( 505 ) 

 5. Hv iiieaiis of llie simple form of the I^lCckkk's fonimlao we 



iiiv now enixl)ie<l k» .show more ele.uiy llial nallv all curxes To 

 heloiiu' as tliev should lo I ho same uoiiiis. Vov I his we |)rove the 



O'l (/-fli 



('(|iialit\' of llie iionera (/^ aiKl //^^i of ( '^ ajid ('2 



Accord iii.u' to the rcdations (1') extended I o (], \\(\ lia\e 



2/7/ r= {»,-!) (ni— 2) _ 2(./,+/.v) = (yv-l)(v7-2) — 2 (.// + /•/ _, ) 



and therefore 



2 (///+1-//J 3Z: (/-^^ _ _ ;.^ ) -2 (,//+ 1 -./,) - ^] (/•/+! -/•;) -2 (r; r;_i) . .(3). 



Moi-eo\-er the lirst of the three ecpiations {'2'] for /' and / -j- i uMves 

 bv means of subtraction 



Vi^o— i-i^i — ( r'.^^ — i'. ) — 2 (r/;^i — ,/,) — (,.;^i — /-•) _ 8 (r; - /•/ ^ 1 ) . (4) 



Thus by snl)ti'actio]i of (4) from (Ü) we ,ü,'et 



2 (//z+i — ƒ/;) = (/7+0 — r/^i) — 2 Ov_(_i"'yv) + (/•/— /'/ -l) 

 — (;v^2 — -'V— 1) — 3(/7-f I— /'/) 

 and from lhi> e((iiation the second mombei- (lisapi)eai's in cojise((nence 

 of the third of tlu^ (Mpiations (2'). 



r.et us observe by Ihe w^ay tliat th(Mi umbers of rank ;vi, ^'1. ^'o. .. r„_[_i 

 of ^,'i form the first terms of a recurrent series Avilh tlu^ thii-d of 

 the expiations ('2') as e((ualion of comlitiou and tlnis — for ,/• as 

 the variable — with (i — .i;)'' as deuonnnator of the uenei-aliuu' tVaclion. 



In order to cause the representation to remain as simple as |>ossible 

 we Jiave sui)posed the cnrve (], to Jack all hiuliei- siniiidarities. For 

 the influence of the latter we refer to tlie abo\e-uientioued essay of 

 Yekonese. 



The Plücker's formnlae given liere shall be apj^lied elsexvhei-e \o 

 tlie ca?e of the cnrve C„ of order 2"'' foj-ming in S„ the section of 



o 



/i— 1 qnadratic spaces Q," . 



Mathematics. - ''(>// s(/.^f<'ins 0/' ('<>)'/''■'< Ix'toinjiiiii to 'lucolniions on 

 ratimuil riirres.'^ Uy Prof. -Ian df, YinEs. 



I. We suppose the points of a rational plane ciir\(' T'" lo be 

 afranged in the ,u'i-oups of an inxobition /', .v^5, ajid bi-ing a conic 



('- through each quintuple of |>oiuts belonging to a selfsame 

 group. The system [/''] formed in this way has evideidly no doui)le 

 right lines, so that r^ = 0. So l)etweeji the characterising mnnbers 

 >i,r,<i exist the relations 2 v = (i -\- ö and 2 ;t = r ; so we have 

 = 2 n and d 1= 3 f/- 



