PROF. W. K. CLIFFORD ON MR. SPOTTISWOODE’S CONTACT PROBLEMS. 
711 
system there is only one conic whose plane passes through an arbitrary point, viz. the 
conic determined by the plane through that point and the axis. 
There are eight conics of the system which meet an arbitrary line. 
The number of conics which meet an arbitrary line is clearly the same as the order of 
the surface which they trace out. Now, since through every point on the axis can le 
drawn six conics of the system (as proved in the last section), the axis is a six-fold line 
on the surface. The section of the surface, then, by a plane through the axis is made 
up of the axis taken six times over and the conic in that plane ; or it is of the order 
eight. Q.E.D. 
V. Conics not subject to any condition. 
In order to get rid of the restriction of meeting a fixed axis, we must again modify 
our fundamental equations. We have to put them into a form in which they will 
represent any conic touching the surface u n at the point A. For this purpose it is 
necessary and sufficient that C should be movable over a plane passing through A; 
since every conic through A must cut the plane in one other point, but this may be any 
point of the plane. We attain this analytically by substituting for aC in the second set 
of equations, XC+^D, where C and D are now fixed points not in the tangent plane 
and not in any straight line through A. This is equivalent to still considering the conics 
which meet a given axis, but allowing that axis to move over a fixed plane. 
The transformed equations are : — 
0=(B 2 )+X(C)+g,(D). 
( B 2 ) = (B 4 ) + A(B 2 C) -F^(B 2 D) X 2 (C 2 ) + 2fyz( CD) + ^ 2 (D 2 ) 
(B 3 )=(B 5 )+X(B 3 C)+^(B 3 D)+x 2 (BC 2 ) + 2^(BCD)+^ 2 (BD 2 ) 
(B 4 ) - Y(C 2 ) - 2ty&(CD) - ^(D 2 ) = (B 6 ) + /v(B 4 C) + <=<B 4 D) + Y(B 2 C 2 ) + 2ty,(B 2 C~ ' 
0=(B 3 )-f-x(BC)+/z(BD). 
(n 
m 
( 6 '") 
+^ 2 (B 2 D 2 )+ Y(C 3 ) + 3\>(C 2 D) + 3tyo 2 (CD 2 ) +^ 3 (D 3 ) J 
Locus of poles of axis in given plane in regard to sextactics. 
From the first two of these equations we obtain 
l:X:p=(BD)(C)-(BC)(D) : (B 3 )(D)-(B 2 )(BD) : (B 3 )(C)-(B 2 )(BC) 
—n , : l 3 :m 3 , say; 
here n x — 0 is the equation to a straight line passing through A, while l 3 =0, m 3 =0 are 
cubics touching the chief tangents at A. 
Substituting these values in (5 W ) and (6'"), we obtain 
