Mathematics. — "On Satellite Points on Curves, Given by the 

 Equations: x = ati' , yzzzbti". By Prof. W. A. Versluys. 

 (Communicated by Prof. .1. Cardinaal). 



(Communicated in the meeting of September 29, 1917). 



^ 1. If p and q are positive integers, liaving no common factor, 

 q being greater tlian p, the ecjuations 



a; = ati' , y =z bt9 . . . . . . . . (I) 



lepresent for any value of a and b, if t is supposed variable, a 

 curve of order q, \yhicli will be indicated as a curve C{p,q). 



The curve is completely determined by the point P{a,b), but 

 any other point of the curve, for which the parameter t is not zero 

 or infinite also determines I he curve completely. 



If a or b or both are given all possible values, the equations (1) 

 represent a pencil of curves of order q. Let the curve of the pencil 

 determined by the point P be the curve Cp{p,q). 



§ 2. The tangent in a point P{ati', bt'i), to the curve CpI/.»,*/) has 

 the equation : 



x—atP y — ht9 

 patP '^ qbt^f-^' 

 Let *S be a point in which this tangent intersects the curve 

 Cp{p,q) and let vt be the parameter of this point S, v is then a 

 root of the equation : 



a {vt)P — atv b {vt)'/—bt<) 



paV—^ qbtU-^ ' 



or after simplification 



uP — 1 v<j — 1 



(2) 



P 9 



This equation is of order q in v, but possesses a double root 

 v = ^, so that the q^ — 2 remaining roots correspond to q — 2 

 intersections or satellite points of P. 



It is to be seen at once that the equation (2) possesses at most 

 3 real roots, if q is odd and 2 real roots at most, if q is even. 



If therefore q is odd and v the real root of the equation (2), 

 different from 1, the point with parameter vt is the only real 



