( r)()4 ) 



of („ by r,,, r. /'„, ''/-+i ^vc liiid llic iM|ii;ili(»us (1') oxteii<lc(l IVoiii 



(/^1,2) lu (/ = J. 2, . . . . // — J), wliicli (M|ii;ili(»)is cjuise llic ;il)(»ve- 

 meiitioned 3(y/ — 1) equations to pass into tlie (Miuations (2'j, in liie 

 same war extended from (/ =: 1, 2) to (/= 1, 2, . . . , n — i). 



4. Accordinu' to this iiolalioii lli(> system of the oy/ Pi.ïckkr's 

 iiiiiulxM's of ('„ consists of three uiroiips : a Liroii|) of // -|- 2 tpiantilies 

 r (numhcrs of rank),. a .ui'oiip ofv/ 1 (|iiaiilili('s r/ (iiiimhers of (h)iii)le 

 points), a group of // — 1 (|iianlities / (uiuuIxm-s of (htuhlc taugeuls). 

 We shall now indicate what is the exact siuiiilicalion (»l'lho>e t|iianlilies. 



Xnmhi'is of rdiiL We consich'r >\,, r^,, /';,_i sepai'ately. 



\\\ /•,, we understand the numltei- of siationaiy points of ('„ i.e. 

 the mnnher of the points Ihrou.uli which //-j- i successive spaces 

 N,) -1 through // successive poiids of the curve |)ass. 



1^'or yy = 1, 2, . . . , // we liiid ihal y^, iu(Hcale> ho\v many spaces 

 ;S^,_i throngii /> successive ixtinis of ('„ cut any space \^,; of these 

 numlters /'i is th(^ order and r,, the class of ( \t. 



The Jiumher of stationary sjjaces /S,_-i of ('„., i.e. the mnnher of 

 spaces S,^ \ ihrmigh // -[- 1 successive poiiils of (_\, is indicated 



by y'„-|_i. 



Xiinihers of (loiil)l(' poii/fs. 'j"he (pianlily r/,, is the nund)er of 

 double points of th<^ section ( '^ of the hicus of the spaces .S^,_i 

 through j) successive points of (-],-\-\ with a plane situated in the 

 space ,s^,-|-i of that curve. So by returning from the projection ^),-fi 

 to the given curxe ( \, we find the foHowing: If Ave ])rojecl th(^ 

 single inluiite number of spaces ,S^, i through /> successive ])oin1s of 

 Cn out of any space ^„^,,-2 ^v(' hn<l -^ single intinile luunber of 

 spaces ;Si-2 i^ii<l therefore a twofold inlinite number of intersections 

 /S'„_4 of two non-successive spaces »S,-o . The locus of those spaces 

 ;S', 4 is a curved siiace ^vilh n — 2 dimensions, cut in a certain 

 number of poiiUs by any plane. This numl)er of points, at the same 

 time the order of this cnr\ed s|)ace, is d^,. 



Xiniihi'i's of «loiihic taiKjciits. The (piantity t^, is the minibei- of 



(p) 

 <louble tangents of C-, . By ascendmg trom f^.^-i to f„ we ai-rive 



at the following: By projecting the single intinite mimbei' of spaces 



,s^, through 7>+l successi\e points of Cn out of any space >S'i y;-2, 



we lind a single infinite number of s[»aces ,S'„_i euveloi»ing a ciu-ved 



space of n — 1 dimensions. The mnnher of doidile tangents of any 



plane; section of this envelope is /,. 



For the rest it is easy to see that the lumdters d^, and /„_^, refer 



to ([uanlilies dnalistically o[)posite iji the space ;S'„. 



