Roberts — 0)i Bicurml Curvea. 67 



But the degree of the curve is 2m and couse(iueiitly the degree of 

 Z, M, JV, considered as functions of the parameter, will be of the 

 (2m - l)??!*" degree involving i^/Jt which is of the n"" degree. 



And we infer, as before, that the degree of A is 



{2m - \) m - {2m + ?< - 2), 

 or 



i{(2w* - l)2»i- 4«i- 2?i + 4}. 



Hence the degree of A is 



Ki?-l)(i>-2)-(«-l), 

 where p = 2m. 



It must be remembered, bowever, that as A involves \/ R, the 

 equation A. = must be rationalised to solve it ; and consequently 

 the number of roots of this equation, when freed from radicals, is of 

 the degree (^-l)(^-2)-2(«-l), and admits of as many 

 roots. But, since each double point has two values of the parameter, 

 corresponding to it according to the branch on which it lies, the 

 number of the double points of the curve in question is 



^(p-l){p-2)-{>i-ll 



that is to say that the deficiency of the curve is {n - \). 



The degree of the curve may, however, be reduced if A^, A2, A, 

 contain a common factor which is also common to H. If the degree 

 of the factor be r, the degree of the curve will be reduced by r. 



This is easily seen by determining the values of the parameters 

 corresponding to the points of sections of the curve with an arbitrary 

 right line, when it will appear that the common factor will divide the 

 equation when made rational. The degree then of the curve is 



p - 2m - r. 



Such a curve is represented by the system of equations 



X = AiU + £1 .\/uS. 



y = Ai'U + £2 ■v/w'S. 



2 = A3U + £3 ^y uS, 



where A-^^ Ao, and A^ are of the {m - r)"* degree in X and /x. 



u is of the r"" degree and ^i, B^^ £3, of the {m - n)"* degree, the 

 function under the square root being of the degree 2n. 



