456 Mr Dixon, Note on Plane Unicursal Curves. 



the w-ic curve : then W cannot be a multiple of U, for it only 

 contains terms of n different degrees, whereas in U there are n + 1, 

 namely 0, 1, 2 ... n. 



We have, then, after proper choice of a multiplier, 



Hence 



The curve 



W (x, y) + \i#i + X 2 <£ 2 + . . . + h P 4>p = 



therefore meets the given curve in the points 1} 2 ... mn , which 

 proves the proposition. 



In the case of a cusp the form of the condition is modified. 

 Suppose a to be the value of at the cusp, and (h, k) its 

 coordinates, then 



xe-h.ze, Ye-k.ze 



have the factor (0 — a) 2 , and thus the equations (1) are replaced by 

 Ta_Y'a = ^a 

 Xa ~ Yol " Za ' 



Hence, if F{6) =f(X0, Y0, Zd), 



F'a Z'a 



Fa Za 



^ 1 Za 



or 2, -£- = m . -^~ 



a — 6 r Za 



is the condition to be satisfied by 6 1} 2 ... 6 mn , when the point a is 

 a cusp. There is no difficulty about the converse theorem. 



Suppose now that the curve has a triple point, the three 

 values of being a, /3, 7. Then we have by the same reasoning 

 as before 



Fa/(Za)™ = F/3l(Z/3) m = Fr / /(Zy) m , 



but there is also a further condition. Let A, B, G be so chosen 

 that 



F0-A (Z0) m - B (Z0) m ~ 1 X0 - C(Z0) m ~ 1 Y0 



has the factors (0 — a) 2 (0 — /3) 2 , that is, that the curve 



f(x, y,l)-A-Bx-Cy = 



has a double point at the triple point of U. Then (0 — <y) 2 will 

 also be a factor, so that 



F0-A (Z0) m - B (Z0)™- 1 X0 - C(Z0) m -' Y0 



and its first derivative both vanish when = a, /3, 7. 



