712 PROF. W. K. CLIFFORD ON MR. SPOTTISWOODE’S CONTACT PROBLEMS. 
<B 2 )=<(B 4 )+w 1 4(B 2 C)+^ 1 m3(B 2 D)+^(C 2 )+2Z 3 m3(CD)+mi(D 2 ) 
=u 6 , say; 
^(B 3 )=^?(B 5 )+w,/ 3 (B 3 C)+w 1 m 3 (B 3 D)+^(BC 2 )+ 24 m 3 (BCD)+^(BD 2 ) 
=v 7 , say. 
Here the curves u 6 = 0, ^=0 are of the sixth and seventh orders respectively, each of 
them having one branch at A in each of the chief directions, and one other branch 
different for the two curves. 
The equations 
n%B 2 )=u 6 , 
n 2 (B 3 )=v 7 
must hold for six-point contact, but they are not homogeneous. Eliminating n 2 , how- 
ever, there results 
(B 2 ).v 7 =(B 3 ).u 6 , 
a curve of the ninth order , locus of the 'poles in regard to the sextactic conics of an axis 
moving in a fixed plane. This curve has two branches at A in each of the chief direc- 
tions, and one other branch; and consequently is met by the plane ACD in five points 
coinciding with A, and in four other points. Now the plane ACD does not in general 
contain a sextactic conic ; the pole of the axis can therefore only be in this plane when 
the axis itself is in the tangent plane. In this case there is a certain number of proper 
sextactic conics in planes through the axis, and it is clear that the pole of the axis in 
regard to any such conic is the point A. These conics, therefore, correspond to the five 
intersections of the plane ACD with the locus of poles which coincide with A; or 
through an axis in the tangent plane can he drawn five proper sextactic conics. In the 
tangent plane itself there are four improper conics having six-point contact ; viz. the 
pair of chief tangents, which (as a sharp conic or line-pair reckoned among conics given 
tangentially) counts for two, and each chief tangent doubled. 
Number of septactic conics. 
There is a finite number of septactic conics at the point A ; each of these meets the 
plane ACD in a determinate point A-j-AC-|-|wJ), and fixes thereby a position of the 
point B. These positions of the point B must necessarily lie in the 9thic locus just 
investigated; it remains only to find a homogeneous relation which shall determine 
another locus for B, and to count the number of their intersections. 
To this end we must first substitute for 1 : X : g> their values n x : l 3 : m 3 in equation (7"'). 
The result is 
n^B^—nffC 2 )— 211^37^(0))— n 1 ml(B 2 )=n 3 (B 6 )-i-nil 3 (B 4 C)-{-n 1 m 3 (B 1 D) 
-f w 1 Z 2 (B 2 C 2 ) + 2n l l 3 m 3 (B 2 CD) +^m 2 (B 2 D) 
+Z 3 (C 3 ) + 3l 2 3 m 3 (C 2 D) + 3Z 3 m 2 (CD 2 )-f-m 3 (D 3 ), 
or 
n 1 t 5 =w g , say. 
