( 508 ) 



J J J., willi ('" (leteriuiiies a ai-oiij) of the f' of which llic i-cmaiiiin^- 

 \\\{) [)(>iiil> furnish the second ri,uii( lijie of ihc <l(',u-eiie)-;ile(l ("' : 

 thirdlv eacli of ihe poijils J^, .1, is coliiiicai- willi 2 (// — Ij pairs of 

 tli(^ /•'. So w (' ha\e cf = // — 2 -f y/ -|- 4 (// — J ) = I) (// — 1). 



For .v = 4 we ^^allt Iml one li\(>d point J,. There are 3 (// — 2) 

 colHnear Iri|)le1s and '.\ (// —.1) |)airs of lines, ^\•hel•e eacli (tf the 

 ri^hl Hnes hears \\\i\ points of a uronp of V\ so here \ve liml 

 Ó = 'A in. — 2) + 3 (// — -[) — 3 {'hi ~ 'M. 



(i. The correspoiKhMice lA', A'') has foi- n < 5 ihe cliaracterizing 

 nnnilxM' (2// — s){'hi — .v — \): in the correspon(h;'nce (/^A') eacli 

 poiiil P is c(nijnf>'ate to (2// — x) poinls A', each point Alo.v(2/v — -v) 

 points l\ As /■'•■ conlaijis 2 (.v — 1) coincidences the nnnihcr of c(tnics 

 of rf'-"] touching' ^ '" is now re|)resejited bv 2;2// — .y)(2y/ — .v — 'i) + 

 + 2 (2y/ — .v) {s + 1) -f 2 (.s- — 1 j = 2(2// — 1 )(2// — s-\- 1 ), whicli corres- 

 ponds to the value the nnmber {l-\i -f- /'»') yiossesses here. 



The cori-espondence (A. A') f<»i' an /"-' is of ordei' (2// — 2) (2// — 3): 

 it has (2// — 2) (2// — 3) jiairs in coniinon with P. 



So die Msteni [T"'] contains (// — J; '2// — 3) c(niics hcarin^j^- eacli 

 two jtairs of the ({uadraUc involulion. 



Mathematics. — •' FunddDu'tital niritlntions on fiifion((l ciirri's of 

 orih-r firc^' l»v Prof. J.vN DK \'kiks. 



1. If the points of a rational curve of order five, ('\ with six 

 double points JJk (/• = 1. 2. 3, 4. 5, (>j. are arranged in the pairs P', 

 P" of an involntion /-'. the line P' P" enveUipes a directing curve 

 of class four. For, the indicated involntion has four |>airs in coninion 

 with the central involution of order live, which is deteiiuined by the 

 rays of a pencil. If a paii' of the /' is formed by the jioints D\ 

 and }y\ lying in I), on Ihe two branches of the C\ the directing 

 curve breaks up into a curve of (dass three and the pencil of i-ays 

 having its vertex in ]),,. If a second paii' consists of the points //^ 

 and //',, the real directing curve will be a conic. Then we evidently 

 gel an /' given with P'' : we shall therefore call it -a /nnddiiiciital 

 involution. It is determined by the pencil of conies with the base-p(Mnts 

 /A, a, 7J„ />,. 



2. The bearer of a pair of points of the fundamental involntion 



P meets P' in three points more 7", T" , 7"" forming a group 

 t»f a cubic involution. For, of the tangents out of a point 7" = 7^ 



