( 86 ) 



with node Ok, sending but four tangents through O'. o , so that in that 

 plane 0\Ok replaces two edges of the cone. 



So if 6 has with an arbitrary cubic curve through the four points 

 Ok ten points in common lying outside Ok- By applying our trans- 

 formation we find from this that the curves of T touching <p form 

 a surface «2» 10 of order ten. 



A right line through O x cuts <p' e in four more points ; on its image 

 therefore rest four curves </■ touching (p. From this ensues that <2> 10 

 has five sixfold points Ok- 



The right lines OkOi lie therefore on <Z> 10 ; it can as follows become 

 evident that they are nodal rig ktlines. A right line resting on O x O* and 

 O s 4 has six points in common with <f'\ So its image must have 

 on O t O, and O t O t four points in common with </> 10 . 



The section of <b with 1 0,O t consists of the right lines O x 0„ 0,0 3 , 

 OJJ x to be counted double and a curve of order four, having nodes 

 in O lt O a , 9 and in the point of intersection of the nodal line 4 & ; 

 thus it consists of two conies. These conies form evidently with 4 O t 

 two cubic curves of -T, touching </?. 



Consequently there lie on </> lu ten nodal lines and twenty conies. 



When we regard the tangential cones out of 6 to two quadri- 

 nodal cubic surfaces tf>' 3 it follows readily that T contains twenty 

 curves touching two given planes. 



§ 4. To determine how many curves r> 3 can be brought through 

 four points Ok having the right line b as bisecant and resting on 

 the right lines c and d, we have but to find the number of right 

 lines r' which cut /2' 3 two times and y' 3 and d' 3 one time, when these 

 three curves have four points Ok in common. 



Now the chords of /3' 3 resting on a right line /' form a biquadratic 

 scroll on which p?' 3 is nodal curve, having thus with y' 3 besides the four 

 points four more points in common. From this follows immediately 

 that the right lines cutting ft* twice and y' 3 once also form a biqua- 

 dratic scroll J£'\ The cones which project these curves out of a point of 

 ji 3 ' having two edges in common, not containing one of the points 

 0, the curve /?' 3 is also nodal curve on S'\ With ó n this scroll has 

 besides Ok four points in common; so on y" and d" rest four chords 

 of i? /3 , and by applying the transformation we find that the curves 

 q s which cut b twice and c once form a surface 2* of order four. 



If we bring d through O x , then its image d has with ^" 4 two 

 more points in common ; consequently d cuts the surface ^ 4 in two 

 points lying outside lt so that O x is a node. Therefore the surface 

 2* has four double points Ok. 



