254 MR CHARLES TWEEDIE ON THE 



The conies MjN 2 — M 2 'S 1 = ; NjLg - N 2 L 2 = ; L x M 2 - L 2 M. 1 = have three points 

 in common, and this is the characteristic of the quadric Cremona transformation (v. 

 Salmon). 



Conversely, any three conies having three common points may be so represented. 

 Let ABCPQ, ABCQR, ABCRP be three such conies (where no two of the 

 points P Q R are coincident). Let L x - 0, L 2 = denote two lines through P ; M x = 0, 

 M 2 = two lines through Q ; N x = 0, N 2 = two lines through R. 



The conic ABCPQ may be represented as the intersection of corresponding rays 

 of the pencils 



L x -aL., = 

 M x - a M 2 = 0. 



Similarly, the conic ABCQR may be obtained from 



Mj-aM^O 

 N 1 -aN 2 = 0, 



Nj and N 2 being lines suitably chosen through R. 

 Now the two pencils of lines 



L.-aU = 



N.-aN^O 



furnish a conic which clearly passes through A, B, C, also through the centres P and R, 

 and therefore through the five points ABC PR, i.e., they furnish the conic ABC PR. 

 The values of a corresponding to A, B, C are found by expressing the condition that the 

 three equations in x and y 



L^-aL^O; M x -aM 2 = 0; 1^-aN^O 

 be consistent. When the cubic for a has two equal roots, two of the points coincide 

 and the conies have contact of the first order ; when all three roots coincide the conies 

 have contact of the second order — all at a common point. A point P may also coincide 

 with A, say, in a given direction, but Q cannot coincide with A at the same time. 



It may be noted that a linear construction for any number of points on the third 

 conic is hereby indicated, the first two conies being given. Let S be any point on the 

 first conic, and let Q S meet the second conic in T. Then R T and P S intersect on the 

 third conic in U. 



§ 4. Let the bilinear equations in x, y, £, >; be 



( 1 ) HAjX +B 1 ij+C 1 ) + , ] (A.^ + B,t/ + C 2 ) + A 3 x + B. iV + C 3 = 



(2) ^a^ + foy + yj + v (a.yX + ft0 + y. 2 ) +a s x+8& +y s = 0. 



If these equations are iuvolutive, then the interchange of £ and cc, >] and y, must 

 give two equations which lead to identical solutions with (1) and (2). Hence the result 

 of the substitutions must be to replace (1) and (2) by two equations, 



7<1)+Z(2)=0 



m(l) + v(2) = 

 where kn — lm is distinct from zero. 



