( 344) 



P^ and 0„. If we combine these results, we arrive at tlie following: 

 Iheoi-eni: 



Tiie section of ^A, and I*, is a dejienei'aled cni-AC of order nine 

 broken n[) into a plane ciiliic and six right lines. On this section are 

 situated twelxe nodes, points of intersection of the principal axes 

 two by two; nioreovei six nodes are situated on it, formed each 

 time bv one of the points of intersection of a principal axis with 

 ( .,, and six points of contact, which are the remaining points of 

 intersection. So /', is a sixfold tangent plane of O^. 



So we come to the conclusion that ()„ possesses besides the three- 

 fold cm-\e (f ,, still a nodal cni've of ^\■llicll for the present we caimot 

 make out how it is composed, bnt of which the total order is IS. 



The numlter of points of intersection of this ciirNC with one of 

 the generati'ices of ()„ can he determined. Let <i he one of the J'ight 

 lines connecting a poijit .1 of </., with one of the corresponding 

 points .1', on T.,. An ai-hitrary plane ^^ throngh r/ cnts r/-, in two more 

 jioints I) and (' to which cor)-es|)ond on (',, tw(t triplets of poijits 

 B\, B'.„ ll\ and ('\, (".,, C\. In like manner a plane (/ throngh 

 ii cnts the curve ( \, in two more [toints to which corresi)ond 

 two points oji <ƒ,; so there exists between the |)encils of planes 

 (^ Jmd (/ a ((i,2)-correspondence and the number of planes of coin- 

 cidence amounts to S. So all togethei' a is cut bv S |)rincipal axes. 

 As in the general case this numi>er must be diminished Iw 'A, 

 for now too the three points of interseclictn of <f,, and (\ must be 

 taken into accoimt -. so x is cut by tive principal axes. Each gene- 

 ratrix of ( K, has thus lixc points in common with the nodal cin'X'c. 



From the preceding is apparent that the gejieral section of the 

 surface possesses J8 nodes and o triple jtoints: if we have in mind 

 that the hitter are eipdvalent to il nodes we see that the general 

 section is not rational, as a cur\e of order nine can lia\'e 2S ]iodes 

 and the cur\e imder investigation possesses ojdy 27 nodes. 



6. We shall consider a single case, where the surface O^ is 

 simplified. We have already noticed that the cone of axes is of order 

 three without nodal generatrix; there w(tuld be one if the net of 

 conies possessed in /'^ a ]»oint, common to all cojucs. As however 

 to this pencil belongs the isotropic circle this case is excluded; it 

 may however happen that the cone of axes breaks u[) into a <pia- 

 dratic cone and a [)lane, or into three planes. 



7. We choose an example of the first case. When the cone of 

 axes breaks up into a cpiadratic cojie and a plane, then the .lacobian 

 curve in /\ must degenerate into a right line (\ aiul a conic C^. 

 This happens: 



