( ^51". ) 



that of the uormal curve of the space S^' that is of the rational 64 of 

 the fourth order, represented by the equations : 



A2 





where j-q, rf-j, .rj, .r^, .^4 are the homogeneous coordinates of tho 

 point L of the curve belonging' to the parametervalue A. 



2. If Ai and A^ are the parametervalues of any two points ^i 

 and ij on C4 , the coordinates of any point A of tlie line joining 

 -^1 and Z/2 are given by the relations 



^'o = ^'i + Pi 

 a■■^ =pi>.i + Pi h 



'■'■3 = Pi ^1' + P2 V / (1) 



''•3 = P\ h^ + Pi h^ 

 Xi = pi A,* + Pi A2* ;' 



and now by eliminating the four quantities Aj, A^, pi, p.^ we find 

 the equation of the locus required. 



The result of this elimination is the cubic curved space 



For if 



P\K +Pih 



**'o '^1 ^^ 

 .Tj a-3 arj 



*2 H iJ"* 



P\ Al + Pi Ao Px Xi" + P2 ^2' 



P\ ^1" 4- P2 ^2" Pi ^1^ + ^2 ^2^ 



(2) 



P\ h^ + i02 ^^2- P\ ^l^ + Pi h^ Pi ^1* + Pi h* 



is written down, it is immediately evident that every combination 

 of partial columns vanishes after easy simplifications, two of the 

 three columns being equal to each other ^). 



1) As I already stated above, I nowhere met with this simple deduction of the 

 invariant of Sylvf.steh which can be pursued through all spaces S~". In the original 



