1881.] Prof. Caylexj, Note on AbeVs Theorem. 121 



If the variable curve be a cubic, n = 3, and we Lave between 

 the displacements of the nine points the relation 



dw l + day.;, ... + dwg = : 



here eight of the points do not determine the cubic <f>, but they 

 nevertheless determine the ninth point, viz. (reproducing the 

 reasoning which establishes this well-known and fundamental 

 theorem as to cubic curves) if <f> = be a particular cubic through 

 the 8 points, then the general cubic is cf) + kf— 0, and the 

 intersections with /= are given by the equations <f> = 0, /= ; 

 whence the ninth point is independent of k, and is determined 

 uniquely by the 8 points. There should thus be a single relation 

 between the displacements, viz. this is the relation just found. 



And so if the variable curve be a quartic, or curve of any 

 higher order, it appears in like manner that there should be a 

 single relation between the displacements ; this relation being in 

 fact the foregoing relation %dw = 0. 



But take the fixed curve to be a quartic, ra = 4 : then we have 

 between the displacements day the relation 



X (x, y, 1) dco = 0, 



that is the three equations 



Xxdco = 0, 'Sydco = 0, Xdco = 0. 



If the variable curve is a conic, n = 2, then there are 8 points 

 of intersection ; 5 of these taken at pleasure determine the conic, 

 and they consequently determine the remaining 3 points of inter- 

 section : hence there should be 3 equations. And so if the 

 variable curve be a curve of any higher order, then by conside- 

 rations similar to those made use of in the case where the first 

 curve is a cubic it appears that the number of equations between 

 the displacements da should always be = 3. 



But if the variable curve be a line, n = l, then the number 

 of the points of intersection is = 4 : 2 of these taken at pleasure 

 determine the line, and they consequently determine the re- 

 maining 2 points of intersection ; and the number of equations 

 between the displacements dto should thus be = 2.. But by what 

 precedes we have the 3 equations 



dco l + dco 2 + dco 3 + dw i = 0, 

 x l dco l + cc„do) 2 + % 3 da) 3 + a^dco^ = 0, 

 y x doy t + y 2 da> 2 + y z d<o z + y 4 da> 4 = ; 



here the 4 points of intersection are on a line y = ax + b; we have 

 therefore y^ax^ + b, ... y t = ax t +b ; the equations between the 



