Mr Dixon, Note on Plane Unicursal Curves. 455 



In particular, take / to be the (n — 8)-ic through all the nodes 

 but one, say (a 1} /3j). Its n(n — 3) intersections with U correspond 

 to the parameters a 2 , f3 2 , oc 3 , j3 3 ... a p> /3 P . 



Hence ft (* " "J fo ~ &) - (^T~* 



There are p relations of this form. 



Conversely, if lf 2 ■■• 0mn satisfy the p conditions 



inn a . _ Q / 7rj.\m 



the points 1} 2 ... 6 mn on U are its intersections with a curve of 



mn 



degree to. Write F{6) for U (0 - r ), and let r = (r = 1, 2 . . . p) 



r=l 



be the curve of degree n — 3 that passes through all the nodes 

 but (a,., /3 r ). Thus we have 



(K V0\ v 



Now by proper choice of the coefficients \ 1} X 2 ... the ex- 

 pression 



F x (0) = F{0)- (Z0)"+* U(0- a,) (0 - fr) i *r 



r=l r =i{t> — a r ){t> — p r ) 



can be made to vanish when = a 1} a 2 ... a p . The value of \ is 



F{a,) (Za^-t 



(Za \™>' p > 



y ^ } n(a 1 -« r )(o 1 -/8 r ) 



r=2 



and this is unchanged in value by hypothesis when /3 X is put in 

 the place of a 2 . Hence F 1 {0) also vanishes when = /3 X and 

 similarly when = ft 2 , ft 3 ... ft p . If then to Ij> ?i — 3, .F, (0) must 

 vanish identically, since it vanishes for more than n (n — 3) values 

 of 0. Thus the curve 



X.J0J + A. 2 ^>2 + ••■ + ^p(f>p = 



consists of the line at infinity n — 3 — to times, with a curve of 

 degree to that passes through the mn given points. 



If to ^n — 2 take a curve of degree m, having, unless to = ?i — 2, 

 a multiple point of order to — n + 1 at an arbitrary point, say the 

 origin, not lying on the curve U: its equation will contain 



\m (to + 3) — \ (to — n + l)(m — n + 2), 



that is mn — ^(n — V){n — 2) arbitrary constants, and it can there- 

 fore be made to pass through this number of arbitrary points : 

 choose the points on U whose parameters satisfy the equation 

 F 1 (0) = 0, of which points there are mn — \{n — l)(n — 2), since 

 each node has two parameters. Let IT=0 be the equation to 



