t Bó ) 



Ok (& = 1,2, 3, 4, 5) is now transformed into a sheaf of ray s having as 

 vertex (he point 0\ conjugate to t . 



To the bisecant b' through 0\ of the curve o'* brought arbitrarily 

 through O x ,O tJ O t ,0 A corresponds a f through O lf ,. O t , 4 , <>,, 

 having the right line s as chord. 



The following will show thai the indicated transformation enables 

 us to deduce by a simple method a number of well-known properties 

 of systems of curves a> 3 . 



§ 2. Let us consider the curves co 3 of the congruence r cutting 

 the right lines / and in. They are transformed into the right lines 

 through 0' ., resting on two curves X n and ii'\ Now the cubic 

 cones, projecting these curves out of 0' 6 have besides the right lines 

 0\ Ok (k = 1; 2, 3, 4) live edges more in common, which are the 

 images of as many twisted curves belonging !o l\ 



From this is evident that the curves of r cutting a given right 

 line / form a .surface A b of order five. 



The image of A' a is a cubic cone, projecting X n out of 0' s and 

 having the bisecant b' out of O' 6 as nodal edge. Therefore the curve 

 ji 3 of r having / as bisecant is a nodal curve of A'\ 



If we bring the right line m through 1 its image is a right line 

 m' passing likewise through (J x and having therefore with the above 

 mentioned projecting cone of A' 8 besides Ö, two points in common. 

 From this we conclude that A h has five threefold points Ok> 



So the section of A b with OkOiO m consists of the right lines 

 QkOl, OiO m , O m Ok and a conic through (.)/„ Oi, O m cutting O^O y 

 and forming with this right line a cubic curve of r. Consequently 

 eleven right lines and ten conies lie on A\ 



§ 3. The curves (> 3 of r touching a given plane g>, are trans- 

 formed by the correspondence into tangents t' through 0\ of a 

 cubic surface </>' 3 having conic points in 0fc(£ = 1, 2,3,4). The 

 polar surface of O' 6 passes through the four double points 0, so 

 it has as image a quadratic surface through those points. The section 

 of the latter with <p is the image of the locus of the points, in 

 which <b n is touched by the right lines /'. This conic contains 

 therefore the points in which <j is touched by the curves q*. 



Through <ƒ. pass six principal tangents of 0' 3 , the congruence r 

 contains therefore sii: curves, osculating <p. 



The enveloping cone </"'' out of O' s to &' has four nodal edges 

 O h Ok', for a plane through 0'.f) k cuts <7>' 3 according to a cubic curve 



