PROF. W. K. CLIFFORD ON MR. SPOTTISWOODE’S CONTACT PROBLEMS. 709 
0 = (B 2 )+(C) (3) 
0=(B 3 )+(BC) (4) 
0==(B 4 )+(B 2 C)+(C 2 ) .. (6) 
0 = (B 5 ) + (B 3 C) + (BC 2 ) (6) 
0 = (B 6 ) + (B 4 C) + (B 2 C 2 ) +(C 3 ) (7) 
III. Conics through a fixed ^oint. 
Combining equation (3) successively with (4) and (5), so as to obtain results homo- 
geneous in B, we find 
0 = (B 3 )(C) — (BC)(B 2 ), 
0 = (B 4 )(C) 2 - (B 2 C)(B 2 )(C) + (C 2 )(B 2 ) 2 . 
If we regard C as a fixed point, these are equations of a cubic and a quartie curve in 
the tangent plane, on each of which B must lie if the conic K 2 has five-point contact. 
But these curves have a common node at A, and common tangents at it ; for every term 
in each has at least one factor of the form (B m ), which we know to represent a polar of 
A in regard to the intersection of u n by the tangent plane — that is, a curve touched by 
the chief tangents at A. Of their 12 intersections, then, 6 coincide with the point A ; 
and there remain six conics having five-point contact at A which pass through an arbitrary 
point C. 
IV. Conics meeting a fixed axis through A. 
If the point C, instead of being altogether given, is movable on a fixed straight line 
through A, we may represent it by A-f-XC ; where now C is really a fixed point, and X 
a quantity to be determined. When we make this substitution in our equations, they 
become* 
0 = (B 2 )+*(C), 
0=(B 3 )+X(BC), 
0 = (B 4 ) + (B 2 )+X(B 2 C) +2x(C) + a 2 (C 2 ), 
0 = (B 5 ) + (B 3 ) + A(B 3 C) + 2a(BC) - f a 2 (BC 2 ), 
0 = (B 6 ) + (B 4 ) -f- x(B 4 C) + ( B 2 ) + 2x(B 2 C) + x 2 (B 2 C 2 ) 4- 3x(C) + 3\ 2 (C 2 ) -f- X 3 ( C 3 ) . 
The last three admit of obvious simplifications by aid of the previous ones ; and the 
system may finally be written 
0=(B 2 )+x(C) (3') 
0=(B 3 )+a(BC) (4') 
(B 2 )=(B 4 )+x(B 2 C)+X 2 (C 2 ) (5') 
(B 3 )=(B 5 )+a(B 3 C)+^ 2 (BC 2 ) (60 
(B 4 ) - /v 2 (C 2 ) = (B 6 ) + a(B 4 C) + A 2 (B 2 C 2 ) +x 3 (C 3 ) (70 
* It must be remembered that (B) and terms involving A only vanish by hypothesis. The formula; have 
also been simplified by the omission of certain coefficients depending on n which do not affect the final results. 
5 C 2 
