254 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
44. We have X=0, Y=0, Z=0, W=0 for the equations of the planes 
(12.34.56=®); (42'=?/), (14'=z), (12'=w); 
ancl representing by f=£X+^Y+^Z-j-W=0 the equation of any other plane (41'=f), 
the equation of* the cubic surface may be presented in the several forms : 
0=U=Wff 
+k£YZ 
=Wgg 
-fksjZX 
=Whh 
+1<XY 
=WM 
+k|?j^ 
=WlT 1 
+kyzx 
=Wmm, 
-j-kzxy 
=Wmq 
+kxyz 
=WlJ 
+kyzx 
=Wm,m 
+kzxy 
=Wn,n 
+kxyz 
=W PP i 
+k|yz 
=w qqi 
-j-k^zx 
=Wir 1 
+k^xy 
=Wpp"j 
+kfyz 
II 
1 
tP| 
-j-k;?zx 
=Wrr x 
+k&y, 
which are the 16 forms containing W, out of the complete system of 120 trihedral-pair 
forms. 
45. The 27 lines are each of them facultative ; we have therefore V—^= 27 ; £'=45 ; 
moreover each of the lines is a double tangent of the spinode curve, and therefore 
0'(=2 ? ')=54. 
46. The equation of the reciprocal surface is not here investigated; its form is 
S 3 — T 2 =0, 
where S=(#]£r, y , z, w) 4 , T =(#]£#, y , z, w) 6 ; wherefore n'= 12. 
The nodal curve is composed of the lines which are the reciprocals of the original 27 
lines (b 1 =27, £'=45 ut supra). It may be remarked that the reciprocal of a double- 
sixer is a double-sixer. Hence the 27 lines of the reciprocal surface may be (and that 
in 36 different ways) represented by 
