266 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
which shows that there is at B 3 an eightfold point, the tangents being given by 
(X+Y+Z)(JX+mY+»Z)=0, 
(/V 2 , p 2 , -pv, -vX, -XpJYZ, ZX, XY) 2 =0. 
Each of the facultative lines is a double tangent of the spinode curve.; whence j3'=18. 
Reciprocal Surface. 
66. The equation may be deduced from that for 11=12 — C 2 , viz. writing 
(a, b, c,f g, hJX, Y, X) 2 =2(X+Y+Z)(ZX+mY+wZ), 
that is 
(a, b , c,f , g , h)=(2 1, 2m, 2 n, m-\-n, n-\-l , 
we have , 
(A, B, C, F, G, H)=-(X 2 , p 2 , v\ pv, vX, Xp) ; K=0. 
Writing also 
X, p, v—m—n , n — l, l—m as before, 
xsc-\-py-\-vz—t 7, 
Imnxyz—b , 
l(m + n)yz +m(A+ l)zx m)xy = y , 
ZA yz + mpz x + nvxy — A, 
(w+Z)y+(Z+m) z=Z, 
we have 
U=2y, Y=2(n|/, W=-4^ 2 , 
and then 
~L=h 2 w 2 —2Jdw-\-a 2 , M=2(kwv J r G$), N = Mflmnkxyzw — 4 / 2 ) ; 
so that the equation is 
0=L 3 N = 4(W - 27cwt + 
+L 2 M 2 + Y{Tc 2 w 2 -2Jcwt +<?)\Jcwv + <4) 2 
— 1 8/^wLMN — 1 44 icw(k 2 w 2 — 2 Jcwt-\- a 2 )(Jcwv -f a-ty)(JcwQ — 4 /2 ) 
— 16^wM 3 —Y2%Jcw(Jcwv -j-(r\p) 3 
—27ti 2 w 2 N 2 - 432&W(«-^ 2 ) ; 
or reducing the first two terms so as to throw out from the whole equation the factor 
Jew, the equation is 
4L 2 { £L 4- ( y 2 - ^ 2 )lcw + + va) } 
-18LMN-16M 3 — 27&wN 2 =0, 
or, what is the same thing, it is 
{Tc 2 w 2 - 2Jcwt + c 2 ) 2 {k 2 w 2 0+kw(- 2 td+v 2 -^ 2 ) -f ^+2^+21^} 
— ?»6(lc 2 w 2 — 2 \kwt+a 2 ){Jcwv + c$)(fkw6 — \f/ 2 ) 
-%2(lwu-Sr^y 
— 10 8 Jcw{ ilcwb — 4/ 2 ) 2 = 0 . 
