llf»! 



8. Let 1 7'') lie a pencil bcloiigiiij;' \o the net ['/^]. wliieli is pro- 

 duced by the intersection of tiie net [</>"] with the plane q . The 

 locus of tlie points which have the same polar line with regard to 

 a curve 7/' and the curves of a pencil (7"), is a curve xp of order 

 2?i -\- p — 3'), hence a curve of order 9, if for y/' the curve ot 

 coincidences 7" is taken. In the points Si- tf?" like 7", has nodes 

 and there the same tangents as 7"; so the two curves have 30 points 

 in common in 6\. Further both of them pass through the 12 nodes 

 of the pencil [^p^]. In each of the remaining 12 common points JJ, 

 y* is touched by ip'*, which means tiiat there tiie curves of a pencil 

 belonging to [<f.-^] have three-point contact. In [Q'') occur therefore 

 twelve groups, in which every time tkree points have coincided. 



In each of the 12 points D, 7" is touched b}^ the complementanj 

 curve y^'\ into which 7" is transformed by {Q,Q); the latter is the 

 locus of the pairs of points which complete the coincidences oï {Q^) 

 into quadruples. The tigure of order 48, into which 7" is transfor- 

 med, consists of y'' itself, of the 5 curves cii^, each counted twice, 

 and the complementary curve; the latter is consequently indeed 

 of order 12. With t^ it has four points in common, arising from 

 the 4 coincidences of the P lying on it ; the remaining 20 lie in 

 the points Sk» From this it ensues that 7^^ has quadruple points in 

 the 5 singular points >S'. 



In Sk, 7^^. and 7" have therefore 5 X 4 X 2 = 40 points in common, 

 they further toucfi in the 12 points D. The remaining 8 intersections 

 arise from quadruples of which twice two points have coincided; so 

 {Q^) contains four groups, wiiich consist each of two coincidences. 



Mathematics. — "O/i Wv.Rm'YvIs functions." By Prof. W. Kapteyn. 



(Communicated in the meetings of March 28 and April 24, 1913). 



1. The n'^^^ derivative of «~^'' may be |)iit in this form, first given 

 by Hermit E 



where 



nin — 1) nin — \){n — 2)(?i — 3) 

 H,, (..) == (2^0" - \y-- (2.t-)"-2 + -'^ ^^-^ (2.^')"-^ - • (I) 



These polynomia satisfy the Ibllowing relations 



1) See e.g. Gremona-Gurtze, Elnle.itung in eine geometrische Theorie der ebenen 

 Curven, p. 121. 



2) Exerc. de Tisserand, 1S77, p. 26, 27 and UU. 



