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VI. 



ON BICURSAL CURVES. 



Br EEV. WILLIAM RALPH WESTROPP ROBERTS, M.A., 



Fellow Trinity College, Dublin. 



[Read January 26, 1903.] 



It is well known that the coordinates of any point on a curve which 

 possesses its maximiim number of double points can be expressed 

 as rational algebraic functions of a variable parameter. The converse 

 theorem is also true, namely, that if the coordinates of a curve are 

 expressed as rational functions of a parameter, such a curve possesses 

 the maximum number of double points. Curves of this nature are 

 termed unicursal curves, and to each value of the parameter 

 corresponds one and one point only lying on the curve. 



We propose to consider in this paper curves which we shall call 

 hicursal, since to each value of the parameter, in terms of which the 

 coordinates of the curve are expressed, correspond two points lying 

 on the curve. 



We suppose then, that the coordinates of such a curve are ex- 

 pressed in terms of a parameter in the following manner : — 



(1) x= A, ^ B,yR. 



1/ = Aj + £z -y R. 



% = A^ + B, yR. 



Where Ai, A.,, A3, are binary quantics of the m"' degree in 

 two variables X and fx, the ratio of X to /a being regarded as the 

 parameter determining the points on the curve, R is a binary quantic 

 of the 2n*'^ degree, and £1, B2, B3 binary quantics of (m - «)"' 

 degree in A. and /x. 



Such e(juations obviously remain unchanged in form for any linear 

 transformation of the variables \ and /jl. 



