258 ME CHARLES TWEEDIE ON THE 



conies cut, so that X, Y, Z are the intersections of pairs of opposite sides of this 

 quadrangle. 



To a straight line 



.-^ 



px + qy + rz — 



corresponds the conic 



*Z.p(bn — cm)>]£ = 0, 



i.e., a conic through the principal points X, Y, Z. This conic is not degenerate unless 

 a coefficient is zero. (This would always happen if b/c = m/n.) 



If £> = 0, i.e., if the line passes through X, the conic breaks up into the line YZ 

 (which corresponds to X) and a line through X. Thus, if to the point P corresponds 

 P', to X P corresponds X P', and inversely ; so that the lines through a principal point 



X are paired in involution, the two self-corresponding lines 

 of the involution being simply the two sides of A B C D 

 that pass through X, viz., AB and CD. Any two corre- 

 sponding lines through X are therefore harmonically separated 

 by XABandXDC. 



If a line X P cuts B C in P, the ray X P' corresponding to X P therefore cuts B C in 

 P', which is the harmonic conjugate of P with respect to B and C. Hence Beltrami's 

 theorem : To a straight line not passing through a principal point corresponds a conic 

 through the following nine points — the three points X, Y, Z, and the harmonic conjugates 

 of all such points P in which the straight line cuts the six sides of the quadrangle 

 ABCD. 



The discussion is simplified by noting that the conies of reference may be replaced 

 by any two conies of the system through A B C D, and in particular by two pairs of 

 opposite sides of A B C D, especially when all these points are real. 



Beltrami also proves that to a curve of degree n, passing a times through X, 

 iS times through Y, y times through Z, there corresponds a curve of degree n' , passing 

 a! times through X, /3' times through Y, y' times through Z, where 



il =2n ■ — a — ft — y ; 

 nf — a — a — a ; n' — {3' = n — fi ; n! — y' = n — y ', 

 ii. — 'In! — a' — fl' — y . 



One case is worthy of note. To a conic through two principal points corresponds 

 a conic through the same points. If the conies intersect in P and P', P and P' are 

 corresponding points and are coincident only when the conic passes through a vertex of 

 A B C D, in which case the conies touch at that point. Hence any non-degenerate 

 conic through two principal points and through two of the points ABCD must 

 correspond to itself, for two conies cannot have four common points and contact at 

 two of these points without coinciding. The points on such a conic are paired in an 

 involution, and therefore the joins of corresponding points are concurrent, the centre of 

 the involution being the intersections of the tangents at these two self-corresponding 

 points through which the conic passes. 





