(343) 



The loens of the principal axes (if (|iia(lratic snrfaces belonging to 

 a pencil is a skew snrface of wiiicli one of the dirccliix curxes is 

 a skew eiihie ^/-., possessiiiu a direclor cone; racli point ol' \\\v skew 

 cnhic is homologons to a ti'iedei' of ravs of the cone. 



4. The or(kM' of tlie sui-face can Itc (Iclerniined l»\- in\-esiig;itinn- 

 bv how nuinv pi'inci[)al axes an arbiti'ary rii>iil line / is cnl, oi- how 

 many planes possessing a principal axis can be made to pass through 

 /, which comes lo the same thing. 



• Let ^1 be a point of (f.^, lo which tliree poiids A\, .1',^, J'., cor- 

 respond on the Jacobian cnr\e ( '., in the |)lane at inlinity /\ . 

 Let moreover /"* be a plane tlu'ongh /; then this cuts (/.^ in three points 

 A, B, C, to which correspojid again in I\ the [joints A\ A'.^ . . . ( ".,, 

 so to the plane /* coi'res[)ond throngh / nine [)lanes /*\ l*\, . . . /■",,. 



If reversely we assnme a [)oint .1' on ( ',^ unly one ])oint A 

 on (p.^ corresponds to it. If we now make a plane /-*' pass through 

 /, it cuts C\ in three points A, B , 6", to which correspond thi-ee 

 points .1. B, C; so to a plane B' correspond three [)lanes /*. From 

 this ensnes : 



The two coaxial pencils of planes J* and J*' have a (3,9)-corre- 

 spondence. So the number of elements of coijicideiice amonids to 

 12. From this reasoning, however, we may not conclude that the 

 order of the skew snrface is to be 12; this number must be dimi- 

 nishe<l by the nunibej' of points common to <ƒ,, and ('.^. The three 

 points of intersection of </)., and Bj, are namely situated on (4; if 

 we call one of these }>oints S, then >S\ eoincides with >S' quite in- 

 depemlenlly of the jjosition of the assumed right line /. So of the 

 12 planes of coincidence 3 pass through the [)oints of intersection of 

 <f^ and ('s ; so 9 remains for the order of the skew snrface. 



5. A full in\estigation of this surface ^A, is a very extensive 

 one; however, we can consider some proj»erties and ti-ai'c some 

 particnlarities. From the plan of the problem ensues that tVom 

 each point of ^, three generatrices can be drawn meeting /\ in 

 the three corresponding j)oijits; so y., is a threefold curve of ^>„. 



The section of /*^ and (),, possesses some particidarilies which 

 we shall look into. In the xery tii-sl jdace lie on it the three centres 

 'S' '^2' '"^'n "' ''"^' paraboloids of the pencil su|)|)osed to be real for 

 the present. ()ut of each of those |)oints two |>riu('i|)al axes can be 

 drawn lia\ing therefore twehe common points of intersection. Moic- 

 ovei" each of these axes cuts T., in two more points, which can thus 

 be regai'ded as (loid)le points. ( )iie of these points belongs howcxer 

 lo a triplet ol' points corresponding to a point of intersection of (f ^ 

 aiKl ( \ ; so it can be i-egarded as a point of contact of the plane 



