458 Mr Dixon, On the Order of Certain Systems of Conditions. 



On the Order of Certain Systems of Conditions. 

 By A. C. Dixon. 



[Received 28 March 1904.] 



Let there be m + 1 expressions u ly u 2 ..,u m+1> rational and 

 integral and of the degree 2m in x: then constants a x , a 2 ... a m+1 

 can be so chosen that the roots of the equation 



QrjWl ~\~ a 2 U 2 + . . . + $771-1-1 Uqn+l = " 



shall be equal in pairs. The object of this note is to shew in the 

 first place that this can be done in 2 m ways. For put 



ofa + a * u 2 + . . . + a m+1 u m+1 = (p x m + p x x m ~ Y + . . . + p m ) 2 . 



Then on equating coefficients we have 2m + 1 expressions linear 

 and homogeneous in a u a 2 ... a m+1 , each equal to a homogeneous 

 quadratic expression in p , p x ... p m . 



Eliminating a ly a 2 ... a m+1 we have m homogeneous equations 

 of the second degree to give p : p 1 : p 2 ..., and the number of 

 solutions is therefore 2 m , as stated. 



More generally, let u-^ , u 2 ... u m+1 involve two variables x, y 

 connected by an algebraic equation f (x, y) — of degree n and 

 deficiency p. Suppose all but 2m of the points, where 



a^u^ -J- a 2 u 2 -\- . . . -f- a nv -^i u m ^.j 



vanishes, to be fixed independently of a 1} a 2 — Then when m^p 

 a l} a 2 ... a m+1 may be so chosen that the other 2m points coincide 

 in pairs and the number of ways in which this can be done 

 is 2 m+ P. 



To save complications, let the curve f(x, y) = 0he brought by- 

 rational transformation to a form in which it has no multiple 

 points except ordinary nodes. Let the constants in it be changed 

 continuously to such values that the curve has p new nodes. 

 Then a curve touching the original in m points may become 

 a curve passing through q of the new nodes and having m — q 

 proper contacts, q being any whole number from to p : such 

 a curve will, according to the usual convention, be counted 

 2« times among the solutions of the problem. The number of 

 such curves is easily found since the curve f(x, y) = Q is now 

 unicursal, and x, y can be expressed rationally in terms of a single 

 parameter 6. 



The conditions that 



$!«! + a 2 u 2 + ... + a m+1 a m+1 = 



