PROF. W. K. CLIFFORD ON MR. SPOTTISWOODE’S CONTACT PROBLEMS. 715 
The first two of these show that B and C are points on the chief tangents at A. If 
we regard the absolute values of the coordinates of A as given, then it appears from the 
third equation that B and C may be chosen arbitrarily on the chief tangents and D 
anywhere in space, the equation giving the relation between the absolute values of their 
coordinates which determines the particular surface AD — BC=0. 
For four-branch contact we have the additional equations, 
0=A n-3 B 3 , 0=A" -3 C 3 , 
0 =2 A”" 2 BD + (n- 2) A M ~ 3 B 2 C, 
0=2A”~ 2 CD + (w— 2)A” _3 BC 2 . 
The first two of these indicate that B and C lie on the polar cubic of A in regard to 
the section of u n by the tangent plane. Now this polar cubic has a node at A, whose 
tangents are the chief tangents. Each of these lines therefore meets the cubic in three 
points at A, and cannot have any other point on the curve unless it be itself a part of 
the cubic. But the points B and C have to lie one on each of the chief tangents. In 
order, therefore, that all the equations may be satisfied, the polar cubic in question must 
break up into the two chief tangents and some other line. 
This condition may be put into another form. For if we seek the points in which 
the line AB meets the surface, by substituting the coordinates of A -f-AB in u n — 0, the 
conditions A”w re =0, A” -1 Bi^=0, A n-2 B 2 w„=0, A” -3 BX=0 indicate that four roots of 
the equation are equal to zero, or that the line meets the surface in four consecutive 
points at A. We find, therefore, that 
Those points of a surface at which a quadric may have four-branched contact are the 
points at which each chief tangent meets the surface in four consecutive points , or, 
which is the same thing, the points whose polar cubic contains the chief tangent. 
The number of these points has been counted by Clebsch, Crelle, lxiii. 14*, and turns 
out to be 
rc(41# 2 -162w+162). 
A point of this nature being given , one quadric surface having four-branch contact at 
it may be drawn through another arbitrary point. 
The coordinates of the point A being given as to their absolute values, let us substitute 
for B and C, A+xB, A-J-^C ; where now B and C are fixed points on the chief tangents, 
whose coordinates are given absolutely. This being so, the following equations are 
satisfied ex hypothesi : — 
AX,=0, A ra-1 Bw„=:0, A” -1 Om„=0, 
A”- 2 BX=0, A»- 2 CX=0, A»- 3 BX=0, A re - 3 CX=0; 
from which it follows at once, for example, that 
A”~ 3 ( A -f XB)X= 0. 
* The investigation is given by Salmon, Geom. Three Dim. p. 444. 
5 D 
MDCCCLXXIV. 
