716 PROF. W. K. CLIFFORD ON MR. SPOTTISWOODE’S CONTACT PROBLEMS. 
For D also let us substitute id), where the coordinates of D are now given absolutely. 
Our three remaining equations are (omitting for shortness the mention of A) 
0=j/D+(rc-l)^.BC, (1) 
0=^D+(w-2)^.BC+^.BD+i(w-2».B 2 C, (2) 
0=»T)+(n-2)X(* . BC+^ . CD+i(^-2)^ 2 . BC 2 ; (3) 
and it remains only to show that these equations determine uniquely X, v. 
If we subtract (1) from (2) and (3) successively, we obtain 
0=— ^.BC+*.BD+|(w— 2)ty,.B 2 C, (4) 
0=-x.BC + t'.CD+i(w-2)^.BC 2 , (5) 
the values X=0, ^=0 not being admissible. But if we substitute in (4) and (5) the 
value of Xp derived from (1), we obtain two simple equations which determine the 
ratios X : p : v ; after which the absolute values are uniquely determined from (1). 
It is otherwise obvious that ifO— g 2 be the point-equation to a four-branch quadric, 
and 0=^ to the tangent plane, §=g 2 -\-gt\ will be the equation to a pencil of quadrics 
having four-branch contact, one of which may be made to pass through an arbitrary 
point. 
Special investigation for n=3. 
In tbe case in which u n is a cubic surface, the points in question are clearly the 135 
points of contact of the 45 triple tangent planes, namely, the intersections of the 27 lines. 
This case may be conveniently studied by means of the representation of that surface 
on a plane. The plane sections of the surface here correspond to a system of cubics 
having six common points ; the quadric sections therefore to sextics having nodes at 
these points. The problem is then to draw a sextic curve having six given nodes and a 
quadruple point elsewhere. 
The twenty-seven lines of the cubic are represented by 
the 6 fixed points, 
the 15 lines joining them, and 
the 6 conics each through five of them. 
It shall now be proved that the sextic must include two of these ; and that for each 
pair that meet there is a singly infinite number of sextics. 
The sextic cannot be a proper curve; for six nodes and a quadruple point are equi- 
valent to twelve nodes, and a proper sextic can have only ten. Nevertheless we may 
apply a quadric transformation to it, whose principal points are the quadruple point 
and two of the nodes. The sextic is thus reduced to a quartic, passing 2, 0, 0 times 
through the principal points respectively and having four other nodes. But a quartic 
having five nodes must be made up of a conic and two straight lines. Now, if the 
node at a principal point is the intersection of the two lines, the original sextic was 
