-¥'Cz), 
in order to find I', yj, t! it will be sufficient to find the coefficients of y^, z^ in the 
