( 345 ) 



a. When tlie conies of the net pass iliroiif^h two fixed points. 



I). When the net possesses a double right line. 



We restrirl ourselves in Ihis conimunicalion to the lirst of these 

 cases; then the base curve of the pencil of siu'laces is circidai'. 



It is in the first place necessary now that (he cone is degenerated 

 into two j)arts to consider the distribution of the axes on cone and 

 plane, if the base curve of the |)encil of sui'faces is circular, tiiere is 

 a system of parallel planes so that each plane is cut accoi'ding lo 

 a pencil of circles. Of each surface of the pencil one |)i-incipal axis 

 runs parallel to these planes. From this ensues : 



When in consetpience of the existence of a cii-cular base curve the 

 cone of axes degenerates into a (luadralic cone and a |)lane, then 

 of the three points x4'i, .l'., .1',, homologous to a point .1 on <p., one 

 lies on the right line (_\ in I* ^ and two on the conic ( \,. So the 

 skew^ surface (>„ degenerates into two other skew surfaces intersecting 

 each other in their common directrix (f,\. Vov one skew surface (p^ is 

 a nodal curve, for the other it is single. This already suggests that the 

 former of the two skew surfaces is of order six, the latter of order 

 three. This can be reasoned more minutely in the following manner : 



Let / be once more a right line; a [)lane P through / has three 

 points ^1, B, ('in common with <ƒ., to which six points ^1',, ^1'.^ . . . (\,( ", 

 on C'a correspond; so six planes V correspond to /^; if reversely 

 we make a plane P' to lie through /, it cuts ('^ in two points to 

 which on r/., two points corres])oud, so that between the i)lanes /•'and 

 P' a (2, (i)-correspondence exists. However (f\^ has a poiiit in common 

 with (_\, as C'l contains the poijit of contact of a hyperbolic para- 

 boloitl of the pencil with /^ ; so tiiei-e remain foi' (f ,^ two ])oints of 

 contact with 6'^ and the order JS, which woidd arise on account of 

 the (2, (^)-correspon(lence, must be <liminished by 2; so we get a 

 skew surface ( f,. of oi'der six. 'J'lie second skew surface is of order three. 



In the general case the section of /^, and (),, consisted, besides of 

 T',, of three [)ciirs (jf right lines, to be called n^a.,. I'J»-,^ <\<--,- ïf ^A, 

 degenerates in the uianner described above these I'ight lines will also 

 be distributed themselves oJi ''>,. and ^A,. liel .1' agaiji be the point 

 where </^., cuts the right line ( \, thus the |toinl of contact of a 

 hyperbolic [)araboloid of the pencil; through J' pass the two prin- 

 cipal axes (i^<i..^ and these belong to <}^,, whilst the principal axis not 

 lying ill /', through A' belongs t(» ( 1.^. To ^A, belongs thus one 

 princi[)al axis of each of the |)airs hj).^ and <\<'.,, so I', is a double 

 tangent plane of i>.. ■a\u\ the section of (),^ and l\ consists of the 

 conic ( \^, the pjiir of axes 'i^^a., and the priiici|)al axes h^ and (\. 

 Of a^ and <i.^ the point of intersection r/^r/^ is the no<le in the curve 



