( ;^i4 ) 



liypergcometry. For all this we refer to two important papers by 

 Mr. W. Fe. Meyer. The first, published at Tubingen 1883, is entitled ; 

 ^^Apolaritat und Bationale Ctirven^^ ; the second, inserted as '^Bericht 

 iiher den (jegemvartigen Stand der Tnvariantentheorie" in the 1^' Vol. 

 published in 1892 of the ^^Jahrcsbcn'clit der Deutschoi Mathematiker- 

 Ve)-eini(jiing\ is an invaluable report about this branch of Mathematics. 

 On page 365 of the former work a theorem appears under /.j. 

 which is closely connected with our subject. 



It goes without saying that it must be possible to reach con- 

 verfely the above quoted theorems of Sylvester and the higher invo- 

 lutions connected with them starting from the theory of poly- 

 dimensional space. Indeed, Mr. f'LiFFORD has stated already in 

 1878 in his important treatise "0« the classification of loci'' {Phil. 

 Trans., Vol. 169, part 2°^, p. 663—681) that in this direction a 

 geometrical interpi'etation of any invariant of a binary form is to be 

 found. So in trying to determine a certain locus in space with an 

 even number of dimensions I have fallen back upon a geometrical 

 interpretation of the invariant of Sylvester; however examining 

 the above mentioned literature I soon discovered that this interpre- 

 tation had already been found. 



Yet I wish to publish my study. In the first place because it may 

 prove that the geometrical way is at least equally simple as the 

 algebraical. Secondly on account of its containing a method of eli- 

 mination I have as yet nowhere met with in this form. Tiiirdly 

 because it is not quite impossible that entirely corresponding inves- 

 tigations may lead to a geometrical interpretation of other general 

 invariants ^). 



1. A curve allows of a twofold infinite number of chords, con- 

 taining together a threefold infinite number of points. If this curve 

 is situated in the space S'^ with three dimensions these points will 

 fill the whole space and one or more of these chords will pass 

 through any given point. If the curve is situated in the space 'S'^' 

 with four dimensions the locus of the points through which chords 

 pass, i.e. the locus of those chords themselves, is a curved space of 

 the thii'd order. The point from which we start here is the investi- 

 gation of this curved space for the simplest possible case, namely 



*) I think e. g. of tlie invarinnt (aJ)"" {cdf" (ac) [id) of a general binary form a^"'*'^ 

 of odd degree (compare Salmon 1. c, p. 129, problem 2""^) forming an extension of 

 Ibe discriminant of at'', of wliicb as yet no general algebraical interpretation seerag 

 known. 



